X-RAYS FROM SELECTED QUASIMOLECULAR TRANSITIONS

EFTHYMIOS P. LIAROKAPIS

1979

ABSTRACT

X-RAYS FROM SELECTED QUASIMOLECULAR TRANSITIONS

Efthymios P. Liarokapis

Yale University 1979

An (x-ray)-(x-ray) coincidence technique has been employed to isolate,

for the first time, selected quasimolecular transitions in symmetric and a-

symmetric atomic collisions. The method exploits the cascade relationship

between a 2pa—- Isa MO transition and the K x-ray which follows from the

filling of the ensuing vacancy in the separated atom. In this connection the

background contributions to the cascade spectra from sequential collisions

and from double inner-shell vacancy production have been also studied in

detail experimentally, by studying the (K x-ray)-(K x-ray) coincidence yields

in asymmetric atomic collisions and, theoretically, by using the quasistatic

approximation to calculate the expected emission spectra. It was deduced

that this, background constitutes only a small fraction of the coincidence

events detected and that they cannot alter the salient interpretation of the

coincidence spectra as being associated with the cascade type MO transitions.

Background contributions to the MO x-rays in coincidence with K x-rays

from the dynamic rearrangement of vacancies among the inner-shell orbitals,

have been evaluated from the relatiye importance of each vacancy production

mechanism in the Isa MO. It was found that such background contributions

cannot be excluded in the symmetric collisions, but they w ill be absent in

the asymmetric system. A special feature was observed in the spectra of

MO x-rays in coincidence with K x-rays which might reflect a characteris­

tic in the dynamic evolution of the MO's. In the case of asymmetric colli­

sions, the close , similarity of the two MO x-ray spectra in coincidence with

either atom K x-ray has proven that the vacancy sharing mechanism in­

volves molecular orbitals and occurs at large internuclear distances. The

new technique provides a way of studying the structure and evolution of a

few MO's so it can be used to test the existing theories on them.

X -R A Y S FROM SELECTED Q UASIM O LECULAR TRANSITIONS

A Dissertation

Presented to the Faculty of the Graduate School

of

Yale University

in Candidacy for the Degree of

Doctor of Philosophy

by

Efthymios P, Liarokapis

December 1979

I would like to thank Prof. D. Allan Bromley for his continuous en­

couragement, and invaluable aid in completing this thesis. I would like to

show my appreciation to my advisor Prof. Jack S. Greenberg for the idea

that this thesis is based upon and for helpful suggestions and comments; to

Dr. John O'Brien who taught me much of the experimental technique used in

this work and for many varied and informative discussions on all facets of

this thesis; to fellow graduate student Theodoros Zouros for many helpful

discussions and for reading and correcting my thesis. I would also like to

thank John Schweppe and Jim Thomas who helped me run some of the ex­

periments and analyze the data and Mike Ritter for his help in correcting

my English.

I am grateful to W. Betz, J. Kirsch, G. Soff, and B. Muller from

the University of Frankfurt, for providing us with theoretical calculations

on the molecular orbital evolution in the Nb+Nb atomic collisions.

This experimental work would not have been carried out without the

aid of Accelerator. Engineer Kenzo Sato and the accelerator staff of Dick

D'Alexander, Phil Clarkin, Ted Duda, and Bob Herrington. My appreciation

should also extend to A1 Jeddry, Joe Cimino, and Ray Comeau for their

craftsmanship.

I also wish to acknowledge Prof. Karl Erb for his helpful conversations

and suggestions, Charles Gingell and Jack Burton for their help on the nec­

essary electronics, Sandy Sicignano for the many drawing she made for this

ACKNOW LEDGEM ENT

thesis, and particularly my wife, Evangelia, for the typing and correcting

of the manuscript and for her love and encouragement during this difficult

and unpleasant time.

T A BLE OF CONTENTS

Abstract

Acknowledgements

Introduction.......................................................................................................... 1

Chapter I: Theory................................................................................................9

1.1 Introduction................................................................................... 9

1.2 Molecular orbital formation........................................................11

1.3 Non-characteristic x -ra y s .......................................................... 14

1.4 2pa MO vacancy production........................................................16

a) The coupling to other orb ita ls............................................ 18

b) The 2pa-2p7T rotational coupling...........................................19

c) 2pa vacancy production cross section..................................23

I. 5 The I sct vacancy production....................................................... 25

- Direct inner-shell excitation.................................................. 26

1.6a MO x-ray production in the static approximation....................... 31

1.6b Dynamic theory of MO x-ray em ission...................................... 34

I. 7 Vacancy Sharing.......................................................................... 37

a) Single vacancy..................................................................... 37

b) K -L matching.........................................................................39

c) Double vacancy sharing.........................................................41

Chapter II: Experimental Considerations....................................................... 55

II. 1 Introduction............................................................................... 55

II. 2 Beam l in e ............................................................................................ 57

3a) (X-ray)-(X-ray) coincidence measurements.......................59

3b) K x-ray cross section measurements............................. 61

H.4 Detectors............. 63

4a) Cascade measurements......................................................63

Correction for efficiency and absorption............................64

4b) (K x-ray)-(K x-ray) coincidence measurements.............. 66

4c) K x-ray cross section measurements...............................66

II. 5 Electronic Set-up....................................................................... 69

5a) (x-ray)-(x-ray) coincidence measurements.................... 69

5b) K x-ray cross section measurements...............................70

n. 6 Data Collection.......................................................................... 71

Chapter HI: Data and Analysis........................................................................85

III. 1 Introduction.................................................................................85

HE. 2 Symmetric co llis ions.................................................................89

^Presentation of data..................................................................89

-Analysis....................................................................................105

HI. 3 Asymmetric system (Nb+Sn)................................................... 116

-Data presentation.....................................................................116

Analysis ....................................................................................119

HI. 4 KX-KX Coincidence Experiment........................................... 134

Introduction.............................................................................. 134

Cross section for KX-KX production.................................... 136

V

H.3 Scattering chamber and ta rg e ts ................................................ 59

Data and co rrec tio n s .....................................................................139

vi

The Ni+Sn system ................................................................ 143

III. 4 K x-ray cross section measurements...................................161

Introduction..............................................................................161

Data Analysis.......................................................................... 165

Chapter IV: Discussion................................................................................ 181

IV. 1 Background Contributions...................................................... 181

a) Multiple co llis ions............................................................181

b) One collision background contributions.......................... 184

IV. 2 Nb+Nb...................................................................................... 198

IV. 3 Nb+Sn co llis ions.....................................................................208

IV.4 KX-KX coincidence............................................................... 214

a) Nb+Sn collisions................................................................215

b) Ni+Sn system .................................................................... 218

IV. 5 Characteristic K-vacancy production........................ 229

Introduction............................................................................ 229

a) Vacancy sharing region ......................................................231

b) Comparison of our data with theory............................. 233

Chapter V: Summary.................................................................................... 250

Appendix I: Error Analysis in the coincidence spectra............................. 253

Appendix II: Efficiency and Absorption.......................................................255

Appendix HI: Direct excitation calculations............................................... 259

The Nb+Sn sy s tem ...................................................................... 141

R e fe ren ces .............................................................................................................. 261

During the last ten years an increased interest has been shown in study­

ing atomic processes related to heavy ion collisions (Da 74, Sa 72, Me 731,

Vi 77). This interest originated from the idea (Ge 69) of testing the validi­

ty of QED in the strong electric field of superheavy elements. Since, at

present, superheavy elements cannot be produced in the laboratory, it was

suggested (P i 69, Ge 69) that heavy ion collisions could provide the strong

electromagnetic fields needed. According to this idea, during the short time

of the atomic collision, the combined charges of the two nuclei will produce

an intense field around both atoms. For combined atomic numbers larger

than 170 (Mu 741), the potential energy of the most inner-shell orbital (Isa )

will be sufficient for the decay of a vacancy in this orbital and the production

of a positron. Experiments aiming to detect the emitted positrons in heavy

atomic collisions (U + U) are now under way in Germany (Ko 79). These ex­

periments will provide a direct test of QED under extreme conditions (very

strong fields). But in order to study the positron production in the heavy

ion collisions, one-has to understand the vacancy production mechanism in

the inner-shell orbitals and the evolution of molecular orbitals during the

atomic collisions. So an extended study has been initiated on atomic colli­

sion processes.

According to the present theories in slow atomic collisions, the most

inner-shell electrons associated with each atom have sufficient time to ad­

just their motion around the two attracting centers, forming (L i 67) quasi-

1

INTRODUCTION

molecular orbitals (MO's). With non-adiabatic redistribution of electrons

in the new orbitals, vacancies may be created in the 2pct, Isct MO's. The

latter are the lowest MO's, and they are correlated to the K-shells of the

separated atoms. The filling up of these vacancies during the collisions,

from higher MO's or the continuum, can produce x-rays with a continuous

spectral profile. Depending on the MO in which the vacancy is created

(and subsequently filled), the emitted x-rays appear, in the upper part of

the photon energy spectrum, as two distinct continua. The lowest energy

continuum (which has been referred to as Cl radiation, F r 75) is due to the

transitions filling vacancies in the 2po MO, and the higher energy continuum

(designated C2 radiation, Fr 76, Da 74) is due to transitions filling the Isct

MO vacancies. These continua extend from the K x-ray energy of the

light atom to (and beyond) the United-Atom (UA) limit.

The first direct experimental verification of the quasi-molecule for­

mation was provided by spectroscopic analyses of the MO x-rays filling 2p7r

states, which are emitted during the atom-ion collisions (Sa 72). Later,

the same method was used for the detection of Cl and C2 radiation (Me 731,

Da 74). Although the detection of a continuum x-ray spectrum above the

characteristic x-ray energies provided the first indication for the existence

of MO's, significant contributions from other effects (nucleus-nucleus

Bremsstrahlung, T r 76), radiative electron capture (Bez 75, e tc .) led to

an uncertainty in the' interpretation of the observed x-rays. Besides, the

dynamic broadening of the MO x-rays (L i 74, Br 74, Gr 74) extended

their spectra to higher than the UA-limit energies, rendering the spectro-

2

3

scopic analysis even more doubtful.

In an attempt to clarify some of these difficulties, an indirect method

for identifying the quasi-moledular origin of the continuum was proposed

(Mu 74) and pursued successfully in an experiment at Yale University (Gr

74). The theoretical proposal consisted of utilizing a directional aniso­

tropy in the emission of MO x-rays, which was predicted to be unique

for this source of continuum radiation alone. Although the measurements

confirmed the existence of such an anisotropy in the x-ray emission spec­

trum, which peaked close to the UA-limit (in close agreement with the

theory), the magnitude of this anisotropy was found to depend slightly on

the projectile energy, contrary to theoretical prediction. Moreover,

according to the theory, the transitions from different MO's contribute

different amounts of anisotropy (positive or negative), and the net result

greatly depends upon the population (alignment) of each level. In the

absence of an exact theory for the anisotropy, and mainly due to the

inability of detecting selectively few MO transitions, this method has

yet to become a useful spectroscopic tool.

One way to discriminate MO x-rays from background and at the same

time selectively detect one transition alone (the 2pa-4so) was proposed by

J.S. Greenberg, a few years ago (Gr 76). According to this proposal,

the electron transitions to the Isa MO from the next orbital (2po), will

create a vacancy in the 2pa MO, which in the exit channel w ill end up as

a K-shell vacancy in one of the colliding atoms. Therefore, in principle,

one should be able to detect the MO x-ray in coincidence with the associated

K x-ray. No other transition is strictly correlated to the K x-ray so that

the measurement selects these particular MO x-rays from background or

other transitions. One of the principal difficulties in carrying out such a

measurement is the large ratio of K x-ray to MO x-ray cross-sections

6(10 to 1). This large difference in cross-sections necessitates low beam

intensities (~0. 2nA) and introduces difficulties with the accumulation of

data. But, as demonstrated in this thesis, the basic idea has proved to

be workable, and selected transitions have been observed in both symmetric

and asymmetric colliding systems. By choosing selectively specific MO

transitions, and suppressing the background, this technique provides not

only a convincing proof for the existence of molecular orbitals, but it has

opened the way for a test of phenomena closely related to the formation

and evolution of the MO's.

One such phenomenon studied herein is the sharing of vacancies

existing in one MO between two (or more) levels of the separated atoms.

According to the theory (Me 73), the sharing mechanism is due to radial

coupling between the orbitals at large internuclear distances. Using the

coincidence of the. cascade MO x-ray with the K x-rays of the heavy or

light atom in slightly asymmetric collisions (Nb on Sn), a direct verifica­

tion of the sharing mechanism is obtained. As we shall see, the close

resemblance of the spectra in the two cases (coincidence of Nb or Sn

K x-rays with MO x-rays), generically links the vacancies in the 2pa MO,

via the sharing mechanism, to the vacancies in the Isa MO, and it estab­

lishes that the coupling occurs at large internuclear distances in agreement

Another process that can be investigated with the coincidence technique

more meaningfully, is the directional anisotropy in the emission of the

cascade x-rays (2pa-»lso). According to recent theoretical investigations

(An 7811), it should be much larger than the total anisotropy obtained by

averaging over all transitions. Since the anisotropy results from the

motion of the nucleus, its measurement will reveal some very interesting

features of the dynamical evolution of the MO's.

In addition to the MO K x-ray coincidence, the extension of the method

to the study of .multiple vacancy production in asymmetric systems, can

provide more information on the formation and sharing of single and double

vacancies in the 2pa or higher orbitals (3da). In fact, a study of double

vacancy production is not only of general interest for the understanding of

vacancy production via heavy ion collisions, but as we shall see, it con­

stitutes an important aspect of the MO K x-ray coincidence technique. In

connection to-this, the formation of double vacancies in the same or different

orbitals was studied_both theoretically and experimentally in this thesis. In

addition, the coupling between the orbitals, which result in level mixing and .

an exchange of vacancies, was also examined in this manner.

Closely linked to the problem of MO x-ray emission are the questions

relating the formation of Isct MO vacancies (through Direct Coulomb excitation

or from multiple collisions). As we shall see, the assignment of the x-ray

emissions obtained in the coincidence studies depends on the degree of con­

tribution from multiple collisions to Isct MO vacancy production. One way to

5

with the theory.

6

study these two vacancy production mechanisms, is by measuring the K x-

ray cross sections for various collision systems. At the same time, using

the theoretical estimates for the contribution from Direct Coulomb excita­

tion, the importance of each mechanism can be delineated. Such measure­

ments have been carried out at different projectile energies. Besides the

valuable information provided by those studies on the I sct MO vacancy pro­

duction mechanisms, they have yielded an accurate normalization for the

cross section obtained from the coincidence data.

The thesis has been organized as follows:

Chapter I discusses the theoiy necessary for the understanding of the»

data and is divided into five parts. The first part describes the method of

using atom-ion collisions to study the inner-shell ionization of heavy atoms.

The second part describes the quasimolecular picture, which is the most

appropriate for explaining the continuum x-ray production in symmetric and

near-symmetric systems, while the third examines the emission spectra of

MO transitions. The fourth part treats the vacancy production probability in

the 2pa , Isa MO's, which are the main source of K and MO x-rays, respec­

tively. The fifth examines the existing theories predicting MO x-ray cross

sections. The last part is devoted to the discussion of the theoretical pre­

dictions concerning the sharing mechanism for one and two vacancies.

Chapter II presents all the details concerning the experiment. It includ­

es general considerations for each particular experiment, the beam and tar­

gets used, as well as the detection devices and the whole experimental appa­

ratus. In addition, the description of the electronics is included together

7

with an explanation of the data collection and presentation through the IBM

360/44MPS computer system presently installed at WNSL (Yale Un.).

In Chapter HI, the data are presented and analyzed, and are divided

in parts according to their contents. The primary data from the coincidence

MO x-rays are presented in the first two parts. The symmetric case (Nb+Nb)

is shown first, followed by the asymmetric system (Nb+Sn), where the effect

of the vacancy sharing mechanism is explained. In both parts, a detailed

analysis has been carried out to correct the data for contribution arising from

double vacancy production in single and sequential collisions. (K x-ray)-(K x-

ray) coincidence studies have provided the experimental information necessary

for evaluating the background effects and are therefore presented in the next

part. The last section is devoted to the analysis of K x-ray cross section

measurements which are utilized to estimate the effect of the multiple colli­

sions on the Iso MO excitation.

The following-.chapter is devoted to a discussion of the data. The pos­

sible background contributions are examined first in detail. Then each case

of MO-K x-ray coincidences for symmetric and asymmetric systems is dis­

cussed applying the general considerations of the first part. The (K x-ray)-

(K x-ray) coincidence cross sections for asymmetric collision systems fol­

lows with a detailed theoretical treatment of double K x-ray production,

which includes the effect of the K -L level matching. In the last part, the

theoretical predictions concerning the contributions from Direct Coulomb ex­

citation to the production of K x-rays (in the heavier atom) are presented.

In all cases, the measurements are compared with the theoretical predictions.

The last chapter summarizes the results from all the experiments

and discusses future experiments.

9

1.1 Introduction

In this thesis, the dynamic structure of the most inner-shell quasimole-

cular orbitals (MO) formed in slow atomic collisions (i.e . for projectile ve­

locities smaller than the atomic orbital velocity) is studied by separating

out selected MO-transitions to the Isa MO. The signature for the selection

is provided by the unique correlation of the 2pcr MO with the K-shells of

the colliding atoms (fig. 1.6). Thus the creation of a Isa MO vacancy, and

its filling by a 2pa—- Isa radiative transition during the atomic collision, will

necessarily produce a K-shell vacancy in one of the two colliding atoms, and

so a K x-ray will be emitted which will be correlated to the MO x-ray.

The successful selection of particular MO transitions, by the application

of the MO x-ray, K x-ray coincidence technique, provides a straightforward

proof of MO formation. With the additional use of the K x-ray, K x-ray coinci­

dence measurements, the coincidence technique supplies important information

on the single and double vacancy production mechanisms involving the most

inner-shell molecular orbitals. The study of the inner-shell (2pa, Isa MO's)

vacancy production and evolution is essential for the interpretation of the cas­

cade spectra and for their association with MO transitions. As we shall see

later in this chapter, the two orbitals (2pa and Isa) can be excited, via either

one or two collision process. In the process involving two collisions (Me 74),

an L-vacancy (or K-vacancy) is created in a first collision, and in the second

collision is promoted to the 2pa MO (or Isa MO), through rotational (radial)

coupling at small (large) internuclear distances. Since the mean life of a K-

I. THEORY

vacancy is much shorter than the L-vacancy mean life, Isct MO excitation in the

two collision process is much less likely to occur than the excitation of the 2po

MO. Of course, what is really important is the relative contribution of each

vacancy production mechanism, which will differ for the two levels. It should

be also noted that, for slightly asymmetric systems, the K-vacancy which is

produced in the first collision, w ill be shared in the second between the 2pci

and lscr-MO's. So, only a fraction of these vacancies w ill end up as Isct MO

vacancies, and the contributions from the two collisions w ill diminish. In the

same way, a 2pcr MO vacancy will be shared among the K-shells of the two

colliding atoms (fig. 1.7), increasing (as we shall see) the K-vacancy pro-i

duction of the heavier atom by many orders of magnitude. Therefore, the

vacancy sharing mechanism also has to be studied in detail.

In the one collision process the fast moving nuclei can directly excite

the electrons in each level. The one collision process can be treated approx­

imately as an interaction of the projectile nucleus and the target electron

bound in the increased potential of both atoms (UABEA,SCA, Fo 76, Ban 59).

More accurately, the nuclear motion can be treated as a perturbation, which

couples the quasimolecular orbitals and produces inner-shell vacancies (PSS,

Ba 7311). A ll the above vacancy production mechanisms will be applied for

the theoretical prediction of the single and double excitation probabilities of

the inner shell orbitals.

Closely related to the lscr MO vacancy production is the possibility of

a 2pCT+lsCT radiative transition, in which we are interested in this thesis. The

transition probability can be calculated theoretically in a static (Me 74) or

dynamic approximation (Ma74, An 781). In the first case, the emission prob-

10

abilitj7 , at each internuclear distance R, is considered independently from the

emission at any other distance. In other words, the distance R simply specifies

the energy of the emitted x-ray (through the transformation of the molecular

orbitals with R), and its time dependence does not affect the emission spectra.

In the second case, the nuclear motion is incorporated into the calculations,

and a dynamic broadening of the spectra is predicted (L i 74, Br 74, Gr 74).

In accordance with the above introductory remarks, the theory section

has been organised in the following way:

The first part deals with the molecular orbital formation in slow atomic

collisions, as it was proposed by Fano and Lichten (Fa 65, L i 67). Then, the

single and double vacancy production mechanisms for the 2p<y, lscr MO's are

presented and discussed. The presentation of MO x-ray production mechanisms

(static and dynamic approximation) follows. Finally, the vacancy sharing mecha­

nism (Me 73) is discussed with application to three different cases (2pcr->lscr

Me 73, 3dTT-»2p7r Le 76, 3da-+2po-Me 78), for single and double vacancy transfer.

Whenever necessary, an early application of the theory to the collision

systems we have used, or a reference to data presented later, are given, for

the better understanding of this section.

I. 2 Molecular orbital formation

For atomic collision velocities (v^) smaller (Mad 75) than the electron orbital

velocity of the separate atoms, the electronic motion w ill be adjusted adiaba-

tically to the moving attracting centers. Then, in the limit of complete adiaba­

tic ity, one can calculate the evolution of the atomic levels during the collision,

simply by assuming that the wavefunction of the system is a direct product

11

12

of nuclear and electronic parts and ignoring any coupling between the electrons

(Born-Oppenheimer approximation). This method has been extensively used in

the study of molecules (Hu 27). With the help of the Hartree-Fock approxima­

tion, one can calculate the evolution of the levels, for all internuclear distances

from two distant points (separated atoms-SA) to the distance of closest approach

between ions (united atoms-UA)%. then, in the level correlation diagram, when two

adiabatic levels with the same symmetries meet, they repel each other (avoid­

ed crossing, La 58).

Figure 1.1 (L i 67) represents the homonuclear system Ar+Ar, in which

the molecular orbitals have been labeled with the atomic number (n) of the

united atom states, and the projection of the angular momentum (m), on the

internuclear axis (for a, it, & orbitals m^=0,1 ,2 , . . . . correspondingly). Even

parity (gerade or g) orbitals correlate to the even parity (s ,d ,g ) in the unit­

ed atom limit, and odd parity (ungerate or u) to the odd ones (p,f,h). Accord­

ing to the complete adiabatic picture, MO's with the same and parity should

not cross i.e . in figure 1.1 3da (or 4fo) should not cross with 3sa (or 4po).

But the nuclear motion (including the rotation of the internuclear axis) can

produce crossing (at least for the lower states) forming diabatic states, and

then the Ar+Ar correlation diagram appears as in fig. 1.1. As it was pro­

posed by Lichten et al. (L i 67, Bar 72), the way to correlate the SA levels

to those of the UA is that of conserving the radial nodes (n-1-1) in both cases

starting with the lowest energy. For example, the Is state in the SA in fig.

1.1 (Ar+Ar) which has n-l-l=0 (m =0) will be correlated to Isa ,2pa , the1 g u

2s to 2sa ,3pa , and so on. In all cases, ^states are correlated with states g u

of the same total quantum number (n), while a states to those of greater

or equal (n).

The passage from a symmetric to ari asymmteric system, is not straight­

forward because, levels which for symmetric systems are degenerate in ener­

gy , now have to cross, and the dynamic evolution of states into the molecu­

lar orbitals has to be calculated, incorporating the effect of the level Cross­

ings (fig. 1.2). In asymmetric systems, there is no distinction in even and

odd parity orbitals and the only good quantum number is the magnetic quantum

number (m). Again, the nuclear motion is the necessary perturbation to make

the levels cross. At the crossing point of two levels we might have a vacancy

transfer from one level to the other which in connection with the promotion

mechanism can end up as an inner-shell vacancy, i.e . , with the crossing of

the 4fo MO or higher orbitals (fig. 1.1), vacancies can be promoted from

higher states to the 2p-shell.

For very asymmetric systems, one might have a matching of one level

of the. heavy atom with another of the light one, and then an enhanced exci­

tation will occur in the K-shell of the lighter atom because the levels are

strongly mixed, and .the excitation is very probable (swapping effect, Me 78).

In the very asymmetric case, the K-shell of the light system is well outside

the L-shell of the heavy one, and so, the probability of the inner-shell exci­

tation through the molecular orbital mechanism for the heavy atom is negli­

gible. Those inner-shells (i.e . Isct) can only be excited via direct excitation

resulting from the nuclear motion. For the lighter atom, there may still be

level matching and an enhanced K-shell excitation.

The nuclear motion, besides the level crossing, will produce two kinds

of perturbations: one from the rotation of the nuclear axis (rotational coupling),

13

14

another from the change of the internuclear distance (radial coupling). Mathe-

matically, this can be expressed as -rr = + • The rotational coupl-01 o d o R

ing (first term) generates transitions (in first order) with Anij=±l, g— g, uA-g,

u-u (Li 67). The transitions caused from the second term exhibit the selec­

tion rules S~*S> u-»u, u/>g (L i 67). In both cases, the perturbation

1 vis of the order AE ~ -=- ~ ——- where \ is the approximate distance of inter-

I Z X

action (L i 63).

1.3 Non-characteristic x-rays

The x-rays which arise from transitions to vacant inner-shell states of

a quasimolecule produced in the short time of the atomic collisions, can be

roughly divided into two categories depending on their energy and the shape

of their spectra. The first region, which is extended above the characteristic

K x-rays (Da 74), was named by the Dubna group (Fr 75,76), the continuum -

one (C l) radiation: while the following region, extending up to and beyond

the characteristic energy of the UA-limit (Da 74, Me 731),was called contin-

nuum two (C2) radiation (fig. 1.3). The two regions can be easily distinguished

from the different, slopes they present in the x-ray spectra. Heinig et al. (He 76,

77, An 771), based on the energy level diagram .suggested that the C l radia­

tion is the result of transitions to an intermediate L -K molecular orbital

(ILKMO) which are due to the 2pa vacancies. From our MO x-ray, K x-ray

coincidence experiments (fig. 1.4), it was proven that the C l radiation is

not associated to K x-rays. So, except for a small part produced by double

vacancies or sequential collisions, the large amount of C l radiation (fig. 1.4,

15

broken line 23-35 KeV region) disappears in the coincidence spectra (data points

in figure), supporting the idea that this photon energy region is produced by

transitions to higher orbitals than the lscr MO. Additional confirmation by the

Doppler shift analysis of those x-rays, indicates that they are emitted from the

quasimolecules (F r 76, Vi 77,78).

Appreciable background contributions in the C l region originates with elec­

tron capture (REC), and the secondary electron bremsstrahlung (SEB). The

REC hjLS been calculated theoretically (K1 75, Br 77) and it is an important

contribution for large projectile energies. Its average photon energy of radia-

2tion is > = | hco |+m v /2 (where v is the projectile velocity and hitl6 1 6 J.

the energy of the ground state). In our collision systems it extends few KeV

above the K x-ray peak and so it contributes only on the low energy part of

the C l radiation region.

The C2 radiation has been identified by its extension to the united atom

limit (Gi 74, Me 731, Da 74) and its associated anisotropy, peaked close to

the UA limit (Gr 74). It has been described as a radiative transition to a

vacant Isa MO (fig. 1.3). Dynamic broadening effects extend the transition

energies above the K , K energies of the united atom and background con­ey &

tributions (mainly from nucleus-nucleus bremsstrahlung and ambient background)

and complicate the identification of the high energy limit. Besides, in those

experiments (Gi 74, Me 731, Gr 74), the transitions to the Isa MO from all

higher levels have been accumulated and the special characteristics of each

level are obscured because of the energy dependent inner-shell ionization. In

the case of the anisotropy, the multi-level transitions would not allow a straight

analysis of the measured values, resulting in large errors and uncertainties.

Finally, the contribution from multiple collision for the Isa MO vacancy

production in the low Z region, and the direct Coulomb excitation for higher Z,

as well as, the dominance of relativistic effects in the high Z region complicates

the unified description of the MO x-rays.

In the following discussion, we will concentrate on lower Z systems, thus

relativistic effects are not very important. Initially the vacancy production me­

chanisms in the inner-shell orbitals of interest (2pa, Isa) will be examined. Then

the x-ray emission cross-section will be calculated in the static (Me 74) and

dynamic (Br 74, Mu 74) approximation.

1.4 2po MO vacancy production

A 2pa MO vacancy can be created in three different ways:

a) by direct excitation to the continuum or to an excited state (one-step me­

chanism Ke 73, fig. i.5a)„

b) by rotationaL coupling to a vacant 2pi/ state produced earlier in the colli­

sion (two-step mechanism, Ke 73, fig. 1.5b), and

c) by multiple - collisions (Me 77, fig. 1.5c).

In the last case, a 2dtr vacancy, produced in some way, joins the L-shell of

the separate: atom. This vacancy can live long enough to enter into a second col­

lision, where the 2pTf-*2pa promotion can operate. So, the total 2pa MO vacancy

cross-section will be expressed as

a(2po):=a1 (2pa) + a2 (2po) + a. (2pa)SO sc xuc

where,: the subscripts (sc), (fnc), refer to single and multiple collisions, and

17

the indices (1), (2), to one and two-step mechanisms respectively.

Since the mechanism of direct excitation is presented, in detail, in the

next section, it will not be considered here.

The second vacancy production mechanism, consists of a 2pir MO vacan­

cy production in the first half of the collision (before the point of closest

approach), and in the transfer of this vacancy to the 2pcr MO, via rotational

coupling;, at small internuclear distances.

The 2pTt MO can be excited via radial coupling to vacant higher tt states,

or directly to the continuum. In the first case, the coupling between the 2pn

and other it states can be approximated as a vacancy sharing mechanism be­

tween two 7t orbitals if we assume that the vacancy transfer occurs at large

internuclear distances where there is a constant energy gap between the two

2orbitals. In this approximation the o (2pa) cross section w ill be expressed

S C

as (Fas75, Me 761)

2cr (2pa)~N(v )o sc 1 rot

where is the rotational 2pir-»2pa coupling cross section per incident 2p?tx

vacancy, is the projectile velocity and N(v^) the probability of exciting the

2pn MO. Based on the vacancy sharing mechanism formulas, one can express

the 2prr excitation probability as,

e lwhere F (H) is the 2p-shell binding energy of the heavy atom. More general-

2p

ly one expects that, tx

N(v^)«e f

where a depends upon the collision atoms.

1 8

Experimentally, one finds that, the exponential law is followed well for

atomic collisions with Z , Z <10 (Fas75, Me 761). There also exists a semi-JL ^empirical formalism by Lennard et al. (Le 78) valid for all collision systems

which treats both (a) and (b) mechanisms in a unified way. They use an empir­

ically determined function of a few parameters of the collision systems. The

advantage of this procedure is that it can reproduce all available data on 2pa

one-coll.ision excitation within 40%(Le 78). Based on this formulation (Le 78),

we have: calculated that for our collision systems (Nb+Nb and Nb+Sn), the con­

tribution of both (a) and (b) processes is less than 30% of the total cx(2po) cross

section.

Since, the third mechanism is the most significant, it w ill be studied in

detail. At'the beginning,. the couplings of the 2pc MO with other orbitals will

be considered. Then the simple model of Briggs et al. (Br 73) for the 2p7T-»2pa

rotational coupling will be presented. Finally, the a (2pa) w ill be calculated.me

a) The couplings to other orbitals

From the correlation diagram of the molecular orbitals for symmetric

collisions (fig. 1.6), it can be seen that the K-shell of the separate atoms is

associated with the 2pcr and Isa MO’ s. The 2pa MO later in the collision cross­

es the 2sa MO and meets (in the non-relativistic limit) the 2pn orbital at very

small internuclear distances. For symmetric or asymmetric systems (Br 75,

Ta 75) it has been shown that there is no appreciable transfer of vacancies

into the 2sa MO at the crossing. For small internuclear distances a vacancy,

initially in the 2prr MO, will pass to the 2paMO through the rotational coup­

ling. Actually, the rotational coupling transfers vacancies from the projec­

tion 2p rrx of-the 2piT MO, on the collision plane xz (where z is along the

19

internuelear axis).

In slightly asymmetric collisions in which we are also interested , in

connection with the cascade MO x-rays, the K-shell excitation of the light at­

om results from a 2pcr MO vacancy, while the K-shell of the heavy element

can receive its vacancy either from the Isa MO (fig. 1.7) or the 2pa MO

through the vacancy sharing mechanism (Me 73). Since a(2po) » a ( ls a ) , the

K-shell vacancies are mainly produced from the 2pa MO in symmetric or

slightly asymmetric systems (M 77).

Actually, all four MO's Isa, 2sa, 2pa, and 2ptt are mixed, but it can

be shown (Br 75, Ta 75) that the main part of the 2py MO vacancies is trans­

mitted to the 2pa level; in any case, the 2sa-2pir does not interfere with the

2pa-2piT coupling since they occur at different internuelear distances. The

2p7r—2pa vacancy transfer probability, at small distances of closest approach,

is largely independent of the coupling to other MO's. Experimentally, this

means that the transition from symmetric to asymmetric systems, is not

accompanied with any abrupt change in the K x-ray production for the light

system. In the following part the 2pT7-2pa rotational coupling is considered.

b) The 2pir-2pa rotational coupling

The rotational coupling can be calculated by using one electron wave-

functions in the Hartree-Fock approximation, or the H* like MO's and from

them, the 2ptt—2pa vacancy transfer probability, at each impact parameter,

can be calculated. The solution (Br 73) is two coupled differential equations

for the amplitudes of the 2pTTx > 2pa states respectively in the total wave­

function.

where b is the impact parameter, R the internuclear distance, e , e , theJL

MO energies, and f(R) the matrix element of the rotation operator between

the two one-electron wavefunctions.

The solutions for C , C do not depend strongly on the details of theJL c t

path of the projectile, and the straight line approximation can give results

accurate to 10% compared to the actual Coulomb trajectory (Br 73). But they

do depend greatly on the phase factors, and the detailed calculations of the

MO energies are of great importance.

The probability of a single vacancy transfer P(b)= |C (+°°)| depends

greatly on the impact parameter (Br 72). As seen in fig. 1.8a at lower ve­

locities it has only a smooth peak and the effect of the Coulomb repulsion

reduces appreciably the probability function. Such repulsion leads to an ef­

fective threshold in the cross section for the K-shell vacancy production. At

higher velocities, the probability splits into two parts. The low peak is a

very pronounced one (fig. 1.8c) reaching almost unit probability at impact z 2

parameters (M is *be reduced mass), corresponding to 90° scattering

in the CM system, that is, at half of the distance of closest approach in

head-on collisions. The other part can be approximated using a straight line

projectile trajectory and it is this part that contributes most to the total

cross section (averaged over all impact parameters).

The almost perfect interchange of the two MO's, at 90° scattering angle

(CM system) can be explained from the degeneracy of the 2pcr, 2ptr levels in

the norirelativistic limit. Therefore, the rotation of the internuclear axis will

merely interchange the quantization, without changing the internal energy of

the system (Br 73).

The double or single vacancy transfer probability can be easily calcu­

lated because the one-electron operators do not mix the spin states (Br 751).

If we define p* as the average number of 2p7r vacancies in the two spin sta-X

1 2tes ( ± ) , P0 and P_ as the number of single (1) or double (2) 2pcr MO 2p<r 2pcr

2vacancies, and P(b) (=|C (+®)| ) as the vacancy transfer probability due to

the 2prr-*2pcr rotational coupling, we have

P* = p+P+(l-p “p ‘ ) + p "p "(l-p +P+)2pcr

+ + - - - + - += p P + p P - 2 p p P P

and

2 + - + -

» p p

Since, p+=p =p and P+= P = P(b), we finally have

P 2o<T 2Pp (b)(1-Pp (b»

2 2 2p2p/ p p (b>

The total number of vacancies in the 2pcr MO will be;

P „ = +2P? = p+P++p“p ”= 2pP(b)2jD<r 2pcr 2pa

The 2p77 vacancy occupation probability per spin state p, can be connectedx

to the number of 2p-shell vacancies available in the projectile, through a sta­

tistical factor (Br 72). Since the 2p-shell has 6 states, the probability of a

2ptt vacancy per spin state will be 1/6 (assuming an equal distribution of

vacancies in each orbital) for asymmetric collisions (fig. 1.2) or 1/12 for

the symmetric case (fig. 1.1). So, if rj is the number of 2p-shell vacanciesLin the p r o je c t i le , p=7j /12 (s y m m e tr ic c a s e ) o r p=n /6 (a s y m .) .

l

These results are used in Ch.IV in connection with the K x-ray, K x-

ray coincidence cross sections.

A scaling of the system from the H*'molecule to a heavy molecule, canCt

be carried out provided it is possible to scale the energy difference (e - £0)X 2

and the rotation factor f(R). Bates et al. (Bat 53) proved that this energy

2difference scales approximately as Z where Z is the shielded charge of the

s s

nucleus. The matrix element f(R), was shown (Br 73) to scale approximately

as f (R)-*f (Z R); therefore, the H* like system can be used to calculate theA n S 2

cross section and the probability function for a heavier atom (disregarding

relativistic effects). According to this theory, length scales as Z 7 time ass

-2 -2 2 Zg , and velocity as Zg . Cross sections and energies scale as Zg ,Zg where

Z-1<Z <Z-0. 5. s

Extending the scaling law to asymmetric systems, two scaling parame­

ters can be used (Ta 76). Again, it is required that the internuclear potential

and the energy splitting of the two levels 2p7T, 2pa, scale with the various

systems (Ge 61). As a result, the scaling of the cross section with two

parameters depends on the screened charges Z =Z -1, Z =Z -1 (Ta 76).S1 1 ®2 2

The spin orbit interaction will split the two levels 2p7T, 2pcr; thus,

they will no longer be degenerate. For instance, in the Nb on Nb collision

system, the energy splitting of the two levels at the UA limit is about 2.2 KeV.

23

This energy splitting will have the tendency to reduce the rotational coupling

between the 2q7t, 2pcr MO's, decreasing the probability of vacancy transfer.

A complete relativistic treatment of the process can be carried out by

using two-center Dirac equations (Mu 73). In a simple approximation, the

spin orbit, interaction effect on the rotation coupling has been calculated as

a function of the projectile velocity and some characteristic of the colliding

atoms (An 77III). With increasing projectile velocity, the total .effect on the

vacancy transfer probability becomes smaller. In the Nb on Nb system, the

change in the rotational coupling due to the spin orbit splitting is less than

5%.

c) 2pa vacancy production cross section

As stated previously, the main vacancy production mechanism for the

2pct MO, is expected to be the 2pT7—2pcr rotational coupling. So, in order to

calculate the 2p<x vacancy production cross section, one needs an estimate

2of the average number of 2pttx vacancies. For thick targets ( 2: 2 0 0 jjg/cm ),

the repeated.excitation of the projectile electrons produces an equilibrium

distribution of vacancies among the projectile inner-shells. The 2071 orbital

is correlated to the 2p-shell of the projectile (fig. 1.1, 1.2); hence, a calcu­

lation of the equilibrium distribution of 2p-shell vacancies is needed.

Following Meyerhof et al. (Me 77), one can express the equilibrium

value of 2p-shell vacancies per projectile in solid targets as:

^ “(V'ViVproj^'T1 '

where v is the projectile velocity, rj the atomic density, t 1 the decay 1 2t lx

24

constant for x-ray emission from a projectile with one 2p-shell vacancy and2p

a .(ML,v J the projectile 2p(L +L„) x-ray cross section, proj 1 2 3Then, the 2po vacancy production cross section from multiple collisions

will be:

/o v 1 eq a ■ (2pa) = — T) w a , , me 3 2p rot

where is the 2prrx— 2pa rotational coupling, w is the 3dir— 2pir vacancy

sharing probability (see 1.7a), and the factor (1/3) is the previously derived

statistical factor for both atoms and spin states.

The final formula will be:1 2d

a (2po) = — ?70v t w n a .a . . me 3 2 1 lx 2p proj rot

. This formula will be used later in this thesis (Ch. IV) for the check of

the K x-ray cross section data. It will be shown that it fails to reproduce

both the absolute values of the cross sections and their projectile energy de­

pendence. So, it seems that the above formula does not account correctly

for the distribution of vacancies in the 2p7r MO. If the contribution from the

other two processes is also included, the total cross sections will not change

by more than 40%. Thus, there will still be a discrepancy between theoret­

ical predictions _and experimental values, for the atomic collisions of interest.

1.5 The Isa vacancy production

Since the M O x-rays of interest are the result of transitions to a va­

cant Isa MO, it is necessary to study the different mechanisms for produc­

ing a vacancy in this orbital.

The Is a M O vacancy production can occur in solid targets through

one or two collision process. In the one collision process the only way to

excite the Isa M O is through direct excitation due to the nuclear motion.

It has been proven (at least for heavy systems) by Greiner et al. (So 78)

that this occurs mainly after the UA limit. The two collision process con­

sists of the initial production of a 2pa M O vacancy in a first collision. This

can then enter into a second collision where it can follow the Isa M O (fig. 1.9).

Since the 2pa M O vacancy production cross section, a(2pa), is much greater

than the cross section a (Isa) from a single collision, it is expected thatSC

the two processes will compete with each other and their relative contribu­

tion will vary with energy and colliding atoms. In fact, the contribution to

Isa M O from multiple collisions (a (Isa)) will depend on the average meanmelife of the ls(SA) vacancy and the projectile velocity. Since this mean life

4scales as l/Z and the K-shell electron velocity as Z, it is obvious that

for very heavy systems (Pb or U) with the same ratio of Vproj vei *

cross section (a (Isa)) from direct excitation can dominate, while for ourSC

collision systems, both mechanisms will contribute appreciably. In the fol­

lowing section, some approximate methods (Mer 58, Ga 701, Brd 6 6, Ba 78,

An 78III) for predicting the Isa M O excitation through direct excitation are

presented. Since the same methods can be applied to other orbitals as well,

25

26

they will not be specialized to the Isa M O alone. After that discussion, the

static approximation for the M O x-ray production in one or two collision

process will be described (Me 74).

Direct inner-shell excitation

In the case of very asymmetric atomic collisions, or velocities larger

than the electron orbital velocity, the inner-shell electrons will be mainly

excited via direct Coulomb interaction. The Coulomb interaction is the result

of momentum transfer between the two particles. To ionize an electron, theU0momentum transfer must be greater than q , where = (°e *s tbe

binding energy of the electron and v the projectile velocity). This means

that, the impact parameters, which contribute most to the ionization, have - 1 vvalues q =-- (ftco is the binding energy of the electron). Therefore, for0 co e

e -islow collisions (when q^ <<aj?;)> tbe ls<T ionization occurs at distances

much smaller than a (K-shell radius).KThe ionization cross section can be calculated using the plane wave

Born approximation (PWBA), the semiclassical approximation (SCA), or the

impulse approximation (BEA). In the PWBA (Mer 58), one ignores the re­

pulsion of the two nuclei and treats the projectile-electron interaction in the

first order, assuming that the electron levels are those of the unperturbed

target. The nonrelativistic PWBA, predicts a universal form for the ioniza-2tion cross section divided by the Z . (the strength of the Coulomb field),pro;

n 2a~~2 = t ~ 2 fP W B A (0s’V ’Z Z 77proj s s

27

where 9 = -- -s z2 is proportional to the binding egergy predicted by thes ghydrogenic wavefunction ( c =---),

s oJ l nis proportional to the ratio of the incident energy to (e )

sZ is the effective charge of the state (s) accounting for screening.s

Since the PWBA approximation does not account for the disturbance of

the target electrons (increased binding) and the nuclear repulsion, it deviates

higher energies, second order polarization effects (Ba 731, Ba 78) dominate2. 5and give a characteristic Z ’ dependance on the cross section.

The direct excitation can also be viewed as a direct energy exchange

between a free electron and the incoming particle. The effect of binding to

the target nucleus is then simply taken into consideration by weighting the

distribution of electron momenta associated with the bound states. This is the

Binary Encounter Approximation or impulse approximation (BEA) developed

by Gryzinski (Gry 65) and Garcia (Ga 701, 7011, 71). The actual difference

between Born and impulse approximation is that the first estimates the effect

of the Coulomb interaction with the electron as a perturbation to the atomic

states of the target, while the second estimates the effect of the nucleus

only as supplying a distribution of initial momenta to the free target electron

colliding with another particle.

If hydrogenic velocity distributions are used for the bound electron,

the BEA predicts that the product of the cross section with the square of

the binding energy is a universal function of the incident energy i.e:

2from experimental values in the low energy region (17 /8, <0.1), and it reach-K K2es asymptotically the data at the high energy region (0.1 <77 /0 <1. 0). ForK K

where Z^, X are the charge and mass (in electron mass units) of the projec­

tile. Comparing the PWBA and BEA, we note that:,2,4 ,2 2 \

andzi

®k m i uk xukIn the low energy region, the BEA deviates from the data, because at such

velocities quasi adiabatic changes of the electronic energies occur (Brd 661,11,

Ba 731). One semiempirical way (Fo 76) to include those adiabatic effects,

is to replace the electron binding energy by the value it has at the UA limit,

and the projectile charge by an effective charge which considers also the ef­

fect of the target nucleus (UABEA).

Thus, according to UABEA,

Zeff° = - 2 - W E/X U UA>

UAwhere

Z 2 f f = [ Z £ + Z 2 ] f o r t h e 2 P a m o

and

Z = Z for the Isa MO.eff 1

In the semiclassical approximation (SCA), Bang and Hansteen (Ban 59)

have treated classically the effect of the nuclear repulsion on the total cross

section. For large energies (large 77 ), the effect of the repulsion is negli-sgible, so that the results of SCA coincide with those of PWBA, but for small

29

77, the SCA cross section is lower compared to the cross section calculated

using the PWBA method.

The SCA approximation in its first formulation fails to take into account

the increased binding of the excited electron in the combined field of the two

nuclei . This increased binding will result (Brd 66, Ba 731) in considerably

smaller K-shell ionizations than those calculated from the SCA theory. It was

suggested that the replacement of the K-shell binding energy for the isolated

atom by an augmented energy, considering the effect of the projectile charge,

should give a more successful description of the K-shell ionization cross

section. Later (Ba 7311) this theory was explained theoretically in the frame­

work of the perturbed stationary state approximation (PSS). The same pro­

cedure was extended for the L-shell (Brd74) and the K-shell cross section

for intermediate projectile velocities (v1«v where v is the orbital veloc-1 2K 2Kity of the K-shell electron in the heavier atom). Hence, the theory can now

explain well the experimental data of K-shell ionization cross section; for

collision atoms with Z /Z , ranging from 0.03 to 0.3 and v /v , ranging1 2 1 2Kfrom 0.07 to 2.0 (Ba 78).

The PSS theory consists of the expansion of the total wavefunction in

terms of time dependent adiabatic wavefunctions. Therefore, it incorporates

the quasimolecular effects in the atomic model, and tries to bridge the two

different approaches (molecular and atomic model).

Finally, there is a semiempirical method developed by Anholt and Me­

yerhof (An 7711, 78III) along the line of the PSS approximation. The big ad­

vantage of this method is its attempt not to explain the total cross section

30

theoretically, but to supply some factors that will correct the theory so it fits

the data. Those factors have been derived by comparing the deviation between

experiment and prediction (App. Ill); they account for relativistic corrections,

Coulomb deflection, and the increased binding. The method does not consid­

er the contribution from the charge exchange, the target atom recoil, or the

polarization effects. Polarization is the result of the next order perturbation2 3term which deviates from the Z . law and it is proportional to Zproj proj

Finally, when the increased binding and the relativistic effects are included,2 5the approximate projectile charge dependence law is Z .. Since the contri-proj

bution of the polarization of the wavefunctions at high velocities is more

important than the increased binding effect, the theory is not expected to

reproduce the experiment well in that region. The contribution from recoil

and charge exchange effects is expected to be less than 10% of the total

cross section. The method seems to reproduce the data within a factor of

two for various atomic collisions and for projectile velocities v^ < v . It

also suggests a highly increased Isa vacancy production for the super heav­

ies in qualitative agreement with other predictions (Bet 76, So 78) and

recent experiments (Gr 77). But due to the many correction factors it in­

cludes, it is still uncertain how well the theory fits the data in our colli­

sion systems. More detailed calculations from the theory are presented in

Appendix 111.

31

I. 6a M O x-ray production in the static approximation

In the static approximation, the production of a Isa M O vacancy and

its filling with the emission of a M O x-ray, are considered two completely

independent processes. The emission of a M O x-ray at an internuclear dis­

tance R is also considered to be unaffected by the nuclear motion. Under

these assumptions the differential cross section from one collision Isa exci­

tation will be (Me 74):

, 1°MO ^ dR 1_ = 2 j;db(2*)p a» w —x x R x

where plsa.(b) is the probability (per spin state) of exciting the Isa M O at

the impact parameter b, E is the photon energy, r the Isa M O vacancyX X

mean life, and v the radial component of the projectile velocity v at theXv 1distance R. Since,

22Z Z e 1 2where D = ------ (distance of closest approach)mVj

m im 2m = ----- (the reduced mass)■ V m 2

we get:

1MO= a(l8^dR/dE^ F(i.) _ beoause j,)

where

F (b)=lf b f b\ [JdbbPflD)]"1 , b- =Vr2-RD .Vl-b /b» 0

From the two collision Isa M O excitation the differential cross section

will be (Me 74):

32

M O . \ R dRX X X

where a is the K vacancy cross section, w the 2pa— Isa vacancy sharingK

factor, r] the number of target atoms per unit volume, and r. the meanK

life of the projectile K-shell vacancy.

Dividing these two differential yields, we find that the relative contri­

bution of each mechanism (t ) is:Xv

[—x b’2

V — r ~ =M <V a(lscr)]<w ’!Tkv1) j W ) •

L dE J xApproximating the single vacancy production P(b)~(l + e^^a) 1 (Han 75) and

assuming (Me 74) that a«a (a is the Bohr radius) we get after some manip-K Kulation that:

2

The value of the ratio t depends upon the internuelear distance R atXV

the moment of the x-ray emission, and thus upon the x-ray energy (t =t(R)=XV

= t(E )) through the parameter b' = ,/fc2-DR.xSince,

2-I f — l + exp(b'/a) ,

l / l + exp(|-/LV ) j

we can easily set limits on the value of t for different collision systems.XV

4 6We know that r, ~ l/Z and if a^n, , then a~l/Z so that the ratio t,~l/Z .k k RThis supports the reference made earlier that while in light systems the

33

two collision mechanism might dominate in the Isa M O vacancy production,

in much heavier systems exactly the opposite will happen. Applying the above

formula (t )in three systems of interest (assuming a^a,) and considering

M O x-rays produced at impact parameters b'<a we get:Ka) For the 95 MeV Ni+Ni data (Vi 77) 14<t <25; therefore, the mul-K

tiple collisions are dominant,

b) For 190 MeV Nb+Nb (our data), 0.7<t <1.3 and the two mechanismsKseem to contribute equally,

c) For 190 MeV Nb+Sn (our data), . 08<t <0.12, the one collisionRprocess is dominant.

For much heavier collision systems (i.e. Pb+Pb), the ratio t is evenR

smaller (t <0.01). It should be noted, that towards the UA limit (b'—■ 0), itthe relative importance of the one collision process increases. Besides, the

value of a in the exponential can be considerably smaller than the Bohr ra­

dius (at least for the heavy atoms), causing the importance of the two col­

lision process to be further reduced. In any case, for the symmetric systems

we are interested in (Nb+Nb), we cannot ignore one of the two processes

and both have to be considered.

The formulas of this part, together with some extensions to include

the probability of double vacancies, will be used in Ch.IV for the theoret­

ical estimate of the double vacancy contributions in the coincidence yields.

34

I. 6b Dynamic theory of M O x-ray emission

An important part of the M O x-rays, is produced by transitions from

the low lying states. In the dipole approximation, the differential cross sec­

tion integrated over all impact parameters is (Ma 74):00

„ lDcj<“> I2 •0

where a is the fine structure constant, and the continuum dipole velocity

matrix element D (to) is equal to the Fourier transform of the time depen­

dent dipole matrix term, D ,(R(t)) = <ls | v |j>= ico<lscr|r |j> (j=2p<j, 2p77, 3d a,CJ.... ). Assuming that ^^ft) *s probability amplitude of a lscr vacancy

at the time t, the matrix element D . is (Ma 74, Mu 74, An 781):CJ00

v w>= I Di <R(t)) ai s / > exp (‘ d * + ^ ,dt 0 °

where r is the total (x-ray plus Auger emission) decay rate of the initial

MO. Together with the time dependence of the varying states it is respon­

sible for the collision broadening and it will not be considered here. For

the purpose of this section it can be omitted.

Using the IF-like MO's scaled for the heavier systems, Briggs and

Dettmann (Br 77) calculated the cross section and the anisotropy for tran­

sitions from the 2pa and MO's. As we have seen before, the 2po and

2p77'x levels are highly mixed at small internuclear distances because of the

rotational coupling and they can be described by two coupled first order

differential equations. In the more general solution, the four MO's 2pa,

2pTT, 3dcr, and 3d7t should be used (An 781, II) to calculate the one electron

35

wavefunctions of the MO's with the HF procedure. This method results in ma­

ny coupled differential equations, analogous to those of Briggs, which can

be solved with the help of a fast computer. In general, the consideration

of the four MO's should give good results for symmetric and asymmetric

systems, provided the right initial conditions have been supplied. But, it

is obvious that there is little possibility that one can find out exactly the

number of vacancies in each level, as they change with the projectile ener­

gy. So, it seems that there is no easy way to predict the M O x-ray spec­

trum with precision, unless one selects just one transition, as is done in

the coincidence experiments of this thesis.

For the x-ray emission anisotropy, the situation is even worse, since

the transition from each M O exhibits different (positive or negative) aniso­

tropy (An 7811). Thus the total anisotropy depends only on the vacancies

present in each MO.

Recently, Anholt (An 781) has proposed that the M O x-ray yield apart

from a normalization factor should have, for various colliding atoms and

projectile energies, the same shape for x-ray energies normalized to the

UA limit. The normalization factor defines the probability of an initial lscr

M O vacancy and it should include the effect of multiple vacancies in the

outer MO's, so it depends upon the projectile energy. This scaling does

not apply for x-ray energies above the UA limit, because of the dynamic

broadening, or for systems where the direct Coulomb excitation is the main

mechanism of the Isa M O vacancy production, and the multiple collision

effects for producing Isa M O vacancy are negligible.

36

In the one collision process, and especially for the dominant part of

one-step excitation and subsequent de-excitation, the two steps cannot

be treated incoherently and the whole mechanism has to be considered as

a second order process (Th 77). For the multiple collision Isa M O excita­

tion mechanism, the inner-shell vacancy production and its filling from high­

er orbitals can be effectively considered as separate processes. Thus, it

is safe to calculate the transition probability first, assuming the presence

of one vacancy, and then multiply the total yield by the average number of

vacancies in the inner-shell MO.

As noted above, for the atomic collision systems we are interested

in, both processes may contribute to the total cross section, but for very

heavy systems, the one collision process will be the main Isa vacancy pro­

duction mechanism. For such heavy systems the relativistic effects are

very important and. the whole process of M O x-ray production should be

considered from the very beginning relativistically (Sm 75, So 78, Mu 78,

Hei 78).

It seems that a unified method is needed which will incorporate rela­

tivistic effects together with single and multiple collision contributions. For

this to be done, the relative importance of each process has to be determin­

ed.

37

1.7 Vacancy Sharing

a) Single vacancy

In the asymmetric collisions there is a finite probability that a 2po M O

vacancy will end up as a Is vacancy for the heavy atom (Me 73, fig. 1.7).

This vacancy sharing can happen because the two levels are mixed due to the

nuclear motion. Thus, the 2pa, Isa MO's can be mixed by radial coupling

and the coupling is strong in the region where the energy difference between

the two MO's is constant i.e. , for R > a , + a (a., are the K-shell atomicIK IK

radii). Since this region is well separated from the region where the 2pn-

2pcr rotational coupling is operative (R<a ), the two effects .will act inde-Kpendently (Br 75).

An apparent coupling between the 2p7r, Isa MO's has been shown (Br 75)

to be a two-step process (2p7T— 2pa rotational coupling at small internuclear

distances, followed by 2pa—- Isa radial coupling at large distances). The pres­

ence of the 2saMO does not change the 2pa—- Isa radial coupling appreciably.

So, it seems that the two-level one electron system (Dem 64) is the most

appropriate description of the 2pa— Isa vacancy sharing mechanism (Me 73).

According to this theory (Dem 64) if we express the Hamiltonian interaction as:

H12 = 0 exP(_kR) = /Sexp(-yt) , the probability of a 2pa-*-lsa vacancy sharing is:

w(2po-~lso) = {1+ exp(y A E ) ] 1 ,

where AE is the energy difference of the two coupled levels.

Because

/Tf« (/T^ + /I^)/2 (1 , I the K-shell binding energies of the two

atoms),we conclude that:

w = {1 + exp(2x)}_1 , = exp(-2x) ,

and the approximation

where

77/5- . ^2x =

Similar relations have been established for the vacancy sharing between

the levels 3dfl- , 2p7T or the 3da, 2pa (K-L matching).

In the first case, the vacancy sharing factor is:

w 2p = [ 1 + exp(2x') ] _1 ,

where

2x» = 0.89vi

I|j , 1^ are now the. L-shell binding energies of the heavy and light atoms

and the factor 0. 89 is empirically determined (Le 76).

Since it is unknown how well those higher orbitals (3d77). follow the mo­

lecular orbital paths, it is not clear how applicable the Lennard formula

(Le 76) is.

The two level Meyerhof formula (Me 73) has been shown to be correct

for a variety of colliding atoms and intermediate energies. For very large

39

velocities the 2pa-*-lsa radial coupling interferes with the 2p7r— 2pa rotation­

al coupling and the Meyerhof formula loses its validity (Ta 75). This also

happens for very asymmetric systems, but in such cases the sharing ratio

is very small, so the discrepancy is not important. For very low energies,

a deviation from the above theory is also expected (Jo 75).

As will be explained later in detail (Chs HI, IV), the validity of the

assumption that the vacancy sharing mechanism is a coupling between MO's

has been tested using the M O x-ray, K x-ray coincidence technique in a

slightly asymmetric system (Nb on Sn). By comparing the two M O x-ray

spectra, which are in coincidence with the Nb or Sn K x-rays, it has been

demonstrated from the similarity of the two spectra at all x-ray energies

that the mechanism does involve MO's and occurs at large intrnuclear di­

stances all in agreement with theory.

b) K-L matching

In the case of very asymmetric collisions, the potential energy of the

K-shell of the light atom can match the potential energy of the L-shell of

the heavy atom. Then the 3da M O lies very close to the 2sa, 2pa, 2p7r

MO's and it is correlated either to the 2s (heavy) in the unswapped case

(fig. 1.10a, Me 78), or the Is (light) in the case of swapping (fig. 1.10b).

Because of the nuclear motion (resulting in radial and rotational coupling

between MO's) these four MO's are highly mixed and an exchange of vacan­

cies can take place in a way similar to the 2pa— Isa radial coupling.

Meyerhof et al. (Me 78), based on the two state coupling model of

Nikitin (Ni 62), developed a theory for the vacancy sharing probability

40

between the 3da M O and each of the four exit channels: Is (light), 2s (heavy),

2Pf/2(heavy), and 2p^2(heavy). Assuming that there is a relatively weak

coupling (rotational or radial) at large intemuclear distances between the

2pa, 2pir, and 2saMO's (fig. 1.10a), one can use the Nikitin model and

express the vacancy sharing probability between the 3dcrMO and each of the

three levels separately i.e:exP {2 X. (l + cos0.)}-l

Poi exp(4 .) -1

where

2Xj = it (y r ls- / y / p /2 , i=l, 2, 3 corresponding to the 2sa,

2pa, and 2p7rMO's.

0. are experimentally determined angles (for Ni on Sn systems they are

119* 72°, and 54“).

Then >

a.P = --- -— = P f P /(l-P )}i a (3do) o oi/K oi

where

pQ =

41

c) Double vacancy sharing

The problem of the double vacancy sharing is still open because the

experimental error is usually too large to rule out one specific theory.

By double vacancy sharing one means the existence of a double va­

cancy in one shell (which will be completely empty), and the subsequent

sharing of one or both vacancies between two orbitals. Because the radial

coupling (responsible for the sharing of vacancies) does not mix the two

spin states, each vacancy can be treated independently. But the increased

ionization due to the second vacancy will produce a small shift in the energy

levels and so, the vacancy sharing probability will be different in the sharing

of the first or the second vacancy. How much this small shift in energy

changes the vacancy sharing ratios is still under dispute.

Defining as w^ the transition probability of one vacancy transfer when

there are two in the 2pa M O and w the transition probability of one vacancy

transfer when there is one vacancy in each 2pa, Isa MO's, we have for

the sharing factors (Le 78, Ri 78, Mac 78),

W k k (H )=W l W 2

'vkk<LiH)*w 2(2‘w r w 2)Some recent theoretical research (Os 78) and experimental evidence

(Le 78, Ri 78) supports the simplest model admitting that the two vacancies

follow the single vacancy sharing factor, so that the double vacancy sharing

factor should be expressed as (Le 78, Ri 78):

w kk<H>=w 2

w (L, H)= 2w(l-w)KKMacdonald et al. (Mac 78) suggest that instead of the \v «w «w1 2

approximation, more accurate values should be used. Taking as binding

energies I , I their difference E can be calculated with the Hartree-FockJl Ct

approximation and then the values of the two factors vvj>W2 can deduced.

For instance, for 32 MeV Sulfur on Argon they claim that w W 1.2w«1.4w ,1 2giving an overall 10-20% error in the double vacancy w (H)/w (L) com-KK KK

2pared to the approximate formula (w/l-w) .

As we shall see, in our case the question of an exact determination

of the sharing factor is not decisive, so the simplest model by Lennard

et al. (Le 78) has been followed in all calculations of the double vacancy

sharing factor (sections III.4 and IV.4).

Up to now, we have discussed the vacancy production and sharing in

the most inner-shell orbitals. In this thesis the 2pa—-Isa M O cascade transi­

tions, have been selected to supply information on the M O formation and

evolution. In order for this cascade mechanism to be uniquely correlated

to the special characteristics of the MO's involved in the transitions (2pa,

Isa), other mechanisms producing M O x-ray,K x-ray coincidences should

be carefully considered. As we shall see later (Ch IV) in detail, three other

mechanisms can contribute to the cascade M O spectra:

1) sequential independent collisions of the same projectile with different

target atoms

2) multiple inner-shell vacancies in the 2pa, Isa MO's which will be

42

W kk(LH 1"W)2

43

filled from the higher orbitals (or the continuum) emitting M O or K x-rays.

3) various couplings between MO's (rotational or radial) which can alter

the association of the cascade M O x-rays with the 2pcr—-lscr transitions.

The importance of all these background contributions to the cascade

spectra will be discussed in chapter IV in detail. Especially for the first

two mechanisms (double vacancies and sequential collisions) an extended

analysis, based on the static approximation (1.5, Me 74), will be presented.

Approximately, the contribution from double vacancies and sequential collisions

will be calculated from the K x-ray, K x-ray coincidence yields in slightly

asymmetric and very asymmetric collision systems (KX-KX measurements).

In the next chapter the experimental method is outlined.

Fig. 1.

Fig. 1.

Fig. 1.

Fig. 1.

Fig. 1

Fig. 1.

Fig. 1.

Fig. 1.

Fig. 1.

Fig. 1.

1 The energy levels of the diabatic molecular orbitals of the Ar+Ar

system from the separate atom, to the united atom (Kr) limit.

2 The energy levels of the diabatic MO's of an asymmetric system.

3 The non-characteristic x-rays for symmetric collisions (Nb+Nb).

4 M O x-rays from 160 MeV Nb+Nb collision system.

Data: X-rays in coincidence with K x-rays, broken line: Singles

spectrum normalized to the data at 40-50 KeV.

5 The three different processes of producing a 2pa M O vacancy in

symmetric atomic collisions, a) 2pcr M O direct excitation, b) 2p:7

excitation followed by 2pi7— 2pa vacancy transfer, and c) via

multiple collisions.

6 The low lying M O ’s of a symmetric collision system and the cor­

responding relativistic levels.

7 The lowest MO's in asymmetric collisions and the region of vacancy

sharing mecjianism.

8 a) The projectile's Coulomb trajectory (Br 73), b) and c) The impact

parameter dependence for low and high projectile velocities (Br 73).

9 The M O x-ray production in one and two collision processes.

10 The correlation diagram in the case of very asymmetric collision

systems (K-L matching), a) unswapped case, b) the case of swapping.

IO O O « -.01

Kr

i i I I I______ I— I. 0 2 . 0 5 .1 .2 .5 1.0 2 5 10

A r 4- A r

Figure 1.1

U N IT E DA T O M

S E PA TO M S

5 F

5 D

5 p 7T

5 P < ^ 5 per 5 S

4 F

4 0

4 P4 S

3 D

3 P7T 3 P < T 3 p o -

3 S 3 s c r

2p7T 2 P < T p o -

Figure 1.2

CO

UN

TS

Figure 1 .3

CO

UN

TS

ENERGY (keV)

Figure 1.4

PROCESSES OF Zpcr MO VACANCY PRODUCTION

c)TWO STEP PRO CESSFIR S T COLLISION SECOND COLLISION

R=-cO R=0 R= + CO R=0 R=+0O

Figure 1. 5

BIND

ING

ENER

GY

SYMMETRIC SYSTEM

2p cru - — - 2 p y z O /2 )

lscrg -— -Isi/2 0/z)

g: even parity state

u: odd parity state

Figure 1.6

ENERGY LEVELSUNITED ATOM SEPARATE ATOM

R = 0 R =00

H: Heavy atom U Light atom

F i g u r e 1 . 7

a) THE CLASSICAL TRAJECTORY OA IS THE INTERNUCLEAR LINE

Impact Parameter b (au) Impact Parameter b (au)

b ) IMPACT PARAMETER DEPENDENCE FOR (A) THE COULOMB TRAJECTORY AND

(B) THE STRAIGHT -LINE TRAJECTORY

Impact Parameter b (au)

c) SAME AS (b) BUT HIGHER IMPACT VELOCITY

Mo K X-RAY PRODUCTION

IN SYMMETRIC COLLISIONS

ONE COLLISION PROCESS

TWO COLLISION PROCESS

1st Collision 2nd Collision

R = — CO R=0 R=+00 R =—00 R =0 R=+CD

BINDING ENERGY BINDING ENERGY

Figure 1.9

K -L MATCHING

a)Unswapped case

UNITED SEPARATEATOM ATOM

b)Case of swapping

UNITED SEPARATEATOM ATOMR=0 R=co

H: Heavy atom L L i g h t atom

Figure 1.10

55

II. EXPERIMENTAL CONSIDERATIONS

II-l. Introduction

As it was briefly stated in the Introduction, the main concern of this

thesis is to isolate and study inner-shell M O transitions by requiring

coincidences between M O x-rays and K x-rays. Then, not only specific

M O transitions can be separated from the total spectrum, but also the

background contributions from other secondary effects (NNB,REC,AB,etc.)

which may contribute considerably to the M O spectrum in the photon

energy of interest (Vi 77) can be considerably reduced.

Several factors contribute to the difficulty of such measurements. Due

to the very small cross section for producing M O x-rays (order of /tbarns),

the coincidence rate is expected to be small (less that 1 count/sec). More­

over, the large number of K x-rays compared to the M O radiation (roughly 0

10 to 1) produce a large accidental coincidence counting rate. But in

spite of these unfavorable experimental conditions, the coincidence studies

proved to be workable. Although the counting rate in the K x-ray counter5had to be kept at ~10 Hz for a reasonable data accumulation rate, the

real councidences exceeded the accidental event rate typically by a factor

of four to six (fig. 2.1).

More important to the viability of these coincidence studies are the

background contributions from a number of processes. For example,

coincidence events are produced by sequential scattering of the projectile

on different target atoms in a solid target. The contribution from this

effect can be appreciable since the cross section for a K x-ray production

- 2 1 2is quite large (order of 10 cm ).

Another important background effect comes from multiple vacancy

production in molecular orbitals. Since the double vacancy cross section

increases faster with projectile energy than the cross section of M O x-rays,

the contribution to the M O coincidence spectrum from multiple vacancies is

expected to grow larger at higher energies where it may dominate the

C2 spectrum. Moreover, the M O x-ray production per K x-ray increases

with projectile energy because the Isct M O vacancy production cross section

asymptotically reaches the vacancy production cross section in the 2pa M O

with increasing projectile energy. Thus, by decreasing the projectile

energy to reduce the contribution from multiple vacancies, the detection

of M O x-rays above the accidental background becomes more difficult.

To explain these important points, the M O x-rays were detected at different

projectile energies.

In connection with these considerations it should be noted that

theoretically, it was shown that the contributions from double vacancies and

sequential collisions can be approximated, by normalizing the single x-ray

spectrum to the K x-ray peak of the cascade spectra. Experimentally this

important aspect of dealing ivith the background was checked by measuring

the KX-KX coincidence yields in slightly asymmetric and very asymmetric

systems (KX-KX coin, experiments).

In addition, the relative importance of the one and two collision

processes in IscrMO vacancy production has been explored by measuring

the cross section for K-vacancy production for various collision systems.

56

57

The Nb Projectiles have been used at three projectile energies. To

minimize the uncertainties arising from unknown target thicknesses or the

beam current integration, the K x-ray cross sections have been normalized

to the Rutherford cross sections for the scattered ions detected at a

constant angle with the beam axis.

11-2. Beam line

The measurements involve the use of Nb and Ni projectiles. The

beams were obtained with a Universal Negative Ion Source (UNIS) modeled

after a Middleton type sputter source. In the Nb case, a NbO beam was

utilized as the most intense component produced by the source (~150 nA),

while in the Ni case, the Ni ions provided considerable amount of beam

(~700nA).

The beams were accelerated through the Yale MP-1 Tandem Van de

Graaff accelerator and the projectile energies ranged from 100-200 MeV

on target. In all x-ray, x-ray coincidence measurements, small beam in­

tensities (<lnA)had to be employed to limit accidental coincidence events

in comparison with the real coincidences. Only in the measurements of

the K x-ray cross sections involving Nb projectiles on various atomic

number targets, the maximum beam intensities were required (typically

a few nA), to reduce the relative importance of Ambient Background (AB).+Even for the lowest projectile energy used (100 MeV, 9 charge state),

the gas stripper at the Terminal could not provide sufficient beam inten­

sities, so in all cases the foil stripper was used. For the case of the

+largest beam energies (200 MeV, 18 charge state), the use of the second

foil stripper was necessary. The terminal voltage typically ranged from

9 to 11.8 MV.

In the x-ray, x-ray coincidence experiments, the two detectors were

very closely positioned to the target to maximize the total number of real

coincidence counts. In this geometry, small changes of the beam on the

target position introduce large changes in the total efficiency and additional

uncertainties from any inhomogeneity of the target material. For the K x-ray

cross section measurements, a similar wandering of the position of the

beam on target can produce large uncertainties in monitoring the Ruther­

ford scattering cross section by two particle detectors. So, it was important

for the beam to be well centered and focused. This was accomplished

by the use of the two sets of carbon slits (fig. 2.2) which were 90cm

apart. They were adjustable to an accuracy of 0.001" from outside the

vacuum by mounted micrometers. Thus, by monitoring the current on

each set of the eight insulated C-slits, the beam was centered on the

target. Downstream of each set of slits, a graded shielding arrangement

shadowed the scattering chamber from background produced by the slits.

Following the second set of slits, magnets (~300G) were used to prevent

the electrons from reaching the chamber and distorting the beam current

integration (BCI). Magnetic steerers were used to balance the beam on

the slits and to adjust it on the center of the target. The beam was

stopped in a continuation of the beam line beyond (~lm) the scattering

chamber containing the targets. The chamber and the latter part of the

58

59

beam-line were insulated from the rest of the beam-line. Together

they provided the beam integrating Faraday cup.7The vacuum in the line was typically 3-5* 10 torr. Three in-line

cold traps (fig. 2.2) were used to ensure clean target conditions during

measurements.

11-3. Scattering chamber and targets

3a). (X-ray) - (X-ray) coincidence measurements

In the coincidence measurements, where the real coincidence

yield was proportional to both solid angles, the Al chamber was de­

signed to allow the two large detection crystals to be positioned 1"

from the target. The chamber possessed two openings (lvf in diameter),

one in each side located at +90° with respect to the beam line. Both

openings were covered by 1 mil mylar window epoxied to the chamber.

The windows were thick enough to stop most electrons produced in the

target from reaching the detectors without attenuating appreciably the x-rays

(less than 1% at 10 KeV). The targets were fitted on a target ladder

with four places available (one of which was secured for a quartz beam

monitor), and they were inclined 45° to the beam line.

X-rays from the surroundings (Ambient Background-AB) were

found to contribute appreciably to the singles counting rates in the x-ray

detectors. This background can also produce coincidence events between

the x-ray counters, and contribute to the accidental coincidence rate and

the real coincidence events due to cross scattering of photons. This

60

type of background is particularly troublesome at the high energy end of

the photon spectrum where the intensity of the events being sought is

especially small. To reduce this background, the whole apparatus,

chamber and detectors, were shielded carefully, using 2" thick lead bricks.

By doing so, the AB was reduced by a factor of 10.

The targets used (Nb or Sn) in the M O x-ray, K x-ray coincidence

measurements were monoisotopic and self-supporting, since any backing

would contribute in the continuum x-ray spectrum through the NNB

(Tr 77, Gr 77), or to the K x-rays from the K x-ray production from

the backing. Moreover, the targets were sufficiently thin so that the

projectile energy dependence being examined was not obscurred by the93 120energy loss in the targets. In particular, Nb and Sn monoisotopic,

2self-supporting targets of thickness varying from 340 pg/cm to 1200

pg/cm , were used for the M O x-ray, K x-ray measurements and for part

of the (K x ray) - (K x-ray) coincidence measurements.

For the remaining (K x-ray)' - (K x-ray) coincidence experiments. 2involving the Ni on Sn system, the need of very thin Sn targets (50 pg/cm )

required a C-backing. In that case, the Sn targets were fabricated by2evaporating the 99. 999% pure Sn material on a 3 pg/cm C-backing. Since

the C-backing can contribute Ni K x-ray intensity, especially for the thinnest2targets used (50 pg/cm ), it was important to keep the C-backing thin.

During the run, it was observed that the C-backing was deteriorating quickly,

leaving very little carbon at the beam spot. It is concluded that the contri­

bution from the C-backing was negligible.

61

3b) K x-ray cross section measurements

During the measurements of the K x-ray cross sections, the K x-rays

were detected by two hyperpure planar Germanium detectors, while two

surface barrier silicon particle detectors, positioned inside an aluminum

chamber, provided the measure of the Rutherford scattering cross section.

The particle detectors were located at 30° and -30° with respect to the beam

line, ~2" from the center of the target (fig. 2.3). The front face of each

particle detector was covered by a collimator with a window 4mm*6mm,

providing typical counting rate 300cts/sec. Magnets (~500G) were used

to prevent the numerous electrons from the target to reach the particle

detectors.

Since the targets were inclined at 45° with respect to the beam line,

the increased target thickness in one direction (Left 30°) resulted in rather

poor resolution for this detector. For the other detector (Right 30°) the

energy resolution was a few MeV. Of course the energy resolution depends

upon the target thickness and its homogeneity, so that in some cases, im­

provement was achieved by changing targets.

The K x-rays were detected as in the coincidence experiments through

two 1.50" windows on each side of a cylindrical chamber. A mylar window

of .001" thickness was used to cover the holes, and it was thick enough to

prevent the electrons from reaching the x-ray detectors. The particle detec­

tors were anchored to the top cover of the chamber, while the bottom of the

chamber was the target ladder, with 8 places for the targets.

The targets used on these experiments were of various thickness , from

2 250 pg/cm to 500 /zg/cm . In most of the cases, they have been made by e-2vaporating the monoisotopic material on 30 jzg/cm Carbon backing. In this

case, the contribution from the C-backing had to be determined and subtract­

ed from the total Nb K x-ray cross sections. Their thickness has been meas­

ured by using an alpha particle gauge, which is expected to measure with

an accuracy of 15%. An independent check of their thickness was obtained

by using the two particle detector yields.

62

4a) Cascade measurements

A characteristic feature of the M O x-ray, K x-ray coincidence measure­

ments was the. large difference between the cross section of producing a5K x-ray and the cross section for a M O x-ray. Thus for every 10 K x-ray

cts, less that 10 were accumulated from M O x-rays and of these, only

1-2 cts were really associated with K x-rays from the cascade mechanism.5Therefore, the counter designated to detect M O x-rays was receiving 10 cts/

sec (K x-rays) for every 10 cts/sec (MO x-rays). Since such a high counting

rate was expected to induce pile-up located spectrally in the continuum x-ray

region, a combination of the Al and Cu absorbers was used in front of the3M O x-ray counter to.attenuate the K x-rays by a factor of 10 , while corres­

pondingly attenuating the M O x-rays by ~20% at 50 KeV. With this absorber,

the counting rate in the M O x-rays channel was 30-50 cts/sec and no pile-up

was observed.

In the K x-ray counter with the highest counting rate (-10 cts/sec),

the pile-up was ~1% and it was negligible for the symmetric collisions. In

the asymmetric collisions (Nb+Sn), the pile-up ( 1% of Nb K x-rays) was

located close to the Sn K x-ray peaks contributing 20% to the total Sn K

x-rays.

The M O x-ray channel consisted of a hyperpure planar Ge detector2(PGI, IG 1010). Its active detection area was 1000 m m and the thickness

10mm. With energy resolution 56OeV at 6KeV and 680eV'at 122KeV, it

could resolve the K-,,K_ lines of Nb and show any structure of the x-rayor a

63

11-4. Detectors

64

continuum. Its bias voltage was-2500V.

Another hyperpure planar Ge detector (IG1910) was used for the K

x-rays in all M O x-ray, K x-ray coincidence measurements, except for the

160 MeV Nb+Nb case. In that case a Bicron Nal (Model lxMO 40BP) was

used, which was 1" in diameter and it was 1 mm thick (it had 5 mil Be

entrance window). Its energy resolution was 42% at 6 KeV and 25% at 322KeV. The IG1910 detector consisted of a Ge crystal of 190Cknm active

area and 10mm thick. With energy resolution 770 eV at 6 KeV and 880

eV at 122 KeV, it would not resolve the Nb K^,K^ lines, but it would easily

resolve Sn K x-rays from those of Nb.

Typical x-ray spectra obtained with these detectors are presented in

figures 2.4 - 2.6.

Correction for efficiency and absorption

To correct:; the M O x-ray coincidence spectra for efficiency and absorp­

tion, a double correction is needed involving both x-rays and their respective

counters. Since we are interested in M O x-rays associated with the K

x-rays which have a specific energy, the correction for efficiency and absorp­

tion can be carried out, first for the K x-ray and then for the M O x-rays.

Thus, the total efficiency (efficiency and absorption) of the Kx-rays was

calculated at the mean energy of the K , K . lines. After correcting theCL p

coincidence x-rays yield for this mean K xrray total efficiency the yield

at each x-ray energy was corrected for efficiency using the other counter

total efficiency.

The total efficiency had been' measured for both detectors using

calibrated sources at the place of the target. Then a theoretical expression

for the total efficiency (described in App. II), was used to fit the measured

efficiency. The error from such least square fit is about 5%. In addition,

there is an error resulting from a possible fluctuation and uncertainty of the

beam spot position on target. Because of the large detection angle and the

small target-to-detector distance used, changes of 2 mm in the beam spot

off the target center introduces a 4% error in the total efficiency. The

total error (including statistical uncertainties) is ~ 10%.

For the 160 MeV Nb+Nb case, an additional uncertainty was introduced

in the efficiency determination by the discovery after the measurement

that the Nal detector was non-uniform. For unknown reasons, the crystal

of the detector had generated a dead area, which covered almost 50% of

the total surface. The whole surface of the crystal was scanned with a

series of calibrated sources, carefully collimated, and the dead layer was

mapped. The total efficiency was also measured using calibrated sources.

It was determined that the error associated with determining the efficiency

of this detector had to be enlarged to 20%.

Fig. 2.7 and 2.8 present the total efficiencies, of the two Be detectors

(IG1010 and IG 1910). The data points are the measured values with their

statistical errors and the continuous curves the theoretical least square fits

to the data (App II). In the case of the IG1010 detector (fig. 2.7), the

statistical errors are of the size of the spots.

65

4b) (K x-ray) - (K x-ray) coincidence measurements

The(K x-ray)- (K x-ray) coincidences were measured with both slightly

asymmetric (Nb+Sn) and very asymmetric (Ni+Sn) collision systems. In

all cases the detectors should be able to resolve the K x-rays of the two

elements. Besides, the difference in the cross section of the two K x-rays

(projectile and target) necessitated the use of absorbers in the detector

which was predominently used for the heavier atom K x-rays. In this

higher Z channel, the typical counting rate was 00 cts/sec and no appreciable

pile-up effects were observed. In the lower Z channel, the counting rate4was higher (~10 cts/sec), but the pile-up (<1%) did not introduce any un­

certainty in the integration of the K x-ray peaks.

Forthe Nb+Sn collision systems, the previously described IG1010 and

IG1910 detectors had been used. In the Ni+Sn case, the higher Z K x-rays

were.detected with the same IG1010 detector, while the NiK x-rays with a

Nal crystal of 1.5" diameter. Its energy resolution was 45% at 6 KeV

and 25% at 122 KeV. The correction for the total efficience (efficiency

and absorption) was calculated by considering the value of the theoretically

best fit to the data. (fig. 2.7, 2.8) at an average x-ray energy. The average

energy of the two K x-ray lines (K^K^) , was obtained using the relative

intensity (Wa 59) of each line.

4c) Kx-ray cross-section measurements

In this experiment, the measured cross section varies by. many orders

of magnitude on the different targets, i.e. for the lighter collision atom,

the KX cross section can be many orders of magnitude larger than the heavier

66

67

atom K x-ray cross section. Such variation in the cross section from

target to target necessitated the use of two x-ray detectors, where one

(with the best energy resolution) was predominantly used for the K x-rays

of the lighter colliding atom, while the other was for the K x-rays of the

heavier partner. Besides, absorbers were used when it wasnecessary

to reduce the number of the lower-Z K x-rays in favor of the higher-Z

K x-rays. Such reduction gave the advantage of increasing the beam

intensity and, in that way, reducing the importance of the AB background,

which, for the heaviest targets (Sm, Er), was competing with the K X

peaks, (fig. 3.41-3.43)

For the detection of the lighter atom K x-rays, another intrinsic Ge

detector was used (Model Ortec, 1113-10210), with a crystal of 10mm in

diameter and 7 m m thick (6 mil Be entrance window). Operating bias was

-1000V and its energy resolution was 177 eV at 6 KeV and it could resolve

the K^,K^ lines-of the atoms in most of the collision systems. The heavier

atom K x-rays were detected by the IG1010 planar Ge detector.

There was a limit on the maximum useable beam intensity because

the silicon surface barrier particle detectors (Models 18-551Iwith thickness

of 100)jm and.l7-151G of thickness of 150jj m) positioned inside the chamber at

a constant distance, could only be used with low ( 1000Hz) counting rates to

avoid radiation damage. Thus, for the heaviest element, we used the small­

est x-ray detector-to-target distance to achieve the maximum possible

counting rate. For similar reasons, the same distance variations were used

with the small crystal detector. The use of two detector-to-target distances

for each x-ray detector with the combination of some absorbers helped us

68

to keep the counting rates within limits. But at the same time, it intro­

duced some uncertainty due to the variation of the total efficiency (eff. +

abs.). To minimize this uncertainty, a careful determination of the total

efficiency for all combinations of distances and absorbers of each detector

was necessary. Thus, careful measurements were conducted, after the

experiment, and the experimental values were fitted to a theoretical

curve. It turned out that the.' measurements agreed quite veil with the

theoretical values for all the x-ray energies above llKeV where the error

is 10%. Below 11 KeV, the uncertainty from the small absorbers whenever

used, or the dead layer of the. Ge crystal (App. II), resulted in error of

about 20%.

The two particle detectors for the Rutherford scattered projectiles

were carefully positioned to be at 30° relative to the beam line and on either

side. The distance of the detectors to the target was measured so that the

only uncertainty would result from the change of the position of the beam

on the target. Because of the small distance of the detectors from the target

(2.15'r), small changes of the beam spot on the target (~lmm), resulted in

a 20% difference in counting rates for the two detectors, Left30° (L30) and

Right 30° (R30). Since we have used three projectile energies, the careful

collimation of the beam at the center of the target was necessary at the be­

ginning of each run. Still, the L30-R30 anisotropy was in a few cases 40%,

which means that the beam spot was 2mm off the center (§111.5).

69

II. 5 Electronic Set-up

5a) (x-ray)-(x-ray) coincidence measurements

Fig. 2.9 presents a simplified electronic diagram of the circuitry used

in the coincidence studies. Only the principle units are shown. The signals

from the two preamplifiers (or directly from the anode in the case of Nal

detectors) are fed to the two Timing Filter Amplifiers (TFA,Ortec 454) and

then to Constant Fraction Timing Discriminator (CFTD, Camberra 1428).

The two fast signals from the CFTD’s are connected to the Time to Pulse

Height Converter (TPHC, Ortec 437A), where the M O x-ray counter with

a low counting rate, defines the start, and the K x-ray counter with the

high counting rate provides the stop (this way the dead time of the TPHC is

decreased). The output signal of the TPHC, which was delayed 1 sec with

respect to the start signal (the TPHC was externally strobed by the M O x-

rays), was fed to a Timing Single Channel Analyzer (TSCA, Ortec 420A).

The output from the TSCA provided the logic pulse (coincidence event) for

the gating in the Linear Gate Stretchers (LGS, Ortec 442) of the three

linear signals (MO x-ray energy, K x-ray energy, and Time from TPHC).

These linear signals were obtained from two Tennelec 205A Amplifiers and

the TPHC.

An independent circuitry supplied the singles spectrum from the M O

x-ray counter, both for the computer and the Northern Scientific Multichan­

nel analyzer. The Multichannel Analyzer was used for a rapid comparison

and monitoring of data. Since the counting rate of the singles M O x-rays

70

was more than an order of magnitude larger than the coincidence rate, a

prescaler was used (not shown in fig. 2. 9).

The four linear signals (three from coincidence and the singles

Xc-rays) were connected to the Analog to Digital Converters. They were

delayed lfj, sec with respect to each other sequentially at the computer inter­

face.

5b) K x-ray cross-section measurements

In the K x-ray cross section measurements, the x-rays were measured

together with the scattered ions via two particle detectors located at +30°

(R30, L30). The circuitry consisted (fig. 2.10) of the two Tennelec 205A

Amplifiers for the x-rays, which also supplied (through a SCA) the event

signal needed for the computer. Two Ortec-125 preamplifiers were used

for the particle detectors in connection with the Ortec-450 and. Camberra

2010 amplifiers. Therefore, four linear independent signals were pro­

cessed by four ADCs, with four corresponding event signals, generated

by SCAs.

Since in this experiment absolute cross sections were measured, the

dead time in the electronics or in the computer had to be monitored. The

total dead time for the system was obtained by monitoring the pulses from

a pulse generator powered by the counting system with the pulses introduced

at the preamplifier stage. The pulse generators externally triggered through

a prescaler from the BCI counts in order to include the varying event rate

which reflects the Instantaneous beam intensity.

71

The IBM 360/44MPS computer system installed at WNSL (Yale Uni­

versity) was used for data storage and analysis. This multiprogramming,

multilinear system performs data acquisition, data analysis and at the same

time, it can process other independent jobs.

Data were converted in the ADCs, andfrom there,they were transferred

to the tape and to the data input buffer, in the computer memory. With the

computer interface available, one ADC with multiple events or multiple ADCs

with one event, can be associated. For the (x-ray) - (x-ray) coincidence

experiments, the three correlated signals, M O x-ray energy, K x-ray

energy, Time, converted in the three ADCs, correlated with a special event

signal in the computer. The correlated signals were stored in the tape and

in the computer memory as three-dimensional events. Besides the singles

x-rays signal from the M O x-ray counter after converted to the ADC was

correlated to another event and was stored also inthe computer memory

and on tape.

In the case of K x-ray cross section measurements, the four indepen­

dent linear signals were associated with four independentevents and were stored

in the computer memory. A Cathode Ray Tube Display (Type Fairchild 737A)

with the light pen and keyboard functions provided a powerful way to analyse

the data on-line.

The data presented in this thesis are one-dimensional histograms of

associated events. In the case of three-dimensional coincidence events,

11-6. Data Collection

72

the histograms were the projection of all (or part) of the events in one di­

mension. In this case, they were named QM, KX, and Time spectra which

correspond to the Quasi-molecular x-ray energy spectrum from the IG1010

detector, K x-ray energy from the other x-ray detector, and the time spectrum.

By using different windows on two of them, we could separate and observe

part of the events in the third one. Fig. 2.11 presents examples of QM,

KX and Time which include all data. By setting windows on the time peak

of the time spectrum and the K x-ray peak of the K X spectrum, we could

obtain the real and accidental Q M spectrum. By setting another window

outside the time peak in the time spectrum and using the same window in the

KX histogram, we can define the Q M spectrum of accidentals (fig. 2.1 ,

#2). The subtraction of the two spectra will define the spectrum of real events

namely the spectrum of x-rays in true coincidence with K x-rays. The data

were also stored in the Northern Scientific multichannel Analyser (NS-630)

using 1024 channels.

Fig. 2.1 Real and Accidental coincidence spectra, for 160 MeV Nb+Nb

collisions.

Fig. 2.2 Schematic diagram of the end of the beam line.

Fig. 2.3 Diagram of the chamber used for the K x-ray cross section meas­urements.

Fig. 2.4 Typical M O x-ray spectrum of the IG 1010 detector (160 MeV Nb+Nb).

Fig. 2.5 Typical K x-ray spectrum of the Nal counter (160 MeV Nb+Nb).

Fig. 2.6 Typical spectrum (K x-ray channel) of the IG 1910 detector.

Fig. 2.7 The measured total efficiency (efficiency and absorption) for the

IG 1010 detector with 6mil Cu and 6*4.8mil Al foil absorbers.i

The continuous line represents the best theoretical fit to the data.

Fig. 2.8 Total efficiency of the IG 1910 detector with 4mil Al foil absorber.

Fig. 2.9 Simplified electronic diagram for the x-ray, x-ray coincidence

experiments.

Fig. 2.10 Electronic diagram for the K x-ray cross section measurements.

Fig. 2.11 Typical spectra of the three analyzers QM, KX, and Time prior

to any conditions imposed.

CO

UN

TS

10 20 3 0 4 0 50 6 0 70 8 0 90

X-RAY ENERGY (KeV)

Figure 2.1

U P S TR E A M DO W NSTREAMSLIT S S L IT S

F ig u r e 2 .2

Figure 2.3

2 0 4 0 6 0 8 0 100 120 140 160 180 2 0 0

X-RAY ENERGY (KeV)

Figure 2.4

CHANNEL NUMBER

Figure 2.5

CO

UN

TS

10,000

1,000

5 0 1

CHANNEL NUMBER

Figure 2.8

— Energy (keV)

Figure 2.7

20 40 60 80 100 120 140 160--E n e rg y (keV)

Figure 2.8

Q.M. C H A N N E L K X-RAY C H A N N E L

COINCIDENCE E V E N TS

Figure 2.8

B io*'2500v t tBEAM

Bias'IOOOv

IG IOIO / Ortec 1113 -10210 <

R -3 0IT-1SIG

BNC PULSER

+ lOOv

L -3 0 18-5511

ORTEC125

ORTEC K T,25 r v+ 70v

4 chn PULSER

AMP TENNELEC

205A

JAMR

ORTEC 450a:UJ-i4OV)Lia:a.

AMR CAMBERRA

2010

4 in 4 inain

ID_ i

Oin

ID_i

o

1-zLI

Ua

»-ZLJ

Oo

1-ZLJ

Oa

o4

>LI

4 >LI 4 UJ 4

X-RAY PARTICLE PARTICLE X-RAY

Figure 2.10

CO

UN

TS

160 MeV Nb on Nb

SPECTRA OF ANALYZERS

10,000

1,000

Nb —I- ,N k„

OM-spectrum

\ ' CI

c*1 ✓ 2

| ijU*

P1

4,000

3 0 0 0

2,000

1,000

lOl 201 301 401 SOI 601 701 801

Time

1

spectr•urn

a*L ^ W v [W v f

5 kn i:il 2 (H 251

CHANNEL NUMBER

F ig u re 2.11

85

HI. DATA AND ANALYSIS

III. 1 Introduction

In this chapter, which is divided into five parts, the data are presented

and analysed. The second and the third part deal with the cascade M O x-rays

in symmetric and asymmetric collisions detected in coincidence with the K x-

rays (fig. 3.1). The data are corrected first for the contribution from acciden­

tal . coincidences and then for the background contribution from sequential inde­

pendent collisions or double vacancies. The contribution from double vacancies

results mainly from two vacancies in the 2pcr M O or one in each level 2pc and

Isa. The contribution of these effects to the real coincidence spectra requires

a detailed investigation, which will be presented in the next chapter.

In the fourth part of this chapter we present experimental tests on the back­

ground contributions in the cascade M O x-rays from sequential independent col­

lisions or double inner-shell orbital vacancies, which have been performed by

checking the K x-rays, K x-rays coincidences in slightly asymmetric (Nb+Sn),

and very asymmetric (Ni+Sn) atomic collisions. These KX-KX measurements

have shown that, while the contributions from sequential collisions can be treat­

ed theoretically without any dificulty, there is an uncertainty in the evaluation

of contributions from double vacancies created in one collision. In any case,

the contributions from sequential collisions and double vacancies subtracted from

the cascade M O x-rays approximately (within a factor of two), by normalizing

the singles spectrum to the KX-peak of the cascade x-ray spectrum. It should

be noted that the contribution of the background events to the Cl region can

be evaluated exactly from the singles spectrum even for double vacancies;

in this case the transitions in questions fill vacancies in the 2pcr M O (Me 76,

77), which is the main source of the K vacancies as well.

The last part of this chapter is devoted to the presentation and analysis

of data from the K x-ray cross section measurements’-.

86

Fig. 3.1 a) The cascade mechanism of M O x-ray production in symmetric

atomic collisions.

b) The cascade mechanism for slightly asymmetric collisions

top: without 2po—Isa vacancy sharing

bottom: including sharing

CASCADE C2 MO X-RAYS

a ) SYMMETRIC COLLISIONS

b) ASYMMETRIC COLLISIONS

ls(L)

ls(H)

ls(L)

ls(H)

Figure 3.1

89

Presentation of data

The quasimolecular x-ray coincidence experiment was carried out with93 93Nb projectiles of energies 100, 160, and 200 MeV (lab), colliding with Nb

2target of thickness 516 pg/cm . As noted above the IG 1010 hyperpure detector

used for the M O x-ray channel had an absorber consisting of 6mil Cu foil and

six Al foils of 4. 8mil each. The K x-ray channel consisted of a IG 1910 detect­

or during the measurements at 100 and 200 MeV, while the measurement at

160 MeV employed a Nal detector for this purpose. In all cases a 4mil Al

foil was thick enough to stop the L x-rays while attenuating the K x-rays less

than 10% at 17 KeV.

The singles x-ray spectra obtained in the M O x-ray channel, without a

coincidence requirement, are presented in fig. 3.2, 3.3 and 3.4 for the three

projectile energies. In all cases, the two x-ray regions Cl and C2, can be

easily discerned due to their different slopes. The C2 region asymptotically

reaches a flat continuum, which is the ambient background (AB). Even though

the AB was suppressed from 10 cts/sec to less than 1 count/sec, by a 2" Pb

shielding, it was still large enough to preclude any clear identification of the

asymptotic high energy region.

Fig. 3.5 presents the typical ambient background accumulated in two

days with the Pb shield in place. We see that it is rather flat, with the excep­

tion of some high energy peaks (>60 KeV). These are also present in the

singles spectra of fig. 3.2 to 3.4, originating from Pb and other materials

present in the shielding and the surrounding walls.

III. 2 Symmetric collisions

90

Nucleus-nucleus Bremsstrahlung (NNB) produces another background

contribution to the singles spectra. In the symmetric case, dipole excitation

is absent but the quadrupole term exists and extends beyond the UA limit.

The results of the Bremsstrahlung calculations corrected for absorption and

efficiency are shown in the same figures with the singles spectra (the conti­

nuum line). The singles spectra corrected for AB and N-N Bremsstrahlung

are also presented. The error from such corrections is small (<10% ) for

x-ray energies less than 50 KeV, but increases towards the UA limit. We

will be using the singles spectra for the subtraction from the real coinciden­

ce events of those coincidences due to multiple M O vacancies and sequential

collisions, the error in the high energy part is not important (see below). The

low x-ray energy peaks (to the left of the Nb K-x-rays) of the singles spec­

tra are escape peaks of Nb.

The three-dimensional (MO x-ray energy, K x-ray energy, and Time)

data accumulated under the coincidence requirement are presented through their

one-dimensional projection histograms, named after the piece of information

they reflect (QM, KX, and Time). Such one-dimensional histograms of QM,

KX, and Time including all data are presented in fig. 3.6, 3.7, and 3.8 re­

spectively for the 160 MeV case. The Q M histogram (fig. 3.6), includes not

only the real coincidence events in time peak of the Time histogram (fig. 3.8),

but all other events that constitute the flat BG. It is seen to be similar to the

singles spectrum of fig. 3.3. By setting gates on the time peak of the Time

histogram (fig. 3.8) and the K x-ray peak of the KX histogram (fig. 3.7),

selected coincidence events can be extracted. Such spectra for the three

91

different energies are presented in fig. 3.9, 3.10, and 3.11.

The subtraction of the accidental coincidences from the above real coin­

cidences has been carried out by averaging out all data not included in the

peak of the Time histogram (fig. 3.8) over the number of channels in the gate

of this peak. The spectra from these data are presented in the same fig­

ures (3. 9 to 3.11) and constitute the accidental background. We can see that

in the x-ray energy of interest, the real coincidence events are 4-6 times

larger in number than the accidental counts. This ratio of real to accidental

counts at each x-ray energy, is a function of the target thickness, the beam

current intensity, and the ratio of the (MO x-ray)-(K x-ray) coincidence cross

section to the singles x-rays cross section (at that energy, App. I). In fig. 3.12

typical time spectra for three different gates (KX, Cl, and C2) on the Q M

spectrum (160 MeV case) are presented.

It is obvious that the peak to background ratio, which is a measure of

the real counts to accidental ones, increases from the Cl to the C2 region.

This strongly supports the idea of a special mechanism for quasimolecular

x-ray production, in coincidence with K x-rays, which is not present in the

Cl region. Such a mechanism is the production of an M O x-ray from the

2pcr— Iscr transition in coincidence with a K x-ray.

The KX-spectra at each energy are presented in fig. 3.13 and the poorer

resolution of the Nal detector (160 MeV) compared to the IG 1910 detector at

the two other energies can be easily seen. In the last case (IG 1910) pile-up

causes a second peak, which is about 1% in counts. The KX-spectra associat­

ed with the different regions of the QM-spectra, show no difference in form.

Fig. 3

Fig. 3

Fig. 3.

Fig. 3.

Fig. 3.

Fig. 3.

.2, 3.3, and 3.4. Singles spectra of the Q M channel corrected for

efficiency and absorption, at the three projectile energies 100, 160,

and 200 MeV respectively (Nb+Nb). Presentation of the NNB theo­

retical calculations and the singles corrected for AB and NNB.

,5 Ambient Background (AB) for the Q M channel.

6, 3.7, and 3.8. QM, KX, and Time spectra for 160 MeV Nb+Nb.

9, 3.10, and 3.11. Real and accidental Q M spectra (spectrum 1) and

accidental (spectrum 2), at three projectile energies 100, 160, 200MeV.

12 Time spectra of the three regions a, b, and c of the Q M histogram

at 160 MeV Nb+Nb.

13 Characteristic x-ray spectra for the three energies.

Figure 3. 2

Figure 3.3

CO

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— X-RAY ENERGY(KeV)

Figure 3.4

CO

UN

TS

X -RAY ENERGY (KeV)Figure 3.5

CO

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TS

20 40 60 80 100 120 140 160 180 200

X-RAY ENERGY (KeV)

Figure 3.6

CHANNEL NUMBER

Figure 3. 7

CO

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4 000

3000

2000

1 0 0 0

1

160 MeV Nb+Nb Time Spectrum

I7nsec

> 4 lr

50 100 150 200CHANNEL NUMBER

Figure 3. 8

10 2 0 30 4 0 50 60

X-RAY ENERGY (KeV)

70 80 9 0

Figure 3.8

CO

UN

TS

1M ■■■ ■■ I. - .. “—l.vmjyiUfllil 1U1 « l» "1 110 20 30 4 0 50 6 0 70 8 0 90

X-RAY ENERGY (KeV)

Figure 3.10

3 0 4 0 5 0 6 0 7 0

X-RAY ENERGY (KeV)

8 0 9 0 100

CO

UN

TS

500

V) I— 2 Z> O o

400-

300-

200-

100 -

GATE A

— 17 nsec

1 I I I--50 100 150 200 250Chonnel Number

600

500-

400-V)H 2g 300- O

200-

100 -

GATE B

— i----1----1---- 1--0 50 100 150 200 250Channel Number

4000-

3000-

2000-

GATE C

1000-

Channel Number Channel Number

Figure 3 .1 2

CO

UN

TS

Eproj =100 MeV Det: IG 1910

101 201 301 401 501

Nb on Nb

CHANNEL NUMBER

EprojDet:

=200

IG 19MeV

10

^ Nb •VK/E

_ pile-up

101 201 301 401 501 601

Figure 3.13

105

Analysis

From the coincidence data presented in the spectra of figures 3.9, 3.10,

and 3.11, minus the contributions from the accidentals (also shown), we get

the real coincidences which are the events causally related to the K x-rays.

They are presented in figures 3.14, 3.15, and 3.16 averaged over 4 KeV sli­

ces (above 25 KeV x-ray energies), for the three projectile energies.

In principle, the coincidence technique should separate selected quasimo-

lecular transitions from all the M O transitions which contribute to the forma­

tion of the C2 radiation region. Such selected transitions (2po-—<-lscr) could pro­

duce a K x-ray, which will be' detected in coincidence with the associated

M O x-ray. On the other hand, no Cl radiation (from single 2pcr vacancies)

could produce a K x-ray, so in the coincidence spectra the Cl radiation re­

gion should be absent.

A comparison of the Cl region in the single spectrum with the same re­

gion of the real coincidences is presented in fig. 3.17 for the 160 MeV case,

and the two spectra are normalized in the C2 radiation region. The great reduc­

tion of the Cl radiation is obvious and the conclusion straightforward.

In fig. 3.14 to 3.16, we see that the Nb K x-rays are still present, in

addition to a very small shoulder of Cl radiation. In the discussion section

it is established that these K x-rays or the Cl radiation are due to multiple

collisions and double vacancies, and their relative shape is approximately that

of the singles. Thus, in order to calculate the contributions from sequential

independent collisions or double vacancies, the single M O x-rays (corrected

for NNB and AB) had been used. These single spectra have been normalized

to the Nb K x-ray peaks of the coincidence spectra. Thus, in fig. 3.14 to

3.16 the K x-ray peaks and the broken lines have the shape of the singles

spectra. It can be seen that the Cl region coincides (within error) with the

similar part of the singles spectrum; therefore, it will be absent from the

final spectra of M O x-rays (correlated to K x-rays). It is clear that the

uncertainty from the subtraction of the contributions of double vacancies and

sequential collisions is not very important in the case of 100 MeV, but in­

creases in importance with projectile energy (fig. 3.14 to 3.16). This means

that the double vacancy probability increases with projectile energy faster

than the M O x-rays.

Subtracting from the coincidence spectra (fig. 3.14 to 3.16 data points)

the calculated part from double vacancies and sequential collisions (broken

lines) from the singles-to-coincidences normalization, one finds the final spec­

tra presented in fig. 3.18 (for the three energies normalized to the same

number of projectiles and total efficiences).

Correcting for total efficiency and fluorescent yield of the K x-ray and

averaging over a few KeV slices, one finds an estimate of M O x-ray (cascade)

cross sections/KeV, which are presented in fig. 3.19. For the estimate of

the fluorescent yield an approximate expression has been used (Gr 771):

w = w Q/ [l- y (1-w q) ] where w^ is the neutral atom fluorescent yield and i the number of vacancies

in the L-shell. Since the number of vacancies was not measured in the pres­

ent experiments, an approximate value (Vi 77), i=2, is assumed. By making

this correction, the value of the fluorescent yield becomes 0.80, compared

106

107

to the value 0.75 of the neutral atom. The error in the fluorescent yield is

expected to be less than 10%. The errors shown in fig. 3.18 and 3.19 are only

statistical. Additional errors are expected from the total efficiency (10% for

100, 200 MeV and 20% for 160 MeVj, the target thickness (15%) and the BCI

reading (<10%). So that, the total additional error (above statistical) is esti­

mated to be 20% for 100, 200 MeV and 25% for the 160 MeV. This uncertain­

ty will affect all data in the same way and it will not change the x-ray energy

dependence of the cross sections (fig. 3.18, 3.19). This x-ray energy dependence

of the cross sections would be affected only by the uncertainty in the contribution

from double vacancies and sequential collisions in the coincidence spectra.

Table 3.1 presents the differential cross sections (jjbarns/KeV), for the

three projectile energies. The errors shown in this Table are only statistical

and systematic and they do not include the uncertainties from the singles sub­

traction. These M O x-ray cross sections correspond to the emission of M O

x-rays at 90° with respect to the beam axis. For the other angles of emission,

they will be different, due to the directional anisotropy of their emission

(Vi 77).

Fig. 3.14, 3.15, and 3.16. The true coincidence spectra and the contributions

from double vacancies and sequential collisions (broken line) for

the three projectile energies.

Fig. 3.17 True coincidence spectrum and the singles (upper broken line),

normalized in the high energy region. Diagram on the right pre­

sents Cl (to the 2pcr MO) and C2 (to the lscr MO) transitions.

Fig. 3.18 True coincidences corrected for double vacancies and sequential

collisions for 100 (lower data points), 160 (middle data points), and

200 MeV (upper points). Error bars present statistical errors only.

Fig. 3.19 The final differential cross sections (pbarns/KeVj for the three

energies.

CO

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IOOO

100

NbK

k 0 k

1 0 -

escape peak

Nb93+Nb93E|ab = IOOMeV

+ Real Coincidences Singles Normalized to the

Nb KX-peak

t-C I

V\ * *

\ \

C2

N,\

t

t\

\\

\\

I

10 20 30 4 0 50 60 70ENERGY(KeV)

Figure 3.14

CO

UN

TS

ENERGY (KeV)

Figure 3. 15

CO

UN

TS

IOOONb

Kp

100

Ka

escape peak

10 b

I

Nb93+Nb93E|ab=200MeV

4 Real Coincidences — Singles Normalized to the

Nb KX-peak* - C I

%

V ♦ ♦

\\

\\

> r c z

S\

\\

tN

\t

4 - i -\

\\

- t

\\

\\A.

10 20 30 40 50 60 70ENERGY (KeV)

Figure 3.16

ENERGY (keV)

Figure 3.17

COU

NTS

/CH

AN

NEL

100

1 0

Nb+Nba 200 MeV 1a 160 MeV > Target 516 u g /c m 2 o lO O M eVJ

[True(Singles normalized to the K X-peak)] normalized to the same# of proj. a efficiencies.

♦ t t

U.A. limit 1 I

30 40 50 60 70 80-X-RAY ENERGY KeV

Figure 3.18

.......... "1 'N 1b+Nb 'i I N

♦ t

▲ 2 0 0 MeV a 160 MeV

______ *o 100 MeV

11 1 " A i1 T * it T

* fU t +T T

t t Ji -jl aIa

1t

1

ii

\i

oII

" I

20 3 0 4 0 50 6 0 7 0 8 0- X-RAY ENERGY (KeV)

Figure 3.19

SYMMETRIC COLLISIONS (Nb+Nb), Target=516pg/cm2

x-ray

energy

(KeV)

Effic.

Q M

Channel

M O X-RAY CROSS-SECTIONS (pbarns/KeV)

at 90° to the beam line

E=91MeV E=150MeV E=188MeV

28 .008 580:± 160 4670 ± 660 4900 ± 180031 .0415 267 ± 60 1790 ± 240 3250 ± 810

35 .024 215 + 34 1415 ± 127 2520 ± 320

39 .033 205 ± 25 1275 ± 127 2140 ± 225

43 .040 140 ± 19 820 ± 70 1450 ± 167

47 . 046 86 ± 15 496 ± 52 1110 ± 130

51 .051 78 + 12.5 360 + 41 770 + 105

55 . 0555 38 ±8.7 142 ± 27 637 ± 86

59 .059 16.4 +6.2 87 ± 21 315 + 64

63 .0615 54 + 17 237 + 55

65 .062 11.5 ± 3.7

67 .0625 25 + 13 115 + 39

71 .0631 14 ± 10 65 ± 35

Table 3.1

116

III. 3 Asymmetric System (Nb+Sn)

Data presentation

The transition from a symmetric to an asymmetric system, according

to the quasimolecular picture, is expected to be continuous, at least for the

lighter element. For the heavy collision partner, the correlation of the Is

level of this atom with the Isa M O of the quasimolecule (fig. 3.1), will mod­

ify the process of Isa M O excitation. In the coincidence experiment, the M O

x-ray (from the 2pa— lsa excitation) is expected to be associated with K x-rays

of the light partner, since the 2pa M O in the separate atom (SA) limit forms

the K-level of that atom (fig. 3.1 bottom). Thus, in principle the same coin­

cidence- mechanism of the Nb+Nb system can be observed in the asymmetric

case, as well. But, because of the presence of the heavy element, new phe­

nomena might appear. The vacancy sharing mechanism, presumably occuring

at large internuclear distances, would transfer vacancies from the 2pa M O

to the K-shell of the heavy atom as well. Then, one would detect M O x-rays

of a cascade type, associated with K x-rays of the heavy element. Actually,

the detection of M O x-rays in coincidence with K x-rays of the heavy element

would supply a straightforward check of the theory of the vacancy sharing

mechanism, which considers the vacancy sharing as a process resulting from

orbital coupling, and occuring at large internuclear distances.93 120We have used Nb projectiles on two Sn monoisotopic self supporting

2targets of thickness 345 and 520pg/cm , at beam energy 200 MeV (lab).

The experiments have been performed with two target thicknesses in order

to determine the importance of multiple collisions. The detector and absorbers

117

used on the M O x-ray channel were the same as in the 200 MeV Nb+Nb case,

and the total efficiency is expected to be very close to the one of the sym­

metric collisions. For the K x-raj'5, the IG 1910 detector was used which

could easily resolve (fig. 2.6) the characteristic K x-rays of the two collid­

ing atoms but with a thick Al foil (8mil) to stop the L x-rays of Sn.2Figure 3.20 presents the typical singles spectrum for the 520pg/cm

target thickness case. As a result of the thick ahsorber, the Nb K x-lines,

which should be about 20 times more pronounced than the Sn lines, are pre­

sented in the figure about 10 times weaker. The two additional lines, close

to the Nb K x-rays, are the escape peaks of the Sn characteristic lines. A-

symptotically, the spectrum reaches an almost flat background, which is a

combination of Ambient Background (AB) and Nucleus-Nucleus Bremsstrahlung

(NNB). The small peaks, in the high energy region (>60 KeV), are due to

the surrounding and are also present in the symmetric case.

NNB, in the asymmetric case, is more important because of the pre­

sence of dipole radiation, and the interference between dipole and quadrupole

radiation. The total amount, in addition to the AB, dominates the singles

spectrum near the UA limit. The corrected spectrum for Bremsstrahlung

and AB is presented in the same figure with the singles. The low lying con­

tinuous curve is the calculated N-N Bremsstrahlung radiation corrected for

the total efficiency of the detector.

The total uncertainty in the subtracted spectra in the high energy region

( >60 KeV) is large because of the NNB and AB uncertainty and the subtraction

of the two large numbers.

118

The three time spectra which are obtained by setting gates on the Nb K

x-rays in the KX analyzer and on three different x-ray energy regions (Nb

K x-rays, Sn K x-rays and M O x-rays) in the Q M analyzer, are shown in2figure 3.21 for the target of thickness 520 jxg/cm (for the other target the

relative time spectra are similar). The great improvement of the peak to

background ratio in the spectrum w’ith the gate on M O x-rays compared to

the other two spectra (gates on Nb or Sn K x-rays) is clear, indicating a

special selection rule working in the M O region of the x-ray spectrum./ 2The typical K x-ray spectra for the target of thickness 345 j/g/cm

are presented in figure 3.22. Each figure shows the K x-peaks of each region

associated with different gates in the other counter, i.e. total, Nb, Sn,

and C2.2Typical real and accidental spectra for one target thickness (520^g/cm )

of x-rays which are in coincidence with Nb or Sn K x-rays, are presented

in figures 3.23 and 3.24. The spectra of the accidentals, which are an aver­

age of the flat BG randoms on the time spectrum, are also presented in

the same figures. The differences of the two spectra, (real and accidental)

define the real coincidences with the K x-ray of each colliding element

(Nb, Sn), and is shown in figures 3.25 (gate on Nb) and 3.26 (gate on Sn).

for the same target.

119

Analysis

The coincidence spe'ctra for Nb and Sn gates, presented in figures 3.25

and 3.26, include, besides the x-rays from the cascade mechanism (2pa— lso-

in coincidence with the resulting K-shell transitions), other x-rays from se­

quential collisions or double vacancies. As was stated before, the coinciden­

ce x-rays from double vacancies and sequential collisions resemble appro­

ximately in shape the singles spectrum and so a good estimate of those x-

rays is normalizing the singles spectrum to the K x-ray peaks of the coin­

cidence events.

Such a normalization has been plotted in fig. 3.25 and 3.26 (the lower

discontinuous curve). It is clear that there is a net contribution from other

effects, besides the double vacancies and sequential collisions. Some small

contribution from Cl radiation around 35 KeV in the coincidence events

seems to be the result of double vacancies and sequential collisions, as

it agrees quite well with the singles spectrum after the normalization pro­

cedure cited above is used. Such Cl radiation should be absent, if the pro­

posed (2pcp— Isa, K x-ray) cascade mechanism is correct, and the results

seem to support the hypothesis.

Assuming that the above normalization procedure correctly reproduces

the multiple collision and vacancies background one can subtract from the

coincidence events these background contributions (fig. 3.27, target 520^g 2/cm ). The error bars seen in this figure are only statistical. As in the case

of Nb on Nb collision system, additional errors appear from the uncertainty

in the subtraction of the background effects. There is also an additional

120

error, due to the pile-up on the K x-ray counter (fig. 3.22), which contri­

butes a part in the Sn K x-rays. Measurements have shown that it is about

20% of the total Sn counts.

Finally, we can correct the cascade spectrum (fig. 3.27, target 520 . 2pg/cm ) for the total efficiency and the fluorescent yield, and find the appro­

priate cross sections (pbarns/KeV, figures 3.28 and 3.29 for the two target

thicknesses). Instead of calculating these cross sections in a straightfor­

ward manner using the BCI and the target thickness, an indirect way was

chosen, i.e. the differential cross sections of the M O x-rays were deduced

by normalizing the M O x-ray yields to the total yield of K x-rays from Nb

and Sn, and by using the characteristic x-ray cross sections measured in

this thesis (HI. 4 and IV. 5). This is equal to:

_ _ (# of C2) WNb)ef4+ o(Sn)ef4V 1 (# of KX) „2 „1 a

a C2 w k2 2where (a) denotes the gate on Nb or Sn, sff^ (°r e^gn) is tbe tota* effi­

ciency of the IG 1910 detector at the KX-peak of Nb (or Sn), eff ,2 indica­

tes the total efficiency of the IG 1010 detector (MO x-ray channel) at the

M O x-ray energy, and w is the fluorescent yield of the (a) element. ForK

the calculation of the fluorescent yield the same method with the symmet­

ric case was applied, assuming i = 2 for both Nb and Sn. For a(Nb) and

cr(Sn) the values obtained in the K x-ray cross section measurements were

used (a(Nb)= 4000b, a(Sn)=200b, III. 4 and IV. 5).

With this method we avoid the errors both due to target thickness and

BCI, but include the absolute error from the uncertainties in the K x-ray

121

cross sections. The total error includes the statistical one, (shown in fig.

3.28 and 3.29) the uncertainty from the background subtraction, a 10% un­

certainty in the total efficiency, and 25% error from the values of the single

K x-ray cross sections. So, the final cross seetions/KeV are expected to

be accurate to ~ 40% in absolute value.

In table 3.2, the values of the cross section in ^bams/KeV, are shown

together with their statistical errors. Some other values are also presented

in the same table. As in the Nb+Nb collision system, the above differential

cross sections (per x-ray energy) are given at 90° to the beam axis.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

3.20 Singles spectrum of Q M channel, for the case of Nb+Sn, for the2target of thickness 520^g/cm (continuous curve), and the singles

cori’ected for AB and NNB (broken line), are presented.

3.21 Time spectra of the three regions (Nb K x-ray), (Sn K x-ray),

and (C2 M O x-rays), for the case of 520^ig/cm2.

3.22 KX-spectra of Nb, Sn K x-rays, C2 x-rays, and total events2presented for the target of thickness 345 ig./cm .

23.23 QM-spectra for the target of thickness 520^ig/cm .

#1 gates on Nb K x-rays and the time peak of the Time spectrum.

#2 gates on Nb K x-rays and outside the time peak.

3.24 Same as fig. 3.23 but setting gate on Sn K x-rays instead.

3.25 Presentation of real coincidence events (gate on Nb K x-rays),

and the contributions from double vacancies and sequential col-2lisions (broken line), for the target of 520^g/cm .

3.26 Same as fig. 3.25 but setting gate on Sn K x-rays.

3.27 Cascade M O x-rays (corrected for sequential collisions and dou­

ble vacancies) in coincidence with Nb and Sn K x-rays (520^tg/cm2).

3.28 and 3.29. Calculated differential cross sections (^barns/KeV)

for cascade M O x-ray production in coincidence with Nb and Sn

K x-rays (both target thickness).

X - R A Y E N E R G Y (KeV)

Figure 3.20

CO

UN

TS

2 0 0 MeV Nb on Sn Target 5 2 0 /ig /c m 2

TIME SPECTRA5 0 0

400

300

200

100

101 201 301

ATE on <:2

- —4 4 ns oO)

W w ti

Figure 3.21

CO

UN

TS

200 MeV Nb on Sn Target 345 yu.g/cm2

KX Spectra

101 201 301 401 501 101 201 301 401 501CHANNEL NUMBER

Figure 3.22

10 20 30 40 50 60 70 80 90 100

X-RAY ENERGY (KeV)

Figure 3.23

CO

UN

TS

5 10 15 2 0 2 5 3 0 3 5 4 0 45

X-RAY ENERGY (KeV)

Figure 3.24

10 20 3 0 4 0 50 6 0 70 80 90X-RAY ENERGY (KeV)

Figure 3.25

CO

UN

TS

X-RAY ENERGY (KeV)

Figure 3.26

100

oo*

x

1 0

.1

n i i2 0 0 MeV Nb on Sn

- Target 520 g/crrvOGATE on Nb ,

i

d& GATE on Sn

-o--o-1

+ + +*

*n

1I t t j 11 1

T <► <> d kn

i>

30 4 0 50 6 0 70 8 0 9 0X-RAY ENERGY (KeV)

Figure 3.27

Cros

s Se

ctio

n/Ke

V (/

xbar

ns/K

eV)

1 1

DO MeV Nb onSn - irget: 340/j.g/cm 2 _

® Gate on Nb o Gate onSn

OicST(

o I ♦ 1* +

k

t

♦ ii 1

< w

([> T i - i

} | |} 1

Q .11 1 1 |u 1 ■-30 4 0 5 0 6 0 70 8 0 9 0

X-RAY ENERGY(KeV)

Figure 3.28

CROS

S SE

CTIO

NS/

KeV^

ba

rns/

KeV)

t100

1 0 t

t\

2 0 0 MeV Nb on Sn Target:520 / i.g /c m 2o Gate on Nb a Gate on Sn

I

i

I

II ¥

I*

I30 4 0 50 60 70 80

X-RAY ENERGY (KeV)90

Figure 3. 29

ASYMMETRIC COLLISIONS (Nb+Sn)

x-rayenergy(KeV)

M O X-RAY CROSS SECTIONS (^barns/KeV) (90°)E=194 MeV, T. Gate on Nb

2irget=345 /ig/cm Gate on Sn

E=191 MeV, Ta Gate on Nb

2rget=520 jig/cm Gate on Sn

39 150 ±25 6.7 ±3.4 145 ±23.7 16.9 ±6.943 128 ±15 7.9 ±2.3 95 ±12.5 8.5 ±3.447 136 ± 13.8 7.5 ±2 96 ±12.5 9.7 ±2.351 97 ±8.1 4.7 ±1.5 75 ±12.5 8.4 ±2.855 86 ±7.5 4.1 ±1.4 65 ±8.7 6.6 ±2.4

59 49 ±5.6 3.9 ±1.3 54 ±10 2.8 ±1.6

63 44 ±4.8 2.47 ±0.9 43 ±6.2 3.6 ±1.6

67 30 ±4.0 2.02 ±0.74 26 ±6.2 3.7 ±1.6

71 16 ±3.2 1.05 ±0.64 21 ±4.5 3.1 ±1.2

75 10 ±3. 2 13 ±4.5 1.38 ±0.9

79 5 ±2.5 10 ±4.4

(#ofC2) 0(Nb)effNb + a(Sn)effsn , „ ,0a = (trfiaC) ----„2 „1 a------ (a meanS ^ °r Sn)

a C2 W k

o(Nb) = 4000 (b)

CT(Sn) = 200 (b)

Table 3.2

III. 4 KX-KX Coincidence Experiment

Introduction

The coincidence x-ray spectra after correction for the accidental counts,

include the K x-ray peaks of both colliding atoms in addition to the M O x-rays.

These peaks correspond to (K x-ray)-(K x-ray) coincidences, which can be

produced, as it was explained before, from sequential independent collisions

or through double K-shell vacancies in a single collision. Both mechanisms

will also contribute to the production of C2-KX coincidences and, as noted

earlier, they will constitute a possible background in the coincidence spectra .

To check their importance quantitatively one has to study a) the vacancy pro­

duction probability in the most inner-shell orbitals 2pcr, lsn and b) the decay

probability of these vacancies during the atomic collisions. The second part

is studied analytically in the Theory section, while for the first some theo­

retical calculations and measurements have been performed.

The contributions from sequential collisions can be checked easily, since

the production of the first x-ray is not correlated in any way to the second

x-ray, and therefore, the relative ratio of the K x-rays to the M O x-rays

will be the same as in the singles spectrum. Its quantitative contribution

can be computed from the target thickness dependence of the KX-KX ray yield.

Double vacancies produced in a single collision will also give K x-rays

in coincidence with other K x-rays (fig. 4.2). Since the K x-ray peak in the

cascade spectra is mainly the result of 2pa M O double vacancy while any con­

tribution from double vacancies in the C2 energy region will be produced

from 2pcr, Isa vacancies, one should investigate the production probability of (2pa, Isa) double vacancy compared to the (2pa, 2pa) double vacancy.

134

135

Theoretical arguments support the idea (Ch. IV), that the normalization of the

singles spectrum to the K x-ray peak of the coincidence spectrum is a good

approximation (within a factor of two) of the contribution, from double vacan­

cies in the region of interest (C2 radiation). For this idea to be tested,

the KXtKX ray production cross section for the light-light and light-heavy

combinations has been measured in sufficiently asymmetric collision systems.

The need for highly asymmetric sj'stems results from the vacancy sharing»

mechanism, which is mainly responsible for the KX(heavy)-ray production

for systems close to symmetry. Since a Isa M O vacancy is needed for the

production of a C2 x-ray, one can measure the double Isa, 2ps vacancy pro­

duction from the KX(light)-ray, KX(heavy)-ray coincidence yield only in a

sufficient asymmetric system where the KX(heavy)-rays are produced from

the Isa M O alone. But in such highly asymmetric systems (Z /Z <0.6),1 £

the L-shell of the heavy atom is close in potential energy, to the K-shell

of the light atom, resulting in K-L matching (fig. 1.10). In that case, the

KX(light)-rays can be produced from the 2pa M O excitation, or from the 3da,

2pa vacancy sharing. So, in order to compare the KX-KX data with those

of the symmetric case, the relative importance of each mechanism has to

be estimated. In fact, it can be deduced from the coincidences of the Cl

M O x-rays to the K x-rays, and thus more important information becomes

available from the KX-KX coincidence experiments.

136

The total cross section for KX-KX double x-ray coincidence can be ge­

nerally written as the sum of two terms, one from single collision excita­

tion of a double vacancy and the other from sequential collisions (which de­

pend on the target thickness). Defining o*V the cross section for single K2v 2vvacancy production, a and a the multiple and single collision doubleme sc

inner-shell vacancy cross section respectively, t, the mean life of the KKvacancy, N the number of projectiles, n the number of target atoms per

gram, and specifying as Kl, K2 the two vacancies one has:

P(y)=e ^ VTk the probability that a K vacancy in the projectile will

survive through a distance (y) after the collision.

At a distance x inside the target, the number of single K vacancies

that will be produced in the thickness (dx) will be,

dV1V(Kl)=NalY(Kl)ndx or dV1V(K2) = N a1V(K2)ndx

and the double vacancies (from single collisions),

dV2V(Kl, K2) = N o-^Kl, K2)ndx sc scIntegrating over x we get

lv lvV (Kl)=NcT (Kl) na

V^CKl, K2) = N a2V(Kl, K2) na SC scwhere a is the target thickness (gm/cm2).

A K vacancy can live long enough to enter into a second collision, or

it can decay and later be reexcited to produce another (independent) K x-ray.

In the first case, the double vacancies in the projectile will be:2 2v lv 2v

d v m c ( p r o j ) = *dV * p fr ) (ndy) ffm c =

Cross section for KX-KX production

137

lv 2v= Nna a dx(P(y)dy) (for the projectile)in 0

d2vsIq(K1 ’K2) = dylV(Kl)(1-p (y)> ndy a1V(K2)= Nn ct1V(K1) a1V(K2) dx(l-P(y)) dy

Integrating the second,a a

and in the second,

lv 2 lv lvV (Kl, K2) = Nn a (Kl) a (K2) seq2 T% V T® V 1+ Y<ae K +T^v(e K -i))j

while for the projectile (the similar terms for the target are small since

recoil effects are negligible), the doable vacancies from multiple collisions are

_ _ 2v XT 2 lv 2vV (proi) = Nn a a T, v me me k

Since usually r,v«a, we getK

Tlvr, v(l-e ) - ae k

a

2V^(proj) « (K1,K2)= Nn2 (^-) a1V(Kl) a1V(K2)me seq 2

But, the order of production of Kl, K2 vacancies is unimportant for Kl/K2,

so finally

(Kl, K2) = N (na)2 a1V(Kl) o1V(K2) if Kl / K2

= N-^- (o1V(Kl))2 if Kl = K2Ci

Then,

total _a (Kl, K2) = - = ( K l , K2) + (na) cr V(K1) a V(K2) if Kl / K2

= a2V(Kl, Kl) + (-^) (a1V(Kl))2 if Kl = K2S C ci

The corresponding cross section, for the K x-ray production, will be the

above multiplied by the two fluorescent yields. For K1=K2, one can appro­

ximately assume (error <10%), that the fluorescent yield is the same for both

138

vacancies of the atom. We therefore obtain,

V*V (Kl) = (Nna)wKieffKia1V(Kl)

vf(Kl,K2) = WKi'VK2e< i ef42v lv lvor (Kl, K2) + (na) o (Kl)cr (K2) s c

for Kl / K2

(Nna)

“ 2<WK l)2e4 i e0L ©(K1,K1) + (-y-)(o’1V(Kl))2

for Kl = K2

(Nna)

the factor 2 in the last equation results from the detection of identical x-rays

by the two detectors.

Normalizing the above yields to the single K x-ray yields, we get

2v,V (K1.K2) /V (K1.K2) lv =------= w eff I------- + (na) a (K2) > for Kl ? K2V (Kl) K2 K 2 ( tr (Kl)

2vV (K1.K1)A

lv V (Kl) x '= W KieffKl

f v ^ )

I a1V(Kl)+ (na)

-}

a1V(Kl)| for Kl = K2

So, the normalization to the singles K x-rays defines a quantity linear­

ly dependent oil the target thickness, and the slope is proportional to the

singles K x-ray cross section.

Since, the single K x-ray cross section can also be obtained from other

more direct ways, the accuracy of the above formulas can be checked (within

experimental error), using a least square fit of the (double K x-rays) to

(singles K x-rays) ratios.

139

Data and corrections 93 120The Nb on Sn system was used as a test of the vacancy sharing

mechanism as well as to determine the contributions from sequential colli­

sions to the coincidence spectra. Three monoisotopic targets of thicknesses/ 2 0 345, 520, and 1200 j/g/cm were used. They were inclined 45 to the beam

2axis, so their actual thicknesses were 488, 735, and 1697^tg/cm .

Ni projectiles on Sn targets at two energies (100, 200 MeV) were used

to measure the double vacancy production. The two atoms have been chosen

so, that the total atomic number was similar to the one for the symmetric

case (Nb+Nb), since the lscr M O vacancy production probability scales rough­

ly as the UA binding energy (UABEA). Besides the Isa, 2pa M O sharing fac­

tor is quite small (0.0003, Me 73) and the Is level of the heavy atom (Sn)

will receive vacancies only through the Isa M O excitation (responsible for

the C2 M O x-ray radiation as well). So by measuring the K x-rays of Sn in

coincidence with Ni K x-rays the Isa, 2pa double vacancy production proba­

bility can be calculated.

In both cases, the IG 1010 detector was used (mainly) for the detection

of the KX(heavy)-rays. A combination of Al and Cu absorbers decreased the

K x-rays of the light element in favor of the K x-rays of the heavier atom.

In the other channel, the IG 1910 (or Nal) detector was used for the Nb+Sn

(or Ni+Sn) system with 4mil Al foil absorber, to stop most of the L x-rays.

Typical x-ray spectra (not corrected for total efficiency) are presented

in fig. 3.30 and 3.31. For the Nal detector (fig. 3.31b), the L x-rays of

Sn could not be resolved from the K x-rays of Ni, causing a small uncertain­

ty (<10%) in the separation of the Ni K x-rays from this spectrum. For the

140

Ni+Sn system (fig. 3.31a, IG 1010 detector), there is a continuum x-ray extending

between the two K x-rays of Ni and Sn. It is attributed to transitions to the

2pa M O (Cl radiation).

Fig. 3.32 (3.33) presents the time spectra for the three combinations

Nb-Nb (Ni-Ni), Nb-Sn (Ni-Sn), and Sn-Sn (Ni-Cl). In the Ni on Sn time spec­

tra the peak to BG ratio increases from the Ni-Sn to the Ni-Ni case, while

for the Nb on Sn system no large change is observed.

The data from the different targets were actually accumulated at diffe­

rent mean projectile energies. In all cases, the double vacancy data had been

normalized to the singles K x-rays, and the energy dependence of their ratioerC\C 1depends on the energy dependence of the other K x-ray cross section ( —a(Kl)

~ o(K2)).

In the Ni+Sn measurements, the targets were attenuating the low energy

(Ni) K x-rays because they had been inclined 45° to the beam line and the de­

tection angle was quite large (~0.5rad). The absorption has been measured,

using calibrated sources, and the data appropriately corrected, though the max­

imum absorption was less than 7%. In addition, the target thicknesses, which

were known to ~15% from the alpha particle thickness gauge measurements,

have been determined more exactly by an iterative procedure. Thus, the (Ni K

x-rays)/BCI and the (Sn K x-rays)/BCI had been plotted as functions of the

target thickness. Then, small corrections were applied to the target thick­

nesses, until the two sets of (K x-ray)/BCI yields (for Ni or Sn) could be

fitted by two linear expressions of the form y=ax, minimizing the deviations

of the data to the linear fit. Before the small corrections in the target thickness

141

could be applied, the data had been corrected for the difference in the aver­

age energy in each and for the absorption of the Ni K x-rays by the tar­

get With these corrections applied and the iterative procedure the target

thicknesses were known within 5%.

The Nb+Sn system

In slightly asymmetric collision systems (Nb+Sn), the sharing of vacan­

cies between the 2pa, Isa orbitals in the exit channel is expected to be the

dominant mechanism for vacancy production in the K-shells of both colliding

atoms (Me 77). Then, the relative ratio of the K x-ray yields (for heavy

and light elements) will depend only on the sharing factor (Me 73), and the

K-shell fluorescent yield of each atom. For small variations of the projecti­

le energy (~5%), no appreciable change in the fluorescent yields will be

expected. So, any change of the (Nb) to (Sn) K x-rays ratio should be pro­

duced from the change of the vacancy sharing factor with projectile energy.

Table 3.3 presents the targets used, the average projectile energy in

each target, the theoretical vacancy sharing ratio (Me 73) and the experimen­

tal ratio of the two K x-ray yields. Normalizing the theoretical sharing/ 2ratio to the experimental value at the 345 jig/cm target (last line in table

3.3), we observe that the other two experimental ratios (for targets of thick-/ 2nesses 520 and 1200^g/cm ) are reproduced quite well. So, the projectile

energy dependence of the K x-ray yields, of the two colliding atoms, can

be explained, using the theoretical variation with energy of the vacancy shar­

ing factor. Thus, all Sn K x-ray yields had been corrected for the slight

142

variation of the vacancy ratio with projectile energy and for the energy de-2 2 ± 0 1pendence of the Nb K x-ray cross sections (a(Nb)~ E ‘ ' , IV. 5). The

error from the energy correction is expected to be less than 5%.

Fig. 3.34, 3.35, and 3.35 present the three cases of (Nb-Nb)/(Nb),

(Nb-Sn)/(Nb), and (Sn-Sn)/(Sn) double K x-ray yields normalized to the single

K x-rays. In these figures, the (Nb-Nb)/(Nb) ratio has statistical errors of

the size of the dots, while for the other two the error bars are shown in the

figures. Besides these errors, there is some uncertainty in the target thick­

ness. An independent check of the (Nb) and (Sn) K x-ray yields/BCI have prov­

en, that the thicknesses were known to an accuracy of 10%.

From the least square fit of the three set of data (Nb-Nb)/(Nb), (Nb-Sn)

/(Nb), and (Sn-Sn)/(Sn) one can get the best estimate of the slope and the

intercept. Correcting the slope for efficiency according to the formulas of thelxprevious section, the cross section for the production of a K x-ray (a ) or

lva K-shell vacancy (a ) are obtained. The intercept also has to be corrected,

according to the same formulas, for efficiency and fluorescent yield, and then,

the ratios of double to single vacancy production cross section can be extract­

ed. The results are shown in Table 3.4.

According to the theory, the slopes of the two ratios, (Nb-Sn)/(Nb) and

(Sn-Sn)/(Sn), should be the same, which is in agreement with our data. To

calculate the (Sn) K x-ray cross section an average value of the two cross

sections was estimated by fitting the two sets of ratios according to the total

number of counts in each. Finally, one should notice that the ratio of the

two intercepts of (Nb-Sn)/(Nb) and (Sn-Sn)/(Sn) is very close to one (0.95±. 2).

143

A direct comparison, of these values with the theoretical predictions is

presented in section IV. 4.

The Ni+Sn system

In figures 3.37, 3.38, and 3.39 the (Ni-Ni)/(Ni), (Ni-Sa)/(Sn), and

(Ni-Cl)/(Cl) ratios are presented as functions of the target thicknesses for

both projectile energies (100 and 200 MeV). According to the theory presented

the ratios should depend linearly on the thicknesses and their slope should

define the Ni K x-ray cross section. They indeed fit (within statistical error)

to a straight line, and from their slope one has1 x

o (Ni) = 10000 ±500 b for 100 MeV

a1X(Ni)= 45500 ±3000 b for 200 MeV

Comparing these values with other measurements, one observes a dif­

ference of a factor 2 (Ku 73), and 25% (Johnson 79), our data being larger

in both cases. It would seem'that the most recent measurements of Johnson

et al. are more accurate, as the others disagree by a factor of 3i in the

Ni+Ni case from the measurements of P.Vincent et al. (Vi 77).

One major problem, in the Ni+Sn collision system, is the unknown (but

very important) contribution from the 3do— 2pcr vacancy sharing (K-L matching).

In figure 3.40, which reproduces the data from Kubo et al. (Ku 73), one can

observe the Important contribution of both processes since the Sn target is

at the bottom of two valleys, one from each process (rotational coupling-

left side and K-L matching-right side in the diagram). Theoretical estimates

have been provided for both the 2pcr M O vacancy production cross section via

144

rotational coupling (Me 77) and the K-L matching (Me 78, 79). Unfortunately,

these calculations are valid to a factor of two only (or more), and there is

an even greater uncertainty in the 3dcr M O vacancy production cross section.

So, these calculations can only be used qualitatively (IV. 5). Since, for the

the theoretical estimate of the 2pcr, lscr M O (or 2pcr, 2pcr) double vacancy

production probability, the relative importance of each vacancy production

mechanism in the 2pcr M O is needed and it is not known, an alternative way

can be used. It will be shown in Ch. IV, from the ratio of the two intercepts

of the (Ni-Cl)/(C1) line (fig. 3.39), compared to the (Ni-Ni)/(Ni) line (fig.

3.37), the relative contribution of each process for the K-shell (Ni) vacancy

production (3do— 2oa vacancy sharing or 2pcr M O excitation) can be deduced.

Table 3.5 presents the cross sections for (Ni) K x-ray and K vacancy

production, and the ratios of double to single vacancy cross sections, obtain­

ed from the intercepts of the straight lines in fig. 3.37-3.39, after correc­

tion for total efficiency and (Ni) fluorescent yield (w (Ni) = 0. 50).

Fig. 3.30 X-ray spectra, not corrected for efficiency, in the case of 2002MeV Nb+Sn (345^g/cm target). Left side, the spectrum from

the IG 1910 detector. Right side, the spectrum from the IG 1010

detector.

Fig. 3.31 X-ray spectra for both detectors in the case of 100 MeV Ni+Sh2(target 75^g/cm ). a) From the IG 1010 det. , b) From the Nal det.

Fig. 3.32 The time spectra of (Sn-Sn), (Nb-Sn), and (Nb-Nb) double K x-2ray coincidences in the case of 200 MeV Nb+Sn (target 345/ig/cm ).

Fig. 3.33 The time spectra for (Ni-Ni), (Ni-Sn), and (Ni-Cl) double K x-ray2coincidences in the case of 100 MeV Ni+Sn (target 75/ig/cm ).

Fig. 3.34, 3.35, and 3.36. The (Nb-Nb)/(Nb), (Nb-Sn)/(Sn), and (Sn-Sn)/(Sn)

ratio of the yields as functions of the target thickness (200 MeV

Nb+Sn).

Fig. 3.37, 3.38, and 3.39. The (Ni-Ni)/(Ni), (Ni-Sn)/(Sn), and (Ni-Cl)/(C1)

ratio of the yields as function of the target thickness at both pro­

jectile energies. Continuous line: 100 MeV; broken; 200 MeV.

Fig. 3.40 Projectile K x-ray cross sections in the collisions of Ni on va­

rious targets and for different energies (from H. Kubo et al. ,

Ku 73).

2 0 0 MeV Nb on Sn,Target 345ftg/cm2

C H A N N E L N U M B E R

Figure 3.30

Figure 3.31a

Figure 3.31b

CO(-zZ5o(_>

Tdi 201 301

2 0 0 MeV Nb on Sn

Torget 345 /ig /cm 2

TIME SPECTRA 150]------ 2,000

100

50----

1,500

1,000

500

101 201 301

CHANNEL NUMBER

Figure 3.32

CO

UN

TS

100 M e V Ni on Sn

Target 75 /A g/cm 2

TIM E SPECTRA270i

180

I 301 101 201 301 101 201 301

CHANNEL NUMBER

Figure 3.33

(Nb

-Nb

)/(N

b)

-4XIO

Figure 3.34

(Nb-

Sn)/

(Nb)

Figure 3.35

(Sn-

Sn)/(

Sn)

Figure 3.36

100

MeV

— Target thickness (fig/cm2)

Figure 3.37

200

MeV

100

MeV

Figure 3.38

200

MeV

100

MeV

xi0‘4

20

15

10

(Ni-Cl) » 100 MeV200 MeV

loo 250 300

—■ Target

Figure 3.39

40

"4 0 0 s5o ‘

thickness (/ig/cm2)

20

200

MeV

Pr

oje

cti

le

K x

-ra

y

cro

ss

-se

cti

on

(b

ar

ns

)

t i l lH n t n f r r \ m M t f n h n d I n l

( K u 7 3 )

1

v▼ T V *

T . • ▼ : • . !

- • ® V

• J » V T T T T •

o . B - * T

■■ * ■ ■ £ * # —

“ ■ a■

9

m• • •

•

■■

■■

)Ji p r o j e c t i l e s

1f 9 4 M e V

► 61 M e V

1 4 5 M e V

«

i

2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

T a r g e t A t o m i c N u m b e r

Figure 3.40

93 120200 MeV Nb on Sn

K X -K X coincidence

2Target thickness (jug/cm ) 345 520 1200

2Total thickness (/Ug/cm ) 488 735 1697

Average energy (M eV) 194 191 180

Sharing ratio (1-w/w) 20.731 21.242 23.494

( Nb)/(Sn) 2.506 ± 0.020 2.556 ± 0.022 2.802 ± 0.025

(Nb)/(Sn) from sharing 2. 506 2.568 2.840

Table 3.3

93 120200 MeV Nb on Sn

Xx

lva (Nb) = 5540 ± 500 (b) (K-vacancy cross section)

lxa (Sn) = 225 ± 40 (b) (K x -ra y cross section)

lvo (Sn) = 253 ±4 5 (b) (K-vacancy cross section)

o (Nb) = 4430 ±400 (b) (K x -ray cross section)

S L l f i S ^ U 1.64 ± 0 . 2 <%)ct (Nb)

2v2. (Nb-Sn) = o. 14 ± 0. 015 (%;

c (Nb)

2va (Sn-Sn) = 0 074 ± 0 02

ct (Sn)

Table 3.4

M - 5 8 C 1 2 0Ni on Sn

KX-KX coincidence

a) 100 MeV

lxO (N i) = 10000 ± 500 (b)

a 1V(N i) = 20000 ±1000 (b)

- (NL N l> = 1. 0 ± 0. 05 (%) a (N i)

2va (N i-Sn) _

a 1V(Sn)10.67 ±0 .8 0 (%'

? <N L .? 1> = 4.8 ±0 .4 5 (%;

a (C l)

b) 200 MeV

a 1X(N i)= 45500 ±3000 (b)

o 1V(N i) = 91000 ±6000 (b)

2vCT N l* = 2.67 ± 0.3

a 1V(N i)

a_ (N i_Sn) = 16 50 ± 1 0 q q a (Sn)

-— *Nl Cl = 11.1 ± 1. 20 a l v (C l)

Table 3.5

161

The coincidence experiments provided us with a powerful technique of

separating selected quasimolecular transitions. Y et, some uncertainty remains

in the identification of the spectra as transitions coming from one leve l only

(the 2pcr).

In the theory section, it was explained how the two collis ion mechanism

contributes to the creation o f vacancies in the Is a MO fo r the sym m etric

system . It is also c lea r that the creation o f a Is a MO vacancy early in the

collision (before the distance of closest approach), can lead to a 2p 7r-,-lsa ra ­

diative transition followed by the transfer o f vacancy to the 2po MO, v ia r o ­

tational coupling at small internuelear distances (fig . 4 .1 ). These 2pfl-~lsa

transitions w ill be associated with K x -rays , contributing to the rea l coinci­

dences. Because of the sharing of vacancies between the 2pa and Isa MO’ s,

the same phenomenon is observed in the asym m etric system s, resulting in

vacancy creation in the Is a MO o f the heavier colliding atom. This vacancy

creation from the sharing mechanism w ill be superimposed on the vacancy c re ­

ation through d irect excitation, and in near-sym m etric collisions it can dom i­

nate the K x -ray production. Since there are theoretical methods of calculating

the Is a vacancy production through d irect excitation (one collis ion process), a

comparison of the theoretical values with the experimental data w ill help under­

standing the rela tive contribution of the two mechanisms. Besides, the abso­

lute normalization of the final cross sections was doubtful due to many e rro rs

associated with the large solid detection angles, the thick absorbers, as w ell

DI. 4 K x -ray Cross section measurements

Introduction

162

as the possible e rro rs in the BCI reading and the target thickness. Thus, a

m ore accurate way of measuring the characteristic x -ra ys cross section and

consequently norm alizing the coincidence data was needed.

Th ere fo re , the characteristic K x -ray production o f both pro jectile and

93target was studied, using Nb beam on targets o f atom ic number from Z=6

to Z=69. The experim ent was perform ed at three p ro jectile energies 100,

160, and 200 MeV (lab. energy).

F igures 3.41, 3.42, and 3.43 present characteristic spectra o f the two

detectors and fo r the two heaviest targets used (E r, Sm ). It is c lear that

with the highest beam intensities used, the K x -ray peaks o f the heavy target

just discrim inate against background (fig . 3.41, 3 .42 ). The d ifference in the

energy resolution of the two Ge detectors is also c lea r in the K x-ray lines.

Tables 3. 6-3 .8 present the targets used, together with a f irs t estimate

o f their thicknesses, and also the average p ro jectile ve loc ity on the targets.

The thickness of the targets was measured with an alpha gauge before and

a fter the run, and in many cases it was d iscovered that a significant change

o f the thickness had occured during the experiment. In some cases this dete­

rioration of the targets makes the thicknesses presented in Tables 3 .6-3 . 8

only a rough approximation of the rea l values. In the cases o f Nb and Sn,

we have used the same targets as in the coincidence experim ents, and their

thicknesses are known to have less than 10% e rro r . The d ifference in the

average pro jectile energy on the targets, theoretica lly , can make the com ­

parison of the various cross sections difficu lt, but in p ractice the variation

in the mean energy is less than 5% and the maximum expected e r ro r from

such variation is around 10%.

Instead o f calculating the K x -ray cross sections in a straightforward

manner, using the number o f pro jectiles and target atoms, an indirect way

was chosen. As it was stated before the K x -ray cross sections had been

extracted from the Rutherford cross sections of the scattered ions. With this

normalization, the e rro rs in the K x -ray cross sections from the uncertainties

in the target thickness or the BCI reading are avoided. But, an e r ro r remains

from the possible variation of the beam spot on the target, which leads to an

uncertainty in the solid angle subtended by the x -ray detecter. To measure any

such variation two particle detectors w ere used, located sym m etrica lly to

the center o f the target (±30 °). Any difference in the counting rates of the two

partic le detectors produced by (sm all) changes of the Rutherford cross sections

defines this variation of the beam position. With such L30-R30 d ifferences, the

importance o f a careful correction which would include the L -R asym m etry,

was obvious. F ig . 3.44 presents the d ifferences we might have in the total

counting rates from sm all changes in the beam spot on E r target (other cases

are very s im ila r ). In this figu re, d denotes the small change o f the beam on

target (which is inclined 45° to the beam ax is ), 0 , 0 a re the new anglesK Lj

163

(lab) o f the particle detectors and 0 , 0CM the corresponding angles inL

the CM system , gd^, Sd^ are the small changes o f the target-to-detectors

distances, d fZ jV d fi^ the ratio of changes in the detection angles of the two

detectors, cr /a the le ft-righ t asymmetry in the Rutherford cross sections,L R

and Y /Y the L -R asymmetry in the total y ields of the two detectors. From L R

this figu re, we see that we can estimate the correction to the Rutherford

164

cross section on one detector (R30°) from the L -R d ifference, in the number

o f o f scattered pro jectiles or approximately from the total counts in each

counter whenever the separation of the pro jectile counts was not possible.

Such correction has been used in the analysis o f the data and it is believed

to be accurate to 5%.

The d ifferent slope of the target re la tive to the two partic le detectors

a lso introduced a serious effect. The target thickness that the scattered p a r­

tic le passed through before reaching the L30 partic le detector was about 4

tim es that o f the other detector, and resulted in a large spread in the energy

o f the scattered partic les. In most cases it was difficu lt to separate the peak

o f the scattered pro jectile from the reco il or the C backing (usually less than

5%) and only the total counts could be used (to calculate approxim ately the

L -R asym m etry). In some cases an improvement was achieved by changing

the target and using a thinner one, but in most cases the L30° detector could

be used only as correction fo r the sm all variations of the beam spot on target.

F ig .3.45 and 3.46 present spectra o f the two partic le detectors fo r some

o f the targets used and fo r d ifferent p ro jectile energies. The d ifferences observ­

ed between the two detectors and between different targets is obvious. In the

above figures the identification of each peak is marked, and it is c lea r that

in some cases the separation o f the pro jectile peak from the reco il is not

possible.

A s noted ea r lie r in section II, the dead time was measured accurately

using pulsers driven by the BCI. In most cases, it was less than 1%.

165

The data have been analysed in the follow ing way:

The counts in each KX-peak have been integrated, using a Gaussian line

shape fit on a quadratic background. In those cases where the statistics w ere

ve ry poor fo r a reliab le fit, the average background was subtracted from the

total counts in the peak. Each component o f the K X -line (K , K ^ ) has been

corrected fo r effic iency separately.

In the case o f Y target, the separation o f the two K X -lin es was not pos­

sib le , so that the sum was considered. In this case the correction fo r e ffic ien ­

cy was done using an average value o f e ffic iency at the mean energy o f ,

K Y lines (the average energy evaluated using the strength o f each peak). p

The e r ro r in the x -ray intensities from the Gauss fit in a ll cases was <5%.

Corrections w ere made fo r the dead tim e, i f any. In most cases the dead tim e

was less than 1%, and no correction was required. The uncertainty in the e f f i­

ciency correction was ~ 10% for atoms heavier than Ge and ~20% fo r atoms

ligh ter than Ge.

The peaks in the particle detector spectra w ere identified and integrated.

In many cases, the reco il peak could not be resolved from the main peak so

that the total peak intensity was considered. Especially fo r the L30® partic le

detector, the integration of each peak separately was ve ry uncertain, hence

the total counts o f the detector w ere used. These total counts in each parti-,

c le detector defined the L -R asymm etry. The uncertainty from the separation

o f each peak in the spectra o f the R30° counter was 1-6% depending on the

collision system and the target thickness used.

Data Analysis

166

The counts in each particle detector w ere corrected fo r the measured

efficiency which was considered independent of the partic le energy. Any abso­

lute e r ro r in this effic iency (< 1 5 % ) could result in an overa ll normalization

uncertainty, and it could not affect the re la tive values o f the K x -ray cross

sections.

The theoretical values of the Rutherford cross section w ere calculated

fo r the average energy o f the beam on each target,and they w ere corrected

to account fo r the observed L -R asym m etry according to figu re 3.44.

Since most o f the targets w ere made by evaporating the m ateria l on

/ 230 fj,g/cm C backing, there was a small contribution in the Nb K x -rays from

carbon. By integrating the counts in the C-peak o f the partic le detector (R30°),

and comparing them with the counts in the pro jectile peak, the rela tive thick­

ness of C to the target was deduced (it was also known approxim ately) and from

that the contribution of the C backing on the Nb K x -ray y ie lds was calculated.

Thus:

R( # C atom s) aXTUl. ,v ' Nb+t ( C reco il) .. . ..--------------------- ---------- ------------- (t= ta rget)

. R (Nb ions)(# t atoms)

Actually, the Nb on C cross section should be estimated at d ifferent p ro jecti­

le energy than the energy o f the target, but the d ifference is sm all and in a ll

cases the contribution from the C backing turned out to be less than 5%.

The absorption o f the x -rays on target was negligib le in a ll cases.

F inally the K x -ray cross section w ere calculated using the formula:

IQ R (counts in KX-peak)

°K X ~ °N b , + , R30(counts in Nb peak)

Rwhere IG denotes the Germanium detector and <t _ the Rutherford cross section

Nb

of the pro jectile at 30°. Sometimes, it includes the reco il when the two peaks

are integrated together.

Tables 3 .6 , 3 .7 , and 3.8 present the K x-ray cross section of both the

p ro jectile and the target fo r the three energies 100, 160, and 200 MeV. These

cross sections include a relative e r ro r of 10% from total e ffic iency (in the

case o f largets o f Z <32 the e r ro r is ~20% ), a maximum e r r o r o f 10% from

the integration o f the peaks (both K x-rays and partic le spectra) and an o ve r­

a ll norm alization e r ro r o f 15% from the partic le detector e ffic iency.

167

The K x -rays w ere actually detected at 90 with respect to the beam

line but the em ission of K x-rays is isotropic (Le 761, V i 77) in the em itter

fram e o f re feren ce, and at 90° the difference in the cross sections in the two

fram es o f referen ce (lab. and em itter 's ) is v e ry sm all (< 1 % ). Thus, no e r r o r

is expected from the integration o f the d ifferentia l cross section over 4n.

Figures 3.47, 3.48, and 3.49 present the calculated K x -ra y cross s e ­

ctions fo r the three p ro jectile energies. The higher data points represent the

cross sections f o r the ligh ter collis ion atoms and according to the quasimo-

lecu lar picture, they should vary continuously from target to target. The low ­

e r data points, which are the cross sections fo r the heavier atom, w ill be

compared in the next chapter with theoretical predictions based on the va ­

cancy sharing mechanism and the Is a MO d irect excitation. It w il l be shown

that in the most co llis ion systems the K x -ra y (heavy) cross sections can be

explained with the simple idea of a 2pa MO vacancy shared among the two

outgoing atoms (p ro jectile and target), and that only fo r v e ry asym m etric

collis ions (Z^>52 or Z^<24) the d irect excitation mechanism dominates.

Fig. 3.41, 3.42, and 3.43. Typical x -ray spectra fo r Sm and E r (la rge de­

tector, and fo r E r (sm all detector). In the last spectrum only the

Nb K x-ray peaks can be seen.

Fig. 3.44 Evaluation o f the difference in the counting rates (Y , Y ) o f theLi R

two partic le detectors, resulting from the sm all wandering d of

166the beam, on E r target.

F ig. 3.45 Spectra o f the R30 partic le detector, fo r 100, 160, and 200 MeV,

and the L30 (200 M eV ), in the case of Sm. The d ifference of the

R30 and the L30 spectra is obvious.

F ig. 3.46 BaClg R30 spectra, fo r two different targets. The grea t im pro­

vement in the spectrum, resulting from the m ere change of the

target can be seen. A lso presentation o f two other spectra, where

two peaks overlap, and they have to be integrated simultaneously.

F ig. 3.47, 3.48, and 3.49. Characteristic x -ra y cross sections (in barns)

fo r the three pro jectile energies 100, 160, and 200 M eV , fo r Nb

and target.

101 201 301 401 501 601 701 801 901 1001CHANNEL NUMBER

Figure 3.41

10,000

1,000

01 201 301 401 501 601 701 801 901 1001

CHANNEL NUMBER

Figure 3.42

CO

UN

TS

101 201 301 401 501 6a 701 801 901CHANNEL NUMBER

1001

Figure 3.43

d(mm) V > 6dR e L ( ° ) 6dLE r166

®c m l /r

d — L

d - R°L /oR Y i / Y r

3 33.03 +.78 29.19 -2.90 45.02(50.77)

1.139 1.560 1.77

2 32.02 +.52 29.46 -1.932 45.45(49.27)

1.095 L. 354 1.48

1 31.01 +.26 29.73 -.97 45.86(47.77)

1.046 1.166 1.22

-1 28.99 -.26 30. 27 +. 97 46.68(44.75)

.957 0.855 .820

-2 27.98 -.52 30.54 +1.932 47.10(43.20)

.915 .727 .666

-3 26.97 -.78 30. 81 +2. 90 47.54(41.60)

.880 .615 .540

Figure 3.44

CHANNEL NUMBER

Nb on Sn

CHANNEL NUMBER

Figure 3.45

100 MeV Nbon BaClz R30

0 101 201 301 401 501 601 701 801 9011 l

60 MeV Nb on Ba , R30 ,

Cl2

I er

W 11 c

r i Ba

b

0 101201301401501601 701 801901CHANNEL NUMBER

Figure 3.46

K X-

Ray

Cros

s-se

ctio

n (b

arn

s)

lO5A

I04

lO3

lO2

10

1.0

0.1

—* Target Atomic Number Zf

Figure 3.47

:-------- 1-------- :1 0 0 M e Vo M ho Target

o

O ------------O n.

ooa P — ® -

n tc* 0

©•k A

o---------------------- (i 9 o _

••- A -------------

Am '

o

o

o10 2 0 3 0 4 0 5 0 6 0

K X-

Ray

Cros

s-sec

tion

(bar

ns)

icr

io;

io2

O 160 MeVbirget

° o<D

• NoT<

n°o

• •4fc

-

W

•. . .

•

•

9 ' o •

•• '

<!>

Q

. o .O

10

10 2 0 3 0 4 0 5 0 6 0 70Target Atomic Number —

Figure 3.48

1MeV ~

birget

on 2 0 0

o0

O NoTc

°o

So • 1•

i

• •

*

••

o •

•• ■\

©

• *n

Ou

O

10 2 0 3 0 4 0 5 0 6 0 7 0Target Atomic Number Z t —

Figure 3.49

100 MeV

TAR G E T THICKNESS \ L g / cm 2

BACKINGAVERAGE PROJ. ENERGY MeV

CROSS SECTION (barns )P ro je c t ile Target

6 C12 52 none 97 6.73

13 A1 27 255 ~10 95 5.09

22Ti48 • 410 ~15 93. 2 6.41 25000

24C r 52 280 none 95.5 9.19 10850

26Pe 56 382 none 94 12. 26 7100

28Ni58 311 none 95.5 26.7 6867

30Zn64 280 *** 20 96 85.3 5887

34Se80 ~ 140 ~ 20 98 88 2100

35Br?9 ~100 ~ 20 99 127. 5 2830

39Y89 75 25 97.7 2854

41 Nb93 736 none 91 944.3 944.3

42 M o98 520 none 94 1095 678.3

47 Ag107 226 none 97.4 1068 67.7

50 Sn*29 480 none 95 586.2 8.50

52 T e 130 140 40 98 493 2.97

56 B a 138 ~ 50 ~4 0 100 250 .46

62 Sm144 282 50 97.5 150 .148

68 E r 166 100 50 99

Table 3.6

160 MeV

TAR G E T THICKNESS H g/ cm 2

BACKINGAVERAGE PROJ.

ENERGY MeVCROSS SECTION ( barns )P ro jec tile Target

6C '2 52 none 157 33. 8

13 A1 27 255 ~10 154. 5 40.6

22T i 48 410 ~ 15 152.6

24C r52 280 none 155 92.7 26105

26Fe56 382 none 153.6 117.6 18610

28Nf58 311 none 155 217.7 16245

30z “ 64 280 ~2 0 156 365 12253

34Se 80 ~ 350 ~ 20 155 714 8102

3 5 ^ 79 ~100 20 158 926.6 7307

39y 8 9 75 25 157.5 6150

41 Nb 93 736 none 150 2810 2810

4 2 M o98 520 - none 153.4 3570 2671.4

47a § 107 226 none 157. 2 3332 372

50 Snl2° 480 none 155. 5 1995 72.1

5 2 T e l 3 0

56 Bal38 ~70 ~4 0 ~ 159 870 3.47

62 Sm 144 282 50 157.1 606 .714

E r I®® 68 r 100 50 159 ~ 500 4 .8

Table 3.7

200 MeV

TARG ET THICKNESSp g/ cm 2

BACKINGAVERAGE PROJ. ENERGY MeV

CROSS SECTION (barns)P ro jec tile Target

6C12 52 none 197.1 93

13 A l 27 255 -1 0 194. 5 I l f . 7

22Ti48 410 -1 5 192. 5 181.7 120,000

24 C r 52 280 none 195 264.5 57890

26 Fe 56 382 none 193.5 364.5 41125

28 Ni 58 311 none 195 553.9 29470

30 z n 64 280 - 2 0 196 1009 22760

34Se 80 ~400 - 20 193 1716 13203

35 B r 79 ~100 - 20 198 1891 11605

39 Y 89 184 30 197.5 12470

41 N b93 736 none 18S.7 4960 4960

42 M o 98 520 none 193. 2 5100 4064

47 Ag 107 226 none 197.1 4700 705

50Sn129 480 none 194 4090 203. 5

52 T e *30 190 40 197.7 3233 84

56 B a138 - 7 0 - 4 0 199 1743 14. 5

62 Sm 144 282 50 197 1200 2.87

68 E r 166 100 50 199 736 9. 61

Table 3.8

181

IV. 1 Background Contributions

In Ch. I l l , the considerable change of the MO x -ra y spectra (in coinci­

dence with K x -rays ) compared to the singles spectra was presented (fig . 3.17).

It results from the selection of specific MO transitions by the coincidence

requirement. But, as it has been already stated, part of the observed

x -rays in the coincidence spectra is produced from sequential independent

collis ions o r double vacancies. The study of this background contribution

o r of any other possible mechanism for producing MO x -rays correlated

with K x -ra y s , is of great importance fo r identifiing the observed non cha­

racteris tic x -ra y s , with a particu lar MO transition. In this f irs t part of the

discussion, the careful examination of all transitions competing with the

2po—Iscr transition as a cascade with K x -ra y s , w ill be presented and ana­

lysed. The discussion is divided into two cases. The f ir s t deals with back­

ground resulting from two step I sct MO vacancy production p rocesses, and

the second presents the background contributions to the coincidence spectra

from one collis ion I sct excitation.

a) Multiple collisions

The contributions from multiple collisions w ill be of two kinds. In the

f irs t , the MO x -ra y and the K x -ra y w ill be created in two com pletely inde­

pendent collis ions of the same p ro jectile atom. Then, the two x -ra y produc­

tion p rocesses w ill not in terfere with each other and as we have seen in

a previous section (in. 4) the cross section fo r two x -ra y production is p ro ­

portional to the product of the two cross sections of each x -ra y separately.

IV DISCUSSION

182

2x 1 x 1 xaseq(KX" C2)= (n a )° <KXber <C2>

and

°seq(KX"KX) = (-fK*)22x

where cr define the cross section fo r producing two x -ra y s from sequential

lxcollis ions and o the cross section fo r single x -ra y production.

Then,

2xCT (KX-C2) lx

seqv o _2x 1 x

CTseq(K X -K X ) a (KX)

Considering the double detection effic iency fo r the two identical K x -rays

we w ill have fo r the yields

2xY (KX-C2) l x ___

seq*_________ _ Y (C2)

Y^Xq (K X -KX ) Y 1X(KX)

which proves that the double vacancy x -ra y y ields have identical shape as

the single x -ra ys (fo r all x -ra y energies).

The second kind of BG contribution from multiple co llis ions re fe rs

to the 2pa coupling with the 2pir orbital. A possib ility exists, of creating

a K -shell vacancy in the pro jectile in a f irs t co llis ion , which can enter

into a second collision . There, it can follow the Is a orbital (two collis ion

Is a MO excitation). Then, a 2 p ^ l s a radiative transition before the distance

of closest approach, which w ill be followed by a 2p;r—2pa vacancy transfer

(rotational coupling) at small internuclear distances, would produce a K -

shell vacancy and so a K x -ray correlated to the 2p7r—l s a MO x -ra y (fig . 4.1a).

So, in the case of sequential collisions

183

In the one collis ion Isct MO excitation, this 2pn—+1sct, K x -ray correlations

cannot contribute much to the cascade spectra because the Isct MO vacancy

is created (mainly) after the distance of closest approach (So 78). Since

there is no way to separate the 2ptf— Isct transitions from the 2pcr-4sCT ones,

one has to calculate the relative importance of each transition (2ptHLsct o r

2pcr—Iso ) by measuring the one and two collision Isct MO excitation. This can

be carried out, by measuring the cascade MO x -rays in solid and gas targets.-

Approxim ately, the rela tive importance of each e ffect can also be calculated,

using the M eyerhof form alism (Me 74, Theory section 1.6) and the results

o f the measurements of K x -ray cross sections (section IV . 5).

In asym m etric collisions (Nb+Sn), the Isct MO excitation v ia multiple

collisions w ill be a factor of 10 le s s important, because the probability

that a K -shell vacancy would follow the Is a MO in the second collis ion is

quite small (50% fo r Nb+Nb and 4.8% fo r Nb+Sn). At least in the asymm e­

tr ic collisions the observed cascade spectra (fig . 3.25 and 3.26) are p ro ­

duced from one leve l transitions (2po—Iso ).

Other contributions from higher orb ita ls, due to their couplings with

the 2pcr MO, are unlikely to occur fo r reasons presented in the one c o lli­

sion case (follow ing part).

184

The K x -ra y lines in the cascade spectra (fig. 3 .14-3.16) can also be

created (besides the sequential collisions) from inner-shell vacancies which

in the exit channel end up as two K -shell vacancies. As we have explained

in the theory section (1.4) fo r sym m etric o r near-sym m etric co llis ions, the

2pa MO is the principal source of such K -shell vacancies. Two vacancies

in the 2pa MO w ill produce two vacancies in the K -shells of either collision

atom , o r one C l MO and a K x -ra y (fig. 4 .2 ). In the case of two vacancies

in the Is a MO, o r one in each 2pa, Is a M O 's, there is the possib ility of a

C2 MO x -ray accompanied by a K x-ray (fig . 4 .2 ). Thus, double vacancies

in the low er M O 's (Isa , 2pa) w ill contribute in the cascade spectra, at all

x -ra y energies (K x -ra y , C l o r C2 radiation regions) and their importance

has to be calculated.

According to the theory (I. 4 ), the principal way of producing a 2pa MO

vacancy in sym m etric or near-sym m etric atomic collisions is to bring it

from the 2pn MO, v ia rotational (2p7T— 2po) coupling. In the Is a MO, the

vacancy w ill be produced (in an one collision process) v ia d irect excitation,

and the excitation w ill occur mainly after the distance of closest approach

(So 78). Since the vacancy production mechanism in each orbital is d ifferent,

in a f ir s t approximation, the excitation mechanisms fo r the two M O 's are

not correlated. Then the probability of producing a double vacancy in the

2pc, Is a M O's w ill be the product of probabilities fo r creating a vacancy

in each MO. The subsequent decays of the two vacancies by MO x -ra y , K

x -ray simultaneous em ission w ill also not be correlated because they occur

b) One collision background contributions

185

at d ifferent internuelear distances (during the collision the MO transition and

after the collis ion the K transition). Some small shifts in the binding energies

of the M O 's due to the additional vacancy w ill produce m inor changes in the

fluorescent y ie ld , the mean life of the vacancies, and the probability fo r a

second vacancy production, but these changes w ill not be important. In any

case, it w ill have the tendency to decrease the contribution from double va ­

cancies to the C2 energy region because the additional 2pa MO vacancy reduces

the probability of the MO transition. M oreover, the shift in energy of the

double vacancy MO x -ra y spectra (due to the additional 2pa MO vacancy) is

v e ry small and com pletely unobservable with the detector used.

Analytica lly, the single and double vacancy probabilities (fo r sym m etric

2vcollis ions) can be expressed in the follow ing way: define a (2pa ,2pa ) and

2vo (2pcr, I s o ) the cross section fo r producing two vacancies in the 2pa o r

one in each 2pcr, I s a M O 's, and p 2pCT(k)» vacancy production

probabilities per spin state fo r the same le v e ls as functions o f the impact

param eter. Then we can easily find that:

o 1V(2po) = 2vr Jdb b (2P2p(J(b)) = 4tr Jdb b P 2pCJ(b)

lv.a V(lsa) = 277 Jdb b (2Plg a (b)) = 4tt Jdb b Px (b) (IV * 2>

(IV . 3)2v 2a (2p a , 2pa) = 277 Jdb b ( P g ( b ) )

2 v0 (2pa ,lsa ) = 277jd bb (2P lsCT(b))(2P2po(b)) = 877 JdbbPlgff.(b)P2pcr(b) (IV .4)

To find the corresponding x -ray production cross sections we are interested

in this thesis, one can follow the approach by M eyerhof et al. (M e 74) which

186

is presented in I. 6a, and which treats the MO x-ray production quasistati-

cally. Defining as rn and r the mean life s of the 2pa and Isct vacancies2per Isct-20

respectively , one can easily rem ark that the collision time (order 10 sec)

—16is much shorter than the mean life s of the vacancies (order 10 sec and

lon ger, Scf 74). So, there is no need to exp lic itly include the probability of

a 2po MO vacancy not decaying during the collision in the expression of cross

section fo r a K x -ra y production ( l - *^ f ° r MO x -ra y production

does the mean li fe have to be included in the formulas. So, extending the

M eyerhof form a lism to double vacancies (in a single collis ion ) we get fo r

the d ifferential cross section:

2 „ p b 2 P l s a (b) — „ (from IV .2 )x 0 R Isct x

v is the radial ve loc ity R

of the ion at the position R (which

corresponds to an energy d iffe r ­

ence E (ls o )-E (2 p o )= Ex ) and

the fluorescent yield of the IscrM O .

in

d-a <C2~*P9= 2IT fbdb2P (b) 2P (b) ------- ■- w, w, (from IV .4)dE J 2pc ; Isct v _ r dE lscr kx 0 R Isct x

where we have not included ex­

p lic itly that the 2pcr MO vacancy

does not decay during the collision .

Since

m im 2m = m +m 1 2

187

where v is the pro jectile ve loc ity , D the distance of closest approach, m

the reduced m ass, and

A fte r some manipulation we get

, fx n 4ttw /db b Pd £ _ _ (C 2 )_ Is a Is a

•/ y i"dE f"r" /T=lD/R) <dR/dEx> I /■■ 'J-Yx 1 I s a ' ’ J J l- (b / b ')

(IV . 5)

and

dg2X(C2-KX ) _ 8,rwkWls g / db b P ’ - » P = --< b>/(lb b P (b) P (

^ ,AT>/AT? x l *LSCT 2 P_CT /TXT- CXdE v r / M D / H ) (dR/dEx)l T— ~ -A (IV * 6)

x 1 Is a ' y y / l- (b / b ')0

In the last two formulas the (dR/dE ) factor does not depend on the impactX

param eter (b ), because in the static approximation the MO x -ra y energy uni­

quely defines the internuclear distance.

Taking the ratio of the two d ifferential cross sections which w ill be the

ratio of (MO x -rays in coincidence with K x -rays produced from double inner-

shell vacancies) to the (single MO x-rays ) at a specific x -ra y energy E , we get

/db b P , (b )P „ (b)da (C2-KX ) 2 / -------- .l8 ° %>°

D<Ex> = -------------------- b , A - f e / h ') ------------- w (IV . 7)dq (C2) r db b P lg a (b)

dE /db b

IT-x 1 ^ -(b/b')Z

b* = R / l- (D / R ) , e x = Ex (R)

The value of this ratio depends on the 2pa, Is a excitation probabilities

at d ifferent impact param eters.

Approxim ately

where a is a constant chracteristic of the collis ion system (M e 74, Han 75).

These two functions w ill g ive the same values in the previous integrals (within

because it w ill cancel out in the above ratio , but the value of a has to be

known accurately. M eyerhof had used (Me 74) fo r a the K -shell radius o f the

colliding atoms (sym m etric system ). Using the sem iclassical approximation

(Han 75) or a scaling from ligh ter (And 76) o r heavier (Am 79) collis ion sys­

tem s, we find that the M eyerhof value for a is too la rge . It seem s that

150 f m < a <600 fm (fo r Nb+Nb collis ions). F or the P (b) one can use a

scaling from ligh ter systems (Ta 76) including the low impact param eter

kinematic peak (B r 74), o r an experim entally measured function can be used

(Joh 79).

Some recent experiments (Joh 79, Ann 79), indicate that the scaling

from ligh ter systems (Ta 76) does not reproduce the position o f the main

peak quite w e ll, fo r the systems of interest (Nb+Nb). So, we have used the

experimental functions (Joh 79).

S im ilarly to the above relations can be established fo r the single and

double K x -ray cross sections.They w ill be:

2-3%), so the simplest form can be used. The value of P^ is not important

2pa

(IV. 8) (from IV. 1). Again setting

0

OO(IV. 9) (from IV. 3).

0

189

Here it was assumed, that the fluorescent y ield w does not change fromKthe f ir s t to the second vacancy.

Taking again the ratio of the two cross sections we get:

2x „ Jdb b [p2pa(b) I 'A . - _ £ c j g a . _ i i L ! ! _ (IV.10)

° (KX) J d b b P 2pp<b>0Using the scaling from ligh ter systems fo r the P (b) (Ta 76), o r

the measured values (Joh 79), we find that the value of A depends only on

the value of the fla t peak P (fig . 1.8) so,£t

A « 0 . 3 5 w k P2 (IV. 11)

Since there is a double detection probability in the coincidence measurements

fo r the two identical K x -rays , the observable value of the ratio A w ill be

wk P 2> Comparing the two ratios (2A) and ^ (E ^ ), one can find out how

c lose ly the MO x -ra ys due to double (2pa, I s a ) vacancies norm alize to the

K x -ra y yields as the single MO x-rays . F or 2A = D(E ) at a ll photon energies

Ex> this norm alization would be exact which means that the spectrum of x -

rays in coincidence with K x -rays due to 2 p a , Is a double vacancies w ill be

identical in shape with the singles x -ray spectra. Using the experim ental v a l­

ues fo r the impact param eter dependence of the 2paM O , and d ifferent com ­

binations of the exponential factor a, we find fo r the Nb+Nb system the fo l­

lowing:

a) The single MO x -rays per K x -rays ((M O )/(KX )) are less than the

MO x -rays in coincidence with K x -ra ys per K X -K X coincidences ((M O -KX )/

(K X -K X )) produced from double inner-shell vacancies. The two ratios d iffe r

by less than a factor of two (with (MO)/(KX) < (M O -K X )/ (K X -K X )).

190

b) The singles spectra provide a better approximation o f the double va ­

cancies at low er x -ray energies ( < 50 KeV) and their d ifference increases

towards the UA lim it.

c) With increasing pro jectile ve locity the average underestimate o f the

MO x -rays per K x-rays due to double vacancies from the single MO x -rays

per K x -ray decreases. So, at 200 MeV their ratio ( ) is ~ 1*6(K X -K X ) (K X )

and at 100 MeV ~2. This is very interesting, since the contribution from

double vacancies and sequential collisions increases with p ro jectile energy.

Th ere fo re , the e r ro r indroduced by using the singles spectra to subtract the

BG contributions from multiple vacancy production as cited above even at

higher p ro jectile energies is not v e ry large.

d) The higher the value of the exponent a, the better the approximation

o f the C2 MO x-rays from double vacancies by the singles x -rays norm aliz­

ed to the K x -ray peaks of the cascade spectra. F or a reasonable value o f

a (a=400fm), evaluated from the sem iclassica l approximation (Han 75), the

average underestimate ( ) is a factor o f two (1.3 at 30 K eV ,(K X -K X ) (K A )

2 at 45 KeV , and 2.4 at 60 KeV).

The uncertainty introduced in the coincidence spectra from the subtrac­

tion, w ill be sm aller, because roughly 50% of the K x -rays (fig . 3.34 fo r a

s im ila r system ) in the coincidence spectra had been produced by sequential

collisions. As we have seen in the f irs t part, the contributions from se ­

quential collisions have an em ission spectrum identical in shape with the

spectrum of the singles x -rays . Besides, the distortion of the cascade spec­

tra by differences in shape between the singles and double vacancy related

spectra are la rge ly diminished by dynamic broadening at high x -ra y energies.

A lso , the contributions from double vacancies or sequential collis ions to the

total MO x -ra y yield is quite sm all, at least fo r the 100 MeV case ( ~ T -o f

the real coincidence events). Th ere fo re , even a factor of two cannot s ign if­

icantly modify the association of the largest fraction of the M O -KX coinci­

dences with cascade type MO x -rays .

In the case of Nb+Sn s im ilar results can be obtained if we incorporate

the e ffe c t of vacancy sharing (1.7).

As it was stated before , the C l MO x -rays p er K x -ra y due to double

inner-shell vacancies should be equal to the corresponding C l MO x-rays

per K x -ra y in the singles spectrum. Using the above form a lism , this can

be shown readily. The (C l x -ra y )- (K x -ray ) coincidences arise mainly from

double 2pavacancies (in sym m etric or slightly asym m etric collis ions). Then

the d ifferentia l cross section fo r C l-K X coincidences w ill be:

191

d o"X(C l-K X ) 3 dB_____ w w■=-2 ir^b db [P2po(b> j

dE* / L 2pa J v r 0 dE1 2pa kx q*7 R 2pa x *

The factor 2_ in the right hand side of the above relation is the result o f two

vacancies in the 2paM O , and e ither can decay v ia a C l MO x -ray . E* is

the MO x -ray energy and Wg , w^ the fluorescent y ields of the 2pa, I s

states.

A lso ,

, lxda (C l) „ „ r _ _ ... -i dR

w^ 2 n/ b db[2P2pa<b)]dE i 7 L 2 p a ' ' J v R r 2podE i 2pa (IV . 12)

A fte r some manipulation we get

d o 2x(C 1-KX) . 4 ’rW2 p g '\ / d b b r P 2pa<b)5dE' v r LS/R) / T T (IV. 13)

x i 2pcr J y i- (b / b ')

Defining their ratio by B(E^)

/db 1

do:--(ci-Kx, . b [ P 2pa(b)l2

B(Ex ) S = Wk -------------------- (IV - 14>do (C l) J?

>/l-(b/b ')20

and comparing with the value of (2A) we find that they d iffe r by less than 20%

at a ll x -ray energies E^. This is correc t fo r any function used fo r the ^ p a ^

(Joh 79, Ta 76). So, C l MO x-rays in coincidence with K x-rays from double va ­

cancies are v e iy c losely represented by the singles C l MO x-rays both nor­

m alized to the corresponding K x -ray peaks o f their x -ra y spectra.

The calculations, which have been presented up to now, are based on

the assumption that the Isa MO vacancy is created during the atomic collis ion

(one-collision process). Besides we have assumed that in this one-collision

excitation mechanism the excitation and deexcitation of the Isct MO are two

com pletely independent processes. As it has been pointed out (Th 77), this

last assumption is not correct, and a complete quantum mechanical trea t­

ment is required. Unfortunately, there is no easy way fo r the quantum mech­

anical treatment to be carried out consistently, and we have to base our ca l­

culations on the quasistatic expressions. We have also seen (1.6a), that a two

co llis ion mechanism can create Is a MO vacancies. As w e w ill prove in the

follow ing, the tw o-collis ion mechanism cannot modify the above results and

193

the singles x -ray spectrum provides a good approximation of the spectrum

produced from double inner-shell vacancies (within a factor less than two)

at a ll x -ra y energies no matter how the Is a MO vacancy is created (via an

one o r tw o-collis ion process).

In a tw o-collis ion Is o MO excitation the MO x-ray y ie ld w ill be (I. 6a):

d a 2?C2) /*dE^2 = ° k w 7 db <2" b> T Tk <2 H >

X ^ X R x

T= 477W Tj CJj — R 2/ l- (D / R ) ^

k T dE'X X

and the (2 p a , I s a ) double vacancy w ill be:

- o ^ / d b <2„b) V j Tk (2 f L ) ^ <2P (b )) « kv T? v

d a 2° (C 2 - K X )

d E ' .. . . „ ,x •' x R x

= 4t7w r} a, R 2/ 1 -(D /R )^db b t 2p—k Tx " " 7 ~ V b ’4 - b ' 2 b ~ k

, 2 c ,„ „ , r 2P „ (b)= r d° <P2>_ ) M

1 dEk k / / O T

Using the scaling from lighter system s fo r the P (b) (Ta 76), o r the measured*5pC7values (Joh 79), we find that the value of (C2-KX) x -rays from double vacan­

cies per (C2) x -rays (singles) is within (less than)a factor of two compared

to (K X -KX )/ (K X ) (2A in formula IV . 10). Thus a ll the conclusions derived

above fo r the one-collision Is o MO excitation are valid in this case as w e ll.

Up to now, the 2pa MO couplings with higher orbitals has been ignored,

but they might contribute to the cascade spectra through vacancy transfer

to the 2po MO from higher vacant states. F or example the 2pa MO can be

194

rad ia lly coupled to the 3da MO, or I’otationally, to the 2pir o r higher 77

orbitals. Such couplings exist and they are quite strong fo r highly re la tiv is tic

system s (H ei 78).

If the 2po MO couplings with higher orbitals in the outgoing part of

tra jectory ( f ig .4 . ]b ) are quite strong in our collision system s, C l MO x -rays

(from transitions to the 2pa MO) as well as C2 transitions from higher than

the 2pcr MO w ill be associated with K x -rays v ia these couplings. The absence

o f a significant amount of C l MO x -rays in the coincidence spectra (fig . 3.14

to 3.16) supports the idea that the 2pa m ixing with higher orb ita ls is quite

small and so, the BG contributions from such effects in the cascade MO x -

rays are negligib le.

Concluding, we should rem ark that in one collision I s a MO excitation,

there is an uncertainty in determ ining the contributions from double vacancies

to the coincidence spectra; however, at low bombarding energ ies the uncer­

tainty is small.

Fig. 4.1 2p7T— Isa MO transitions in coincidence with K x -rays due to the

2p7T—2pa rotational coupling at small (top figure) and la rge (bottom)

internuclear distances.

F ig. 4.2 C2 MO x -rays in coincidence with K x -rays due to double inner-

shell vacancies.

SAR - - oo.

( o ) U A R = 0

SA R — cc

2p 2p

R = -ooSA

2 P

( b )

UA R = 0

SAR =

2p (SA)

-RAY

00

X-RAY

Figure 4.1

Figu

re 4. 2

BINDING ENERGYoozoomzoocosc5“0r£o>znm(/) “D3)

oc.mor*mZ3

2p<r-ls«r TRANSITIONS

IN COINCIDENCE

WITH K

X-RA

YS

198

Quasim olecular x-rays

In figures 3 .2 to 3.4, the singles spectra were presented fo r the three

pro jectile energies. A comparison o f those spectra revea ls an interesting

feature concerning the dependence o f the y ie lds of x -rays in the C2 and KX

regions on the p ro jectile energy.

It is c lea r that the rela tive importance of the C2 region , compared to the

K x -rays , increases with p ro jectile energy. This energy dependence is in

agreement with previous results (V i 77). Theoretica lly , it is justified from

the fact that with increasing p ro jectile energy the Is a MO vacancy production

cross section asymptotically reaches the corresponding cross section o f the

2pa MO. Since the K x-rays are mainly the result o f the 2pa MO vacancies,

and the MO x-rays are produced from vacancies in the Is a le v e l, the above

conclusion is straightforward. The same increase o f the Is a MO vacancy

production probability with pro jectile energy' has been observed in the K x-ray

cross section measurements (IV. 5). We see, therefore, that the total K -va -

cancy production in the Nb+Nb system (fig. 4.9 to 4.11) increases with p ro ­

jec tile energy m ore slow ly than the Isa vacancy production v ia d irect ex c i­

tation (one-step process).

Another observation we can make from the figures 3.14 to 3.16 is that

the importance o f the MO x-rays from the cascade mechanism compared to

the BG from double vacancies and sequential collisions decreases with in­

creasing p ro jectile energy, e . g . , at 40 KeV the ratio o f rea l coincidences to

BG from double vacancies and sequential collisions is 9, 6, 3 .3 fo r 100, 160,

IV. 2 Nb+Nb

199

and 200 MeV respectivel}'.

The double vacancy BG contribution to the quasi-m olecular spectral

region is mainly the result of simultaneous vacancies in the 2pa, Is a M O 's,

while the coincidence events come from the decay of a Iso MO single vacancy.

The ratio of the two cross sections (in the case of uncorrelated vacancy

production) w ill be proportional to the probability of producing a 2pc vacancy,

which increases with the pro jectile energy. So, the change with the energy

of the ratio of cascade MO x-rays to double vacancy MO x -ra ys is reason­

able.

Although the final differential cross sections (jubarns/KeV) of the

cascade MO x -rays (fig. 3.19) include appreciable statistical e r ro r in addi­

tion to the uncertainty from the subtraction o f the double vacancies and

sequential collision contributions, many observations can be made for the

x -ray region around 40 KeV , where the re la tive e rro r is small.

A straight comparison of the cascade MO x-rays with the single x -rays

(corrected for NNB and AB , figs . 3. 2-3 .4 ) revea ls a general agreem ent in

shape between the two sets of data (fig. 4 .3). The only region where the shapes

d iffer appreciably is between 28 and 40 KeV. In this region, the cascade MO

x -rays lie lower than the singles MO x-rays. This excess of single x -rays

in the 28-40 KeV energy region can be attributed to e ffects such as REC,

or to transitions to the 2pa MO (C l radiation), In the latter case, the C l

radiation should be considerably broadened because the UA lim it L x -ray

energy is ~ 15 K eV . The d ifferentia l cascade cross-section (fig . 3.19) also

possess a slight dip in the MO x -ra y intensity between photon energies of 28

200

to 40 KeV. This e ffect is quite obvious at 100 MeV pro jectile energy, but

becomes sm aller at higher pro jectile energies. The same characteristic has

been observed in the Nb+Sn collis ion system at slightly higher x -ra y energies

(fig. 3.28, 3.29, fo r x -ray energies of 40-47 KeV ). It is d ifficu lt to be­

lieve that it is a resu lt o f poor statistics or any e rro r in the subtraction

of the double vacancies and sequential collis ions, since it seem s to scale

with the co llis ion system s. Besides, as we have stated b e fo re ( I V . 1) the

uncertainty in the contributions from double vacancies in the low x -ra y

energy region is sm all 30%) increasing towards the higher energies which

cannot explain the magnitude o f the observed dip in-, the MO x -ra y intensity.

It could re flec t dynamic evolution o f the two molecular orb ita ls involved

in the cascade transition (2pcr, lscr)during the collis ion (fig . 1 .1 , diabatic

orb ital diagram ). Thus, the absence of any such characteristics in the single

spectrum (fig . 4 .3 ) would mean that the reduction o f 28-40 K eV x rays is

produced by a special structure of the 2pa orbital, which is re flec ted in the

cascade spectra. Since there rea lly exists a slight dip . o f the 2pa orb it­

al at distances 1000 and 2000 fm (fig . 4 .4 ), it is interesting to see i f such

structure in the 2pa MO potential energy can explain the observed spectra.

Based on the calculations presented in I. 6a and in the previous section o f the

discussion (Me 74), we can estimate the expected shape of the 2pa-*lso 'M O

x -ra y spectrum.

F irs t, it should be noted that considering the re la tive cross-sections

of cascade MO x- rays and the single MO x-rays fo r the photon energies be­

tween 40 and 60 KeV where the two spectra agree in shape (fig . 4 .3 ), we

201

find that the cascade MO x-rays are about 20-30% of a ll MO x -rays produced

from transitions to the Isa MO, and their rela tive importance increases with

p ro jectile energy (19% at 100 MeV, 27% at 200 M eV). From the re la tive in­

tensities o f the K K transitions of the UA (Pb ), it could be expected °1 °2

that our data w ere around 28% of the singles, i f a ll the observed cascade

transitions to the Is a MO w ere from the 2pa MO, and 32% i f they w ere a

m ixture o f transitions from the 2pa and 2p7r M O 's. (two-step Is a MO excita­

tion). Considering the uncertainty in the fluorescent y ie ld and the effic iency

o f the K x -ray counter the disagreement between the theoretical and experi­

mental value is not important. The pro jectile energy dependence o f the cas­

cade MO x -rays compared to the total MO x-rays is expected from the in­

creased ionization of the higher states with increasing p ro jectile energy which

can lead to contributions from low er orbital transitions that increase with

p ro jectile energy.

Concerning the uncertainty from the m ixture of the 2pa— Is a transitions

with the 2pa— Isa cascade transitions due to the 2p77-2pa rotational coupling

at sm all internuelear distances, one can set an upper lim it on this m ixture

by assuming that the two-step Isa MO excitation dominates. Then all the Isa

vacancies are produced in the entrance channel of the co llis ion (fig . 1.9 bot­

tom ). Using the scaled probability fo r 2pir-2pa rotational coupling from the

work of Taulbjerg et al. (Ta 76), one can find that the maximum contribution

from the 2p77— Isa transitions to the cascade spectra is 30% varying with

the MO x -ray energy. On the average this contribution w ill be less than 20%.

In the case o f d irect Isa MO excitation this contribution wi l l be even sm aller.

202

A straight comparison of the measured cascade MO x -ra y cross sections

(fig. 3.19) with the quasistatic expressions fo r the dominant 2pa—- Isa transitions

(1.6a), is presented in fig . 4.5. We have chosen fo r com parison the data at

the 100 MeV because the dynamic broadening is sm a ller in this case than at

higher energies. In this figure the broken line presents the theoretical calcu­

lation o f the cascade MO x -ray y ie ld fo r one-step Is a MO excitation using

b/athe expression ^ l s a = 2P /(1 + e ) with a=400fm (calculated from Han 75).

The continuous curve shows the expected contributions from two-step Is a MO

excitation. The theoretical expressions have been norm alized arb itra rily to fit

the data points. We can easily observe that the one-collis ion process fa ils to

reproduce the data, while the tw o-collis ion calculations f it them quite w e ll.

This is not an indication that the tw o-collis ion mechanism dominates the Is a

vacancy production because by using a d ifferent value o f the exponent a (a=1000fm)

the one-collis ion process reflects the data quite w ell (fig . 4.6) . Actually, fo r

a=1400fm the one and tw o-collis ion processes have the same shape. W e also

observe that both Isa MO excitation mechanisms cannot reproduce the sm all

dip of MO x -rays between 28 and 40 KeV. Thus, the orig in o f this sm all

structure in the cascade MO x -ray spectrum is s till unclear.

Concluding we should rem ark that a complete quantum mechanical trea t­

ment o f the excitation and deexcitation processes is needed, which w ill inclu­

de the contributions from both Is a MO excitation processes. Then, a d irect

information on the MO form ation can be extracted from our one-transition

(mainly) spectra.

Fig. 4

Fig. 4

F ig. 4

F ig. 4

.3 Comparison o f the measured d ifferentia l cross sections fo r produc­

ing cascade MO x-rays with corresponding values o f previous data

(V i 77). The two sets o f data have been norm alized in the 40-50

KeV region.

.4 R e la tiv istic one electron two center MO calculations perform ed

by M uller et al. (Mu 77) for the Nb+Nb co llis ion system .

.5 Theoretica l predictions (Me 74), fo r the 2pa -+lsa MO radiative

transitions in one collis ion (broken line) and two collis ion (contin­

uous line) process fo r Is a MO excitation. Data points are from

100 MeV Nb+Nb (cascade MO x -rays ). Value of a is 400fm.

. 6 Comparison o f the 100 MeV Nb+Nb cascade MO x -ray y ields with

theoretical predictions (M e 74) from one collis ion Is a MO excita­

tion process (a=1000fm).

10"

10-3

K)-4

10-5

I0"6

OOMeV Nb + Nb

Data from Vi 77P .n c rn H p M O v-rnv/c;

|

<DC

1t1u

• 1' t f

4 o

’Io

T o

t ?t «)1

o

o

o

3 0 4 0 5 0 6<0 7 0 80- X-RAY ENERGY (KeV)

Figure 4.3

10

20

3 0

4 0

5 0

6 0

7 0

8 0

9 0

100

110

— Internuelear distance (fm)

2 0 0 0 4 0 0 0 60 0 0 8 0 0 0

Figure 4.4

Diffe

rentia

l cro

ss-se

ction

(yx

barn

s/KeV

)

10,000

1,000

100 MeV Nb+Nb4 Data

Theor. calculations— (two-collision process)— (one-collision process

a = 4 0 0 fm )

3 0 4 0 5 0 6 0 7 0

X - RAY ENERGY (KeV)

Figure 4 .5

Differ

entia

l cro

ss-sec

tion

(/iba

rns/

KeV)

10,000

1,000

100

10

■ - | r - | |

OOMeV Nb + Nb

► Data— Theoretical calculations

(one-collision process

1

4

11

a = iu u u T m )

<>

3<D 4 3 5 0 60 7 0 8 0

X-R A Y ENERGY ( Ke V )

Figure 4. 6

208

F or the case of slightly asym m etric collisions (Nb+Sn), the MO x -ray

cascade mechanism producing K x -rays is s im ila r to the sym m etric case.

The only difference is that fo r asym m etric system s, the 2pa MO is c o r re ­

lated mainly to the K -shell of the lighter atom (fig . 1.2) and so, the 2pa-»

Is a transitions w ill be associated with K x -rays o f Nb. Excluding, fo r the

moment, the possibility o f a simultaneous 2pa, Is a vacancy and the contri­

butions from sequential collisions, we observe (fig . 3 .1 ) that only through

the 2 p a - l s a vacancy sharing mechanism can a 2pa-*lso transition be asso­

ciated with K x -rays o f the heavier collis ion atom (Sn).

Up to now a ll the experimental evidence on the vacancy sharing mecha­

nism is indirect, based on the re la tive ratio o f the K x -ray y ields o f the

two collis ion atoms. So, from the good agreement between theoretical p re ­

dictions (Me 73) and experimental values o f the sharing factor (IV . 5) it has

been assumed that such coupling occurs at la rge intem uclear distances ( la r ­

g e r than the K -shell radii o f both collid ing atoms).

Using the (MO x -ra y )- (K x -ray ) coincidence technique, the vacancy

sharing mechanism can be studied d irectly , by comparing the two MO x -ray

spectra which are in coincidence with Nb or Sn K x -rays . Since there is

no other mechanism that would associate MO x-rays with Sn K x -rays , the

coincidence experiment can provide a straightforward check o f the vacancy

sharing mechanism by tracing the evolution of the vacancy during the co l­

lision and its subsequent decay.

F ig. 4.7 presents the coincidence spectra gate on Nb o r Sn K x -rays

IV. 3 N b + S n collisions

fo r a target o f thickness 345 pg/cm . The spectra gated on Nb and Sn have

been norm alized to the K x-ray y ields and they are compared in the C2 ra­

diation region. We observe that they agree in shape (fo r the other target

s im ila rly ) at a ll x -ray energies. This agreement would be expected from a

coupling o f the 2pa , Isa MO's which would result in a sharing o f the casca­

de 2pcr vacancy among the K shells of the two reacting atoms a fter the co l­

lision. M oreover, this coupling should occur at large internuclear distances

(la rge r than the sum of the two K -shell e lectron rad ii), otherw ise the two

MO x -ray coincidence spectra (gated on Nb or Sn K x -ray ) would disagree

in the low MO x -ray energy region (where the statistical e r ro r is sm all).

Including the contributions from double vacancies o r sequential collis ions,

the previous interpretation wi l l not be altered. As we have proved in section

IV . 1, the sequential collisions scale exactly as the singles x -rays . The doub­

le vacancies also can be found approximately within a factor of two by nor­

m alizing the singles spectrum to the K x-rays of the coincidence spectra.

In any case, the contributions from these two effects lie a factor of 4-5 low­

er than the MO x-rays in coincidence with K x-rays and they cannot account

fo r the observed cascade MO x -rays . These contributions also scale as the

sharing factor in each case (Nb o r Sn rea l coincidences), so their rela tive

contributions wi l l be the same in each coincidence spectrum.

The other BG contributions described in the previous section (IV. 1),

are also present here, with some changes in the re la tive importance o f

each term .

In the present collision system we have:

209

2

a) The contribution from 2pn -» Is o transitions, through multiple co llis ions,

is likely to be less important now, due to the reduced participation o f the

double collisions in the Is a MO vacancy production. While in the sym m etric

case each Nb K -shell vacancy which lived long enough to enter a second co l­

lision had a 50% probability to become a Is a MO vacancy; in the present ca­

se, the corresponding probability is only 4.8%. W e have already found in

the theory section fo r the Nb+Nb case an indication that multiple collisions

should be o f about equal importance as the d irect excitation fo r the Is a va ­

cancy production so that in the present system these BG contributions w ill

be negligib le.

b) The rotational coupling o f the 2p7r, 2pa MO's was a possible sm all

background effect, in the sym m etric case. F o r Nb and Sn co llis ion system

this e ffect w ill not contribute much fo r the same reasons as in the Nb+Nb

collis ion system .

Thus, in the present case the observed MO x-rays in coincidence

with either K x -ray (Nb o r Sn) are due principally to the 2pa-» Is a transi­

tions (one collis ion process). By carrying out calculations s im ila r to the

Nb+Nb collis ion system , we can also prove that the double vacancies are

approximated in shape by the singles x -rays within a factor o f two near

the UA lim it and less than two in low x-ray energies. On the average this

factor is less than two. Considering again the contributions from sequen­

tia l collisions which have the same x -ray spectrum as the singles x -rays ,

and which constitute a good amount of the observed K x -rays in the coin­

cidence spectra (40-60% depending upon the target thickness), we find out

210

211

that the singles spectrum in the present case also approximates in shape the

spectrum of x -rays from double vacancies and sequential collis ions within a

factor of two.

In summary w e should rem ark that cascade MO x-rays have been also

detected in the asym m etric collis ion system . They are produced by 2pa— Is a

transitions and their x -ray spectrum presents a sm all dip s im ila r to the one

observed in sym m etric collisions.

Fig. 4.7 Comparison of the x -ray spectra in coincidence with the Nb and

2Sn K x-rays fo r the target of 345 ^g/cm , The two spectra have

been norm alized in the Sn K x -rays .

COUN

TS

ENERGY (keV)

Figure 4.7

214

In Ch. I l l (section 3), the cross section fo r producing two K x -rays in

one collis ion has been presented fo r slightly asym m etric (Tab. 3 .4 ) and ve ry

asym m etric (Tab. 3 .5 ) collisions. It is very important here to understand

the double vacancy production mechanism in each particu lar case, and to

relate these considerations to the contributions from double vacancies in the

MO x -ra y coincidence measurements (Nb+Sn, Nb+Nb system s).

In section IV . 1 we have shown that in order fo r the singles x -ray spec­

tra to be used fo r the subtraction o f the contributions from double vacancies

in the cascade spectra, one has to establish that the two ratios, D(E ) and

2A (formulas IV . 7 and IV . 10), have very c lose values fo r a ll MO x-ray en­

erg ies Ex> In a f irs t approximation one can disregard the impact param eter

dependence in the IV. 7 integrals and compare instead 2A with

b'f d b b P (b ) P (b )

D' (E )= 2w ---d--------------- 2 p o _x k b*

f d b b P , (b )J lSCTo

How close ly D '(E ) approximate D(E ) can not be generally explained becauseX X

o f their dependence upon the function p ls 0 (h) and the value o f b' and so upon

the MO x-ray energy E . It can be easily seen that fo r values of the impact

param eter b ' , fo r which F lg 0 (b) has s till la rge values (b < a , fig . 4.8 Han 75),

D '(E ) is v e ry close to D(E ) because the main portion o f the integral a risesX X

from values o f b « b ' . F o r la rge r values o f b' D '(E ) is la rge r than D(E ).X X

So, besides the straightforward calculations o f section IV . 1 this firs t approx­

imation can be used as an additional check of the validity of the subtraction

o f the double vacancy x -rays from the coincidence spectra.

IV .4 K X -K X COINCIDENCE

215

The advantage of using the ratio D' ( Ex ) is that it can be easily calcu­

lated experim entally. It w ill be simply the ratio of the K x -rays (due to a Is a

vacancy) in coincidence with K x -rays (due to a 2pa vacancy) norm alized to

the single K x -ray y ie ld (due to Is a vacancies). Since in sym m etric o r near

sym m etric systems there is no way to separate the K x -rays produced from

each MO vacancy, we have to use very asym m etric atom ic collis ions with to ­

tal atomic number close to the sym m etric collis ion system . In this way the

same inner-shell vacancy production probability in each orb ital (Isa , 2pa) is

secured (UABEA).

Since in each K x -ray , K x -ray coincidence system (Nb+Sn or Ni+Sn) a

d ifferent MO is coupled to the same leve l of interest (2pa), the two collis ion

system s w ill be examined separately. W e start with the presentation o f the

Nb+Sn collis ion system , where the 2pa-lsa radial coupling dominates the K

vacancy production o f Sn atoms. Then, the Ni+Sn system is examined where

the K -L matching (3da-2pa vacancy sharing) introduces some interesting fea­

tures not present in the slightly asym m etric collisions (Nb+Sn).

a)1 Nb+Sn collisions

In the Nb+Sn asym m etric collis ions, the 2pa-lsa vacancy sharing ratio

is quite la rge (~ 5 % ), so that the K -shell vacancy production o f both collis ion

atoms is dominated by the 2pa MO vacancy production (IV . 5). The same is

true fo r the double vacancy production in one or both collis ion atoms. Thus,

we should be hble. to predict the measured (one collis ion ) two vacancy production

cross sections based on the simple picture o f the vacancy sharing mechanism.

Follow ing the formulation presented in IV . 1, we can easily find that the

216

single and double vacancy yields are:

(again we assume 1 -r ~ 1 )2pcr

N(H) = 2ttJ db b (2P2^^(b)) w

N (L ) = 2tt J db b (2P2.3(T(b)) (1-w )

N (L -L ) = 2rr j2 (P 2p^(b ))2( l - w )2 b db

N (L -H ) = 277-J* (P 2pcr(b ))2(2 w (l-w )) bdb

N(H-H) = 4ttJ (P 2pa(b ))2 w2 bdb

where w is the sharing factor, b the impact param eter and Pgp the p ro ­

bability of the 2pa MO excitation. The factor 2 in front of the ligh t-ligh t

(L -L ) and heavy-heavy (H-H) double K vacancy productions is due to the

double e ffic iency fo r detecting identical x -rays in two detectors. The factor

2 in the sharing factor (L -H case) is due to the possib ility of e ither of the

2pa MO vacancies to be shared in each K -shell. In the above form ulas we

have followed the approximation that the sharing of a double vacancy follows

the simple relation proposed by Lennard et al. (L e 78, section I. 7c).

The calculation of the measured ratios is straightforward We get,

217

N (L -L ) _ J'*P 2pc/ b^ bdb 1—w N(H-H)

N ( L ) " r p , <b> b d b = w “ <«>J 2pa

Comparing the above calculations with the measured values, we find that

they agree (within statistical e r r o r ) , with the measured ratios o f cross

sections (Tab. 3 .4 ). So,

and

2vp (Nb-Nb)

lv=' —JT— — = —— - = 20.73 (theory) and 22.2 ± 2 .5 (exp .)

(H" H) p (Sn-Sn) Wlv _

a (Sn)

N (L -L )

N(H)

and

N (H -L ) ° 2Y( ^ - a ')

w?H-m ° — 2v (Sn> = 1 <theor« and ° - 95 ± 0 -2 (exp->W 2 ? - ^ S

o (Sn)

The statistical e r ro r in both cases is too la rge , to test small variations of

2the double vacancy sharing factor from the simple law (1-w) (L e 78). As

such the va lid ity of any of the assumptions concerning the double vacancy

sharing factor cannot be examined closely.

As it was stated before, bacause of the la rge sharing factor (4.8% ),

the K X -K X coincidence yields of this case cannot provide much information

on the double inner-shell (2pp, l s c ) vacancy production, but only on the e f­

fect o f sequential collisions. The double vacancy production probability can

be studied only in ve ry asym m etric atomic collis ions (next section).

In the ve ry asym m etric system (Ni+Sn) the L -sh ell of the Sn is close

to the K-shell o f Ni so, the coupling of the 3do, 2 p a leve ls can transfer va ­

cancies from one level to the other (K -L matching). So there is another way

(besides the three processes presented in 1.4) fo r a vacancy to be created

in the 2po MO through radial 3da-*2p«r coupling. The 3dq-MO is correlated

to the M -leve l in the separate atom lim it and so it can be eas ily excited;

thus the number of vacancies transferred to the 2pa MO can be quite la rge ,

even though the transfer probability (3da-»2pcr) is small (^0.3% at 100 MeV

and 1% at 200 M eV, Me 78). But the 3da MO contributes vacancies to the

2p<J orbital only in the exit channel (large internuclear distances) and so,

it does not e ffect the yield of C l MO x -rays (MO transitions to the 2pa MO).

Th ere fo re , the rela tive strength of the C l radiation compared to the Ni K

x -rays yield supplies important information on the rela tive contributions of

each 2pa MO vacancy production mechanism. One has to know the radiative

transition probability to the 2pa MO, and then the calculation of the rela tive

contributions would be straightforward. A lternatively the same information

can be obtained experim entally by comparing the double vacancy production

yield of two N i K x -rays to the double x -ra y yield of a C l MO x -ra y and

a Ni K x -ray . As it w ill be shown below a straight comparison of the two

ratios [(C l x - ra 3'S)/(Ni K x-rays ) compared to (C l-N i)/ (N i-N i)J can delineate

the contribution from each process.

L e t us define as P and P the vacancy production probabilities (perO uspin state) fo r the 3da and 2pa M O 's respectively (through one of the mech-

b) Ni-Sn system

219

1 2anisms presented in 1.4). Besides we define B ^ a n d B>pa the total single

and double vacancy production probability in the Ni K -shell, and w the 3da

->2pavacancy sharing factor. The Ni K-shell which rece ives vacancies from

3da-»2pa vacancy sharing or d irectly from the 2pa orb ita l, w ill be excited

with the follow ing single and double vacancy probabilities (1.4).

P 2 p a= [ P 3 W + (1_W) P 2 tP 3 W + (1" W) P2 2P3W + 2P2(1_W)

P 2pa= [ p 3 w + ( ! - w) P 2 3 [ P 3 W + (1~W) P 2 > [P 3w + (1- w )p 2 ] 2

(± d e fin e s the two spin states).

The two 2pa, Is a M O's are fa r apart in energy and there is no appre-

-4ciable transfer of vacancies between them (~ 1 ’ 10 , Me 74); th ere fore , the

coupling of the 2pa MO with the l s c orbital can be ignored and the K (heavy)-

shell rece ives vacancies only through the excitation of the Is a MO. The p re ­

vious probabilities depend upon the impact param eter b. To find the c o r re ­

sponding cross sections one has to integrate over b and so,

2v 7a (N i-N i) = 2<n J [ p 3w + p 2 1-w ^ bdl:)

lv -a (N i) = 4tt J [P gw + P 2(l-w ) jbdb

sim ila i'ly defining as P the Is a MO vacancy production probability p er spin

state, we find

lva (Sn) = 2it f2 P bdb

1

For the Ni-Sn double K-shell vacancy the probability w ill be the product of

the probabilities fo r each state so,

2v r>o (Sn-Ni) = 2JT J2P (2P w + 2 (l-w )P ) bdbi O u

220

With the above relations we can calculate the experim entally defined intercepts

in Table 3. 5

o2v(N i-N i) i

o (N i) 2 f(P w + ( l -w )P )bdbO Li

SO,

° 2v< N i-s „ ). „—— — — — — ^ —

a V(Sn) J P^bdb

Since w « 1 => l-w « 1 we have

g 2v(N i-N i) _ 1 J < y P 2) 2 M b

o (N i ) 2 J ( P w + P 2 )b d b

o 2v(Ni-Sn) „ Jp 1< V P 2>b®

o (Sn) J P ^ d 15

Special attention is required fo r C l MO x-ray radiation because this

process involves the excitation and de-excitation o f the orb ital. Assuming

that the excitation does not in terfere with the de-excitation, we can treat

this process in a s im ila r way to the static approximation fo r the Is a MO.

Considering firs t the total cross sections we have

a) F or C l MO x-ray radiation

g (C l ) = 2 „ j 2 P 2 bdbT2po

(where T2p a *s Pr °bability of the 2pa MO to be de-excited produc­

ing a MO x -ray )

221

b) For C l MO x -ray , K x -ray coincidence, three possib ilities exist:

1) Two vacancies in the 2pa MO; 2) one vacancy in each o f the two

2pa, 3da MO's; and 3) a 3da—- 2pa transition followed by the 3da,

2pa radial coupling in the exit channel.

The probabilities from each e ffect w ill be:

+ - + - - +From (2pa, 2po) vacancies, probability= P P 0 Tt 0 (1 -T . ) + T rt (1 -T 0 )1

2 2 L 2pa 2pa 2pa 2po J

= 2P2 T 2pa(1“ T2pa)

— + + H- ™ ~From (2pa, 3da) vacancies, probability = P_w (P T J + P w (P T )u 2 2p o o 2 2p O

= 2P2P 3WT2pa

Cascade < 3 d a -2 p a ), p robab ility- ™ (P2

= 2P w T2 2pa-3do

(where T ^ is the 2pa MO de-excitation probability through 3da-*2pa

transition).

Since rn « 0 = » 1 - t . « 1 , we get2pa 2pa

a (C l-N i) = 2tt \ [2Po t _ + 2P w P r + 2P w r „ , Ibdb ' ' J L 2 2pa 2 3 2pa 2 2pa-3da J

and

2

q (N i-C l) _ n P 2T 2po + P 2P 3 ",T 2po4 F 2",T2pa - 3 d q > db

t’ <C1> [ P , T bdbJ 2 2po

To find simple relations between the three (double x -ray to single x -

ray) ratios (N i-N i)/ (N i), (N i-Sn)/(Sn), and (N i-C l)/ (C 1 ) two assuptions have

to be made fo r the impact param eter (C l) dependence o f the term s in the

222

in tegra ls .

i) One can assume that the 3da MO vacancy production probability P3

is nearly constant for impact param eters up to a specific value and

then it goes to zero. This is a reasonable assumption because a 3do

MO vacancy can easily diffuse to the continuum v ia the different c ro s ­

sings with other M O 's. Then the random walk approximation (Brd 79)

can be applied. It predicts an almost constant value o f P up to an im -Opact param eter b ^ and then a smooth fa ll o ff (Brd 79). The value o f

w ill be la rge r than the corresponding value o f the 2pa MO due to

reduced binding o f the 3da MO.

ii) The second assumption is that r „ and r „ „ , a re nearly inde-^ 2pa 2pa-3da J

pendent o f the impact param eter. This is not co rrec t because we know

they w ill vary with b, but an average value can be considered.

A lso define the re la tive contribution o f the two processes fo r the 2pa

MO excitation as (K) fo r the 3dCT-2pa coupling, and (1 -K ) fo r the other ways

o f 2pa excitation.

Then,

lv hlia (N i)= 2tt PgW — + 2 77 JPgbdb

2b_i3 2 k

n m l= 2ttP w ----

= 277 (— ) f P . bdb 1-k J 2po2po

and the three ratios become:

223

g 2v(N i-N i) 1 P 3 w 2 ^ L + 2P3 '> l P 2 M b -f J~P 2bdb

o 1V(N i) 2 b2

P3W ^ + JP2 bdb

b2

1 JP 2 bdbT [k P w + 2 (l-k )P w + (1-k) ----------

JP2bdb

= (1- - ) P w +

o 2, . P ft bdb 1-k J 2pa

2 3 2 fP bdb2pa

2v c P w fp bdb+ fp P .bd b f t3, P o bdbg (Ni-Sn) = 2 3 J 1 1 2 = 2p v , + 2 J Is a 2pa

a 1V(Sn) fP bdb 3 fp bdbJ 1 • I s a

and

2v/ -. tvtx PowTo fP„bdb + wr fP0bdb + T0 fP^bdbq (C l-N i) _ 3 2pa “ 2 2pa-3da J 2 2pa J 2

r fp l 2pa J 2

2^ , r T2pa-3dc n 2pobdb

= P w + w [ — J + ----- ---------

T2p® ■f'Vbdb

Using the scaling fo r the 2pa MO excitation v ia 2p77-2pa coupling proposed

by Taulbjerg et al. (Ta 76) and discussed in the Theory section, o r m ore

accurate functions fo r the total 2pq excitation from the experim ents o f Johnson

et al. (Joh 79), one can calculate the two ratios o f integrals:

fP 2 bdb fP , P n bdb2pa J Is a 2pa

ft> bdb fP , bdb2pa J Isa

Th e ir value is much la rger than the vacancy sharing probability w. Thereto i'e,

224

in a f irs t approximation only the leading term s (the term s including the Pg ^

functions) could be retained in the above formulas. In this way we can obtain

a firs t estimate of the rela tive importance of the two 2pa MO vacancy prod­

uction mechanisms (via a 2p77— 2pc rotational coupling or a 3da— 2pa vacancy

sharing), as w e ll as an approximate value o f the double ratios defined in

the follow ing.

So, using only the leading term s in the above vacancy production cross

sections we get:

C ^ N i -N i ) 1-k^ i P2pObdb o 2v(N i-C l) JP 2pgbdb

o 1V(Ni) ~ 2 JP2pobdb ' a 1V(C l) ~ JP2p<jbdb

c 2V(N i-Sn) 2 JP ls o P 2pgbdb

" JPlsobdb

and then,

f (Ni-Ni) . lv

a (N i) 1-k2v

r a (N i-C l) .l v

o (C l)

fP P 0 bdb 2v J ls a 2pa

a (N i-Sn) ^lv ._ , 3 l*P, bdb

a (Sn) _ 2 J ls a

( } Jp 2Pabdblv .

a (C1) PP bdb2pa

Using the experimental values fo r the double vacancy to single vacancy

225

production ratios (Table 3. 5) we can define the values o f the unknown quanti­

ties m ore accurately. So assuming,

P « 1 andO' 2pq-3da ^ 1

T2po T

(the accuracy of these values is rela tively unimportant because the contribution

o f the re la tive term s in the above ratios is sm all), we get

( V bdb J 2 dct

f P 0 bdb •J 2pa

= 4 .5 ± 0 .5 = 1 0 . ± 1 . 2

fP , P „ bdb J Is a 2pCT

= 5.0 ± 0 .4 % > 100 MeV

lSCTbdb

(1-k) = 36 ± 4 %

= 7 .2 ± 0 .5 % S 200 MeV

= 37 ± 5

From these values one conclusion is that the re la tive contribution o f each

2pc MO vacancy production mechanism does not change (o r changes ve ry little )

with pro jectile energy.

Another important result is that with increasing p ro jectile ve locity the

2double 2pCT MO vacancy production per single 2po vacancy ( J p ^ ^ b d b / J ^ p a

bdb ) ratio increases faster than the corresponding ratio o f simultaneous 2pcr,

I sc t MO's vacancies per I sc t MO single vacancy. As it has been already

stated previously regarding the contribution of double vacancies to the casca­

de MO x -ra y , K x -ray spectra, this different energy dependence o f the true

ratios proves that the singles spectra norm alized to the K x -ra y peaks of

the coincidence spectra, provide a better approximation o f the contribution

of double vacancies in the C2 region at higher p ro jectile energies. Since,

with increasing pro jectile energy the rela tive importance of double vacancy

contribution to the C2 region o f the real coincidence events increases (fig.

3 .14-3.16, compare data points to the broken line in the C2 region, with

increasing pro jectile energy), the above energy dependence o f the double va­

cancies supports the idea that the observed coincidences result from casca­

de transitions.

Finally by comparing D '(E^) and 2A we observe that at 100 MeV the

ratio (D '/2A) is ~ 2 and ~ 1 .5 fo r 200 MeV. This supports the conclusions

made in section TV. 1 that the singles spectra provide a good approximation

o f the x -ray spectra produced from double inner-shell vacancies.

226

Fig. 4 .8 The schematic Isa MO excitation probability as a function o f the

impact param eter. A rb itra ry units have been used (from Han 75).

(arb

itrar

y un

its)

Impact parameter

rK; Bohr radius

rad: adiabatic radius (=a)

Fu'ure 4.8

229

IV. 5 CHARACTERISTIC K -VAC AN C Y PRODUCTION

Introduction

The p ro jectile and target K-vacancy cross section fo r 100, 160, and

200 MeV are presented in figures 4 .9 , 4.10, and 4.11. The values shown

have been calculated from the measured K x -ray cross sections (Tab. 3.7

to 3 .9 ), using the neutral atom fluorescent y ie ld (Bam 72). As it has been

explained before , the ionization o f the pro jectile w ill change the value o f its

fluorescent y ie ld .' Since the degree o f ionization w ill vary with target and

with pro jectile energy and it has not been measured, the neutral atom fluo­

rescent y ie ld has been used. The e r ro r from the change of the fluorescent

y ie ld w ill be less than 10%. Table 4 .1 presents the K-vacancy cross sections

fo r p ro jectile and target at a ll three energies. Typical e rro rs are 15% (r e l­

ative) and 25% (absolute). Based on the work o f M eyerhof et al. (Me 76), we

divide the data into regions according to the special working mechanism fo r

the K-vacancy production.

F or the p ro jectile K-vacancy cross sections we distinguish two regions:

i) The low atomic number area (Z^<22 ), where the Nb K-vacancy cross

section, as a function o f the target atomic number, presents a characteristic

change in slope. In this area (6 < Z t <22), the cross section (p ro jectile ) does

not change much fo r d ifferent targets, and in the 100 MeV case it starts in­

creasing with decreasing Z^. The sudden change in slope is m ore pronounc­

ed in the 100 MeV measurements, but it is also present in the other areas.

In this low Z^ region, the direct excitation is responsible fo r the production

230

of K vacancies in the heavier collis ion partner (Nb). Calculations have been

perform ed on the basis o f some sem iem pirica l relations (App. I l l ) and they

are compared with the data at the end of this section.

ii) The bell shaped area of 22 < <65 , which is the place o f dom i­

nance o f quasimolecular phenomena. With the help o f the sharing factor the

ratio o f the two K vacancy cross sections (p ro jectile and ta rget), can be p re ­

dicted.

F or the target K vacancy cross sections we have:

i) A region which can barely be seen in the 200 M eV data, fo r Z^<22,

which shows up as a change in the slope o f the target cross section (upper

data) becoming m ore strongly dependent on Z^. The strong enhancement in

the target (ligh ter collis ion partner) K vacancy cross section in that region

is due to the approaching o f the L -sh e ll from the heavy atom (Nb) to the K

leve l of the target. Then, because of the K -L leve l matching, the K -shell

of the target w ill rece ive vacancies d irectly from the 2p7T MO and from high­

e r orb ita ls, which result in the sudden increase o f the measured K vacancy

cross section fo r the target. This phenomenon (K -L leve l matching) is also

taking place in the high Z region (Z t >70 ), but we did not have any data in

that region to observe it.

ii) The second region extends from Z «2 2 to Z « 5 2 , which correspondsl» V

to the s im ila r region (ii) of the p ro jectile . The rela tive cross sections (ta r ­

get and p ro jectile ) can be explained on the basis of the theoretical values

of the sharing factor.

ii i ) A t the high atomic number lim it, Z^>55, the measured cross sec-

231

tions cannot be explained on the basis of the quasimolecular picture since

they are produced from other effects. This region is s im ila r to the region

(i) o f the p ro jectile . The main K vacancy production mechanism fo r the target,

which is then the heavier collis ion partner, w ill be the d irect excitation.

Follow ing this indroduction, w e examine the underlying K vacancy pro­

duction mechanisms in both co llis ion atoms. A fte r that, we compare the cross

sections with theoretical calculations based on the d irect excitation and on the

excitation of the 2pa MO (1.4).

a. Vacancy sharing region

According to the theory (Me 77) for sym m etric or nearly sym m etric co l­

lision system s, the main vacancy production mechanism in both colliding at­

oms , is the creation o f a 2pa MO vacancy and its sharing between the K -

shells o f the two atoms. The theory can a lso predict the vacancy sharing

probability in each orbital (Me 73). A straight comparison o f the theoretical

values o f the sharing ratio (Me 73) with the ratios o f the measured K vacan­

cy cross sections (fo r the p ro jectile and the target) at each collis ion system

w ill provide a check fo r the theory. Besides, it w ill help us to understand

the vacancy production mechanisms in those atomic collisions (2 2 < Z ^ < 5 6 ).

F igures 4.12 to 4.14 present the theoretical predictions of the sharing

wratio ( - ^ , where w is the sharing factor), and the ratios o f the K vacancy

cross sections (heavy to light), as functions of the target atomic number. We

observe that our data agree quite w ell with theory (Table 4 .2 ) in a ll p ro je c ­

tile energies fo r the region 3 0 < Z ^ < 5 1 .

F or 100 M eV, the agreement is very good even fo r a sharing ratio of

^_w ~ 0 . 0003 (C r case), but at higher pro jectile energies there is a perm a­

nent deviation which increases with pro jectile energy (fig . 4 .13 , 4.14, Z^ =

22, 24, 26, and 28). It seems that the disagreement is not accidental, but

is related to the pro jectile energy. With increasing p ro jectile energy, the

fluorescent y ie ld o f Nb w ill be increased. But fo r the neutral atom it is

already 0.752 (Bam 72) and so, it cannot change by m ore than 30%. The de­

viation between our data and theory is much la rge r (a factor o f 4 fo r 200 MeV

Nb+Ti). A change in the fluorescent yield of the target cannot be that la rge

(e .g . factor o f 4 fo r T i); Thus, the theoretical predictions by M eyerhof (Me

73) are not correct fo r large pro jectile ve loc ities and very asym m etric co l­

lis ion system s. S im ilar conclusions by Taulbjerg et al. (Ta 75) led them to

examine the effect that higher orbitals (2p-n- MO) have on the 2pa—- Isa shar­

ing. They proposed (Ta 75) some corrected values fo r the sharing ratio. The

Taulbjerg et al. corrections, applied to our collis ion system s, are v e ry sm all

and cannot account fo r the observed deviations. It th erefore seem s that fo r

these very asym m etric atomic collisions, the K -L matching mechanism par­

ticipates in the K vacancy production of the light collis ion partner, because

the 3da MO which is correlated to the 3p-level of the p ro jectile w ill be high­

ly excited at higher pro jectile energies. Since this orbital w ill supply vacan­

cies in the K -shell of the light atom (1.4c), an increased ionization in the

K -shell w ill be observed. In order to calculate the sharing ratio in the case

of K -L level matching, one needs to consider the coupling o f many states

and one also has to know the vacancy production probability in each orbital.

232

\v

233

In the region of heavier targets than the p ro jectile , we find that there

is a good agreement between theory and experiment fo r a ll points up to Ba.

From this target and up, a consistent deviation occurs, with the theory un­

derestimating our measured sharing ratios. Since the K -L leve l matching

should further reduce the sharing ratios, it is obvious that a new mechanism

is responsible fo r the creation o f vacancies there, and the sharing of vacan­

cies (the quasimolecular picture) is not an accurate description. In the case

of E r the K vacancy cross section is la rger than the cross section o f Sm

though Z = 62 < Z =62 (fig . 4.9 and 4.11). This is due to the background am Br

contributions from K -shell internal conversion decay. Since E r has a strong­

ly deform ed nucleus, this effect w ill make a large contribution to the K x-

ray production, while in the other targets such an e ffect w ill not g ive any

appreciable contribution. The contribution from this nuclear e ffect has been

calculated and it was found to account for most of the E r K vacancy produc­

tion. In the case of Sn target the contribution was less than 10%. Due to

the subtraction o f two almost equal numbers the sharing ratio fo r the Er

case is quite uncertain.

b. Comparison of our data with theory

Up to now, we have proved that the vacancy sharing mechanism be­

tween the 2pa and ls a MO's can explain only the cross sections in the r e ­

gion 22<Z^<56 . Outside these lim its it starts to deviate, and the d ifferen­

ce between measured ratio and theoretical value becomes la rg e r fo r heav­

ier o r very light targets. It was proposed that d irect excitation is respon­

sible fo r the K vacancy production in that region. Since there are some

234

sem iem pirica l calculations for the prediction o f the Coulomb excitation, it

is important to compare them with our data. Our calculations have been bas­

ed on the method described in A pp .III, with some sm all corrections (An 78III),

whichdre important only fo r the higher atomic number targets.

The theoretical predictions are presented in the same figu res with the

K vacancy cross sections (continuous lines). We observe that the theoretical

curve reproduces the data quite w e ll in the low region , both in shape and

magnitude; the d ifference being comparable to the absolute e r r o r o f the mea­

surements. This is va lid fo r all three pro jectile energies which indicates

that the agreem ent between theory and experiment is not accidental.

On the other hand, looking at the high Z region, we observe a la rger

deviation, a factor of 3 to 5 in a ll cases, with the theory constantly under­

estimating the cross sections. It should be noted that Behncke et al. (Ben 78),

using heavier systems (Xe p ro jectiles ), observed a deviation o f a factor of

two but in that case the theory was overestim ating the measured cross s e c ­

tions.

We can also see from figs . 4 .9 to 4.11 that the d ifference between the

theoretical values and the measured K vacancy (heavy) cross sections is la r ­

g er at sm a ller p ro jectile energies. This might indicate an e r r o r in the the­

oretica l treatment of the effect o f the Coulomb deflections (A pp .III). In any

case, the theory, though it cannot reproduce the data v e ry w e ll, can be

used as an indication of the re la tive importance of I s a MO vacancy produc­

tion mechanisms. For the case of interest (Nb+Sn) the effect o f Is a MO

direct excitation to the K(heavy) vacancy production seems to be two orders

235

of magnitude sm aller than the e ffect of the 2pa-~ Is a vacancy sharing mech­

anism and it can be ignored in the calculations (III. 2-3, IV . 3-4).

Considering now the probability of a Is a MO vacancy production through

multiple collis ions, we can use the measured K (pro jectile ) vacancy produc­

tion and the theoretical calculations fo r the Is a d irect excitation cross sec­

tion (which is approximately three orders o f magnitude sm a ller ). Based on

the calculations by M eyerhof et al. (Me 74), which w ere presented in detail

in the theory section (I. 6a), we find that in the case o f Nb+Sn the contribu­

tions from multiple collisions fo r the Isa MO excitation are v e ry small.

In the case of Nb target we observe that with increasing pro jectile en­

ergy the contribution o f Isa d irect excitation (continuous curve fig . 4.9 to

4.11) increases considerably. Based again, on the previous calculations by

M eyerhof et al. we find that, while in the 200 MeV case the two mechanisms

(d irect excitation o r v ia multiple collis ions) seem to be of equal importance,

in the 100 M eV the two collision process starts to be m ore important. At

low er p ro jectile ve loc ities the multiple collis ion contribution to the Is a MO

excitation w ill be la rge r due to the strong decrease of the I s a MO d irect

excitation process.

As we have seen, the middle area of 22< Z^<56 can be explained on

the basis of the sharing of vacancies between 2pa and Isa M O 's in the exit

channel. Summing up the target and p ro jectile cross sections, one can find

an experimental value comparing this total cross section with the theoretical

predictions (1.4), and can a rrive at some interesting conclusions.

In fig. 4.15, the total 2pa vacancy production cross section is plotted

236

as a function of the total atomic charge (pro jectile and target) fo r the three

p ro jectile energies. It is observed that they follow w ell defined curves of

s im ila r shape.

F igure 4.16 presents the Nb K x-ray cross section as a function o f pro­

jec tile energy, from the values o f Tables 3.6 to 3 .8 . It turns out that

KX 2. 2 ± 0.1a ~ E Nb proj

In the theory section (1.4c), it was stated that the K x -ra y cross section

from multiple collisions alone can be expressed as:

1 2pa (2pa) = — n v t a . w „ a

me 3 2 1 lx proj 2p rot

For the case of Nb+Nb, the above formula g ives the results presented

in the same figure (4.16) (broken line). The energy dependence o f the theo­

retica l values is g 2-® ± 0 - 3 w hich is very close to the experim ental one. Still, pro)there is a slight deviation which might result from single collis ion contribu­

tions (one or two step) to the 2pa MO vacancy production, as described in the

theory section (1.4). The overa ll normalization between theory and our data

is o ff by 30%. Considering the 25 % total e r ro r o f our measured

cross section the agreement between theory and experim ent is very good.

F igure 4.17 presents the Nb K x-ray cross section as a function of

pro jectile energy fo r the Nb+Sn system . The corresponding cross section

fo r the target (Sn) depends only on the sharing factor, and it is not o f any

special interest. From this figure we can see that the theoretical calcula­

tions based on the multiple collis ion 2pa excitation fa il to reproduce the

measured values both in slope and in absolute magnitude. In fact, the theory

predicts that cx(200 M eV )«1500b and a , ~ E^ ’ 3 ± , while we have m ea-Nb proj

sured it to be a(200)?»4000b and cr~ g 2 .7 ±0 .2 ^proj

S im ilar discrepancies between theoretical predictions (I. 4) and m ea­

sured total 2pa MO vacancy cross sections appear fo r the other targets. F i­

nally the theoretical predictions from multiple collisions alone fa il to rep ro­

duce the data with the exception of the sym m etric case. Since this coinci­

dence o f the two values might be fortuitous, we can assume that fo r this

region also the existing theory (Me 77) does not reproduce the data w ell.

237

Fig. 4 .9 , 4.10, and 4.11. K vacancy cross sections, as a function of the

target atomic number fo r 100, 160, and 200 MeV. The continuous

lines are the theoretical predictions from dix'ect Coulomb excitation.

Fig. 4.12, 4.13, and 4.14. Theoretical and experimental sharing ratio (w / l-w ),

as a function of the target atomic number, fo r the three pro jectile

energies.

F ig. 4.15 Total 2pa vacancy production cross section as a function of the

combined atomic number.

Fig. 4.16 Energy dependence of the characteristic K x -ra y cross section of

the Nb+Nb collis ion system (continuous line). A lso the theoretical

predictions (Me 77, broken line).

Fig. 4.17 Energy dependence of the K x -ray cross section fo r the Nb+Sn

system. Continuous line: our data; broken line: theoretical p re ­

dictions (Me 77).

160 Mo Tare ® Nb

eVjet

O ■ n

OCr)

oo

o

eb ® ' c

oo 9

C0

« ©o

> O '

©o

9 ~'

o

o

\() 2<5 30 4 D 50 6 0

TARGET ATOMIC NUMBER

-vac

ancy

cr

oss

sect

ion

(bar

ns) 1 0 '

10

1 0 '

10

10a :

t

.1

*...... r -D MeV !NbTarget -

'O 20 <©o

kJ onO| o o Jvj

« © 0 ....& o

(» 9------ o----- O-©QO o0 © o

o ------o-o

.

10 2 0 3 0 4 0 5 0 6 0Target Atomic Number —

Figure 4.11

Sha

ring

ra

tio

Figure 4.12

Shari

ng

ratio

Figure 4.13

Shar

ing

ratio

Figure 4.14

(2pe

r M

O) v

acan

cy

cros

s se

ctio

n

(bar

ns)

i o 6

lO 5

io 4

io 3

IO2

1 0

50 60 70 8 0 9 0 100

Total Atomic Number —

Figure 4.15

.................. 1-----------------------

0 ^ 0 0 o 1 6 0 A 1 0 0

M e V nM e V “

M e V j

•

- A ^ O _O

Ao 9

is ^ _O 9

A ALA AA %

C D ® -

......... A , ' O % ..........L J

° 9

AO

9

A A --------------- o-- A

A- A -s y m m e t r y

Nb + Nb

KX-RAY CROSS SECTIONS

o Theoretical predictions (Me 77 )

# Measured values

Figure 4.16

loq(o-

)

3.5

t

3.0

2 .5

o Theoretical predictions (Me 77 )

• Measured values

Nb + Sn Nb KX-RAY CROSS SECTIONS

i

6/

S y ' /

V /V c /

)

>

/

/

S

K1 V

2 2.1 2.2 2.3 2.4

- iog(Eproj)

Figure 4.17

K-vacancy7 cross-section (barns)

lOOMeV

Target

160MeV 200MeV lOOMeV

Nb

l60MeV 200MeV

/ 2 8.47 44.95 123.67

1 3 - 2 7 6.77 53.95 155.21

T i4822 113000 543000 8.52 241.68

~ 52~ , C r24

38340 92240 204560 12. 22 123.3 351. 8

„ 56 ^ Fe 26

20760 54420 120250 16.30 156.4 484.7

XT - 5828

16590 39240 71200 35. 51 289. 5 736. 5

Q Zn6430

12214 25420 47220 113.4 485. 0 1342.

Se8°34

3500 13500 22000 117.0 950. 2282.

T5 7 935

2930 11690 18570 169.6 1232. 2515.

Y 8939

m k93Nb41

1255.7 3736.7 6595. 1255.7 3736.7 6595.

9 8, „M o 428S1. 3469. 5278. 1456. 4748. 6786.

* 10747

81.17 445.9 845.1 1420.7 4430. 6249.

s o - " 09.90 83.89 236.9 779. 5 2652.5 5438.3

b a - " 03.39 96. 655.6 4300.

138Ba

560. 511 3.851 16. 09 332. 5 1157. 2318.

c 144 „ Sm 62

0. 159 0.769 3. 088 200.0 805.85 1597.0

v 166 „ E r68

5. 079 10. 17 665. 0 978.7

Table 4.1

Sharing ratio (w/l-u.)

measured values theoretical values

lOOMeV 160MeV 200MeV lOOMeV 160MeV 200MeVW k

6 ° 12

2 2 - 4 8 .000075 .00045 .00012 .00189 ' 0.221

n 5 224

.00032 .00134 .00172 .00034 .00189 .00373 0. 283

IT 5 6„ Fe 26 .000785 .00287 .00403 .000804 .00379 .00697 0.342

M-5828

.00214 .00738 .01035 .00213 .00800 . 0135 0.414

17 6430

.00929 .0191 .02842 .00545 .0168 . 0260 0.482

q 8°34

.0335 . 0703 . 1037 .0381 .0755 .0992 0.600

TJ 7 935

.0579 . 1054 . 1350 .0604 . 1087 .1374 0.625

Y 8939

Nb9341

0.752

™ 98 ..M o 42

.60502 .73068 .7778 .6162 .6845 .7134 0.770

* 10747 .05714 .10064 . 1352 . 0565 . 1041 .1330 0.834

..S n 12°50

.01269 .03163 .04356 .0124 .0324 .0464 0.859

5 2 - 1 3 0 .00518 .02233 .00500 . 0240 0.875

„ 138 Ba

56.00154 .00333 .00694 .000746 .00337 .00615 0.901

144 Sm

62.00080 .000954 .00193 .000044 .000301 .000717 0.928

v 166 Ei-68

.00076 . 00104 .000029 .000089 0.945

Table 4.2

250

Summarizing, we can b rie fly present the most important resu lts of this

thesis.

The cascade relationship between the 2pa-> ls a transitions and the sep­

arated atom K x -rays provided the idea of isolating specific MO transitions to

the most inner-shell molecular oi’b ital (lsa ) of two collid ing atom s. The MO

x -ra y , K x -ray coincidence technique applied in sym m etric andasym m etric

collis ion systems v e r ified the initial idea. A considerable amount ( ~ 30%)

of the total MO transitions to the ls a MO was shown to have originated from

cascade transitions, which are correlated to K x-rays of the separate atom.

The uncertainty in the total amount of the cascade MO x -rays from the con­

tribution of double vacancies was quite small, especia lly fo r the lowest pro­

jectile energy'used (~15% ), and it could not explain the observed spectra.

Thus, with the x -ra y , x -ray coincidence technique, the cascade type MO

transitions have been separated from the total amount of inner-shell transitions.

Due to the participation of multiple collisions in the l s a MO excitation

fo r the solid targets used, there is the possibility o f the 2 p n -»ls a transition

to be associated with K x -rays in sym m etric collis ion system s. In the asym ­

m etric collis ions, this possibility was excluded by the sm all probability that

a K vacancy w ill follow the ls a MO in the second collis ion . T o clear up the

uncertainty in the sym m etric collisions ease, the K vacancy production

mechanism has been studied in a variety of collis ion system s, both experi­

mentally (K x -ray cross-section measurements) and theoretica lly (App. I I I ) .

The firs t conclusion from those studies was that for the sym m etric collisions

V. Summary

the participation of the 2p77->lsa transitions in the total coincidence spectra

cannot be excluded. On the other hand, the absence o f any considerable

amount of C l radiation in the rea l coincidence spectra indicated that the con­

tributions from the 2prr+Isa transitions to the coincidence spectra in on e-co l­

lis ion processes should be quite sm all and calculations supported this idea.

With the present experim ental set-up, the importance o f the 2p7T->ls a

transitions in the coincidence spectra could not be fina lly reso lved . In order to

c lear up this question, one has to study the same collis ion system using gas

and solid targets. Then, a straight comparison of the cascade MO x -ra y

production in the two systems can revea l the rela tive importance of each

transition, because in gas targets the multiple collisions do not contribute in

the Is a MO vacancy production. In any case, the present measurements

have shown that the MO x -ra y , K x -ra y coincidence experim ents are workable

at least for co llis ion systems not much lighter than the ones used. For

heavier collid ing atoms, the same technique could easily be used because of

the more favorable production of MO x-rays compared to the K x -rays in

heavy atomic collisions.

The coincidence method can supply d irect information on the dynamic

form ation and the evolution of molecular orbitals. Using slightly asym m etric

co llis ion system s, where the production of K -vacancies in both atoms is governed

by the vacancy sharing mechanism, we have shown that the sharing mechanism

is a process between M O 's occuring at large internuclear distances, which is

in complete agreem ent with theory (Me73). The same was found to be true for

the 3da-»2pa vacancy sharing, which does not contribute to the C l radiation

251

252

(transitions to the 2pcr MO), since it occurs at large internuclear distances.

A detailed experimental and theoretical analysis o f the aforementioned

uncertainty' of the contributions from double inner-shell vacancies was accom ­

plished, which has set an upper lim it (factor of two) on the uncertainty of

their contributions to the coincidence spectra. The uncertainty o f a factor of

two in the double vacancy contributions in the coincidence spectra is mainly

the result of the absence of measurements on the impact param eter depen­

dence of the inner-shell vacancy production. An important p ro ject fo r the fu­

ture would be the systematic study of this dependence. Any improvement of

our knowledge in this area would supply enough information fo r the theoretical

prediction of the MO x-ray production. As was shown by our calculations, up

to now, there is a system atic deviation between the quasistatic predictions

and the measured cross sections. Part of this problem is produced by the lack

of a consistent theory which would treat coherently the excitation and deexci­

tation (for one or two vacancies) in the two inner-shells of interest (2 p a , Is a ).

Since the cascade MO x-rays have been observed (beyond any doubt) in

both collis ion systems (Nb+Nb, Nb+Sn) and (mainly) one MO transitions has

been separated, it is important that the new method be applied to study the

transformation of MO's during the atomic collisions and their couplings. Be­

sides, the vacancy sharing mechanism and the directional anisotropy in the

em ission of MO x-rays can be studied for the separated transition. Since an

acceptable theory fo r the anisotropy is s till m issing, the measurement of the

anisotropy from one transition w ill create a new method fo r spectroscopic an­

alysis of the MO x-rays and fo r the testing of existing theories.

253

A PPE N D IX I . E rro r Analysis in the Coincidence spectra

In the QM-channel, the total number of real coincidences, at each x-ray

energy E , is:XN . (E ) = K*c . (E )-(BCI)-d-eff (E )-e ffTW

coin x ' com x v QM x KX

where K is the constant of the collis ion system , and d is the target thickness.

The accidental counts are:

Na c e < V ' T l W Ex>-A KX

where r is the tim e resolution o f the system , A the average counting rateK X

on the KX-channel and N _,,,(E ) the counts in the QM-channel at the E x-QM ' x ' ^ x

ray energy. So,

N . (E ) N . (E )com x .1 com x

N (E ) TA N (E )acc x1 KX QM x ;

From the above we conclude that:

a) Since A is proportional to the beam intensity any increase in theK X

beam intensity results in an equal decrease in the peak to BG ratio of the

tim e peak.

b) F or constant beam intensity, different parts of the QM-spectrum

w ill present different tim e spectra depending on the ratio;

N . (E ) a (E ) coin x coin x

N BG<E x > W V + W V

c) Since a large part of the high energy BG region of the QM-spectrum

is the result of Ambient Background, a decrease of this background ( i .e .

through shielding) im proves the tim ing spectrum greatly decreasing the un-

254

At the same time the statistical eri'o r is:

certainty in the rea l coincidence counts.

l/Ncoi„(Ex> + Naec<Ex> | /p n . (e ) y i

com x v

4 ,

Nacc

-2- =N . N . coin com

N KX 2coin (N . ( E ) )

com x

This ratio decreases as the counting rate increases or as either o f the de­

tector effic iencies increase. The last statement explains the need o f la rge

solid angles and suggests the avoidence of using absorbers. B u t, there are

lim itations on the counting rates in both x -ray detectors. In one channel (KX ),

5there is the maximum counting rate of the detector (usually ~ 1 0 Hz), while

in the other (QM) a maximum rate ~ 150cts/sec is needed to avoid possible

pulse pile-up. So, the use of some absorbers in the Q M -coun ter, is neces­

sary to stop the numerous K x -rays , and w ill not affect the MO x-rays ap­

preciably.

In each run, we had to compromise between a high counting rate in the

KX counter and a low counting rate in the QM counter by adjusting the beam

intensity, target thickness, and absorbers.

255

The theoretical calculations of efficiency are not easy because o f the

large detection angle. The difficulty arises from the fact that the effect of the

absorber on the total efficiency has to be incorporated in the original in legra-

tion over the angle. Thus, the total effic iency measurements perform ed with

d ifferent absorbers cannot be compared d irectly .

The theoretical calculations w ere based on the analysis carried out by

Hansen et al. (Ha 73) extended to include the effect of the absorbers. The

total d ifferentia l e ffic iency (effic iency and absorption) at a specific angle 9

was defined accordingly:

d (e ff) _dO a AB Au DL ES COL CC

where e is the intrinsic efficiency of the sensitive volume o f the detector,&f AT,, f . , f _ T are the correction factors fo r the absorbers, the gold contact AB Au DL

on the detector face, and the dead layer between the Au and the active face

of the detector respectively; f is the correction factor fo r the possibilityESof an x -ray to escape from the crystal without detection; and takes care

o f the collim ation effects. The last factor, f^,^, is a correction factor fo r

charge collection.

From the above correction factors, f^ g , and have been con­

sidered to be independent o f the angle, which is approxim ately true. The

general expression of the energy dependent factors has been integrated over

angles, assuming cylindrical symmetry7. This assumption is considered reas­

onable as special care was taken for the accurate allignment o f the detectors

before each run.

APPE N D IX H. E fficiency and Absorption

256

a) ^(K) ( Mk AL ^kf = 1 ------ ^ Ck f l + — ln( ^ ------ ) ] + k [ l + — — ln ( --- ) ] '

E S 2 » y ) a \ ^ y p % M y 1

where ^ , // are the total attenuation coefficients fo r the x -ray denotedK K _O' Pin the index, jj(K ) the K -shell photoelectric coefficient, k^, k^ are the fra c ­

tions of K x -rays in the K^ and K^ groups, and is the fluorescent y ie ld

of Ge.

The factor f^,Qk is a correction for the increased transparency of the

collim ator with increased x-ray energy. The collim ators are used fo r screen ­

ing o ff the insensitive area of the crystal. This correction factor is difficu lt

to be calculated because it depends on the element used fo r collim ation. I f

it is ignored there w ill be a deviation in the high energy data ( >50 KeV)

which w ill become la rge r with increasing energy. Calculations (St 70) of the

collim ation factor revealed a nearly linear dependence on the energy. By com ­

paring the theoretical efficiency, not including this factor, with the m easur­

ed effic iency, an estimate of each value could be deduced.

The charge collection factor, f , was assumed to be independent o fOOenergy. Each of the integrated correction factors can be expressed analytical­

ly as:

ea = l - e x p ( - MGeV

f. = e x p (- ju.x e )

The escape correction factor can be expressed as:

and

257

where p is the attenuation coefficient of the element, d^ is the thickness of

the crystal and the thickness of the ith absorber. It can be easily under- i

stood that:

d X‘d „ = ----- — and x „ = 1

0 cos0 0. cos 9i

In order to take into consideration the escape from the edges o f the c rys ­

tal we can either use an average depth o f the crystal (Smt 71, Ha 73) o r integ

rate the above formulas incorporating the change in geom etry due to the

edges. Both methods give v e ry close results, so the simplest method of the

average depth was used. The average depth is defined (Smt 71) as:

z <e > - 3

Thus, the integration over angles was carried out from 0 = 0 to

-1 rwhere 0 = tan (----— r is the radius of the crystal and a, is its dista-

max a+Z

nee from the target.

The integration was perform ed numerically with the help of a computer

program . From the detector specifications and our measurements, some approx

imate values w ere obtained fo r the characteristic correction factors. F or

instance, the distance a was measured, but it was also adjusted, within 1 to

2mm, to fit the measurements in the high x-ray energy region. Besides, the

crystal dead layer was completely unknown and it had to be varied to fit a ll

the measurements for each detector. The same was true fo r the other para­

meters which could be varied by a small amount to give the best fit. In spite

of the need of simultaneous calculations for absorption and e ffic iency, which

258

complicated the procedure, the achieved results w ere in a rather good agree­

ment with the measured values (fig . 2.7, 2.8). Considering the fact that only

two parameters w ere largely' varied (distance and dead layer), the success

o f the fit suggested that the e rro r associated with the total effic iency could

not be la rger than 10% fo r x-ray' energies above 11 KeV.

F or the low energy region, the lack of adequate measurements in con­

nection with the sensitivity of that region over small uncertainties in the thick­

ness o f the dead layer and the absorbers, suggests an e rro r ~20%. It should

be noted that once the efficiency o f one energy' measurement has been chosen

the e rro r should not exceed 5% over the rest of the spectrum.

The measurements w ere perform ed by using callibrated sources in place

241of the target. The sources used w ere: Am (13.9 KeV, 17.8 KeV, 20.8 KeV,

57 13726.4 Kev, and 59.5 KeV ), Co (14.4 KeV, 122 KeV, and 136.3 KeV ), Cs

55(32.1 KeV and 36.5 KeV) and Fe (5.9 KeV and 6.5 KeV).

259

R. Anholt and W .E . M eyerhof (An 77 ) have expressed the K -shell

ionization cross section fo r the heavy collision partner as:

W V V - V 1- ZH- v i> z l f b f c f r ■where ct is the experimental cross section fo r proton heavy element c o lli­

sion, v is the pro jectile velocity , and Z , Z denote the charges o f the1 L H

light and heavy elements respectively . F , F , and F are the binding ener-B C R

gy , the Coulomb deflection, and the re la tiv is tic correction factors. They

have the follow ing values:

a) Binding energy factor

This factor has been calculated from the low velocity lim it o f the cross

section and it has been corrected to take into account the other two effects

by iteration.

APPE ND IX III. D irect excitation calculations

+ D I"

JE ' Uk<ZH

n( ?k)= 9 .094-9 .71 6.42 ? k ~7.45 ^

R = .0136 KeV and U (Z ) is the K -shell binding energy of the atom of atomicKnumber Z.

b) R elativ istic correction

The Dirac wave functions divei'ge for large momenta, q » a * re -O K

suiting in an increased efficiency of momentum transfer and in la rger K -shell

cross sections. An approximate expression has been given by Bang-Hansteen

(Ban59) resulting in the following correction factor:

F =R(Zl , v v )

R R ( l , Z H , V l )

R(Zl ; V [l+ (1 q2]

Q = Uk(Z L + Z H > \ Uk(Z L + V 2 2 1/2 q i - , Z )^k V H

Z F = Z H (1 + 0 .0075Zl )

where a is the fine structure constant,

c) Coulomb deflection factor

The effect of the increased Coulomb repulsion, due to the pro jectile

charge, has been calculated by Basbas et al. (Ba 731) and it is:

C (Z L , Z H , v 1)= 9 E 10( . d , o e ) ,

where E is the exponential integral function of order 10 (Ab 72) and

_4 3 /p7Tdqo = 8. 55*10 Z / Mn^ ,

M =a l a h

a l + a h

(A is the atomic mass) ,

C = 1 + TW V

Uk(Z H>- 1

n

Norm alizing the above to the proton-target collis ion we have the factor:

261

Am 79

An 771

7 711

77in

781

7 811

78HI

And 76

Ann 79

Ba 731

7311

78

Bam 72

Ban 59

Ab 72 M. Abram ovitz and I. Stegun, Handbook o f Math. Functions (Dover,

1972)

A. Amundsen, to be published

R. Anholt and W. E. Meyerhof, Phys. Rev. A16, 913 (1977)

R . Anholt and W. E. M eyerhof, Phys. Rev. A16, 190 (1977)

R. Anholt, W .E . M eyerhof, and A . Salin, Phys. Rev. A16 , 951

(1977)

R. Anholt, to be published

R. Anholt, to be published

R. Anholt, Phys. Rev. A17, 983 (1978); ibid. 976 (1978)

J .S . Andersen et al. , Nucl. Instr. and Meth. 132, 507 (1976)

C .H . Annett et al. , Phys. Rev. A19, 1038 (1979)

G. Basbas, W. Brandt, and R. Laubert, Phys. Rev. A7, 983 (1973)

G. Basbas, W. Brandt, and R. H. R itchie. Phys. Rev. A7, 1971

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G. Basbas, W. Brandt, and K. Laubert, Phys. Rev. A 17, 1655

(1978)

W. Bambynek et al. , Rev. Mod. Phys. 44, 716 (1972)

J. Bang and J .M . Hansteen, M at-Fys. Medd. 3_1, No 13 (1959)

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