Copyright 1970. AU rights reserved

PIONIC ATOMS G. BACKENSTOSS

University of Karlsruhe, Germany and

CERN. Geneva. Switzerland

CONTENTS 1. INTR ODUCT ION ................................................. 467 2. BAS IC PR OPERT IES OF E XOT IC AT OMS ......................... 468

2.1 PROPERTIES C OMMON TO A LL E XOTIC A TOMS... .. . . . . . . . . . . . • .. . . . .. 469 Formation of the atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 469 Level scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 470 Short-range effects and finite size ................................. " 470 Vacuum polarization. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 472 Mesonic cascade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 473

2.2 S PECIFIC PROPERTIES OF ... -M ESONIC A TOMS . . . • . . . . . . . . . . . . . . . . . . , . 475 Level shifts..................................................... 475 Level widths. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 477 Intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 477

3. E XPER IME NTA L T EC HN IQUES .... .. .... ............. ........ . 479 4. T HE ORET ICA L A PPR OAC H T O PION IC AT OMS .................. 484

Elastic scattering and energy shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 485 Pion absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 488

5. RES ULTS ON PIONIC AT OMS ............................ ....... . 491 5.1 RESULTS DERIVED FROM THE ELECTROMAGNETIC PROPERTIES......... 491

... -Mass ............................................................ 491 Pionic cascades in chemical compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49 2

5.2 R ESULTS ON THE S TRONG PION-N UCLEUS INTERACTION ... ..... .... . 493 Energy shifts .............. ........ . .. .. ..... ............. ..... . 493 Linewidths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 495 Linewidths from transition intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 497

6. INTER PRETAT ION OF RES ULTS A ND C ONC LUS IONS ............. 500

1. INTRODUCTION Soon after the discovery ( 1) of the 11" meson, the first experimental indica­

tion of the existence of mesonic atoms could be deduced from the observa­tions of Conversi, Pancini & Piccioni (2), who found that negative fJ. mesons decay in light but not in heavy elements. It was pointed out by Wheeler (3)

that this could be explained by the formation of mesonic atoms, since the probability of nuclear capture for a muon bound in a 1s state varies with the atomic number as Z4, thus favoring decay in light nuclei.

At the same time, theoretical studies by Wheeler and by Fermi & Teller (4) led to the conclusion that mesonic atoms should exist because the time

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468 BA CKE NST OSS

necessary for a slow meson of 2 keV to reach the Is level is ",,10-13 sec, a time that is short compared with the lifetime of muons, pions, and kaons.

Various indications for the existence of muonic atoms, such as Auger electrons in photographic emulsions (5) and X rays measured with NaI crystals, have been found in studies with cosmic rays (6) . It was expected that pionic atoms should also exist. However, in this case, information from cosmic ray experiments was of course more difficult to obtain. One indication for Auger electrons caused by stopped 7f' mesons was reported (7). Capture was expected to be so strong that no 7r- decay could be observed. Therefore, even more than for muons, results on pionic atoms could be expected only after the pion-producing synchrocyclotrons came into operation. The first conclusive evidence for pionic X rays was obtained by Camac et al (8) at Rochester, followed by work at Pittsburgh by Stearns and collaborators (9) and at Liverpool by West et al (10). Comprehensive reviews by de Benedetti (1 1), West (12), and Stearns ( 13) describe the experimental results up to 1958, as well as the main features and conclusions drawn from them. The shift of the energy levels due to the strong 7r-nucleus interaction was first observed experimentally, and there exist at this early stage only a few theoret­ical papers ( 14, 15) concerning the pion-nucleus interaction.

The development of solid state detectors as high-resolution 'Y-ray detec­tors activated new interest in the field of pionic atoms. In 1965, measure­ments were begun at Berkeley (16) and CERN ( 1 7), and a little later at Virginia (18). It was now possible to determine natural linewidths, and small energy shifts and intensities for many transitions and nuclei; this work has been described in a few Conference reviews (19-22).

The pionic atom is a system consisting of a pion and an atomic nucleus. One can therefore expect that by investigating pionic atoms, information is obtained about the two components of this system as well as about the inter­action between them. I ndeed, the best measurement of the pion mass origi­nates from measurements on pionic X rays. Basic information about the pion-nucleon interaction at low energies can be derived from pionic atoms, which complements measurements on pion-nucleon scattering. Finally, the pion provides a powerful tool for studying the structure of the nucleus; it is also sensitive to neutrons. Owing to the possibility of strong absorption by the nucleus, the pion is sensitive to high-momentum components or short distances in the nucleus. These facts can be exploited in pionic X-ray studies.

In Section 2 the properties of exotic atoms, including hadronic as well as muonic atoms, will be reviewed. Experimental techniques and theoretical approaches will be given in Sections 3 and 4, experimental results are pre­sented in Section 5, and conclusions are drawn in Section 6.

2. BASIC PROPERTIES OF EXOTIC ATOMS An atomic nucleus (positively charged) in whose field a negative meson

moves is called a mesonic atom. This implies that the properties of such a system are closely related to the properties of a.n ordinary electronic atom,

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PIONIC AT OMS 469

This is so because in both systems the electromagnetic interaction, i.e. the Coulomb field, plays a dominant role. At first sight this might be astonish­ing, because the mesons are subject to the strong interaction which, by definition, dominates other interactions. However, the range of the strong interacting field is much shorter than that of the Coulomb field, and there is a wide range in which the Coulomb field dominates. The meson itself is captured by the atom in a highly excited state. It can undergo many transi­tions into lower states before it enters the range of the strong interactions.

2.1 PROPERTIES COMMON TO ALL EXOTIC ATOMS

The main features of mesonic atoms ('lI"-mesonic and K-mesonic atoms) are therefore common to all exotic atoms such as those formed with muons (p.-) , hyperons (�-), and antiprotons (p). It is therefore convenient to de­scribe first these common properties. This can be done largely by referring to the properties of muonic atoms which have been studied in detail and which have been reviewed recently by Wu & Wilets (23) , and Devons & Duerdoth (24); and by Burhop (25), who also treats all other mesonic atoms.

Compared with the electron, all these particles have heavy masses. This has two important consequences. Firstly, the particles are much closer to the nucleus since the radius of the Bohr orbit is inversely proportional to the mass of the particle in the electromagnetic field. Therefore, these heavy par­ticles are much better suited for testing nuclear properties than are the electrons. Secondly, for the same reason, the meson orbits with principal quantum number n< ymm/me (with mm the mass of the heavy particle and me the electron mass) are located within the innermost electron orbit. There­fore, screening by the electron cloud is negligible for most applications. Furthermore, since present sources for the heavy unstable particles unfor­tunately exclude any formation of exotic atoms with more than one of those particles, one has to deal with a hydrogen-like atom. This two-body problem has, on the other hand, great advantages compared with the many-body problem encountered in electronic atoms.

Formation of the atom.-The formation of a mesonic atom is experimen­tally very simple. When a negative particle is slowed down, it is finally cap­tured by an atom of the stopping material. The interaction with the atomic electron cloud is likely to be of the type of the Auger effect, since at least one electron has to be ejected in order to have an ionized atom that can bind the negative particle. But very little is known about the detailed process of the formation. From the fact that the cascade is influenced by chemical or solid state properties (Section 5. 1 ) , one can deduce that the particle must be initially in very highly excited states, since only there are the energies of the particle low enough to be influenced by the low energies characteristic of the chemical bond or the crystalline structure. It is not known in which initial state the particle is captured. We might learn something about the mech­anism by studying the cascade process more closely. The time necessary for

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470 B ACKENST OSS

the formation and de-excitation of mesonic atoms is about 10-13 sec in con­densed substances so that the mesonic atom can be considered as a stable system. However, the time required for the slowing-down process from usually some 100 MeV to about 2 keV amounts to 10-1°-10-9 sec.

Level scheme.-The atomic levels between which the observed mesonic X rays occur are basically the same as the levels of normal electronic atoms. Already the simple Bohr formulas lead to a good understanding:

_ -p.c2 (za )2 ] En --- --

2 n

1i2 n2 rn = ---

p.e2 Z

1.

where En is the energy of an atomic level with the principal quantum num­ber n, and rn is the radius of the corresponding Bohr orbit; a is thc fine struc­ture constant and p. the reduced mass of the moving particlep. = m/(l +m/A) , where m i s the rest mass of the particle and A the mass of the nucleus.

The actual behavior of the atomic system is described by the relativistic wave equations, i .e. the Dirac equation for spin t particles and the Klein­Gordon equation for spin 0 particles.

The exact solutions of the Klein-Gordon equation for a point nucleus are

which is valid for Z<1/(2a)=68 and with terms of order (aZ)6 and higher neglected. The omission of these terms is well justified because their con­tribution, whenever it reaches some significance, is outweighed by the error introduced by the assumption of a point nucleus. Formally the angular momentum quantum number l in the Klein-Gordon case for spin 0 particles is replaced by j=l±! for spin! particles in the Dirac equation. Whereas two j states always belong to one l value, giving rise to the fine structure of the spin doublet, these doublets do not exist for spin 0 particles. Here the fine structure splitting occurs only between the different l substates belong­ing to the same n. Since the population of the levels l+! and l-! is of com- . parable magnitude, the intensities of the two doublet components are similar and are easy to observe. On the contrary, the population of the different l sub states may be very different so that the I multiplets usually cannot be observed, and only one l state is visible.

Short-range effects and finite size.-An important modification, particu­larly for large Z and low n, is the energy shift due to the finite extension of the nuclear charge. If the meson wavefunction overlaps the nuclear'charge

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PIONIC ATOMS 471

distribution, only a fraction of the charge contributes to the binding; thus the binding energy is reduced. If this overlap is small, the effect can be esti­mated by a first-order perturbation treatment. If V FsCr) is the potential due to the extended charge distribution of a spherically symmetrical nucleus having an electric charge density per) with '�72VFS(r) = -pCr) and with the Coulomb potential of a point charge Vc= -Zelr, the shift of the level energy AEn,z for the state (n,l) is

3.

The finite-size effect is typically short range. Such effects can be treated quite generally with the help of the S-matrix theory (26, 27), where the energy shift in a given state is related to the scattering amplitude of the same state. The scattering amplitude of a particle can be developed into partial­wave amplitudesf(8) = 'Lziz(8) . In the limit of k-+O (k particle momentum) the scattering amplitude fl(O) can be described by the scattering length az [only for 1=0, the dimension is that of a length, while more generally the dimension is VI+! (p-wave scattering volume) ] :

fl(8) = (21 + 1) alk21 PI (cos 8)

If one introduces a short-range pseudopotential VCr), thenf(8) can be calcu­lated in a plane-wave Born approximation,

4.

The use of pseudopotential replaces the true interaction to all orders. The energy shift becomes

AEn.! = f VCr) !l{;n.l!2dr s.

where f".z = YZmc/>".z are hydrogenic wavefunctions. Equation 3 represents now a special case of Equation 5. One thus (28, 28a) obtains relations between the energy shifts AEn.z and the scattering lengths az for 1 = 0

and more generally

2 1 (12 + 1)! Enz n (!nrB) 21+1 (21!!)2 (n - 1- 1)!

where rB is the Bohr radius.

6.

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472 BAC KENST OSS

These relations are quite general and can be applied to the short-range part of the Coulomb field of a finite-size nucleus as well as to the short-range pion-nucleus potential, as discussed in Sections 2 .2 and 4. In the latter case the strong-interaction shift is obtained, whereas for the finite-size shift in the s state:

t:.En,o � 1/6 1 tYn,O(O) 12Ze2(r2)

where (r2) = fp(r)r2dr I fp(r)dr. In practice, calculations of these shifts are performed exactly by the use

of a computer program using the actual potential and the relativistic wave­functions. This shift is of great importance for muonic atoms, and is one of the main justifications for the work done in this field. It is described in detail by Acker et al (29) and in the relevant review articles (23-25). Energy shifts caused by the finite nuclear charge distribution will be given in Table 2 , col. 7 .

Vacuum polarization.-Another effect which cannot be neglected even for larger n, bearing in mind the precision of present-day measurements, is caused by the radiative corrections for which the vacuum polarization domi­nates-an effect contrary to the situation in electronic atoms. The vacuum polarization, a second-order effect with the leading term "'a, describes the virtual production of e+e- pairs in the Coulomb field and results in an increase of the potential produced by the nuclear charge. Since the Bohr orbit radius is proportional to 11m , the meson moves in a stronger field than the electron and the effect is correspondingly stronger. The dominant part in the Lamb shift of the hydrogen atom, the self-energy graph, is proportional to 11m2 of the moving particle while the vacuum polarization is proportional to the mass of the virtually produced particles, i.e. the electron mass, independent of the mass of the moving particle. As a consequence, the relative importance of self-energy and vacuum polarization is interchanged for electronic and mesonic atoms. The vacuum polarization contributes only 2.6% to the hy­drogenic Lamb shift (30) , whereas the self-energy in muonic levels amounts only to a few percent of the vacuum polarization (3 1) .

The characteristic range within which the effect i s present is the Compton wavelength of the electron

tt ;I.e =- = 386.16F

mee

Since the Bohr orbit of a pion is

rs(n) n2

= 194- F Z

the vacuum polarization effect is noticeable in heavy nuclei even for high levels, and decreases less rapidly with n as the finite-size effect.

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PION IC AT OMS 473 The effect can be described to first order in 0: by a potential Vp(r) which

for a spherical charge distribution is (32)

20: f r' V per) = - e2 - ;.;.. per') - [ZI( I r - r'l ) - ZI(r + r') Jdr' 8a.

3 r with

ZI(r) = f 00 e-(2/"1.e)re (1 + _�)v'¥"=1 dt 1 2�2 �3

This potential produces an energy shift ""J l fn I 2VpdT.

8b.

The integral ZI(r) can be approximated by a series in r/'!I.. which by taking into account only the first term ""In (r /'!I..) leads to the widely used formula (23) given by Ford & Wills (33) . However, for light nuclei and the higher excited states of heavier nuclei of interest in pionic atoms, this approxima­tion is not sufficient and one has to consider terms of at least the order (r /'!I..)3 as discussed by Fricke (32) who has also shown that effects of higher order in 0: contribute only a few percent to the total vacuum polarization. The agreement between the quantum electrodynamic calculation of the vacuum polarization and its experimental value has been found for muonic atoms to be better than 1 % (34). Data for the vacuum polarization correc­tion are displayed in Table 2, col. 6.

The screening of the atomic electrons results in a negative energy shift reducing the binding energy. Since even for highly excited mesonic states the mesons are far inside the electron orbits, this shift is very small and, for example for the muonic Sg-4f transition in Pb, amounts to only ",,70 eV. Such small corrections are only important for precision measurements of mesonic masses or electromagnetic corrections where one utilizes higher levels to avoid unknown effects caused by the nucleus. (See Table 1 .)

Mesonic cascade.-The energy levels of mesonic atoms cannot be de­termined directly. One observes the transitions between levels where the energy difference is released as electromagnetic energy eX rays) or is trans­ferred to an electron (Auger effect) . So far, only the de-excitation of mesonic atoms has been observed, excitation being very difficult because only a small number of atoms can be produced, and the power requirements are prohibi­tive in most practical cases. Therefore, the cascading-down of the meson from the highly excited states in which it is created to the ground state is the natural source of X rays and should be discussed while nuclear absorp­tion is neglected.

The transition probabilities for the two processes by which the de-excita­tion of a mesonic atom predominantly occurs are

9.

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474 BACKENSTOSS

and

Wx 10.

for Auger and electric dipole transitions, respectively. Here rep. is the distance between the meson and the electron, 1/Ip.i, 1/I/1/Iei, 1/1/ are meson and electron wavefunctions in the initial and final states and AE is the transition energy. In both c;ases, the selection rule Al = ± 1 holds, while Al = -1 is favored. In Equation 10 the factor (AE)3 strongly enhances transitions with An as large as possible. Since t sn -1, a transition to the smallest n compatible with At = -1 can take place. Thus electric dipole transitions tend to populate circular orbits (n,t = n - 1). The Auger transition rate is more difficult to discuss since it depends on the matrix elements of Equation 9. These are greatest if there is a good overlap between electron and meson wavefunc­tions. We would expect the Auger transition probability to be large when the meson orbits are comparable with electron orbits, that is, for large nand small binding energies at the upper end of the cascade. Furthermore, it implies a preference for An as small as possible.

However, if one assumes that the Auger effect will take place with elec­trons having the same Bohr orbital radius as the meson, the smallest An is given by the corresponding energy release required to eject the electron into the continuum. For example, a pion with the same Bohr radius as an elec­tron with ne=4 has n,,= (vmr/me)ne=66. To eject this electron into an unbound state, I An .. I � 29, whereas At = ± 1. The orbit n" = 37 thus reached corresponds to ne"" 2, and to eject an ne = 2 electron I Lln" I � 13,At = ± 1. If now an ne = 1 electron is ejected I An" I � 10, At = ± 1. The following steps are I An". I � 3, At = ± 1 and I An" I ;::: 2, At = ± 1, provided ejection of Is elec­trons takes place. Only if n,,:::; 9 will Auger transitions of the type Lln = -1, At = -1 dominate. This means that also Auger transitions occurring be­tween high n states tend to increase the population of the high 1 states, i.e. the more circular orbits. In spite of the rather complex cascade processes, the X-ray transitions and their intensities carry information about the initial distribution of the states into which capture took place and therefore about the capture process. Beyond this basic interest, the understanding of the cascade and the population of the levels is of considerable practical impor­tance in view of the interpretation of intensity measurements of pionic atoms (Section 2.2), especially the strong enhancement of the circular orbits (n,l=n-l) which leads to a valuable simplification. Once a meson is trapped in a circular orbit it must stay in a circular orbit, undergoing transi­tions of type An = -1, At = -1 until it reaches the 1s state.

Cascade calculations for muonic atoms, i.e. without absorption, assuming a distribution over the l substates for an initial n (usually n = 14), have been performed by Eisenberg & Kessler (35) and Hufner (36).

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PIONIC ATOMS 475

2.2 SPECIFIC PROPERTIES OF 7T-MESONIC ATOMS

Pions are subject to the strong interaction which takes place between the pion and the nucleus, in addition to the electromagnetic interaction. As mentioned before, this interaction, because of its short range, plays an important role only for mesonic states relatively close to the nucleus. Even for the heaviest nuclei the effects have not been observed for n >6. There­fore, investigations not related to the strong interaction (for example, mea­surement of the pion mass, study of the pionic cascade, and study of the influence of chemical or condensed-state properties) must be performed on transitions that are far from the nuclear range. On the contrary, the study of nuclear effects caused by the strong interactions is best carried out in transitions leading to levels that are as low as possible. Since muons interact only weakly or electromagnetically with the nucleus, it is rather obvious (but nevertheless it should be remarked) that to investigate the deviations of the pionic atoms from the purely electromagnetic behavior, a comparison with muonic atoms is the most straightforward method and a good under­standing of the muonic atom is often vital.

The deviations from the purely electromagnetic level scheme result mainly from two effects. Firstly, a level shift relative to the level energy determined by the electromagnetic interaction occurs. Secondly, the pions can be absorbed by the nucleons. This causes a broadening of the levels out of which absorption takes place, which may be measured in many cases from the natural Lorentzian linewidth l' of a pionic X-ray transition. Since the range of the strong interaction is short, there will be only one level that is strongly affected in a particular atom, the effect in the next higher level (n+1, 1+1) being about three orders of magnitude smaller. This implies that with present techniques neither energy shifts nor Iineshapes can be measured for more than one X-ray transition in one particular pionic atom, and that the energy shifts and linewidths of an X-ray transition are almost entirely caused by the lower level of a transition. However, the width of the upper level of a transition, although three orders of magnitude smaller, may also be determined by a measurement of the intensity of a transition which de­pends on the ratio of the probability for pion absorption Wa =I'a/Ii to the transition probability W", for electric dipole radiation. Therefore, the main emphasis of the experimental work concerning the strong pion-nucleus interaction is on the measurement of exact energies, Iineshapes, and intensi­ties.

Level shifts.-The interpretation of the energy levels and their shift due to the strong interaction is rather complex. The energies of the mesonic levels can be written approximately as a sum

1 1.

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476 BACKENST OSS

EKG is the binding energy of the pion in a nucleus (Z, A) with a point charge as given in Equation 2 which represents a bound-state eigenvalue of the Klein-Gordon equation

1 2a .

where Jj is the reduced pion mass and Vc the Coulomb potential. In all practical cases, EKG is by far the dominating term. The energy shift EFs due to the finite size of the nuclear charge is obtained by replacing Vo in Equation 12a by the potential VFS produced by the extended charge dis­tribution; EFS is always <0. Evp is the correction due to the vacuu m polari­zation (see Section 2.1) and is always >0. EN is the energy shift due to the strong interaction. I t can be described by a strong-interaction potential VN and one obtains from Equation 12a

A crude estimate of the behavior of EN as a function of Z is useful and may be obtained by perturbation theory, where

If one assumes a square-well potential up to the nuclear radius R and Roc A 1/3 with A oc Z, one obtains

13.

Therefore the strong-interaction shift EN is a strongly varying function of Z, in particular for larger l values. Experimentally from Equation 11 EN is determined as the difference

14. of the measured energy value E and the value Eth calculated with all electro­magnetic effects included (Eth =EKG+EFs+Evp). This means, of course, that one must be able to handle all electromagnetic effects before an under­standing of the strong interaction can be attempted. In practice, these quantities are fortunately smaller than EN (see Table 2) and their errors are small compared with the experimental error. This approach is open to criticism, since the contributions on the right-hand side of Equation 11 are not rigorously additive. The strong-interaction shift EN is accompanied by a change in the pion wavefunction which also influences Evp and EFs• A rigorous comparison can only be made between the experimental value and the value calculated by taking all interactions into account simultaneously. But this can be done only if such a theory exists (see Section 4), which was

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PIONIC ATOMS 477

not the case when the first measurements were made. Furthermore, the breaking-down into various contributions shows more clearly the magnitude of the effects involved.

Level widths.-The interpretation of the natural linewidth of a level r is straightforward. According to the relation

r = fiW r(eV) = 6.58 X 1O-16W (sec-I)

Since all transition probabilities of the electromagnetic type re1«rex, where rex denotes the instrumental resolution width of the detecting system, they can be neglected and one measures only the width connected with the strong interaction, i.e. the absorption of the pion out of the lower level of a transi­tion. The width r is also a strongly varying function of Z. The same argu­ments as given above for the energy shifts lead to the same Z dependence as in Equation 10

rCn,l) ex: Z4/3(21+3)

There is one reason for caution. Mesonic X-ray lines can also be split by hyperfine effects caused by electric or magnetic moments. In particular, nuclei with large electric quadrupole moments may produce line splittings for transitions in which a broadening due to the strong interaction is already observed. In most cases, this splitting cannot be resolved experimentally and appears as line broadening. The hyperfine structure of the level can, how­ever, be calculated in perturbation theory (29, 37)

llE = !e2Q(cf>n.l(r) I j(r) I cf>n.l(r» X (ang momentum factors) 15.

where the pionic matrix element may be written as

if one assumes that the quadrupole distribution is concentrated at the sur­face of the nucleus with radius R. This expression replaces the expectation value (1/r3) for a nuclear point charge and leads to a reduction of the energy shift llE of less than 10%. llE must be taken into account in determining the true width r of a line.

Inten sitie s.-The most striking difference observed between muonic and pionic X-ray spectra appears in the intensity ratios of the various X-ray series. Provided the energy is high enough not to be completely dominated by Auger transitions, then all the muonic X-ray series from transitions leading to a high n (e.g. n � 10 for a heavy element, correspondingly smaller for light elements) to the transitions into the ground state (K series) are

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478 BACKENSTOSS

present. In pionic X-ray spectra, however, the yield of a certain X-ray transition decreases with increasing Z, and finally the transition disappears; all succeeding transitions are not present. The K series (2p -1s, 3p -1s ... ) or its dominant line (2p-1s) is observed only for Z �11, the dominant line of the L series (3d-2p) for Z �30, the dominant line of the M series (4f -3d) for Z �S9. The reaSOn is that the pion absorption rate Wa from a given level increases more strongly with Z than does the electric dipole transition rate Wx. Wx is proportional to ZO, whereas Wa is, for the smallest 1 where it can compete with X-ray emission (i.e. 1= 1 according to Equation 13) , pro­portional to Z6.7. Beyond a certain Z the absorption dominates so strongly that the X-ray transition can no longer be observed.

An exact measurement of the transition intensities can be used to deter­mine absorption rates from the initial level of the transition. The method is based upon a comparison of Wa with the easily calculable transition rate Wx.

The yield Y of an observed X-ray transition per mesonic atom formed, be­tween an initial and filial state, is given by

Y(i;f) = Y(n,l;n', 1') = Wx(n, 1; n', l')P(n, 1)

16. Wx(n,l;n',l') + WA(n,l) + Wa(n,l)

where WA(n,l). denotes the transition rate for Auger transition from the initial level (n ,l) . If we restrict ourselves to X-ray transition between circu­lar orbits, there also exists only one possibility for the Auger transition, tln = -1, tll = -1. P(n,l) is the population of the initial level. Hence, one obtains the absorption rate from the level (n,t):

Wa(n,l) = r/ft = Wx(n, n - 1; n - 1, n - 2)

·{[P(n,n -1)/Y(n,n -lin -l,n - 2)] - l } 17.

- WA(n,n - l;n - 1,n - 2)

provided the X-ray yield Y is measured and the population P is known. The best way to determine P is, of course, to measure the X-ray transitions which populate the level, provided filling of the level takes place predominantly by X-ray transitions:

� � P(n,l) = L Yen + i, 1 + 1; n, 1) + L Yen + i, 1 - 1; n, 1) 18.

i=l i=l

The transition energies of the corresponding members of the first and second sums are very similar and are measured as one experimental X-ray line. It is not always possible to measure these yields with sufficient accuracy, and one has to resort to more indirect methods, as described in Section 5.2. The quantity WA is in most cases small, and can be neglected. The quantity Wx

can be calculated according to Equation 10. However, it is apparent that the strong interaction influences the radiative transition probability Wx. As we have seen above, the transition energy tlE is affected by the strong interac­tion. Also the pionic wavefunctions are deformed, as will be discussed later

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PIONIC ATOMS 479

(Section 4), and this causes a change of the matrix element in Equation 10. The two effects cancel to a large extent. Nevertheless, Wx can be altered by the strong interaction up to about 10% (88) . Therefore the inclusion of this effect is required for an accurate determination of Wa. Numerical examples for these changes are shown in Table 5.

This method works best if X-ray transition rates and absorption rates are of the same order of magnitude. Therefore, absorption rates Wa � 1014-1016 seCI can be determined corresponding to r= 10-1 to 10 eV. Since lineshape measurements (Section 3) are limited to r> 10 eV, this method very for­tunately complements the direct width determination, allowing the mea­surement of r over more than six orders of magnitude.

3. EXPERIMENTAL TECHNIQUES The production of mesonic atoms is the easiest part of the experiment,

and the method is quite conventional. Since the mesons must be stopped in the target material, it is advantageous to use low-energy pions to avoid slowing-down material , in which pions are lost by interactions with the nuclei. However, the maximum number of negative pions available from proton bombardment of an internal target in a synchrocyclotron is found at higher energies. As a compromise, at the synchrocyc1otrons so far exclusively used for studies on pionic atoms, a pion beam of ,....,200 M eV /e has been used; fluxes of about 106/sec can then be achieved. This may be different in future work on linear accelerators or meson factories where an external target for pion production is used and therefore low-momentum pions will be used from the beginning. Although the pion beam is momentum-analyzed by a bending magnet, it is inevitable that the beam is contaminated by muons from 7r decay. To a large extent the muons can be rejected by the different range of the muons and pions. In the conventional method, using a scintillation counter telescope as indicated in Fig. 1 , the coincidence ( 1 , 2, 3, 4) signals a charged particle stopped in the target. The range curves thus obtained have a typical width of about 5 g/cm2 and contain about 5-10 X 104 stopped pions. A cross-shaped target geometry allows the simultaneous use of two detectors and leads to a reduction of the self-absorption in the target. Counters 5 pre­vent scattered particles from being registered in the gamma detectors (Ge) . A time resolution of a few nanoseconds can easily be attained by such a coincidence.

The majority of the data have been obtained using a solid state detector to measure the X-ray energies. For energies between about 8 to 50 keV a Si(Li) detector is advantageous, whercas for larger energies a Ge(Li) de­tector is to be preferred. Since most of the pionic X-ray lines of interest are in the energy region of a few hundred keV, the depth of the detector is less important than its surface. The latter determines the solid angle, which should be as large as possible in view of the weak extended source supplied by the stopped mesons. Since the capacity of the detector, which essentially determines the signal-to-noise behavior of the detector and therefore its

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480 BACKENST OSS

.) lEADING EDGE OA ZERO CIIOSSING 00 �st FRACTION DIsc.

FIGURE 1 . Typical counter arrangement and basic electronic circuits for measurements of mesonic X rays.

energy resolution, is proportional to the surface, its size is limited. Therefore it is advantageous to measure with more than one detector simultaneously. A good energy resolution of the detector is essential for precise energy measurements, and even more for measurements of the natural linewidth, in particular for narrow widths. Even for intensity measurements a good resolution is required, since in pionic spectra very frequently nuclear'Y lines occur which appear with the instrumental linewidth and which can be sepa­rated better if this is small.

With low noise charge-sensitive preamplifiers utilizing in the first stage a Ge-FET at liquid nitrogen temperature, placed inside the vacuum next to the detector, and detector sizes of a few cm2, an FWHM of about 1 keY in the 100 keY region can now be reached. After proper amplification by a linear amplifier, the pulses are measured by an analog-to-digital converter CADe). The information may be stored in the memory of a multichannel analyzer or an incremental magnetic tape, or may be immediately processed by a com­puter on line. Under favorable conditions without any limitation in counting statistics, X-ray energies could be measured to about ± 20 eV and natural Iinewidths larger than 100 eV.

The precise measurement of the energy of an X-ray event necessitates a certain time which is of the order of microseconds, not counting the time needed by the ADC. On the other hand, to select events from the gamma detector coincident with a charged particle stopping in the target, a fast signal should be provided by the gamma detector. These two conflicting

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PIONIC ATOMS 481

requirements can only be satisfied by splitting off from the linear branch a fast branch as indicated in Figure 1 in which the best possible time informa­tion is extracted.

To reduce time jitter and the dependence of the timing on the pulse height, which is particularly severe for energies below 100 keY, several methods have been applied. The leading-edge method utilizes a fast nonlinear amplification, driving the pulses into saturation. The rising flank of the pulses becomes steep, and the time at which the pulse rises above a discriminator level set just above the noise is largely independent of the pulse height. The zero-crossing method makes use of the fact that bipolar pulses of very differ­ent rise time go through zero in a very narrow time interval, which then provides the time signal which, however, arrives late, necessitating delays in the other branches. Recently the constant-fraction-discriminator method has been applied in which the time at which the pulse passes through a given fraction of its peak is used for timing. In practice this method gave best results.

With these methods the length of the Ge detector coincidence pulse necessary to produce an efficient coincidence with the beam telescope can be reduced to about 20 nsec. This is important for reduction of the accidental background where one has to bear in mind that the synchrocydotrons are pulsed machines with a duty cycle of less than 50% under the most favorable running conditions.

A considerably higher energy resolution than that of a Ge detector with t:.E/E",1o-2 in the 100 keY region can be obtained with a crystal diffraction spectrometer, whose luminosity can be increased with a crystal bent to a cylinder of a radius equal to the distance between source and crystal (Du­mont) . For a quartz crystal , a resolution of t:.E = 1 .6 X lo-6E2 has been reached (38), which corresponds to t:.E/E"",l(Ja in the 100 keY region. On the other hand, the detection efficiency is very low, and at its maximum near 50 keY it reaches only 2.5 X 10-6, falling off rapidly on either side. Therefore, this method is applicable only in special cases where precision measurements are required (see Section 5. 1) . For exact energy measurements of single lines where a high resolution is not really demanding, the poorer resolution of a Ge detector is justly compensated by the much better statistics attainable where the line position can be determined to about 1 % of the instrumental width, which is not possible for the poor statistics in a diffraction spectrum. However, in view of forthcoming beam intensities produced by meson factories, the low detection efficiency may be compensated and hyperfine structures may be resolved where a high-resolution detector is a necessity.

The calibration of the detecting system is of great importance with respect to the energy dependence as well as to the lineshape reproduced. As calibration lines, either'Y lines from radioactive sources or muonic X rays have been used. Muonic lines produced simultaneously with the pionic spectrum have several decisive advantages, if they have energies close to the lines to be measured. Their "source" is distributed over the target in a way

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482 BACKENSTOSS

similar to that for the pionic X rays; geometrical and absorption corrections are nearly the same. The time distribution of the events, and therefore the radiation load to which the detector is submitted, is the same for both types of X rays. Properly selected muonic X-ray transitions can be calculated to any desired accuracy. The problem is only that suitable muonic X-ray transitions cannot always be found. Line energies with large finite size and nuclear polarization corrections cannot be calculated adequately; nor can transitions be used whose fine structure doublet cannot be resolved, since the intensity ratio between the two components is not sufficiently well known. Therefore, radioactive 'Y lines cannot be dismissed as calibration sources. But for precision energy and lineshape calibration, precautions have to be taken to avoid effects from differences in rate and direction of irradia­tion between X-ray and calibration measurements. Since the shape of spectra can also have an influence, e.g. large overloading pulses, as may occur in the pion beam, the best way is a simultaneous calibration. The radioactive sources are always present. To make sure that calibration pulses are mea­sured only during the time the synchrocyclotron pulses are present, only such

'Y-ray events are registered as are in accidental coincidence with an incoming beam particle and provide a routing pulse which is used to distinguish X-ray from 'Y-ray spectra. With the length of the pulses triggered by the beam particles and the 'Y-ray rate, the frequency of these events can be properly adjusted. Since X-ray events are usually scarce, an "inhibit" pulse gives priority to them (Figure 1 ) . I n Figure 2, for example, the pionic spectra of 160 and 180 are given, in which also muonic "calibration lines" are seen.

From an inspection of Figure 2 it becomes clear that the evaluation of spectra of this kind is a major task. To deduce not only line positions but also lineshapes, a detailed comparison with calibration lines is necessary. Mostly, different shapes of the background underneath the line must be taken into account. This can hardly be achieved without the aid of a com­puter which determines a best fit through the experimental points satisfying the conditions imposed by the properties of the calibration lines. Broadened pionic lines have a Lorentzian shape folded with the instrumental resolution of essentially Gaussian type

(x) exp _ (_X -G_X

'Y --------- dx'

1 + (X' -XO)2

r/2

where G denotes the width of the Gaussian Lineshape and r the Lorentzian width of the line centered around Xo. A best-fit program determines the intensity parameter A, the width r, and the position xv, whereas the param­eter G is fed in from the information from calibration spectra. The back-

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6

5

3

T.

7

6'

5

4

T 140

. ..

y

!

.. - .' ," :..&.- ,,"

PIONIC ATOMS

1J.3p- 1s I

Tt2p- ls lr= 7.56 keY

• 1l4p -1s

I y

I 115p-ls

. I

_._._._._._.-'-'- ......

IJ.6p-ls

I 0 ......... --. ........

y 1J.3p-1s

I !

1l4p-1s

y I 115p-ls

... I

..

--:' • ...:..�:.!. •. :.:.:.�.-.-.- .-.-._.-.-

'. ". I,I6p-1s '. .I ' .. ' -..

150 160 170 180 k4tY FIGURE ·Z. X-ray spectra of "0 and 1·0 in the energy region

of the 1!"-Zp->ls line (from 17).

483

T

T

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484 BACKENSTQSS

ground on the left and right of the line is also fitted by the computer and extrapolated underneath the line. Obviously the precisions to which the line parameters can be determined depend on many things. Besides the obvious influence of the peak-to-background ratio and counting statistics, the avail­ability of nearby calibration lines is important. For high precision, linearity of the system cannot be assumed. A polynomial fit through many calibration lines should be adapted. If r«G, the determination of r becomes less certain and also that of large r where the exact shape of the background is important.

4. THEORETICAL APPROACH TO PIONIC ATOMS Experimental data on pionic atoms are so accurate and ample that a

detailed comparison with a theoretical concept may reveal new knowledge about the pion-nucleus interaction and certain aspects of the structure of nuclei. Consequently, we restrict ourselves to the discussion of the specific effects related to the strong pion-nucleus interaction. One has to start with the basic interactions that pions can have with the nuclei, and then to discuss the modifications which occur, because the pion interacts not with a single nucleon but with nuclear matter or with the specific structure of a nucleus. Secondly, the effects on the mesonic energy levels caused by those processes must be calculated. Hence it is evident that several steps are in­volved in connecting the elementary processes with the observable quantities of pionic atoms. This situation, on the other hand, leaves us with the possi­bility of a variety of approaches. In particular, at different points of the chain of arguments, phenomenological theories have been introduced. Accordingly, comparison between theory and experiment can be made at different points and for different quantities; this depends to a great extent on the aim one has in mind when interpreting data on pionic atoms. In this brief article it is impossible to do justice to the different points of view, and we attempt to indicate only the most frequently used models and those calculations that are the most advanced insofar as they allow the most detailed comparison between theory and experiment.

The most common description of the pion-nucleus interaction makes use of an optical potential

Vopt = Re V + i 1m V in close analogy to the scattering of light by a homogeneous optical medium with a complex index of refraction (39) . First-order perturbation treatment is insufficient and solutions of the Schrodinger equation for a real square-well potential have been given by an approximation procedure (39a, 43) and by an exact solution (39b) , respectively. In the more microscopic models one tries to connect the pion-nucleus interaction with the elementary pion-nucleon processes where one applies the impulse approximation (40) . Since elastic s­and p-wave pion-nucleus scatterings which lead to a repulsive and attractive potential, respectively, are important, the simple local optical potential was extended by Kisslinger (41) to a velocity-dependent potential. The different

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PI ONle ATOMS 485

aspects have been combined by Ericson & Ericson (42 , 43) using a multiple­scattering theory for the pions to derive the constants of the Kisslinger potential from the elementary processes.

The elementary processes of interest are :

the elastic scattering of pions on nucleons

7r- + N � 7r- + N 19.

and the pion absorption

7r- + N + N � N + N 20.

The elastic pion-nucleon scattering is dominated by the (J = !, T = !)

resonance in the 7rN system at 180 MeV. This means that even at low ener­gies, the p-wave scattering plays an important role beside the s-wave scatter­ing. This process is discussed in detail by Hamilton & Woolcock (44) and by Moorhouse (45) .

The absorption process (20) is characterized by the fact that in nuclear matter only absorption on two nucleons takes place. The pion in a bound state essentially has zero momentum, whereas the pion mass liberated in the absorption process amounts to 140 MeV. Conservation of energy and momen­tum demands a momentum p for the single nucleon on which absorption takes place

where M and m,.. are nucleon and pion mass, respectively, Bp is the separa­tion energy of the last proton, and Eex the energy used to excite the nucleus. This means that the nucleon which absorbs a pion needs a Fermi momentum of about 500 MeV/c. Since the Fermi momentum in nuclei is typically lower (46) , the absorption on a single nucleon is improbable. However, if the pion is absorbed at a correlated nucleon pair as suggested by Brueckner, Serber & Watson (47) , the final state can easily have zero momentum when the two nucleons share the liberated energy and leave with opposite momentum. In this case the two nucleons carry away a momentum p = 370 MeV/c, but now relative to their center of mass, which has zero momentum. Such a momentum corresponds to a distance r = n/p of about 0.5 F between the two nucleons. This is much less than the average distance of nucleons in a nucleus of ",2 F. Therefore, the study of the pion absorption in nuclei conveyed by pionic atoms is sensitive to the high-momentum components and to short-range correlations in nuclei. There exists a large amount of work on the pion interaction in the very light nuclei reviewed by ZupanCic (48) . The nucleon-pair model for pion absorption is particularly supported by the work of many authors (49-55) . Calculations (56-62) have been per­formed, but cannot be discussed here.

Elastic scattering and energy shift.-The connection between the elastic

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486 BACKENSTOSS

pion-nucleon scattering in a certain angular momentum state and the energy shift of the corresponding energy level was discussed in Section 2 . 1 .

Since the pion orbits are large compared with the nuclear radius, for a pion in a relative s state to the nucleus, the s-wave interaction between the pion and the nucleon is dominant. Similarly, the 7rN p-wave interaction is dominant if the pion is in a p state relative to the nucleus, provided the latter is sufficiently small. A detailed investigation of the balance between s­and p-wave interaction in the 7r-atomic p state is given in (62a) . I t was shown by Partenski & Ericson (27) that the contributions of higher 1 states can be neglected.

Therefore, following Ericson & Ericson (43) the scattering amplitude of the pion on the ith nucleon can be written in the impulse approximation, which is expected to be very good for low-energy pions:

Here t is the isospin of the pion and 't that of the nucleon, and we have neglected a term (d · k Xk') proportional to the spin of the nucleon. The con­stants in Equation 21 can be expressed in terms of the conventional s-wave scattering lengths a2T, where T is the isospin of the 7rN system, and in terms of the p-wave scattering volumes OliJ!l',2J, where J is the spin of the 7rN system. With the scattering lengths given by Hamilton & Woolcock (44) one obtains in units of the pion Compton wavelength i\ ..

ho' = Hal + 20'3) = - 0.0017 (�".

b1' = 1 (0'3 - 0'1) = - O.086(l\ .. )

Co' = 1 (40'33 + 20'13 + 20'31 + au) = O.208(�,,) 3

Cl' = t(2a33 - 20'13 + an - all) = 0.184(i\ • .) 3

22 .

The scattering amplitude for the pion on the nucleus is then obtained to a first approximation by taking the coherent sum of the 7r N scattering lengths. In this way Deser et al ( 14) have derived an expression for the nuclear po­tential, considering only the first two terms in Equation 2 1 , i .e. s-wave interaction, and obtained :

27rli2 [Z A - Z ] VN(r) = - -- - ap + --- an per)

J.tr A A 23.

This produces the nuclear shift in the 1s state. Here the s-wave 7r-P scatter­ing length is ap = l(2al +aa) , the 7r-n scattering length is an =aa, and per) is the nucleon distribution.

I t is interesting that an = bo' +b1' is nearly equal to ap = bo' -bt', but with opposite sign, which generates an s-level shift for nuclei with T = O (A = 2Z) , which is much smaller than the observed shift. This is caused by the almost complete cancellation of the isospin singlet and triplet scattering lengths in

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PIONIC ATOMS 487

the term bo'. Therefore, the approximation of Equation 23 is insufficient and higher-order effects must be considered. In the case of low-energy multiple scattering, it is possible to scatter more than once on the same particle. This gives rise to a modification of the local field associated with the granular structure of nuclear matter (43, 44a) . This can be described by an effective field cPeff of the incident wave :

24.

which replaces the incident wave cPo Consequently the s-wave scattering length is substituted by

a.rr = a - a2 <�) r Dorr

In practice the linear term is very small. The squared term dominates and leads to a repulsive potential in place of Equation 23. Since (1/r)oorr is larger for long-range correlations, one sees mainly the effect of these (Pauli correla­tions) . The p-wave scattering is also important and depends on the pion momentum k as seen in the third term in Equation 21. The pion momentum varies through the nuclear surface. With the identity

k = - i(veikr)e-ikr

terms of the type k · k'p(r) are transformed to an expression of the type Vp(r)V in the potential obtained by inversion of Equation 4 :

27r1i2f VCr) = - -' f(O)e-i (k-k/)rd3(k - k')

m"

In this way one obtains a nuclear potential of the form (43, 63)

27r1i2 V N(r) = - - { bop(r) + bi [Pn(r) - pp(r) ]

p."

25.

26.

where per) =Pn(r) +pp(r) is the sum of the neutron density Pn and the proton density pp. The effective parameters bo, bl, Co, C1 are closely related with the scattering lengths bo', bI', co', Cl' discussed above. But they include the cor­relation effects mcntioned above. Furthermore they contain contributions due to the nuclear binding and the Fermi motion of the scatterers as well as the real parts of the two-nucleon scattering amplitudes discussed below. The short-range correlations also give rise to the Lorentz-Lorenz effect (43) in the p-wave interaction exhibited by the factor f:

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488 BACKENSTOSS

27.

where � = 0 means no correlations and � = 1 very short-range anticorrelations between nucleons. Expression 27 is independent of the range of the correla­tions. Therefore the short-range correlations which exist between all nucleons are most important.

Note that the p-wave 1r-P scattering volume (co' - Cl') is 14 times smaller than the 1r-n scattering volume (co' +Cl') , which means that there is practi­cally no 1r-P interaction in the p state at low energy.

The potential (24) consists of a local part (first two terms) and a velocity­dependent part similar to that of Kisslinger (41) and leaves the possibility of constructing a repulsive shift for the s level and an attractive shift for the l evels Z>O. The potential operator VN operates on the pion wavefunction 1/1. 1 t therefore becomes clear that the local part dominates for an s wave where the amplitude at the nucleus is large and the gradient is small. The contrary is true for P. d, and f waves where the amplitude is small and the gradient may be larger at the nuclear surface. Hence the parametrization with the Kisslinger potential is very advantageous, since each parameter is deter­mined essentially by one observable effect. as indicated in Table 6.

Pion absorption.-If the nuclear potential VN should also describe the pion absorption discussed above. an imaginary part has to be added to it. It is sensible to introduce an imaginary part in the local part ( 1m Eo) as well as in the gradient part ( Im Co) of the potential.

These parameters appear as the imaginary parts of the constants of a phenomenological two-nucleon scattering amplitude which is constructed in analogy to Equation 2 1 .

h = S(ri - rj)li [r - Hri + rj) ]

. { Eo + Cok' · k + (El + Clk' · k) [(t · �i) + (hj) (hi) ] } 28.

These complex constants Eo and Co are linear combinations of the various amplitudes of the angular momentum and isospin states of the (1r2N) system. The main contribution comes from nucleon pairs with the quantum number of the deuteron. The s-wave absorption rate is then given by Im ((3u) I lf>d(O) 1 2, where If>d(O) is the deuteron wavefunetion at the origin and {311 is the pion s-wave interaction amplitude for the pion-deuteron system. The quantities {3i,t and similarly the quantities "(i,t denoting the p-wave interaction ampli­tude can be determined from the reaction

2N +=! 1r + 2N

The short-range correlation in the deuteron system is contained in the con­stants {3i,to giving them the character of phenomenological constants. Assum-

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PIONIC ATOMS 489

ing that the short-range behavior of the absorption is the same for quasi­deuterons in the nucleus as it is for the deuteron, we find the nuclear absorp­tion rate proportional to

29.

where the factor i accounts for the fact that only i of all np pairs have the deuteronic quantum numbers. With Pn =pp = !p, one obtains from Equation 29 an s-wave absorption proportional to 1m Bop2, and similarly for the p­wave absorption proportional to 1 m Cop2.

The complete nuclear potential used to compare experimental with theoretical data for physical nuclei is :

V N(r) = - 21T 1i2 {bop(r) + b1 [Pn(r) - pp(r) ] + i 1m Bop2(r) 2fJ.

with

+ V + (cop(r) + Cl [Pn(r) - pp(r) ] + i 1m CoP2(r») V} 30.

This potential VN(r) is used to describe the strong-interaction effects of the pion-nucleus system. To calculate the energy levels and absorption rates, the complex nonlocal potential VN(r) must be introduced into the Klein­Gordon equation 12b. Krell & Ericson (63) solve this problem by a substitu­tion

which, when the second line of Equation 30 is abbreviated as V'a(r)V', trans­forms the velocity-dependent part of the potential into a local potential of dipole character at the nuclear surface. They also describe in detail a method of integration for the bound-state eigenvalues of the complex interaction (64) . The wavefunctions taken from (63) and shown in Figure 3 indicate clearly the effects of the various parts of the potential (Equation 30) , es­sentially a repulsion in the 1s pion and an attraction of the 2p pion. It be­comes obvious from this that, for example, the electric dipole matrix element between those two states (Equation 10) will be increased. The theoretical values in Tables 3 and 4 have been obtained by considering the parameters of the potential (30) as free parameters which have been adjusted to the experimental data and taking for per) the charge distribution of the nuclei (65).

As we will sec in Sec. 6, the overall agreement between the values calcu­lated in this way and the experimental data is remarkably good. However, since this theoretical approach is based on the scattering that pions undergo

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490

2

'10-5

2

2

PION

1<1> 12 I :

Ifmr3 IT m : Ill : '1 :Ill :

� / / �// � I I , I I I I I I , /

BACKENSTOSS

PION DENSITY IN THE ls STATE OF '�O I : without strong interacting potential IT : with fu II nuclear potential (standard) m: only bo = - 0 . 030 Ill : only Im(Bel = 0. 035 Y.: be= -0 .030 ; Im(Bol = 0. 035 :Ill: co= 0 . 24 ; Im (Co) = 0 . 1 5

-

r 3 4 5 6 7 8 9 10 1'1 12 [f�

DENSITY I N THE 2 p STAT E OF �gCQ without strong interacting potential with full nuclear potential ( standard ) only Co = 0 .24 only Im(Co) = 0 . 1 5 bo = - O 030 ;' I m ( Bo ) = 0 0 3 5 coo 0 24 ; 1 m (Co l = 0 . 1 5

� ' , , -,

--

r 1 1 12 [fml

FIGURE 3. Pion densities with different parts of the optical potential effective (from 63). Top: Is state of 160 ; bottom: 2p state of (OCa.

essentially in nuclear matter, one cannot expect to see the effects originating from individual properties of the nuclei.

Here a microscopic description by Chung, Danos & Huber (66) proved successful. The absorption rates of the pions from the 1s and the 2p states have been calculated on the basis of an extended shell-model treatment. The classical shell model , being an independent-particle model, cannot de­scribe the pion absorption which must take place on two nucleons. Therefore,

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PIONIC ATOMS

following a suggestion of J astrow (67) a two-body correlation factor

fq(r) = 1 - jo(qr)

49 1

3 1 .

i s introduced, which describes the short-range correlations. I t is a function of the characteristic momentum q exchanged between the two nucleons in­volved in the absorption process, and plays the role of a phenomenological parameter. At a momentum in the region of about 300 MeV Ie, the calculated pion absorption rate in the 1s and 2p states reaches about one half of the observed values.

5. RESULTS ON PIONIC ATOMS According to our earlier discussion of the properties of pionic atoms, we

will first consider the results obtained from investigations of pionic atoms where only the purely electromagnetic properties are relevant and the speci­fic strong pion-nucleon interaction does not play an essential role. In the subsequent section the results concerning the strong interaction will be dis­cussed.

5.1 RESULTS DERIVED FROM THE ELECTROMAGNETIC PROPERTIES The most significant result from measurements in pionic atoms is the

best determination so far of the mass of the negative 7r meson. Also to be discussed in this section is the study of the pionic cascades in atoms of pure elements or chemical compounds which is influenced by the chemical and physical properties of the material in which the meson is stopped.

'Ir-Mass.-As can be seen from Equation 2 the energy of any X-ray transi­tion is proportional to the meson mass. The accuracy Am 1m with which the meson mass can be obtained is equal to the accuracy AEIE with which the transition energy can be measured, provided all corrections as discussed in Sec. 2 . 1 are known sufficiently well. To keep these small, levels are measured corresponding to large Bohr orbits where the nucleus acts essentially as a distant pointIike source. At the same time screening due to the atomic elec­trons increases. In principle, the screening correction can be calculated quite accurately by using a self-consistent potential which takes into account the nucleus, the electrons, and the meson (68). However, it is not clear in all cases how many atomic electrons are present when the mesonic transition takes place, although a detailed study of the mesonic cascade may give an answer. Thus it is important to select an X-ray transition in such a way that the corrections are small enough so that the uncertainties originating from them are small compared with the measuring error AEIE. This latter quan­tity should be minimized, and therefore the transition to be chosen depends also on the properties of the X-ray detection system. The best measurement of the pion mass has been performed on the 4f-3d transitions in 20Ca and 22Ti with a bent-crystal diffraction spectrometer (69, 70) reaching nearly AEI E = 10-4 and yielding for the 7r mass

(139.577 ± 0.014) MeV

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492 BACKENSTOSS

TABLE 1. Characteristic data of the pionic 4f-3d transitions in Ca and Ti ( in keY )

Measured energy Vacuum polarization Strong-interaction shift Electron screening

Ca

72 .352 ± 127 ppm +0. 232 ± 0 . 003 + 0 . 002 ± 0 . 002 - 0 . 001 ± 0 . 001

Ti

87 . 651 ± 99 ppm +0 .301 ± 0 . 003 +0 . 004 ± 0 . 004 - 0 . 001 ± O . OOI

In Table 1, the characteristic data of these measurements are given. The luminosity of a crystal diffraction spectrometer decreases rapidly

with increasing energy. Therefore, measurements at higher energies are less favorable. With the improvement of Ge(Li) detectors it might be possible to reach AE / E :::; 10-4• Since, however, a AE = 8 eV a t 80 ke V can at present hardly be reached, a measurement with Ge(Li) detectors should be more promising at higher energies. This implies somewhat larger corrections for the vacuum polarization and strong-interaction shifts. But their error should be sufficiently small.

Recently a Ge(Li) spectrometer has heen used to measure the pionic Sg- 4f transition in Gal to be (237. 138 ± 0.017) keY and the 6h- Sg transition in slTl to be (30 1 .730 ± 0.01S) keV. This leads to a value for the 1r mass of ( 139.SS3 ± 0.008) MeV (99, 100) . At least for the Tl measurement the experi­mental error is only slightly greater than the error for the calculated transi­tion energy. The agreement between the two energy measurements is better than the corresponding mass values indicate. The calculation of the vacuum polarization in Table 1 was performed for a point nucleus. I f the calculation is done for an extended nucleus as for the Ge(Li) data, all four values agree well within the quoted errors.

Pionic cascades in chemical compounds.-Pionic cascades have been studied for pionic atoms in the pure chemical element as well as in chemical compounds. Since the effects observed and the conclusions drawn are ex­pected to be similar to those from muonic cascades, this subject is treated here only briefly. The measurement of the ratio between the X rays of the components of a chemical compound provides a direct method of investigat­ing the relative capture rates of the meson by the components. The Fermi­Teller law (4) predicts, on the basis of simple arguments, a capture proba­bility proportional to Z. However, for some time, deviations from this law have been found for muons (7 1 ) , and experiments by Zinov et al (72) clearly showed a dependence of the X-ray intensities for the oxides which varies with the groups of the periodic system. Similar results were obtained at Berkeley (73) . A theoretical explanation was given by Au-Yang (74) , who showed that one cannot consider isolated atoms, but that the properties of the solid state in which the ions are bound plays an important role. Another

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PIONIC ATOMS 493

topic studied is that of the intensity ratios between the different X-ray series (K, L, M . . . ) and within the members of one series, e.g. (2p-41s)/(3p-41s) . These ratios depend on the initial distribution of the mesons in the 1 sub­states (35) . It has been shown (75) for muons that these intensity ratios within one series cannot be explained by assuming the same distribution for different atoms throughout the periodic system (e.g. in the region 18 <Z < 25).

For light atoms, the CERN group has found (76) that for sO the initial distribution over the 1 substates for n = 14, which fits best the K series inten­sities, deviates substantially from the distribution required for the neighbor­ing elements. Since sO was investigated in the form of water, the suspicion that an effect of the chemical composition is responsible might be justified. Measurements (77) on various atoms (C, 0, Ca, Ti) in different chemical compounds, as well as Se in two physical modifications, showed clear differ­ences in the intensity ratios and differences between muonic and pionic cas­cades for the same compounds. I t became apparent that there was a trend towards the population of noncircular orbits being enhanced with increasing electrical resistivity of the sample. Thiswould be consistent with a suppression of the Auger transitions compared with the electric dipole transitions, which could occur as a consequence of the solid state structure of an insulator. I n this field, detailed analysis o f the data and, even more, a survey o f the effects guided by the physics of the solid state are missing.

5.2 RESULTS ON THE STRONG PION-NuCLEUS INTERACTION

Here we present the experimental results obtained beyond the electro­magnetic properties due to the strong pion-nucleus interaction. The older data obtained with NaI detectors are contained in the earlier review ar­ticles (12 , 13) and will be quoted here only occasionally for comparison.

Energy shifts.-The determination of the energy shifts is based on precise measurements of the transition energies. The energies measured at various laboratories are presented in cols. 2-4 of Table 2 (Refs. 16-18, 69, 70, 78-86) . The agreement is impressive in view of the rather complex structure of the spectra, which frequently include muonic transitions and nuclear gamma lines.1 As mentioned before, the exact energy shift due to the strong interac­tion can only be determined if we know all other corrections which are, in turn, not completely independent of the strong interaction. The parameters of the strong-interaction potential available (see Section 6) are sufficiently good to determine the vacuum polarization correction given in col. 6 in presence of the strong-interaction potential V N and the finite-size correction given in col. 7 with an error small compared with the measuring error of the transition energies. From these data the strong-interaction shift EN can easily be derived and is displayed in Figures 4 to 7 .

1 Recently, 2p-ls energies on He, Li, and B e became available from an Argonne

group (90) which agree with the data in Table 2.

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494 B ACKENST OSS

TABLE 2. Energies of pionic 2p·1s, 3d.2p, 4f·3d, and 5g-4f transitions

Cols. 2 to 4 show the experimental data, col. 5 the Klein·Gordon point-nucleus value. Col. 8 gives the theoretical values obtained with the parameters of Table 6, col. 3. Cols. 6 and 7 give the calculated data for finite size and vacuum polarization correc­tion, respectively.

Ele- / Eexp(keV) EKa AEvp AEFS Eth

ment Berkeley& CERNb VirginiaC (5) (6) (7) (8) (1) (2) (3) (4)

Zp - I s TRANSITIONS (He 1 0 . 69 ± 0 . 06 1 0 . 7 5 0 . 05 - 0 . 00 1 0 . 70 'Li 2 3 . 9 ± 0 . 2 2 4 . 18 ± 0 . 06 24.492 0 . 1 1 -0 . 05 2 4 . 09 7Li 2 3 . 8 ± 0 . 2 2 4 . 06 ± 0 . 06 24. 578 0 . 1 1 -0 .04 23.95 'Be 42 . 1 ± 0 . 2 42 . 38 ± 0 . 20 42 . 32 ± 0 . 05 43 .925 0 . 2 1 - 0 . 16 42 . 1 5 lOB 64 .9 ±O . Z 65.94 ± 0 . 18 65 . 79 ± 0 . 1 1 68 . 798 0 . 36 -0 .40 65 .67 uB 64 . 5 ± 0 . 2 64 . 98 ± 0 . 18 65.00 ± 0 . 1 1 68 . 891 0 . 35 - 0 . 39 64.98 I2C 9 3 . 3 ± 0 . 5 92 . 94 ± 0 . 1 5 93 . 19 ± 0 . 12 99 . 409 0 . 54 -0 . 8 1 9 3 . 18 "N 1 2 3 . 9 ± 0 . 5 1 24 . 74 ± 0 . 1 5 1 3 5 . 698 0 . 75 - 1 . 57 124. 72 1'0 1 60 . 6 ± 0 . 7 1 59 . 9 5 ± 0 . 2 5 1 77 . 705 0 . 9 8 - 3 . 16 1 59 . 34 1'0 155.01 ± 0 . 2 5 177 . 889 0 . 93 - 3 . 30 1 55 . 18 l'F 196 . 5 ± 0 . 5 195 . 9 ± 0 . 5 225 . 572 1 . 19 - 5 . 89 193.95 "Na 277 . 2 ± 1 . 0 276 . 2 ± 1 . 0 338. 607 1 . 78 - 1 3 . 42 276 . 55 Mg 330 . 3 ± 1 . 0 403 .928 2 . 09 - 19 . 51 324 . 2 8

3 d - 2 p TRANSITIONS a d g

"AI 87 . 53 ± 0 . 07e 87 . 40 ± 0 . 10 86 . 9 2 1 0 . 39 -0 . 06 87 . 4 6 Si 101 . 58 ± 0 . 1 5 100. 862 0 . 47 -0.01 101 . 55 up 1 16 . 78 ± O . 10 1 1 5 . 882 0 . 56 - 0 . 0 1 1 16 . 7 5 S 133 . 2 ± 0 . 3 1 3 3 . 06 ± 0 . 10 131 .919 0 . 66 - 0 . 02 1 3 3 . 0 1 CI 1 50 . 55 ± 0 . 1 5 149 . 049 0 . 77 - 0 . 04 1 50 . 38 K 188 . 6 ± 0 . 3 1 88 . 7 7 ± 0 . 18 1 86 . 41 5 1 . 02 -0 . 07 188 . 43 Ca 209 . 3 �g : iOf 209 . 66 ± 0 . 18 206. 669 1 . 15 -0 . 10 209 . 14 uCa 208.94 206 . 737 1 . 14 - 0 . 1 2 209 . 00 Ti 253 .98 ± 0 . 20 250.473 1 . 48 - 0 . 20 253.81 OIV 278 . 2 ± 0 . 4 277 . 85 ± 0 . 20 273. 959 1 . 65 - 0 . 25 277 .86 Cr 302 . 5 ± 0 . 5 302 . 75 ± 0 . 2 5 298 . 485 1 . 84 - 0 . 36 302 .95 "Mn 328 . 5 ± 0 . 8 329 . 12 ± 0 . 25 324 . 1 1 7 2 . 04 - 0 . 52 328 . 83 Fe 356 . 9 ± 1 . 0 356 . 43 ± 0 . 30 350 . 794 2 . 20 - 0 . 66 356 . 32 nco 384 . 6 ± 1 . 0 384 . 74 ± 0 . 35 378 . 591 2 . 42 -0 . 84 384 . 63 tiSNi 4 1 5 . 23 ± 0 . 7 414 . 1 1 ± 0 . 48 407 .418 2 . 67 - 1 . 05 4 1 4 . 64 "NI 4 1 4 . 08 ± 0 . 51 Cu 446 . 1 ± 2 . 0 437 . 424 2 . 90 - 1 . 40 444 . 24 Zn 478 . 2 ± 3 . 0 468 . 474 3 . 1 5 - 1 . 73 475 . 88 ---

4/ -3d TRANSI'tIONS a h

Ca 72 . 352 ± 0 . OO9i 72 . 1 1 8 0 . 243 0 . 00 72 . 361 Ti 87 . 651 ± 0 . 009i 87 . 349 0 . 31 5 0 . 00 87 . 666 "Y 278 . 2 ± 0 . 3 276 . 16 1 . 42 - 0 . 0 1 278 . 02 "Nb 307 . 6 ± 0 . 3 307 . 7 ± 0 . 2m 305.45 1 . 62 - 0 . 02 307 . 7 1 Mo 323 . 2 ± 0 . 2m 320 . 66 1 . 74 - 0 . 03 323 . 16 Rh 370 . 9 ± 0 . 4 368 . 58 2 . 04 - 0 . 05 371 . 85 In 442 . 1 ± 1 . 1 442 . 9 ± 0 . 5 437 . 82 2 . 56 - 0 . 08 442 . 62 119Sn 460 . 3 ± 0 . 6 456 . 09 2 . 7 1 -0 . 13 461 . 36 1271 519 . 1 ± 1 . 1 520 . 8 ± O .S 5 1 3 . 2 2 3 . 1 7 -0 .20 520 .25 IIIlCS 560 . 5 ± 1 . 1 562 . 0 ± 1 . 5 553 . 25 3 . 50 - 0 . 30 561 . 47 La 603 . 6 ± 0 . 9 604 . 9 ± 2 . 0 594 . 84 3 . 85 - 0 . 4 1 604 . 56 Ce 626 . 1 ± 2 . 0 616 . 22 4 . 03 - 0 . 48 626 . 76 Pr 649 . 5 ± 2 . 0 648 . 1 ± 2 . 0 637 .99 4 . 23 -0 . 54 649 .48 ---

5g-4/ TRANSITIONS a h k

18lTa 453 . 1 ± 0 . 4 453.90 ± 0 . 20 453 . 4 ± 0 . 3 450 . 67 2 . 40 -0.01 453 . 56 Pt 5 19 . 34 ± 0 . 24 5 1 5 . 45 2 . 87 - 0 . 02 519 . 2 1 197Au 532 . 5 ± 0 . 5 533 . 1 6 ± 0 . 20 528 .95 2 .97 -0 . 02 532 .87 Hg 547 . 14 ± 0 . 2 5 542 . 63 3 . 07 - 0 . 03 546 . 85 TI 561 . 67 ± 0 . 2 5 556 . 50 3 . 18 - 0 . 03 560 . 9 3 2O'Pb 575 . 62 ± 0 . 30 570 . 54 3 . 28 - 0 . 04 575 . 24 Ph 575 . 56 ± 0 . 25 570 . 59 3 . 28 - 0 . 93 575 . 2 1 2O'BI 589 . 8 ± 0 . 9 590 . 06 ± 0 . 30 I 584 . 77 3 . 39 - 0 . 04 589 . 75 ,nTh 698 . 0 ± 0 . 6 698 . 4 ± O . 4m 698 . 15 ± 0 . 22 689 . 55 4 . 28 - 0 . 14 697 . 20 "'U 731 . 4 ± 1 . 1 732 . 0 ± O . 4m 730 . 88 ± 0 . 75 72 1 . 16 4 . 54 - 0 . 1 8 729 . 80

& Refs. (16. 78) ; b Ref. ( 1 7) ; c Refs. (18. 79) ; d Ref. (80) ; e Ref. (81 ) ; f Ref. (82) ; g Ref. (83) ; h Ref. (84) ; I Refs. (69. 70) ; k Ref. (85) : 1 Ref. (86) . m Ref. (89).

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PIONIC ATOMS 495

0 5 10 15 20 25 A

-1 o T=O '" T=112

-2 c T=1

0 -3

- 4 � /'

-5 ./' /c'"' ./'

-6

IY -7

E� x 103 (keV) I Z

FIGURE 4. Is level shift EN versus mass number A in a reduced scale. Nuclei with T=O. !. 1 are marked separately. The solid lines are calculated with the param­eters of Table 6, col. 3.

The results clearly show a negative shift for the 2p-1s transitions. This is determined typically to an accuracy of a few percent and originates from a decrease of the binding e nergy of the 1s level caused by a repulsive potential . All other transitions show a positive energy shift originating from an increase in the binding energies of the lower levels of the transitions caused by an attractive potential .

Furthermore, for the 2p-ls transitions the isospin effect is clearly estab­lished. One sees immediately a substantial increase of EN from 10n to llB as well as from 160 to 180. In Figure 4 this can be seen for all nuclei of Table 2 for which 2p-ls transitions have been measured. Since the strong-interaction shift in the Is level is proportional to Z., ENIZ4 is plotted as a function of the mass number A. There are distinct differences between the nuclei with iso­spin T=O, t, and 1. The lines connect the calculated values. Although mea­surements of the 3d-2p transitions in Ni isotopes (83) and of 4f-3d transitions in Sn isotopes (82) have been performed, no direct evidence for the isospin effect in higher transitions has been found.

Linewidths.-The accuracy with which the Lorentzian Iinewidth of a transition can be determined depends strongly on the instrumental resolu­tion of the detecting system with which the natural Iinewidth is folded .

Furthermore, the presence of muonic and nuclear gamma lines may in some

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496 BACKENSTOSS

10 keV 2 P level shift

- CERN - Berkeley .. Virginia

1O-1''-----<l1----l---------'-------'---------'--____ _ 10 l5 20 25 30

FIGURE 5. 2p level shift versus atomic number Z. The solid line is

calculated with the parameters of Table 6, col. 3.

10 keV

3d level shift

- CERN • Berkeley

O.IL.----;4'nO------f4�5-----;f.50;:------�55;:------6;t;O;-----=-Z

FIGURE 6. 3d level shifts versus atomic number Z. The solid line is

calculated with the parameters of Table 6, col. 3.

z

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keV

e

7

6

5

4

3

2

70

PIONIC ATOMS

4 f level shifts

- CERN • Berkeley • Virginia

FIGURE 7. 4/ level shifts versus atomic number Z. The solid line is calculated with the parameters of Table 6, col. 3.

497

z

cases disturb the measurement. The experimental data for r'B and r2p are presented in Table 3 (Refs. 16-18, 21 , 76, 78-80, 82, 83, 87-90) , and for rad and r4! in Table 4 (Refs. 78, 85) . I n strongly deformed nuclei with a sub­stantial quadrupole moment, the data for these widths neglecting hyperfine structure effects are given in cols. 2 and 3, whereas in co!. 4 the hyperfine splitting is taken into account. In view of the difficulties mentioned the agree­ment between the groups is satisfactory in most cases.

Linewidths from transition intensities.-As described in Sec. 2.2 the line­width r of the upper level of a transition can be obtained from its intensity, provided the population P of the upper level, the electromagnetic transition probability Wx, and the Auger transition probability WA are known. In Table 3, col. 6 , the data are given for all transitions for which sufficient information is available to allow deduction of absorption rates. In Table 5 (Refs. 76, 88, 89, 92) the quantities contained in Equation 17 are given. The X-ray yields Y (col. 2a) are obtained from comparison with muonic X-ray yields. In particular the sum of the transitions of the K series presents an ideal way of calibration. This quantity adds up to almost 100% since all muons-except the few reaching the metastable 2s level-reach the 1s ground state via one of these transitions, provided the Auger transition

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TABLE 3. Lorentzian level widths r1o, r,p, and rOd Cols. 2 to 4 show experimental data obtained from lineshapes, col. 6 from intensity measurements. Cols. 5 and 7 give theoretical values obtained with the parameters of

Table 6, col. 3.

rl.(keV) r2p(eV)

Target Berkeley· CERNb Virginiac Theory CERNd Theory (1) (2) (3) (4) (5) (6) (7)

4He 0 . 00 ± 0 . 09 0 . 05 0 . 002 ± 0 . 001h 6Li 0 . 39 ± 0 .36 0 . 15 ± 0 . 05 0 . 13 0 . 007 7Li 0 . 57 ± 0 . 30 0 . 19 ± 0 . 05 0 . 17 0 . 016 ± 0 . 007h 0 . 0 1 1

9Be 0 . 85 ± 0 . 28 1 . 07 ± 0 . 30 0 . 58 ± 0 . 05 0 . 54 0 . 16 ± 0 . 03

0 . 07 0 . 052 ± 0 . 013h

lOB 1 .4 ± 0 . 5 1 . 25.± 0 . 25 1 . 68 ± 0 . 12 1 . 30 0 . 32 ± 0 . 06 0 . 26 l iB 2 .3 ± 0 . 5 1 . 87 ± 0 . 25 1 . 72 ± 0 . 15 1 . 36 0 . 27 ± 0 . 04 0 . 33 12C 2 . 6 ± 0 . 5 2 .96 ± 0 . 25 3 . 25 ± 0 . 15 2 . 88 1 . 02 ± 0 . 29 0 . 98

1 . 25 ± 0 . 20h UN 4 . 1 ± 0 . 4 4 . 48 ± 0 .30 5 . 02 2 . 1 ± 0 . 3 2 . 8 16() 9 . 0 ± 2 . 0 7 . 56 ± 0 . 50 6 . 75 4 . 7 ± 0 . 8 5 . 3 1 80 8 . 67 ±0 . 70 6 . 00 3 . 8 ± 0 . 7 6 . 1 19F 4 . 6 ± 2 . 0 9 .4 ± 1 . 5 8 . 22 1 1 . 2 ± 1 . 9 10 23Na 4 . 6 ± 3 . 0 10 .3 ± 4 . 0 6 . 2 ± 1 . 2 1 5 .98 34 .6 ± 7 . 6 37

r'p(keV) r.d(eV)

Target Berkeley· CERN- Virginia! Theory CERNg Theory (1) (2) (3) (4) (5) (6) (7)

27AI 0 . 1 1 ± 0 . 08 0 . 1 1 0 . 02 Si 0 . 18 ± 0 . 08 0 . 17 0 . 13 ± 0 . 08 0 . 03 up 0 . 20 ± 0 . 08 0 . 25 0 . 18 ± 0 . 1 1 0 . 06 32S 0 . 8 ± 0 . 4 0 . 79 ± 0 . 15 0 . 36 0 . 25 ± 0 . 15 0 . 10 CI 0 . 89 ±0 . 25 0 . 52 0 . 17 ± 0 . 1 7 0 . 19 K 1 . 9 ± 0 . 15 1 .45 ± 0 . 15 1 . 04 0 . 49 40Ca 2 . 29 ± 0 . 13 2 . 00 ± 0 . 25 1 . 53 0 . 7 ± 0 . 3 0 . 7 1 44Ca 2 . 07 ± 0 . 15 1 . 38 0 . 83 Ti 2 . 89 ± 0 . 25 2 . 66 2 . 5 ± 0 . 7 2 . 01 6 lV 3 . 66 ± 0 . 25 3 . 51 2 . 1 ± 0 . 8 3 . 2 1 Cr 4 .46 ± 0 . 35 4 . 28 4 . 8 ± 1 . 1 4 . 28 66Mn 6 . 38 ± 0 .40 5 . 0 1 6 . 0 ± 1 . 3 5 . 95 Fe 6 . 0 ± 2 . 5 8 . 65 ± 0 . 60 6 . 38 1 0 . 9 ± 2 . 5 8 . 06 69CO 7 . 37 ± 0 . 70 7 . 94 10 . 1 ± 2 . 1 1 1 . 55 HNi 7 . 6 ± 1 . 4

1 2 . 7 ± 3 . 0 9 . 69 14 . 7 ± 4 . 2 14 . 76 6°Ni 8 . 5 ± 1 . 5

Cu 15 .9 1 4 . 0 1 1 . 20 20 . 8 ± 7 . 0 2 1 . 04 Zn I 16 . 8 ± 6 . 0 13 . 45 32 . 5 ± 1 1 . 0 28 .96

• Refs. (16, 78, 82) ; b Ref. (17) ; c Refs. (18. 79. 87) ; d Refs. (76, 88) ; • Ref. (80) ; ! Ref. (83) ; 8 Refs. (89, 2 1) ; h Ref. (98).

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PIONIC ATOMS 499

TABLE 4. Lorentzian level widths rOd and r.! obtained from Iineshapes

Cols. 2 to 4 show experimental data (in col. 4 the hyperfine structure splitting is taken into account) .

Berkeley' CERNb Target

I without hfs without hfs with hfs (1) (2) (3) (4)

r3d (keV)

::y 0 . 8 ± 0 . 6 93Nb 0 . 6 ± 0 . 4 0 . 52 ± O . lOc :�Mo 0 . 56 ± 0 . 10' '�!Rh 1 . 2 ± 0 . 6 "In 2 . 8 ± 0 . 6 2 . 6 ± 0 . 6 '��Sn 1 .9 ± 1 . 2 l:�I 4 . 6 ± 1 . 5 4 . 4 ± 1 . 5 ':!Cs 4 . 2 ± 1 . 8 3 . 3 ± 1 . 5 .7La 6 . 2 ± 2 . 0 6 . 2 ± 2 . 0 ':�Ce 5 . 8 ± 3 . 8 ':!Pr 6 . 7 ± 2 . 8 5 . 4 ± 2 . 5

r4J(keV)

'��Ta 0 . 5 ± 0 . 2 78Pt 1 . 8 ± 1 . 0 '�:Au 1 . 1 ± 0 . 3 1 . 0 ± 0 . 3 80Hg 1 . 4 ± 0 . 5 81Tl 1 . 0 ± 0 . 2 2:Pb 1 . 2 ± 0 . 4 n;�Pb 1 . 1 ± 0 . 3 2�!Bi 1 . 7 ± 1 . 0 1 . 7 ± 0 . 5 1 . 7 ± O . 5 2:�Th 6 . 0 ± 0 . 9 4 . 6 ± 0 . 8c 2:!U 6 . 1 ± 1 . 0 6 . 8 ± 0 . 8o 3 .53± O .46d 2!:PU 9 . 1 ± 2 . 5 4 . 05 ± 0 . 86d

• Ref. (78). b Ref. (85). 0 Ref. (89). d Ref. (86) .

Theory

(5)

0 . 28 0 . 4 1 0 . 51 0 . 88 2 . 01 2 . 19 3 . 67 4 . 71 6 . 24 7 . 05 8 . 1 1

0 . 27 0 . 59 0 . 70 0 . 77 0 . 90 1 . 01 1 . 06 1 . 22 2 . 6 3 . 2 4 . 0

probability is small compared with the X-ray transition probability at least for the last, the 2p-1s transition. Except for the very lightest nuclei this is true. The population P(2s) can be obtained with sufficient accuracy from cascade calculations which can be adapted to give the correct populations for the p levels, i .e. P(2p) , P(3p) , etc, and which yield for P(2s) about 5% for light nuclei and 2-3% for heavy nuclei.

The fraction of muons stopped in the target can be obtained in different ways (91) . Comparison of muonic line intensities with those pionic lines for

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500 BACKENSTOSS

which one is sure that no absorption takes place can be used. The best way is, however, direct measurement of stopped muons by stopping the beam in an active target, i .e. a scintillator, and measuring the muons by the signature of their decay electrons. The comparison with muonic lines provides other advantages. I n most cases it is possible to choose for comparison muonic lines with energies similar to those of the pionic line in question. Therefore, the energy-dependent corrections which must be applied such as detector efficiency, self-absorption in the target, and the geometrical factors are only subject to relative errors. I n col. 3 the population P of the upper level is given, which was obtained by an indirect method where P is measured in the corresponding muonic atom as the sum of all transitions fitting this level. In this case an "initial" distribution of the muonic states over the 1 substates at the level n = 14 is constructed, which yields the correct value for P. As­suming the same 1 distribu tion for pions in the state n = 17 , one can calcula te P via a cascade calculation for pionic atoms. This method is less certain than the direct method where P is obtained according to Equation 18. But in cases where both methods were applicable the results agreed within the errors. For data obtained with the direct method the ratio ( Y / P) is shown for some examples only in col. 4 of Table 5. Finally, the electromagnetic dipole transition rates Wx are needed. The pure electromagnetic values (col. 5) are given as well as the values modified by the influence of the strong interaction (col. 6) (88).

6. INTERPRETATION OF RESULTS AND CONCLUSIONS The results on pionic level shifts and absorption widths presented in

Sec. 5 can be discussed by means of the optical 1r-nucleus potential VN (Equation 30) derived in Section 4, although the optical potential does not take into account all the properties of individual nuclei. The six parameters of this potential, listed in Table 6, have been treated as free parameters, adjusted to give good agreement with the experimental data (89) . This task is facilitated because the parameters are only mildly dependent on each other and each parameter is essentially responsible for one observable quantity, as indicated in Table 6. The experimental quantities thus obtained are given in col . 3 of Table 6. For comparison, the theoretically predicted parameters (63) are displayed in col. 4. The energies and level widths obtained by solving the Klein-Gordon equation (Equation 12b) , with the potential parameters of col. 3 of Table 6, are the ones given as theoretical values in Tables 2 to 4 and shown as lines in Figures 4 to 8. The general trend of the data is described quite well by the nuclear potential chosen. For example, r2p and r3d are shown in Figure 8 to follow the data reasonably well over six orders of mag­nitude, where only the parameter 1m Co has been adjusted. It is remarkable that the same set of parameters is valid for the entire periodic system. The multiple-scattering approach seems to be successful as a framework for the description o(the experiments.

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TABLE 5. Data required for the determination of absorption widths from intensity measurements

Col. 2 shows experimental yields; col. 3 gives the level population of the upper level obtained from muonic cascade data. Col. 4 gives the directly determined ratio Yip. In cols. 5 and 6 the electric dipole transition probabilities are displayed without and with strong interaction taken into account.

Element Y(%) P (%) YIP Wz X I014 eLm. strong

(1) (2) (3) (4) (5) (6)

2p-> Is TRANSITIONS a b a a

9Be 10 . 5 1 1 . 4 1 5 . 0 ± 1 . 6 67 0 . 44 0 . 44 lOB 1 1 . 0 ± 1 . 6 61 1 .07 1 . 06 11B 12 . 5 ± 1 . 0 10 .9 ± 1 . 4 62 1 .07 1 . 04 12C 7 . 5 ± 2 . 0 7 . 6 ± 0 . 9 62 2 . 22 2 . 14 UN 6 . 8 ± 0 . 8 4 . 0 ± 0 . 8 62 4 . 12 3 .95 160 4 .9 ± 0 . 7 3 . 1 ± 1 . 0 57 7 . 02 6 . 70 160 5 . 6 ± 0 . 8 55 7 . 02 6 . 48 19F 4 . 1 ± 1 . 0 70 1 1 . 2 1 0 . 6 UNa 3 . 3 ± 0 . 7 78 25 . 1 22 . 0

3d->2p TRANSITIONS C a

Si 0 . 77 ± 0 . 1 1 6 . 80 6 . 86 S 0 . 75 ± 0 . 1 1 1 1 . 60 1 1 . 72 Ca 0 . 74 ± 0 . 1 1 28 . 3 28 . 8 DlV 0 . 62 ± 0 . 1O 49 . 6 52 . 0 66Mn 0 .44 ± 0 . 06 69 . 2 73 . 1 69CO 0 . 39 ± 0 . 05 94 .0 96 . 5 Zn 0 . 24 ± 0 . 06 144 155

A Refs. (76, 88) ; b Ref. (92) ; • Ref. (89).

TABLE 6. Effective potential parameters

In col. 1 the parameter and its unit is shown; col. 2 denotes the observable quantity to which the parameter is mainly related ; col. 3 shows the value obtained by fitting the data, and col. 4 the values obtained from multiple-scattering theory.

Parameter Theory

(units) Related to Exp

(1) (2) (3) (4)

bo(� .. ) EN(lS) : T=O - 0 . 03 -0. 03 bt(� ... ) EN(ls) ; T�O - 0 . 08 - 0 . 087 cO(�T)3 EN(2p) ; T = O 0 .22 0 . 2 1

and d , j, . . . states Cl(!\ ... )8 EN(2p) ; T�O (0 . 18) 0 . 18

and d, j, . . . states 1 m BO(!\ .. )4 r13 0 . 04 0 . 01 7 1 m CO(!\ .. )6 rzp, rad . . • 0 . 08 0 . 073

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502 BACKENSTOSS

10'

103

jJ i 102

• CERN D Berkeley

10'

FIGURE 8. Absorption widths r2p and rad versus atomic number Z. r2p for Z::; 11 from intensity measurementsj for Z ? 13 from lineshapes. rad for Z ::;30 from intensity measurementsj for Z ? 39 from lineshapes.

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PIONIC ATOMS 503

The pion wavefunction, i .e. the pion density at the nucleus, is determined essentially by the real part of the potential , as can be seen from Figure 3 . This means that the measurement of the energy eigenvalues performed in pionic atoms conveys the knowledge of the pion wavefunction at the nucleus. Figures 4 to 7 indicate that in particular the energy shifts, i .e. the real part of the potential, are well determined, and hence the pion density at the nucleus, which is an important quantity for many low-energy pion phenomena, can be deduced from energy measurements on pionic atoms. From a comparison of the effective potential parameters obtained by a multiparameter fit to the data with the theoretically predicted values, some conclusions can be drawn. The strength parameter of the isospin scalar interaction bo is experimentally determined to better than 10%. About 70% of this value can be attributed (28) to the correlation term (Equation 24) , and hence the measurements con­firm the importance of the long-range correlations. Since correction terms origi­nating from the dispersive effects of nuclear-pion absorption can be roughly estimated (15) and the effect caused by the Fermi motion can be taken into account, the coherent sum bo' can be deduced to be <0.01 (lI.,,) . Because this quantity, as a consequence of the large cancellation of al and aa, is diffi­cult to deduce from pion-nucleon phase shifts, giving values2 of -0.002 (44) , -0.007 ± 0.003 (93) , and - 0.023 (93a) , the value obtained from pionic atoms is an important contribution to the pion-nucleon problem. The iso­vector interaction strength b\, on the other hand, is not subject to cancella­tions and the interpretation is therefore straightforward. The value deduced from pionic atoms is in excellent agreement with data from pion-nucleon scattering of -0.086 (44) or -0.096 ± 0.004 (93) .

Similarly, the real part of the gradient term, the parameters C o and CI, is in good agreement with the predicted values (94) ; however, CI has been fixed as the theoretical value. It would be of great interest to prove or disprove the Lorentz-Lorenz term (Equation 27) in the p-wave interaction, since this is directly connected with the short-range correlations in the nuclei. Detailed analysis is in progress (89) .

The agreement for the imaginary part of the potential connected with pion absorption is considerably worse. There is a disagreement in the s-wave absorption (1m Bo) , between the predicted and the measured value, of about a factor 2 .5 , where the uncertainty in the experimental value is about 10%. On the other hand, the disagreement for the p-wave absorption (1m Co) de­duced (63) from incomplete data has disappeared and the data seem to give no sufficient support for a saturation of the s-wave absorption with in­creasing atomic number Z as suggested elsewhere (78, 95) . However, the dis­crepancy for 1 m Bs is severe enough to suggest that essential features are still missing in the description of the phenomena. The comparison is based on the deuterons, and the deuteron wavefunction has no p-wave contribution. The selection rules for two-nucleon absorption of pions (96) , do indicate that the

2 A recent compilation is given in (94).

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504 BACKENSTOSS

contribution of nucleon pairs in a relative p state might be important. This may reduce the mentioned discrepancy.

The experimental data are sufficiently ample and accurate to justify a theoretical approach which goes beyond the multiple-scattering description and which takes individual nuclear properties into account. A good example is given by the 1s and 2p absorption rates in 160 and 180 . Any simple optical potential description adjusted to produce the increased 1s level shift of the heavier isotope will reproduce, as a consequence of the more strongly repul­sive potential, a reduced s-state absorption, in contradiction to the experi­ment. Here a microscopic approach based on the treatment indicated by Equation 31 succeeded in reproducing (97) the experimental ratio r18(160) /r1s(180) as well as r2p(160)/r2p(180) . This can be understood in terms of the additional d neutrons in 180, which provide more relative momentum in a correlated pair than s or p nucleons, and a correspondingly smalIer momen­tum mismatch with the two-nucleon pion absorption.

However, the most direct way in which nuclear structure enters the in­terpretation of the pionic atom data is associated with the nuclear matter distribution per) , which determines the shape of the potential in Equation 30. As soon as the strength of the interaction is known sufficiently welI, the inves­tigation of pionic atoms should provide an excellent tool for the study of the matter distributions. Whereas the charge distribution is measured very pre­cisely with muonic atoms and electron scattering, the new feature connected with pionic atoms is the dependence also on the neutron distribution. Since the absorptive parts of the potential are proportional to p2 or PnPp, the level widths should be rather sensitive to the matter density. Unfortunately, the theoretical description of the absorption seems to be the most difficult part. A first fit (98) to alI existing energy shifts and level widths between lOB and 209Bi in pionic atoms has been made, where the difference between the rms radii of the neutron and proton distribution .6. = V (rn2 } - V (rp2} was treated as a free parameter. No significant difference [.6. = ( - 0.01 ± 0. 16) F ] was found on the average through the periodic system nor did a comparison (83) between the 3d-2p transitions for the isotopes 58Ni and 6°Ni reveal such a dif­ference. But it is conceivable that more detailed studies utilizing the full knowledge of the pion-nucleus interaction are more conclusive. The energy shift in the levels with greater l is strongly dominated by the p-wave interac­tion, which is strongly dominated by the 7I"-n interaction and could lead to a measurement of the neutron radius. The success of the treatment of Chung et al (97) however indicates the necessity for careful consideration of nuclear structure effects.

To conclude, one may say that the field of pionic atoms has reached a stage where experimental data provide a reasonably complete survey of the experimental facts and where the theoretical treatment is sufficiently ad­vanced for most of the essential features to be understood. There is the need of more work on the pion absorption process. There is ample room for new and more precise experimental work, in particular on separated isotopes, to

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PIONIC ATOMS 505 make full use, in conjunction with refined calculations, of the new features and of the potential power inherent to pionic atoms. These result essentially from the very desirable properties of the pion as a probe particle in a well­defined state, which include its sensitivity to all nucleons, the relative weak­ness of its strong interaction with the nucleons, and its property of being the lightest boson from which originates the dominance of the absorption process on two nucleons.

ACKNOWLEDGMENTS

I wish to thank all members of the C E R N Mesonic Atom Group, par­ticularly Drs. H. Schmitt and H. Koch for calculations on the data and Dr. M� Krell for many valuable comments. I acknowledge with pleasure the many stimulating discussions and constructive criticism given by Drs. M. and T. E. O. Ericson.

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506 BACKENSTOSS

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