Copyright

by

Federico Francisco Buersgens

2003

Neutron emission studies in laser driven fusion

experiments

by

Federico Francisco Buersgens

Thesis

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Master of Arts

The University of Texas at Austin

December 2003

Neutron emission studies in laser driven fusion

experiments

Approved bySupervising Committee:

To my great mentor Prof. Roberto Brie

may he rest in peace

Acknowledgments

The work described in this thesis is based on an experiment that I was able to do

because I had great support from a lot of different people.

First of all I would like to mention Kirk Madison who not only introduced me into

my experiment but also continuously helped me physically as well as intellectually

to run the experiment and to improve the setup. I am glad to say that from this

intense and fruitful collaboration a firm friendship has arisen.

Moreover I would like to express my gratefulness towards Will Grigsby who always

was a shining example of helpfulness. Even on weekends he never refused to help

me realigning or repairing the laser or just to explain its mysteries to me. In this

context I also want to acknowledge the support I got from Gilliss Dyer for all sorts

of vacuum-related questions as well as from Aaron Edens and Greg Hays in all sort

of matters. Furthermore Rene Hartke and Jens Osterhoff did a great job in helping

me setup the vacuum transport of the laser beam into the target chamber.

However what would a scientific experiment look like without custom made machine

shop items? The answer is obvious and that is the reason why I want to thank Allan

Schroeder and his whole crew for always being helpful to built, whatever we needed.

In addition to these persons mentioned above I am also grateful for the

support from the German Academic Foundation and the ’Wurzburg-Program’ which

ultimately enabled me to come to the University of Texas.

Finally I wish to thank Prof. Roger Bengtson for being willing to be the

v

co-reader of my thesis.

Last but certainly not least I owe a great deal of gratitude to my supervisor Prof.

Todd Ditmire who gave me this chance to work in his group and also invested a

lot of time, money and energy in my project. Besides of being a good scientific

instructor he also established a very amicable atmosphere in the entire group.

Federico Francisco Buersgens

The University of Texas at Austin

December 2003

vi

Neutron emission studies in laser driven fusion

experiments

Federico Francisco Buersgens, M.A.

The University of Texas at Austin, 2003

Supervisor: Todd R. Ditmire

Recent experiments on the interaction of intense, ultra-fast pulses with large van der

Waals bonded clusters have shown that these clusters can explode with substantial

kinetic energies, [1-3]. If the clusters are smaller than a critical radius (which is

determined by the atomic species and the available intensity), then at sufficiently

high intensities the laser pulse is able to extract field ionized electrons from the

cluster. This leads to a subsequent explosion of the cluster , due to the Coulomb

repulsion of the closely spaced ions. It was shown, that the resulting kinetic energy

of the ions are sufficient to drive nuclear fusion reactions [4].

Moreover an anisotropy of the neutron yield was reported by Grillon et al. in [5],

which they explained by a significant contribution to the fusion yield from collisions

of hot ions with the surrounding gas and radially directed ions.

These results motivated a more detailed analysis of the angular distribution

of the neutron yield, which will allow conclusions about the prevailing reaction

mechanisms in the plasma filament.

In this thesis a study of the angular emission pattern of fusion neutrons

from the irradiation of deuterium (D2) and deuterated methane (CD4) clusters is

presented. However the results reported here, turn out not to be in agreement with

vii

the data in [5] - a fact, which remains to be completely understood.

For this experimental campaign 40fs pulses at various energies (up to 200mJ)

from the 20 TW Ti:Saphire THOR-laser were used.

Ultimately these studies are motivated by the long term goal of using this kind of

experiments for a table top neutron source to do time resolved neutron damage

studies and to improve the understanding of plasma processes that eventually could

become relevant for fusion reactors.

viii

Contents

Acknowledgments v

Abstract vii

List of Figures xi

Chapter 1 Introduction 1

Chapter 2 Theory of Laser Cluster Interaction 5

2.1 Optical field ionization of atoms . . . . . . . . . . . . . . . . . . . . 5

2.2 Laser interaction with cluster electrons . . . . . . . . . . . . . . . . . 9

2.3 Coulombic cluster explosions . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Expected Fusion Yield . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Thermonuclear cross sections . . . . . . . . . . . . . . . . . . . . . . 20

Chapter 3 Experimental Setup 27

3.1 Design of a neutron detector . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 THOR-Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Gas jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Chapter 4 Experimental results 35

4.1 Systematics of neutron yield measurements . . . . . . . . . . . . . . 35

ix

4.2 Deuterium Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 TOF-Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Energy scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Angular measurement by detector permutation . . . . . . . . 41

4.2.4 Polarization Scan . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 Hetronuclear Clusters: CD4 . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.1 Polarization Scan in CD4 . . . . . . . . . . . . . . . . . . . . 46

4.3.2 Azimuthal Scan in CD4 . . . . . . . . . . . . . . . . . . . . . 48

Chapter 5 Conclusions and Future Directions 50

5.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.1 Angular Measurement . . . . . . . . . . . . . . . . . . . . . . 51

5.1.2 High Energy Neutrons . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Outlook - Future Experiments . . . . . . . . . . . . . . . . . . . . . . 54

5.2.1 Machining Beam . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2.2 Magnetic Electron Confinement . . . . . . . . . . . . . . . . . 55

5.2.3 Solid Density Beam Target . . . . . . . . . . . . . . . . . . . 56

Bibliography 58

Vita 61

x

List of Tables

2.1 Ionization intensities for different atomic species . . . . . . . . . . . 8

xi

List of Figures

2.1 Barrier suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Electron behavior in laser irradiated clusters . . . . . . . . . . . . . . 10

2.3 Comparison of different energy distributions . . . . . . . . . . . . . . 15

2.4 Contributing mechanisms to the total yield . . . . . . . . . . . . . . 16

2.5 Plasma reactivity and expected fusion scaling as a function of tem-

perature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6 Fusion contribution from the Maxwell tail . . . . . . . . . . . . . . . 22

2.7 Differential cross sections in the center of mass and laboratory system 23

2.8 Anisotropy of differential cross section as a function of temperature . 25

3.1 Overview over the THOR-Laser . . . . . . . . . . . . . . . . . . . . . 30

3.2 Sketch of a cryogenically cooled, high pressure gas jet . . . . . . . . 33

4.1 Typical detector event for a 2.45 MeV neutron . . . . . . . . . . . . 36

4.2 Attenuation of neutron flux by matter . . . . . . . . . . . . . . . . . 37

4.3 TOF traces and histogram for the irradiation of D2 clusters . . . . 39

4.4 Energy scan in D2 from a sonic expansion . . . . . . . . . . . . . . . 41

4.5 Angular scan for neutrons from D2 clusters . . . . . . . . . . . . . . 42

4.6 Polarization scan in D2 . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.7 Experimental setup used for CD4 data . . . . . . . . . . . . . . . . . 46

xii

4.8 Polarization Scan in CD4 . . . . . . . . . . . . . . . . . . . . . . . . 47

4.9 Azimuthal scan in CD4 . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1 Fusion probability depending on the ion direction of motion . . . . 52

xiii

Chapter 1

Introduction

The behavior of a wide variety of materials at different states of aggregation have

been studied in the field of ultra-short, ultra-intense lasers. One remarkable obser-

vation was that, the irradiation of clusters, i.e. nanometer-size droplets of van-der-

Waals bonded atoms or molecules, results in a very efficient coupling of the laser

into the plasma. This is explained by the fact, that the electric field of the laser

(which goes as the square root of the intensity) can easily penetrate the cluster (the

plasma-frequency is higher than the frequency of the radiation) and ionize atoms,

since it exceeds the field experienced by an electron in an atom. Moreover such field

ionized electrons can be extracted from the cluster by the external field, and start

an oscillatory motion with an energy determined by the amplitude of the field (for

peak intensities of I0 > 1018 W/cm2 energies of several tens of keV can be achieved).

In fact, clusters can be completely ionized by this mechanism, provided that their

radius is smaller than a critical radius, which is determined by the available inten-

sity. Due to the coulomb repulsion of the remaining, closely spaced ions, the cluster

is going to expand with an energy determined by the electrostatic potential of the

given spatial charge distribution.

Even from this simple picture it is obvious that the resulting ion energies are

1

also going to depend on the ionization time, or more specifically on whether the ion-

ization or the expansion of the partially stripped cluster is going to happen faster. In

this thesis we want to focus on the first case which is referred to as ”coulomb explo-

sion” as opposed to the latter case, which is known as a hydrodynamic expansion. It

turns out that the characteristic response time of ions in nanometer clusters of solid-

density is on the order of a few tens of femtoseconds (fs), which means that for an

explosion in the Coulombic regime pulse durations of less than a 100fs are required.

One particulary interesting field of research on Coulomb explosions explores

ways to exploit the substantial ion energies achieved by these explosions to drive

thermonuclear fusion reactions, [1-3]. If a dense jet of clusters of a suitable atomic

species is irradiated by a fs laser pulse, collisions between ions, emerging from

coulomb explosions, are sufficiently energetic to bring both nuclei close enough that

the strong force overcomes the repulsion of their positive charge.

More specifically, clusters of deuterium or lately also clusters composed of

deuterated, hetronuclear molecules are produced in a high pressure gas jet and are

irradiated with a short pulse laser (40− 100fs). This results in plasma densities in

the filament of the laser focus of the order of np ≈ 1019 cm−3 and the temperatures

of up to 14keV (equivalent to 1.6 ·108 K). The observed nuclear fusion events from

collisions of two deuterium ions can be characterized by the following well-known

reactions:

D + D50%−→ T (1.01MeV ) + p(3.02MeV )

D + D50%−→ He3(0.82MeV ) + n(2.45MeV )

This means that in 50% of the fusion events a neutron with a defined kinetic energy is

going to be emitted. In fact these neutrons can be identified with suitable detectors

using time of flight diagnoses (TOF), such that the fusion yield can be monitored by

2

the number of neutron events in the detector. This is especially promising because

the neutron yield crucially depends on the conditions in the hot plasma filament.

In contrast to this, the measurement of emitted ions, which is a frequently used

diagnostic, may be deceptive, since it is not clear whether they acquired all their

energy from Coulomb explosions or whether they gain their energy from interactions

with electrons1

Moreover an angular scan of the neutron yield done by Grillon et al., [5],

showed an anisotropy, which they explained with a model that suggests a significant

contribution to the fusion yield from collisions of hot ions (from the filament) with

the surrounding gas in the jet. The underlying idea here is, that the differential cross

sections (DCS) for both branches of the reaction shown above are not isotropic2.

This means, that for ion collisions along a given axis, the neutrons are going to be

emitted preferentially in certain directions. But this does also imply, that in return

any preferential emission direction for neutrons, allows to draw conclusions about

the dominant direction of collisions between ions and hence about the motion of

those ions in the hot plasma3.

Furthermore this implies that if hot ions from the filament collide with rela-

tively cold ions in the surrounding gas jet, which can be considered to be at rest in

the laboratory system, the neutron emission pattern is going to be different than in

the case of two hot ions (which hence can be deemed to be identical). This is based

on the fact, that the differential cross sections are not identical in the center of mass

system and the reference frame of the laboratory4. Thus the overall angular distri-1This would happen at a later time, when the filament has expanded and the density hence has

become to low for an appreciable fusion yield.2This is in particular true for the second branch, which shows a larger anisotropy for lower

collision energies.3This conclusion can be drawn, because if there was no preferential direction for the neutron

emission, it would mean that the ion collisions happen under random angles, which would statisti-cally wash out the anisotropy of the DCS.

4For the sake of simplicity the difference can be understood by imagining that in one case the

3

bution of the neutron emission depends on the sum over all probabilities for fusion

events along different axis of propagation of the ions, weighed with the likelihood

of such a path and weighed with the correct DCS. In to sum up above discussion,

neutrons from collisions of hot ions with the surrounding gas (i.e. the so-called

beam target contribution) have a different angular distribution than neutrons from

collisions of hot ions5.

This result obtained by Grillon et al. (that the angular emission of neutrons

reveals information about the prevailing mechanism leading to fusion) motivated a

more detailed study of the angular neutron emission pattern, which is presented in

this thesis.

collision happens between two ions with same but oppositely directed momenta (thus the center ofmass is at rest in the laboratory system), in which case the differential cross section has to be thesame in both frames of reference. Contrariwise, in the case of one ion at rest the differential crosssections in the center of mass system (which is moving in the lab system) is going to be differentthan in the reference frame of the laboratory.

5Moreover the path length through the plume is in general very asymmetric for an ion from thehot filament, such that an additional anisotropy can be expected, due to different fusion probabilitiesfor different directions.

4

Chapter 2

Theory of Laser Cluster

Interaction

In the following different aspects of the ionization of atoms and clusters in the electric

field of an ultra fast, super intense laser are going to be illuminated. With the advent

of pulsed, high intensity lasers with peak intensities in excess of I > 1014W/cm2

the electric field in the focus became comparable to or even larger than the typical

electric fields within an atom. This distortion of the Coulomb potential can not

be treated any more by means of perturbation theory but requires entirely new

concepts. In particular we are going to focus on the process of cluster ionization

and the dynamics that arise from its high charge state.

2.1 Optical field ionization of atoms

The ionization of an atom due to an intense laser pulse can be characterized with two

different concepts: On the one hand (for lower intensities) the tunneling mechanism

prevails whereas on the other hand for higher intensities a multi-photon absorbtion

causes the ionization. For a more quantitative distinction the Keldysh parameter is

5

defined as

γ = (Vi

2UP)1/2 (2.1)

where Vi is the ionization potential and UP the pondermotive potential, this is the

potential an electron has due to its oscillations in the field of the laser. A value for

γ > 1 is an indication for the multiphoton regime and consequently a value γ < 1

indicates tunneling ionization. We can rewrite the pondermotive potential in terms

of the intensity (using I[W/cm2] = 104 · 12 · cε0E[V/m]2 and ω = 2 · 109πc/λ[nm])

such that:

Up[eV ] =eE2

4meω2= 9.34 · 10−20 · I[W/cm2] · λ[nm]2 (2.2)

This means that for hydrogen and deuterium whose ionization potential is Vi =

13.6eV the Keldysh parameter becomes γ = 1 for I = 1.14·1014W/cm2 (for a typical

wavelength of 800nm). More specifically this means that for typical intensities well

above I > 1016W/cm2, as they were used in all experiments described in this thesis,

the Keldysh parameter is about γ < 0.11 and thus certainly in the tunneling regime.

coulomb potential

V(r) ~ – a/r

| R10

|2~ e

-2r/a0

total potential

V(r) ~ a/r – b ×r

external potential

V(r) ~ – b×r

b)a)

radial

probability density:

| R10

|2~ e

-2r/a0

Figure 2.1: The tunneling probability for an electron which is symbolized by theradial probability density |R10|2 of the hydrogen ground state is very low in anundistorted coulomb potential (see a)). However in a coulomb potential that isdistorted by an external field V (r) ∼ r the width of the potential barrier is notinfinitely thick anymore.

6

A very simple model for the so-called above threshold ionization is to assume

that the Coulomb potential is distorted by a static field (which is a reasonable

assumption since the laser oscillation time is long compared to the atomic frequncy).

The Coulomb potential then needs to be modified by the potent −e · Φexternal =r∫0

E dr′ = −eEr leading to the effective potential:

V (r) = − Ze2

4πε0r− eEr (2.3)

Then the local maximum has clearly to be at:

∂V (r)∂r

=Ze2

4πε0r2− eE = 0

rmax =√

Ze

4πε0E(2.4)

By assuming that the ionization potential Vi equals the local maximum of the modi-

fied potential V (rmax) an estimate for the minimum, required electric field Ethreshold

can be obtained:

Ethreshold =V 2

i π2ε0Ze3

(2.5)

Accordingly the threshold intensity is given by:

I0[W/m2] =12· cε30π

2V 4i

e6Z2) = 4 · 1013 · Vi[eV ]

Z2(2.6)

In order to compute actual intensities with (2.6) the effective charge of the nuclei

as experienced by the electorns has to be taken into account1. One way of doing

this, is by using empirical rules for the screening effect of the electrons on inner

shells [15]. Although the theoretical values (s. Table 2.1) are in rough agreement

with the actual values, some experiments seem to indicate that the theoretical values1It may seem counterintuitive to take into account the ionization potential (for each ionization

stage) and the effective nuclear charge, which appears like accounting twice for the same effect.However this simple model relies on the assumption that the ionization potential is equal to themaximum of the potential experienced by the electron. Therefore an alternative model can be used,defining Z as the charge state of the ions which is simpler, but less accurate for some atomic species(e.g. Ar in [14]).

7

charge state Hydrogen Carbon ArgonZ [e] Vi [eV ] I0 [W/cm2] Vi [eV ] I0 [W/cm2] Vi [eV ] I0 [W/cm2]

1 13.6 1.4 ·1014 11.3 6.5 · 1013 15.8 2.4 · 1013

2 24.4 2.5 · 1014 27.6 1.8 · 1014

3 47.9 1.7 · 1015 40.7 6.9 · 1014

4 64.5 3.9 · 1015 59.8 2.1 · 1015

5 392.1 3.2 · 1018 75.0 3.5 · 1015

6 490.1 6.4 · 1018 91.0 5.6 · 1015

Table 2.1: The above table shows typical intensity values at which ionization occurs.Since the ionization rate shows a very steep increase with the intensity usually thereis just about one order of magnitude in intensity between the onset of ionization andsaturation of the same level. In other words though this model is not very accurateit still provides an acceptable however rough estimate.

tend to overestimate the required intensity, [14]. Among other reasons that might be

due to the fact that the ionization potentials of atoms are tabulated for undistorted

potentials. In other words the ionization potentials for distorted potentials are

probably somewhat lower.

However this model does not give any information about the population

of different ionization levels, this is, about the expected ionization rate nor does

it account for potential excitations of the atom. This motivated a more complex

quantum mechanical model that was developed in the 1960′s by A.M Perelomov, et

al. [12]. The equation they have found predicts the probability of tunnel ionization

in an alternating electric field for arbitrary states of the hydrogen atom . About 20

years later V. Amosov, N. Delone and V. Kraınov succeeded in refining this model

in such a manner that it became able to predict the tunnel probability of a complex

atom or ion in an arbitrary state [13]. The tunneling rate Wtunnel in a linearly

polarized alternating electromagnetic field is given by the so-called ADK equation

8

(in CGS units):

Wtunneling = ωa(2l + 1)(l + m)!2|m||m|!(l − |m|)! (

2e

n∗)2n∗ · Vi

2πn∗· ( 2E

π(2Vi)3/2)1/2

· [2(2Vi)3/2

E]2n∗−|m|−1 · exp[

2(2Vi)3/2

3E]

(2.7)

Here n∗ = Z(2Vp[eV ]) stands for the effective principal quantum number of the

complex ion which takes into account quantum effects and the degree of ionization.

Moreover ωa is the atomic frequency with ωa = 4.13 · 1016Hz, l is as usually the

angular momentum and m denotes its projection onto the z-axis. This equation

shows a very steep increase in the tunneling probability and thus in the achieved

charge states of the ions above certain, characteristic intensity thresholds.

2.2 Laser interaction with cluster electrons

In principle there are two conceivable explanations for the observed explosions of

clusters in the field of an intense laser pules. On the one hand a Coulombic explosion

and on the other hand a thermal expansion could be the prevailing mechanism.

In a Coulombic explosion the laser field pulls out the electrons much faster than

the ions can move and hence the cluster can expand leading to high space charge

forces whereas a thermal scenario emphasizes the electron heating i.e. that electron

pressure causes the explosion.

Since recent experiments have shown that the deuterium density inside the clusters

is about n ≈ 1022cm−3 this corresponds to a plasma frequency (for deuterium

ions) of ωp =√

n·e2

ε0m ≈ 9.3 · 1013Hz which corresponds to an ion response time of

1/ωp ≈ 11fs. For a laser pulse of 30fs this means that the ions actually have

time to move however it is considered a reasonable assumption that the ions stay

immobile for a few laser cycles. For this reason in the following the discussion will

be limited to the case of Coulombic explosions, i.e. to the case that the cluster

ionization happens faster than the expansion takes place.

9

It turns out that for uniform spherical clusters an equation of motion can be derived

from some straightforward considerations as demonstrated by B. Breizman and A.

Arefiev in [8]. They assume that if the intensity of the laser is not high enough

(or for the first cycles not yet high enough) that some fraction of the electrons are

extracted from the cluster and the rest remain confined in the cluster. Although

the cluster will clearly acquire hereby a positive charge the remaining electrons are

going to move in such a manner that they compensate the total electric field which

is a superposition of the laser field and the space charge field.

r

d

r1

Laser electric field direction

Extracted electrons

Figure 2.2: Electron behavior in a laser irradiated cluster

The electric field of a uniformly charged sphere that is centered around the

origin is given by

Ei =

ni<qi>|e|3ε0

r for r ≤ R

ni<qi>|e|3ε0

(Rr )2r for r > R

(2.8)

where R is the diameter of the cluster and ni the density of ions with an aver-

age charge state of qi. Analogously the sphere of electrons shown in Figure 2.2 is

displaced with respect to the ion sphere by the vector d. It has an electric field of

Ee = −ni < qi > |e|3ε0

r1 for r1 ≤ Re (2.9)

where the radius of the electron sphere is determined by the time history of the

external field (i.e. the laser pulse). However Re has to be equal to R− d if the laser

10

field grows monotonically. This also means that at the point where both spheres

touch there is a potential leak which enables electrons to be pulled out. Both

equations together lead to a total space charge field inside the electron sphere of:

E =ni < qi > |e|

3ε0r− ni < qi > |e|

3ε0r1 =

ni < qi > |e|3ε0

d (2.10)

Of course this solution only holds for the case, that the complete electron sphere is

situated inside the cluster. If we assume that the ions do not have enough time to

move anywhere during the pulse we can calculate the radius of the electrons sphere

from the maximum internal field which has to be equal to the negative maximum

of the laser field E = −Elaser, max. By inverting (2.10) in order to obtain d the

minimum radius of the electron sphere can be calculated as follows:

Re = R− |d| = R− 3ε0ni < qi > |e| |Elaser,max| (2.11)

This is particulary relevant because once the amplitude of the pulse becomes smaller

the electron sphere is going to move back towards the center of the cluster. In

other words after the external electric field vanished the cluster has a neutral core

surrounded by charged ions. Hence an equation of motion for an ion located at

radius ri can be obtained:

F = mr = qi ·E(r) =qi· < qi > nie

2

3ε0· [(ri

r)3 − (

Re

r)3·]r (2.12)

Here qi denotes the individual charge state of the particular ion sitting atri whereas

< qi > stands for the average charge state inside the entire cluster. Using (2.11)

this equation can be rewritten for a 1D motion of hydrogen like ions:

mr =nie

2

3ε0· (r

3i − [R− 3ε0Elaser,max/(nie)]3

r2) (2.13)

Moreover the energy of an ion that is expelled from ri to infinity is therefore given

by:

Wri→∞ =

∞∫

ri

mr dr =nie

2

3ε0· (r

3i − [R− 3ε0Elaser,max/(nie)]3

ri) (2.14)

11

For the case that Re = R i.e. that the cluster is completely ionized (2.14) simplifies

to

Wmax =e2niR

2

3ε0(2.15)

Since for CD4 and cryogenically cooled D2 the cluster diameter is on the order a

few nanometers the intensity provided by the THOR-laser is sufficient to strip the

cluster completely. For this reason the following discussion is going to be restricted

to this case.

2.3 Coulombic cluster explosions

In this section the question what the ion dynamics in a coulomb explosion look like

is going to be discussed. It turns out that the resulting ion energy distribution is

a convolution of the energy distribution for a given cluster size and the cluster size

distribution in the plasma filament 2. The neutron yield predicted by this model is

found to be consistent with experimental data [9], [10], nevertheless in [6] a more

rigorous approach is presented, which takes into account arbitrary initial cluster size

distributions.

We expect ions that are located at a radius r to have an energy W (r) =

Wmaxr2

R2 . By weighing the inverse radial energy distribution (dW/dr)−1 = r2W

with the radial density distribution dN = 4πr2dr/(43πR3) we can derive an energy

spectrum dN/dW for a single exploding cluster:

dN

dW=

dN

dr

dr

dW=

32

√W/

√W 3

max , if W ≤ Wmax

0 , otherwise(2.16)

2Recent results from Rayleigh scattering experiments have shown that the scattered amplitudefrom clusters in the middle of the plume of a gas jet is constant over several millimeters. Thisstrongly suggests that the cluster size distribution can be considered to be spatially constant onthe length scale of the plasma filament [16]

12

Moreover we can calculate now the average ion energy W and hence the ion tem-

perature T of the mircroplasma created by a single cluster explosion as a function

of the radius of the cluster R:

T (R) =23k−1

B W =23k−1

B

∞∫

0

W · dN

dWdr =

25k−1

B Wmax =25k−1

B · e2niR2

3ε0(2.17)

In a realistic plume of a gas jet however there is a large spread of cluster sizes

which is believed to obey to a log-normal distribution. Nevertheless in the limit of

large numbers a Gaussian distribution is a good approximation especially because

most observed effects depend on the sum of microcanonical quantities. Hence the

distribution of cluster sizes is characterized by

g(R) =1√2πδ

ae−(R−R0)2/2δ2(2.18)

where R0 is the mean cluster radius and δ stands for the variance which is related

to the width of the distribution3. Now we can go one step further and compute

the distribution function for the maximum energies Wmax which depends on the

probability density for the radii (2.18) and (2.15). The probability density for a

function of a stochastic variable relates to the variable’s probability density by:

pY (y) =

∞∫

−∞δ(y − f(x))pX(x) dx (2.19)

where δ(y− f(x)) denotes the Dirac delta function. Next we need to recall a math-

ematical equality for the delta function of a function φ(x):

δ(φ(x)) =∑x0

δ(x− x0)|φ ′(x0)| (2.20)

3The cluster sizes were measured by means of Rayleigh scattering, [20], to be on the order ofa few nanometers for hydrogen clusters. However it is not clear what their distribution functionlooks like.

13

with φ becoming zero for x0 and and φ ′(x0) 6= 0. We can rewrite (2.15) by defining

a constant α ≡ e2ni/3ε0 such that Wmax becomes Wmax = αR2 and the argument

of the delta function in (2.19) becomes Wmax − αR2 where R is the variable over

which we integrate4. Taking into consideration the zero point5 and using (2.20) at

R =√

Wmax/α the integral becomes

p(Wmax) =

∞∫

−∞

δ(R−√

Wmax/α)2α

√Wmax/α

g(R) dR

which can easily be evaluated. The obtained probability density approaches a nor-

mal distribution for σ ¿ µ which independently from α turns out to be properly

normalized. Finally we obtain the probability to find a Coulomb explosion with a

maximum energy larger than W<

P (W<) =

∞∫

W<

g(√

Wmax/α)√Wmax/α

dWmax (2.21)

This enables us to achieve our goal of determining what the energy spectrum from

coulomb explosions from clusters of various sizes looks like. For this purpose for each

energy W we have to sum6 all denominators 1√W 3

max

in (2.16) that are greater than

W and weigh each of them with their probability to exist (2.21). This means, that

apart from a normalization factor, which has to be found numerically, the energy

spectrum is given by:

dN

dW=√

W

∞∫

W

1√W 3

<

∞∫

W<

g(√

Wmax/α)√Wmax/α

dWmax dW< (2.22)

Unfortunately this integral does neither have a straightforward analytic solution nor

is it easily possible to find an analytic connection between the average energy on the4Since the integration happens over R this term in general is not going to be zero5We are only considering the case R > 0.6The word ”sum” is supposed to be understood pictorially: Since Wmax is a continuous variable

this means that we have to integrate from W to infinity.

14

one hand and the average cluster size (and its variance) on the other hand. However

it can be evaluated numerically resulting in numerical values that are very close to

those obtained from a Maxwellian distribution (see Figure 2.3).

100 1000 10000

W [eV]

5. ´ 10- 7

1. ´ 10- 6

5. ´ 10- 6

0.00001

0.00005

0.0001

dN/dW

Comparison of distributions

Maxwell distribution

convoluted distribution

at T=4.3keV

Figure 2.3: Comparison of different energy distributions - The convolution of theenergy distribution (2.16) of a single cluster and the cluster size distribution (2.18)spawns a new distribution function, which comes close to a Maxwellian distribution.The degree of agreement between both distributions clearly depends on the choiceof parameters (here: Average cluster-size µ = 10nm, variance of cluster sizes σ =4nm, α = 60 [ev/nm2] which corresponds to an ion density inside the cluster ofni ≈ 1022 [1/cm3])

2.4 Expected Fusion Yield

For the fusion yield in a gas jet that is irradiated by an intense laser pulse there are

two conceivable mechanisms: On the one hand fusion can occur if two energetic ions

in the hot filament collide with each other and on the other hand fusion can also arise

from collisions of hot ions which leave the filament and traverse the surrounding

plume. While the former mechanism clearly depends on the volume of the hot

plasma the latter one depends on the path length through the beam target. For the

15

overall yield, which clearly has to be the sum of both mechanism, an approximation

can be given by

Y ≈ τd

2

∫n2

i 〈σv〉 dV + Ni

∫n2

i 〈σv〉 dl (2.23)

where ni is the average ion density in the plume and Ni is the number of ions,

that leave the filament and fly through the plume. For the hot plasma filament

we assume a fusion reactivity 〈σv〉 that is determined by the cross section weighed

with the ion velocities given by a Maxwell distribution, whereas for the beam target

the cross section 〈σv〉 refers to an individual ion at speed v hitting another particle

at rest. Moreover the plasma disassembly time τd reflects the fact that the fusion

yield in the hot filament depends on the confinement time of the plasma, i.e. how

long the density remains high enough for an appreciable fusion yield. Even though

the first contribution only depends on macroscopic quantities it is also based on

binary collisions of ions which however are indistinguishable in this case (both have

an energy described by the same distribution function). For this reason the first

addend has a factor of 12 which makes sure that fusion events are not counted twice.

Figure 2.4: Two different mechanisms contribute to the total yield - On the onhand the interaction of hot ions in the plasma filament (red) and on the other handcollisions of fast ions (created by coulomb explosions) with cold ions in the plume(also called ”beam-target”).

The beam target contribution is frequently neglected since the density inside

the plume is relatively low, such that the probability for fusion events is compar-

16

atively low, too7. For this reason we want to restrict our attention in the further

discussion to the prevailing ion-ion mechanism.

The disassembly time can be estimated to be given by τd = γV 1/3/v where

γ is a dimensionless geometry factor, V is the volume of the plasma filament and

v is the average ion velocity. This mans, the plasma is initially confined to a small

volume determined by the focal filament and decays subsequent to the irradiation

on a time scale determined by the time it takes an average ion to leave the filament8

(which clearly depends on the average ion velocity and the geometry of the plasma).

For the sake of simplicity we assume a Maxwellian distribution (which is justified

by the above discussion, s. Figure 2.3) such that v relates to the average ion energy

Ei by the equation

v =√

16Ei/3πmi (2.24)

Besides this we assume for the sake of simplicity that the laser energy is completely

absorbed in the plasma9, i.e. the average ion energy Ei equals the laser energy

Elaser divided by the number of ions Ni. This leads to following expression

Elaser = NEi = V niEi = V ni · 32kBT

V =Elaser

ni · 32kBT

(2.25)

where the relation Eion = 32kbT was used. In summary, using equation (2.24) and

(2.25), an expression for the disassembly time is obtained that only depends on

known quantities7Data presented in this thesis suggest that the beam target contribution could be as high as 2%

of the total yield. However we do not understand yet the mechanism that causes this surprisinglyhigh contribution, which supposedly is significantly smaller.

8A comparison of data from interferometry (showing the shock front caused by the expansionof the filament) and signals from Faraday cups prove in fact that the expansion velocity of thefilament is roughly consistent with the hot ion component (which presumably comes from Coulombexplosions).

9This is roughly true, since ion yields and energies measured with Faraday cups suggest aconversion efficiency of up to 60%. However it may be argued, that these ions acquire a large partof there energy from interactions with electrons that are believed to surround the hot filament andhence after they have left the most likely region of fusion interaction.

17

τd =γV 1/3

v=

γ( Elaser3/2kBTni

)1/3

√16 · 3

2kbT/3πmi

= 0.547 · γE1/3laserm

1/2i

n1/3i (kBT )5/6

(2.26)

For the dependence of the fusion reactivity of two deuterium nuclei on the temper-

ature T an empirical equation can be used, [23]:

〈σv〉DD [cm3/s] ≈ 2.33 · 10−14T−2/3[keV ] e−18.76T−1/3[keV ] (2.27)

which for plasma temperatures below 25keV is found to be in very good agreement

with experimental data.

Next we make an additional approximation by demanding that not only the

ion density10 but also the temperature and hence the plasma reactivity are constant

over the volume of the plasma filament. Then the first integral in (2.23) can be

written as a product such that using (2.27) the dimensionless overall yield Y is

found to have the following scaling behavior11:

Y ≈ 1.662 · 1026 γ · E4/3laser · m

1/2ion · n

2/3D · 〈σv〉

(kBT )11/6(2.28)

Despite of the steady slope of 〈σv〉DD the yield decreases above a critical tempera-

tures when the disassembly time of the plasma decreases faster (owing to higher ion

energies) than the plasma reactivity increases.

In other words for a temperature of T = 15.6keV the fusion yield becomes

maximal (Figure 2.5) since the plasma reactivity 〈σv〉 weighed with the length of

stay of an ion in the plasma has a local maximum there (〈σv〉/(kBT )11/6 = 1.3 ·10−20cm3s−1(keV )−11/6). Moreover we estimate that the ion density averaged over

the volume of the hot plasma filament and the density in the clusters (which is close

to solid density) is related by a geometrical factor of 4/3π(RD )3, where R indicates

10The ion density ni becomes here more specifically the density of deuterium ions nD.11The units in this equation are: Elaser in [J ], mion in [kg], nD in [cm−3], 〈σv〉 in [cm3 · s−1],

and kBT as usual in [keV ].

18

0 5 10 15 20 25 30 35

0

2 ´ 10- 21

4 ´ 10- 21

6 ´ 10- 21

8 ´ 10- 21

1 ´ 10- 20

1.2 ´ 10- 20

Temperature [keV]

Plasma reactivity and fusion yield scaling

as function of ion temperature

fusion yield scanling,

<>/(kT)

in [cms(keV)

]s

vB

11/6

3-1

-11/6

0

1 ´ 10- 18

2 ´ 10- 18

3 ´ 10- 18

4 ´ 10- 18

5 ´ 10- 18

6 ´ 10- 18

7 ´ 10- 18

plasma reactivity,

<> in

[cms]

sv

3-1

b)

a)

Figure 2.5: Plasma reactivity (a) and expected fusion scaling (b) as a function oftemperature - Although for this particular energy range the fusion cross sectiongrows steadily with higher ion energies the fusion yield has a local maximum forenergies T = 15.6keV . This is because for higher ion energies the retention periodof the ions in the filament becomes shorter and hence the probability for them to fusedecreases.

the cluster radius and D the mean distance between two clusters. For a ratio

O(D/R) ≈ 10 the resulting deuterium density is about nD = 1019cm−3 (which has

been experimentally verified by means of interferometry, [4]).

Based upon the rough assumptions made above and assuming furthermore that

the ions have an average energy of 24.4keV (which corresponds to the optimum

temperature) an upper bound for the fusion yield can be set. The fusion yield,

which is twice the number of the emitted neutrons, depends now only on the laser

19

pulse energy and a geometric factor which is on the order of one:

Y ≈ 5.8 · 105 · γ · Elaser[1J ]4/3 (2.29)

Compared with recent experimental data (2.29) turns out to be slightly too opti-

mistic. This might be due to the fact that for this calculation we assumed a mono

disperse cluster size distribution, with a single cluster size (given by (2.17)) of about

30nm (granted that ncluster = 3 · 1022 cm−3).

In order to fully strip such a cluster12, the ponderomotive potential (defined in (2.2))

has to exceed the surface potential of 39keV (on the outside of the cluster, (2.15)),

which provides us with the minimum, required intensity:

Ic [W/cm2] = 3.03 · 1015 ·R2[nm] (2.30)

for λ = 800nm and ni = 3 · 1028 m−3

This means, that for clusters of R = 15nm the required critical intensity is about

Ic ≈ 6.8 · 1017 W/cm2 which in fact can be delivered by current tabletop terrawatt

systems. By using pre-cooled deuterated methane (CD4) the required cluster size

(however not mono-disperse) might become practical in future experiments, such

that the full potential of this cluster experiment can be tapped.

2.5 Thermonuclear cross sections

Since a major part of this thesis is devoted to the measurement of the angular

emission of neutrons a brief discussion about the differential cross sections is given

in the following.12Experimentally some severe difficulties were encountered, when an increase of the cluster sizes

was attempted, [7]. Even though these difficulties remain to be overcome, it might be possible thatthe main problem, namely that the laser gets absorbed either before it comes to a focus or beforeit reaches the high density spot in the middle of the plume, can be solved by a suitable machiningbeam (see last chapter).

20

Since we are only concerned with deuterium we want to focus on the two

thermonuclear reactions:

D + D50%−→ T (1.01MeV ) + p(3.02MeV ) (2.31)

D + D50%−→ He3(0.82MeV ) + n(2.45MeV ) (2.32)

Moreover it is a well-known fact, that most of the fusion events happen in the high-

energy tail of the Maxwell distribution. This behavior can be explained by the steep

increase of the fusion cross section for higher ion energies which overcompensates

the relatively small number of ions at this energy. As shown in Figure 2.6 it turns

out that for a deuterium plasma at a temperature of T = 12keV most fusion events

(≥ 97%) happen at ion energies13 higher than 20keV and 50% of the events even

arise from ions with more than 46.4keV .

This fact underlines the importance of the cluster size distribution discussed

in previous sections, since it shows that a plasma from a mono disperse size-distribution

at the same temperature would have a much lower fusion yield (because there would

be no ions at higher energies than Wmax which relates to the temperature by (2.17)).

Furthermore the contribution from high energy ions is relevant for the interpretation

of the differential cross sections, since it allows us to disregard energies lower than

20keV , which results in a negligible error.

The differential cross sections for both reactions (2.31) and (2.32) were mea-

sured by R. Brown et al., [18], with high precision in the center of mass system

(cm). However, since we are interested in measurements in the laboratory coordi-

nate system we have to convert the cross-sections obtained by Brown et al. using

the following equation

σl(θl, φl) =(1 + γ2 + 2γ cos θ)3/2

|1 + γ cos θ| σcm(θcm, φcm) (2.33)

13In contrast to Figure 2.5, where the fusion yield is plotted as a function of temperature, inFigure 2.6 the contribution of ions at different energies is shown for a given plasma temperature.

21

Maxwell distribution (T=12keV) and

thermonuclear cross section (DD)

[a.u.]

0 20 40 60 80

50% 50%

Maxwell distribution

total cross section

relative fusion

contribution

ion energy [keV]

Figure 2.6: Despite of the small number of ions in the high energy tail of theMaxwell distribution (red) the main contribution to the fusion comes from this regime(turquoise). In this connection the relative fusion contribution is obtained by weigh-ing the nuclear cross section (green) with the Maxwell distribution and normalizingit.

which accounts for the motion of the center of mass in the lab system, i.e. for the

fact that in a moving system the angles are perceived differently than in a reference

frame at rest. Thus in (2.33) the subscript l stands for the laboratory system and

correspondingly cm stands for the center of mass reference frame (for more details

s. [19]). The coefficient γ is given by the ratio of the speed of the center of mass

in the laboratory system to the speed of the observed particle in the center of mass

system. Based upon geometric considerations for the motion of the particles and

the angle between the momenta in both systems γ is found to be given by

γ =

√m1m3

m2m4

E

E + Q(2.34)

where Q is the energy released by the nuclear process and E is the energy initially

associated with the relative motion in the center of mass system, which relates to the

22

initial energy in the lab system El by E = m2m1+m2

El. For m1 = m2 = mDeuterium,

m3 = mneutron, m4 = m3He, an initial ion energy of El = 20keV , and a total release

of energy of Q = 3.27MeV we obtain γ ≈ 0.032.

d/d

[mbarns/str]

sW

Differential cross section

for H(d, He)n (at E =20keV)2 3

l

degrees from incident particle [°]

H2

He3

n

a

0.005

0.010

0.015

0.020

0.025

0.030

cross sectionin c.m. system

cross sectionin lab system

30 60 90 120 150 1800

Figure 2.7: Differential cross sections in the center of mass and laboratory - Thegreen curve shows a fit to the experimentally observed differential cross section forone particle with an energy of 20keV colliding with another particle at rest, [18].The red curve shows how this particular cross section looks like in the lab system.

In Figure 2.7 the two most extreme cases are shown based on a fit to exper-

imentally observed differential cross sections (provided by [18]): On the one hand a

collision of a particle that carries all the energy with another particle at rest, and

on the other hand a collision in which both particles have identical momenta. In

the latter case the center of mass is at rest in the laboratory system and therefore

the cross sections are identical in both frames of reference. Furthermore an exact

differential cross section can be computed for each distribution of energy among

the two particles (where the amount of energy remains fixed), which can be done

23

by imagining a frame of reference that moves exactly at the velocity of either one

of the particles. This obviously is equivalent to the case of one particle at rest for

which we know the differential cross section. By using a Gallileo transformation the

result can be expressed in terms of angles in the lab system.

However we know that the main contribution to the fusion yield comes from

collisions within the hot filament and thus we have no reason to believe that for

these collisions the energy14 is not randomly distributed among the two colliding

ions. This means that for such a statistical ensemble the observed differential cross

section in the laboratory frame is going to approach the shape of the differential

cross section in the center of mass frame, such that we do not need to use Gallileo’s

transformation.

Owing to conservation of momentum, for higher energies of the bombarding

ion15 the differential cross section becomes increasingly anisotropic. In other words,

if the energy of the bombarding ion becomes comparable to the energies released

in the nuclear reaction, then the particles created in that nuclear reaction have to

fly away essentially along the axis of collision. This is because the created particles

have a fixed energy, which also gives them a fixed momentum, which means that

since the total momentum has to be conserved, an appreciable component of their

momenta has to be directed along the axis of collision (which of course is identical to

the direction of the initial momentum). This behavior is shown in Figure 2.8, where

the anisotropy is defined as the normalized difference between the probabilities for

a neutron emission along the axis of collision and perpendicular to it.

This means that for ion collisions along a fixed axis, independently of the ion14Which is of course determined by the temperature of the plasma.15For the sake of simplicity in the following discussion it is assumed that one particle carries all

the energy an the other one is at rest. However the observation that the anisotropy increases is notlimited to this case.

24

30 60 90 120

ion energy [keV]

0.2

0.4

0.6

0.8

1Anisotropy [ ]

Anisotropy of the differential cross

section for H(d,n) He in the c.m. system2 3

50% 50%

best fit:

0.381+0.008 x - 0.00002 x2

0

Figure 2.8: The anisotropy, given by [dσ(0)dΩ − dσ(90)

dΩ )]/dσ(90)dΩ , increases for higher

ion energies. 50% of the fusion events arise from collisions at energies less thanindicated by the dotted turquoise line and 50% arise from energies on the right sideof the line. This refers to a plasma in thermal equilibrium (described by a Maxwelliandistribution) at a temperature of T = 12keV . (Based upon data from [18])

energy16 the probability for neutron emission along the same axis is larger than than

in perpendicular direction. Even though the exact factor describing the discrepancy

between both directions depends on the ion energies the qualitative behavior stays

the same, such that for a distribution of ion energies an integrated anisotropy factor

can be defined.

Returning to the case of a plasma from exploding clusters this means that

if the ions were solely emitted into one direction (e. g. the axis of polarization)

and hence the collisions exclusively happened along this imaginary axis, then the

neutron yield would show an anisotropic pattern similar17 to Figure 2.7 where the16This is of course only true for the energy domain studied here.17In general it wont be identical, because the total cross section strongly depends on the energy

of the ions and hence the differential cross section does, too.

25

ratio of the maximum angular yield to the minimum angular yield is given by the

anisotropy factor plotted in Figure 2.8. If we reverse that conclusion we can set

from the degree of anisotropy of the observed neutron yield an upper bound for

the anisotropy of the ion collisions which ultimately relates to the symmetry of the

cluster explosions. For instance, if the angular neutron distribution for a particular

experiment turns out to be flat, this implies that the collisions between ions are

isotropic, too.

26

Chapter 3

Experimental Setup

3.1 Design of a neutron detector

In order to obtain a quantitative measure of the neutron yield an ideal detector

should be capable of detecting single particles. Furthermore in the particular case

of neutrons from laser driven fusion experiments a fast time response on the order of

nanoseconds is required in order to distinguish via time of flight diagnoses between

neutrons and x-rays. Even though the latter ones are usually only produced in small

numbers for moderate laser energies (≤ 1J) with increasing cluster size (for instance

for cooled CD4) their number can become significant.

A commonly used type of detector is composed of a scintillating material and

a photo multiplier tube (PMT) which are optically coupled together. Pictorially the

working principle of a scintillator is that an incoming charged particle will excite

some of the atoms in the medium to higher atomic levels (for a more accurate

discussion s. [24]). When these atoms decay to their ground state with the emission

of a photon this is called scintillation light (which will be amplified in the PMT).

An important distinction has to be made between plastic scintillators, in which the

scintillating behavior is a property of individual - usually complex - organic molecules

27

and inorganic scintillators for which scintillation is a property of the crystal lattice.

Most of the scintillating materials used for neutron detection take advan-

tage of the fact that the hydrogen-nucleus has a comparatively large cross-section

for neutron interactions and that a high density of protons can easily be achieved

in hydrogen compounds leading to a mean free pass on the order of a couple of

centimeters. Moreover neutrons can transfer a significant fraction of their energy

to recoiling protons which subsequently will excite the medium. It turns out that

atoms in an organic molecule (in case of a plastic scintillator) are excited by recoil-

ing protons much faster than they can move such that not only they will be in an

electronically excited state but also the molecule will be in a vibrationally excited

state. In a suitable material this vibrational excitation will be partially transferred

to other atoms (which may subsequently decay to lower state) and partially dissi-

pated without radiative transition. This is very important because this implies that

the (optical) absorption and and emission bands do not fully overlap which means

that the material is transparent for its scintillation light.

As a rule of thumb it can be assumed that for each 620 eV of depleted

energy (from a proton) one photon (usually of 350-500nm wavelength) is created

in the scintillator. Since the depletion of energy of a neutron in the detector can

essentially happen with any fraction of its kinetic energy (typically 2.45 MeV for

DD-fusion) the response function is essentially flat. Furthermore also the number

of neutrons going through the detector may vary significantly which requires a large

dynamic range of the PMTs.

For the detectors used in our experiment we made the following assumptions:

A typical neutron1 event will produce 2000 scintillation photons within 10 ns. A

highly reflective UV coating reflects 80% of the photons into a conical light guide

that connects the scintillator to the PMT. An index matching oil between the light1This is, a neutron that depletes half of its energy in the detector (1.2 MeV).

28

guide and the front surface of the PMT helps to avoid internal reflections such that

about 50-60% of the scintillation photons hit the photocathode whose quantum

efficiency supposedly is about 25%. That means that 250-300 photoelectrons are

created which allows to make the following estimate

Udetector = nelectrons · e · g · Z/τ (3.1)

where Udetector denotes the expected peak voltage of the detector signal, nelectrons

stands for the estimated number of photoelectrons and e is the charge of an electron.

A typical photomultiplier has a gain of g = 1 · 107 and an output impedance of

Z = 50Ω so that with a neutron burst within τ ≈ 10ns we expect a peak signal of

Udetector ≈ 2V .

3.2 THOR-Laser

In the following a brief overview over the Texas High-Intensity Optical Research

laser faciltiy (THOR) is provided.

The THOR laser represents a kind of table-top multi terrawatt system that was

facilitated by the invention of chirped pulse amplification (CPA) in the 1980′s, [25].

One of the limiting factors, that led to the development of CPA, was that the inten-

sity could not be further increased without exceeding the intensity damage threshold

of the laser optics. That means that for even higher brightness the intensity within

the amplifying stages has to be artificially decreased somehow, in order to avoid

damage within the laser. One way of doing this would be to increase the aperture

of the optics, which is clearly limited by the availability of large diameter optics and

also restricted to large scale research facilities. A more feasible approach is to take

an ultra-short laser pulse and to stretch it in time prior to amplifying it and then

recompress it in time before it is sent to the target (this requires only final optics

with large apertures).

29

Figure 3.1: The THOR-Laser: The pulse coming from the oscillator is stretchedin time and afterwards amplified in three different amplification stages. Finally thepulse is re-compressed in time and propagates through vacuum to the target. (Bycourtesy of Will Grigsby)

By using an optically active material in the cavity of the oscillator which has

a broad transition, various cavity modes with different wavelengthes can achieve

a fixed phase relation (mode-locking). This type of commercially available oscilla-

tors produce very short pulses of a few tens of femtoseconds (in our case 20 fs at

800nm center wavelength). Owing to Heisenberg’s uncertainty principle this small

uncertainty in time corresponds to a significant uncertainty in energy, which for the

radiation is equivalent to a wide spectrum of tens of nanometers (≈ 30nm FWHM

for the THOR laser).

This wide spectrum of the laser pulse can be spread in a dispersing system

containing a diffraction grating and imaging optics (spherical mirror + rooftop mir-

30

ror). The spectrum is dispersed such that for shorter wavelengths the optical path

becomes longer than for shorter wavelengthes which ultimately results in a longer

pulse duration (this broad spectrum is symbolized in Figure 3.1 by a visible spec-

trum from blue to red even though for Ti:Saphire lasers the spectrum is infrared).

This beam, which now has a pulse duration of 600ps instead of 20fs and conse-

quently an intensity that is reduced by a factor of > 104, can safely be amplified.

For this purpose three different multi-pass amplification stages are used which are all

composed of Ti:sapphire crystals2 which are pumped by q-switched Nd:YAG lasers.

The low-energy beam coming from the oscillator is first amplified by a regen-

erative amplifier to about 2mJ , which is achieved by 20 passes through an optically

pumped Ti:sapphire crystal. The pulse is switched out by a high-speed pockels

cell when the gain starts to saturate. Since the maximum energy density that can

be deposited by the pump beam is limited by the fluence point of the amplifying

medium the diameter of the crystal has to increase for higher energies of the seed

pulse (which clearly acquires its energy from the energy deposited by the pump

beam in the material). In order to achieve good spatial overlap of the seed and the

pump beam the aperture of the seed pulse coming out of the regenerative amplifier

is increased from two millimeters to about four millimeters. The seed pulse is then

subsequently amplified to 22mJ in the four-pass amplifier. The final amplification

happens in a five pass amplifier with a Ti:sapphire crystal of 20mm diameter, whose

pump pulses can be intentionally mistimed with respect to the seed pulse in order

to control the output energy over a range of 5− 800mJ .

After the last amplification stage the pulse is sent into a compressor that

again takes advantage of the broad spectrum of the pulse and spreads it with an-

other diffraction grating. However, as opposed to the stretcher, the grating in the2Ti:sapphire is a very suitable material for the discussed purpose since it has a very broad

transition and hence if used for amplification a very homogenous gain for a broad spectrum ofincoming light. This is in particular important since it helps to maintain the bandwidth of thepulse which is crucial for the pulse duration after the compression.

31

compressor is now arranged with the imaging optics in such a manner, that for

shorter wavelength the optical path is shorter. If the geometry of the compressor

exactly compensates for the differences in path length in the stretcher an efficient

compression in time can be achieved. However the pulse duration (FWHM) increases

after the compressor to about 40fs as compared to the pulse from the oscillator,

which is due to the loss of bandwidth due to gain-narrowing and the dispersion

acquired through different materials that are traversed by the pulse3. The pulse is

amplified from initially 6nJ coming out of the oscillator to 1.3J after the 5-pass

such that a final energy of 0.8J after the compressor is achieved.

3.3 Gas jet

Since the cluster size distribution depends very critically on the parameters of the

gas jet, a brief characterization of a cryogenically cooled high pressure gas jet is

given in the following (for more details refer to [20], [21]).

The clustering process relies on the mechanism, that a supersonic nozzle

causes the pre-cooled gas to expand in such a manner, that the molecules within the

gas have very similar velocities. This is achieved by a conical shape of the nozzle

which converts the momenta of the molecules, which are initially directed randomly,

along the axis of symmetry. In the system of the expanding gas this means a further

cooling, such that clusters can form.

It turns out, that as well the onset of clustering as the cluster size can be

characterized by an empirical scaling parameter referred to as the Hagena parameter:

Γ∗ = k · (d/ tanα)0.85

T 2.290

· P0 (3.2)

where d is orifice (in mm), α the expansion half angle (α < 45 for supersonic3This happens even though the dispersion is pre-compensated by a fiber that takes into account

phase effects up to the 5th order.

32

expansion), P0 the backing pressure (in mbar), T0 the temperature of the reservoir

(Kelvin), and k a constant related to bond formation (181 for D2 and 2360 for

CH4). Moreover the onset of clustering is observed to be at Γ∗ > 100− 300 and the

number of atoms per cluster Nc scales like N∝(Γ∗)2.0−2.5. Since the radius of the

cluster clearly scales with the number of atoms per cluster like R ∝ N1/3c it follows

that the radius (which we belive to be crucial in many respects, s. Theory section)

scales with the temperature as R ∝ T 1.5−1.9 (at constant pressure) and with the

pressure as R ∝ P 0.66−0.830 (at constant temperature).

SupersonicNozzle

Poppet

2 a

Indium Seal

Figure 3.2: Gas jet -Typically a backing pressure of 700 − 1000psi is applied toproduce nanometer size clusters of D2 or CD4. The gas is cooled down by coolnitrogen running through the coolant pipes. By magnetic force the poppet is liftedinside the solenoid for variable durations (typically about 1− 3ms, such that the gasexpands, further cools down, and starts clustering. (Picture adapted from [20])

33

Since the same nozzle and the same pressure was used in our experiment

for D2 and CD4 (i.e. (d/ tanα)0.85 in (3.2) is in both cases the same), we would

expect a variation of the cluster size of about RD2/RCD4 ≈ 1.3−1.7 for CD4 at room

temperature and RD2/RCD4 ≈ 1.2−1.4 for modestly pre-cooled CD4 (T0 ≈ 275 K).

The cooling is achieved by flowing gaseous nitrogen, which is in thermal

contact with liquid nitrogen (or icy water for CD4), through the cooling pipe. By

changing the flow rate through the line, the temperature can be controlled over a

wide range.

Typically for our experiments the gas jet is fired for 1ms and a backing

pressure of 1000psi which leads to cluster sizes of a few nanometers.

34

Chapter 4

Experimental results

In the following an overview over the experiments with deuterium and deuterated

methane is given. The main purpose of this experimental campaign was to measure

the angular distribution of the neutron emission in order to gather information

about the prevailing mechanism of fusion. This is motivated by the assumption

that a dominant reaction of two hot ions inside the plume leads to an essentially

isotropic emission1 a significant beam target contribution could cause anisotropies

owing to the non-symmetric irradiation of the plume. However it may be subject

to further theoretical studies whether the cylindrical shape of the plasma filament

imposes a preferential direction of collisions.

4.1 Systematics of neutron yield measurements

In order to make an angular scan of the neutron yield a very thorough understanding

of the systematics involved in such a measurement is crucial. For this purpose

significant efforts had to be devoted to a data analysis that provides stable results

despite of changing noise-levels, strong signal fluctuations, and EMP-induced signal1That is because a fully stripped cluster explodes isotropically and hence collisions of ions from

different clusters are going to happen isotropically, too.

35

oscillations.

0 10 20 30 40 50 60 70 80 90 100-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

arrival time (zero crossing)

peak height

time [ns]

de

tecto

r sig

na

l [

a.u

.]

Typical Neutron Event

+ 1 s

- 3 s

Figure 4.1: Typical detector event for a 2.45 MeV neutron

The EMP induced oscillations2 and high frequency noise is remedied by

smoothing of the data (mauve plot in 4.1) and by subtracting the average back-

ground, which has to be determined from shots without neutron events3. However

the distinction between a signal that was caused by a neutron and one that was not,

always requires some discrimination, which results in a slight error4. As discussed

above, neutrons can deplete variable fractions of their energy in the detector (lead-

ing to widely varying single neutron signals) such that this discrimination becomes

especially significant for very low neutron yields. This problem is in addition to the

intrinsic statistical uncertainty of low neutron yields the dominating contribution2We believe that the discharge of high voltage capacitors in the q-switch of the pump laser is

mostly responsible for that behavior.3Only temporally neighbored shots can be considered since the noise level changes qualitatively

and quantitatively over time.4However the number of signals which are disregarded even though it is not certain that they

were not caused by a neutron is about < 2% of the total number of shots.

36

to the errorbars. Moreover we figured out that in order to determine the number of

neutron events, the pulse height is often a better measure than the integrated pulse,

since the signal slowly decays to zero, which is due to some RC-constant (s. Figure

4.1).

Furthermore for time of flight analysis, a reproducible way to determine the

arrival time has to be found, which we did by defining the arrival time of a neutron

as the firs zero crossing (s. Figure 4.1).

For azimuthal scans (i.e. scans that cannot be done by a change of polariza-

tion) an additional challenge needs to be overcome: The attenuation of the neutron

flux depends on the amount of material the neutrons have to traverse. It turns out

that as shown in Figure 4.2 an aluminum plate as thin as 0.5cm in front of a detector

can attenuate the detected neutron yield by 16% (difference in peak heights).

240 260 280 300 320 340 360 380 400

-16

-14

-12

-10

-8

-6

-4

-2

0

time [ns]

dete

cto

rsig

na

l[a

.u]

integrated signal

with aluminum plate: 1.71 103

integrated signal

without aluminum plate: 1.8 103

Neutron attenuation in Al

Figure 4.2: Attenuation of neutron flux by a 1/4′′ aluminum plate

For this measurement the yield was averaged over 400 shots which corre-

sponds to about 1200 events in the detector, while the overall yield was controlled

37

by other detectors. While a potential change of the overall yield was controlled with

nearby detectors the overall yield was provided by a detector that was far enough

away to be in single neutron counting mode. Basing on the so-obtained overall

yield and furthermore assuming an isotropic neutron emission the number or neu-

tron events in a detector is just given by its solid angle. In this context it might

be noteworthy that the first peak in Figure 4.2, which corresponds to x-ray events,

experiences a much stronger attenuation.

4.2 Deuterium Clusters

In this section we are going to discuss an experimental campaign that intended to en-

hance our understanding of the neutron emission in D2 and to study the systematics

compared to CD4. All of these measurements were performed at the THOR-Laser

using the following parameters: The pulse energy was about Wpulse ≈ 200mJ and a

focal optic with F# = 5 was used which at a pulse duration of τ ≈ 40fs translates

to a peak intensity5 in vacuum of I0 > 1018. All data presented in this section were

taken with a sonic nozzle6 that fired ≈ 750µs before the laser hits the jet and whose

opening angle was about ≈ 120 with an orifice of ≈ 750µm.

4.2.1 TOF-Data

The most important piece of information to identify the particles registered by the

detector is via time of flight diagnosis (TOF). The time of flight histogram shown

in Figure 4.3 shows some x-ray events and at a later time neutron events from5Here the assumption was used that the size of the focal spot of an optic with F#-number 5 is

given by w0 = 1.2·λ [nm]·F# ≈ 5µm which does not account for spherical aberrations. Besides this

the intensity was assumed to be given by the equation for gaussian beam profiles I0 =4√

ln 2Epulse

w20π3/2τ

.6Since the density profile and the cluster size distribution are not identical for a sonic and super

sonic expansion, it is not clear to which extent information obtained for one type also apply for theother type.

38

Figure 4.3: TOF traces and histogram for the irradiation of D2 clusters

DD fusion that correspond to their specific energy. The spread in both peaks is

due to the limited time response of the detectors (about 10ns for neutrons) and

delays by scattering events. Besides these features there are some additional events

in a time window that roughly corresponds to the expected arrival time of high

energy neutrons from DT fusion. This would mean, that the tritium created in

DD fusion (s. equation (2.31)), undergoes a secondary reaction characterized by

D + T → He4 + n(14.1MeV ), which is rather unlikely, despite of the fact that the

fusion cross section for the energetic triton is considerably higher. In fact compared

to the cross section of a single deuterium ion, the cross section for a nuclear reaction

of a triton is higher by four orders of magnitude for its initial energy of 1.01MeV

(and even by five orders of magnitude if it gets slowed down to 160keV , s. [23]), but

39

nevertheless its abundance is determined by the number of primary fusion events.

The number of secondary reactions can be assessed from the ratio of the abundances

(NT /ND) times the ratio of the cross sections (σT /σD). For typical values of 105

neutron events (equal to the number of tritium ions) and about 1014 deuterium ions

at 1J laser energy ( [16]), we would expect the number of secondary reactions to

be 10−4 the number of primary reactions (for the likely case that the tritium ions

get slowed down). This means that the observed events probably are caused by a

small contamination of the deuterium source with tritium. By accounting for the

different plasma reactivities7 and the relative number of events, we can estimate that

a relative abundance of tritium as small as ≈ 0.05% the abundance of deuterium

would be sufficient to explain the observed DT yield.

4.2.2 Energy scan

An energy scan was performed in order to test the assumptions made for the scaling

behavior in (2.29).

In order to plot the yield vs. the pulse energy in Figure 4.4 the signal of each

detector was multiplied with a geometric factor takeing into account their different

distances from the gas jet. The graph shows the change of the average signal height

with the pulse energy for clusters formed in a sonic expansion. The solid line is

the least mean square fit to a power law dependence a1 · Ea2 with a1 = 0.765 and

a2 = 1.60. Even though the power-law dependence obtained here does not agree

exactly with our previous theoretical prediction, (2.29), it is in good agreement with

scaling factors found in similar experiments, [10], [16]. This is because (2.29) does

not account for the fact that higher laser energies lead to a deeper penetration of

the gas jet, which is equivalent to penetrating a region of larger average cluster size7For a temperature of 10keV the plasma reactivity for DT fusion is 〈σv〉DT ≈ 1.1·10−16 [cm3·s−1]

whereas the plasma reactivity for DD fusion is 〈σv〉DD ≈ 1.2 · 10−18 [cm3 · s−1]. Hence the plasmareactivity for DT is larger by a factor of 100.

40

Energy scan in D2

Laser energy [mJ]

neutr

on y

ield

[ ]

40 60 80 100 120 140 160 1800

500

1000

1500

2000

2500

3000

3500

4000

4500

Detector 1Detector 2Detector 3Detector 4

Bestfit: µ E1.60

Figure 4.4: Energy scan in D2 from a sonic expansion

and hence density, such that besides an increased ion yield also higher ion energies

(which lead to increased cross sections) are found. For this reason the theoretical

scaling factor is only valid under idealized circumstances.

The large discrepancies in Figure 4.4 between different data points are due

to the large distance of some detectors from the gas jet and the small number of

200 shots over which we averaged. This means that in particular for lower energies

some data points are based on significantly less than 50 events. A future experi-

ment should intend to enhance the accuracy and moreover to investigate the scaling

behavior over a broader range of energies.

4.2.3 Angular measurement by detector permutation

Another experimental run attempted to measure the angular distribution of the

neutron emission by permuting the position of all four detectors. By doing so we

intend to account for differences in the sensitivities of the detectors, which otherwise

would have needed to be calibrated very thoroughly.

41

Position A Position B Position C Position D0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Positions

rela

tive

co

un

t ra

te [

]

Normalized Neutron Yield fortwo different polarizations

corrected data

Figure 4.5: Angular scan in D2 obtained by permutation of four detectors (repre-sented by different symbols)

Hence the data shown in Figure 4.5 represent the relative yield measured by

permuting the position of four different detectors (represented by different symbols).

In the plot the contribution of the signal for a particular position to the total yield

(for all positions) is plotted for each detector. In other words the total yield for all

four positions was set equal to unity for each detector. We first observed a peak at

position B which is perpendicular to ~k and along the axis of polarization. However

for this position the only material the neutrons had to traverse is a vacuum window

which we assume to have a much longer attenuation length than the stainless steel

chamber walls. Therefore we assume for the red crosses in Figure 4.5 the same

attenuation as shown in Figure 4.2, which means, that we multiplied the original

values with the same factor that was found in Figure 4.2 to relate the peak heights

with and without aluminum plate to each other . In other words: The results for

42

the attenuation of the neutron flux plotted in Figure 4.2 show that even relatively

small variations in the amount of material, which the neutrons have to traverse,

causes different detector signals.

Besides this the window turns out to be essentially transparent for x-rays

even of modest energy, such that the x-rays could have distorted the neutron signal,

too. For these reasons it seems natural to artificially apply the attenuation found

for the aluminum plate to the detector signal in that position, in particular since the

attenuation of this plate presumably comes closer to the attenuation of the stainless

walls of the chamber (which shields the other detectors).

The data corrected by this factor indicate that a possibly underlying anisotropy

of the neutron yield has to be on the order of the errorbars.

4.2.4 Polarization Scan

In order to avoid attenuations of the neutron yield by changing the position of the

detectors we carried out a polarization scan. For this purpose the yield was mea-

sured at three different polarizations (90, 54, 3) that were realized by periscopes.

Besides avoiding difficulties to account for the variation in the neutron flux attenua-

tion a polarization scan provides information about the dependence of ion collisions

on the polarization, whereas a comparison of data from Position A and Position B

(in Figure 4.5) also may depend on asymmetric properties of the gas jet.

Figure 4.6 shows the relative change of the observed fusion yield when the

polarization was changed from 54 ± 3 to 90 ± 3. In other words the ratio of

the detector signals at the initial polarization and at the modified polarization are

plotted, after these values were normalized by the overall yield (measured on detector

D). The polarization was measured with respect to horizontal, i.e. with respect to

the plane shown in the setup.

Since a comparison of data for horizontal and vertical polarization does not

43

0

0.2

0.4

0.6

0.8

1

A B D Cdirection

norm

aliz

ed y

ield

[ ]

Normalized Neutron Yield fortwo different polarizations

Figure 4.6: Comparison of normalized neutron yields for two different polarizations(54 ± 3, 90 ± 3).

show a different behavior we can conclude that the neutron emission is at least

cylindrically symmetric with respect to the axis of propagatiion.

4.3 Hetronuclear Clusters: CD4

The motivation for exploring clusters comprised of hetronuclear molecules in the

context of nuclear fusion experiments was initiated by theoretical work by Last et

al., [17]. Based upon the cluster equations discussed earlier and particle simulations

they predict higher deuterium energies from Coulomb explosions for the case of D2O

under certain circumstances. More specifically they define a kinematic enhancement

factor η = 〈qD〉mB/〈qB〉mD where mD and mB denote the masses of deuterium and

the heavier ion, and where 〈qD〉 and 〈qB〉 stand for the average charge states of deu-

terium and the heavier atomic species, respectively. They predict that the lighter

deuterium ions are going to outrun the heavier ions if η > 1 such that the deuterium

ions experience an outer shell explosion (at an increased average radius compared

44

with their initial position) whereas the heavier species is going to remain longer in

the center of the cluster.

For the case of CD4 this means that while the average charge state of carbon is

smaller than 〈qC〉 < 6 we can expect a kinematic enhancement. In fact consid-

erably higher ion energies were observed for CD4 as compared with D2, [16], but

that may as well be due to a variety of other reasons. So for instance does the final

energy of the ions also depend on the ion density inside the cluster as well as on the

average charge state, which both can easily be larger for methane. Nevertheless this

chance to increase the ion energies even further should be explored because then

an even steeper increase in the neutron yield can be expected, too (for low laser

energies the yield in CD4 was already found to be higher than in D2 even though

the experiment was not in the regime of a kinematic enhancement, [16]).

Besides this advantageous increase in ion energy a surprising anisotropy of the ion

yield was reported by Grillon et al., [5]. They proposed an explanation saying that

lateral collisions of ions from the hot core of the plasma with nuclei in the surround-

ing plume contribute more significantly than initially thought8. The experimental

results shown in the following intend to further understand this interesting phe-

nomenon.

The sketch in Figure 4.7 shows the setup9 that was used for the azimuthal

and polar angular scan in CD4. For the azimuthal scan detector 1 was moved outside

the plane of the flanges of the chamber to avoid differet attenuations of the neutron

flux. Unless stated otherwise the polarization for all experiments is horizontal (i.e.

in the plane of the drawing) however it can be changed from outside the vacuum

by the λ/2 waveplate. In order to avoid losses of intensity due to self-focusing or8The underlying assumption here is, that the hot plasma filament is not in the center of the plume

since the laser can only penetrate the gas jet to a certain extend. This results in an asymmetricpath of the ions through the plume, which presumably leads to an anisotropic neutron emission.

9The setup was changed for the experiment in CD4 because that allows it to benefit from thefull power of the THOR laser.

45

ionization of air, which may happen even with an unfocused beam, the compressed

laser pulse propagates in vacuum.

azimuthal angle

a

spherical mirror(F=5)

Detector 2Detector 1

Detector 3(looking upwards)

Detector 4(normalization)

40 fs pulse coming fromcompressor

l/2 waveplate

Figure 4.7: Experimental setup used for CD4 data

4.3.1 Polarization Scan in CD4

The first experimental campaign with CD4 was devoted to a measurement of the

polar distribution of the neutron emission. If we assume, that all relevant physical

parameters concerning the gas jet (i.e. density and cluster size) do not change over

the diameter of the filament (which can be assumed to be < 100µm) then the only

preferential direction for the hot plasma is given by the polarization of the laser10.

Hence first a polarization scan for the neutron yield was performed to check for a

possibly underlying dependence.

This measurement was carried out at a pulse duration of 40fs and with an energy

of 50mJ on target. The clusters were produced by a cryogenically cooled supersonic

nozzle with an opening angle of 10, an orifice of 750µm, and a backing pressure of

1000psi. In Figure 4.8 the polarization and thus the direction of the electric field10For interactions of ions from the hot core of the plasma with the surrounding plume this does

not necessarily hold. For instance effects of the geometry of the plume are conceivable.

46

Polarization scan for the neutron yield fromcluster explosions of CD4 A

B

± 5%

± 10%

b= °30

b= °60

b= °90

b= °120

b= °150

b=180° b=0°pola

rization

angle

[°]

1.5 1.25 1 0.75 0.5 0.25 0 0.25 0.5 0.75 1 1.25 1.5

normalized neutron yield [ ]

bk

r

E

r

pos.

B

position

A

Figure 4.8: Polarization Scan in CD4 - ”Normalized yield” means the percentage ofthe contribution of each of the two detectors to the total signal (as measured on bothof them together).

( ~E) was varied with a λ/2 waveplate in the plane of the plot (the laser’s direction

of propagation ~k is perpendicular to it).

For each polarization 700 shots were taken, which were subsequently aver-

aged and the average peak height was interpreted as measure for the neutron yield.

Apparently these data shown in Figure 4.8 do not have an obvious dependence on

the polarization of the laser which allows the conclusion that the interactions of the

ions from the coulomb explosion are essentially independent of the polarization, too.

If we infer from Figure 4.8 that the neutron yield does not vary with the polariza-

tion by more than at most 10% and furthermore that hence the intrinsic anisotropy

of the neutron emission from the nuclear reaction11 is largely washed out, we can11The anisotropy of the thermonuclear reaction in the laboratory system would be according to

Figure 2.7 about 60%. This number is obtained by dividing the maximum emission at 0 by theminimum emission at 90.

47

conclude, that the ion collisions happen with an anisotropy12 of < 2%. It remains

left for a thorough theoretical analysis to determine, what kind of upper bound for

the anisotropy of the cluster explosion can be set from this data.

Moreover a measurement of the polar distribution with fixed polarization would

give information about the effects of the density profile and the geometric shape of

the plume. This would be particulary interesting since it would help to interpret

the result of the azimuthal scan discussed below. In other words, while a polar scan

(with fixed polarization) is symmetric with respect to the plasma filament and hence

measures only density effects, an azimuthal scan clearly measures a yield whose com-

position is a convolution of density effects and a possible preferential direction for

collisions that might be imposed by the cylindrical hot core of the plasma.

4.3.2 Azimuthal Scan in CD4

In Figure 4.9 the raw data consisting of 700 shots were smoothed, averaged, and

afterwards integrated from the first zero crossing to the second. At a yield of ≈ 1000

neutrons per shot the detector shown in Figure 4.7 registered for each position about

900 events. The errobars for our data were estimated from the two detectors that

were used for normalization: Since their normalized signal did not vary compared

to each other by more than 5% we assumed the same error applies for the azimuthal

data (however this does not account for potential variations in the effective path

length through matter, which we think are negligible). As shown in Figure 4.9 a

slight anisotropy was observed, which however is not consistent with the pattern

observed by Grillon et al. (blue stars in Figure 4.9). In order to to minimize varia-

tions of the amount of material13 between the plasma filament and the scintillator,

the detector shown in Figure 4.7 was in an elevated position looking down to the12Here we just divide the measured anisotropy by the intrinsic anisotropy of the neutron emission.

13We estimate these variations averaged over the whole surface of the scintillator to be ≤ 1mm.According to Figure 4.2 this would result in an attenuation of ≈ 1.7%.

48

0 20 40 60 80 100 120 140 160 180

0.2

0.4

0.6

0.8

1

1.2

1.4

angle a [° ]

norm

aliz

ed y

ield

[ ]

Integrated SignalData by Grillon et al.

Integrated SignalData by Grillon et al.Integrated SignalData by Grillon et al.

Integrated Signal

Data by Grillon et al.

Neutron yield vs. azimuthal angle

Figure 4.9: Azimuthal scan in CD4 measured by moving the detector shown in 4.7

plane of the experiment at an angle of 45.

Since the filament created by the laser imposes a channel of hot plasma, in

which collisions resulting in a fusion event are more likely than events from collisions

of ions with residual atoms in the surrounding plume, an increase of the measured

yield in ~k direction is not surprising14. The explanation of the exact azimuthal

pattern however remains for further theoretical work, which we think may be based

on the simple model of a cylindrically shaped, hot plasma filament.

14This is again assuming that the intrinsic anisotropy of the neutron emission on a nuclear level,may be partially conserved as a statistical quantity if there is a preferential collision direction.

49

Chapter 5

Conclusions and Future

Directions

In this thesis an experimental study of the angular distribution of the neutron emis-

sion from fusion of deuterium in a high density gas jet is presented. Based upon

the data in the following brief discussion the attempt was undertaken, to illumi-

nate the relevance of this experimental campaign. In addition to this some ideas

how to improve the experiment and especially how to increase the fusion yield are

presented.

5.1 Interpretation

The experimental data presented above are besides the data taken by [5] the first

attempt1 to study systematically the neutron emission from fusion in a dense gas

jet. This may be due to experimental difficulties that arise from the small number of

neutrons typically detected at low laser energies (< 200mJ), which requires several

hundred shots per data point. Moreover a precision measurement of arbitrary angu-1This is true to the best knowledge of the author.

50

lar dependencies would require a special chamber design, in order to avoid unwanted

attenuation in flanges, widows, etc..

5.1.1 Angular Measurement

Despite of these systematic difficulties angular scans were performed in D2 and

CD4, which could set a tight, upper bound on the neutron emission dependence on

the polarization. These data give very little way to a dependence of a - potentially

underlying - preferential ion collision direction on the polarization. It would be

interesting to calculate from this essentially isotropic collision pattern an upper

bound for the dependence of the cluster explosion dynamics on the polarization,

which however is not trivially possible, since there is a large ensemble of spatially

distributed clusters.

Besides this, also an azimuthal scan was carried out for both gases which

however in the case of D2 has large systematic errors (s. Figure 4.5). Nevertheless

the azimuthal scan in CD4 shows a yet unreported increase of the neutron yield

along the axis of propagation of the laser. We believe that this is due to the cylin-

drical shape of the hot plasma filament, which means that the path length through

hot plasma is much longer for an energetic ion along the axis of the laser than per-

pendicular to it. This would mean that more collisions are going to take place along

this axis, which doubtlessly would also lead to more neutrons along that axis (s.

Figure 2.7). A more rigorous analysis of this phenomenon, which also takes into

consideration the density gradient in the filament2 has to be subject to a thorough

simulation.

We suggest a two dimensional model, that takes into account the density

dependence on the position as well as its dependence on time (which accounts for

the expansion of the hot plasma filament). Then it is obvious from Figure 5.1 that2Please refer to the lineout of the Rayleigh scattering amplitude in [16].

51

for any ion the collision probability and thus the fusion probability depends on the

direction α of its trajectory. Moreover Figure 5.1 shows, that for an ion flying from

the left to the right (i.e. from the low density area to the high density area) the

collision probability is higher than for an oppositely directed ion.

a

hot plasma filament(initially laser focus)

beam targetcontribution

randomly chosenion

cold ionor atom

hot ion

Fusion probability depending on the direction of motion

Figure 5.1: The probability for a randomly chosen ion (here the green one) to collideand hence to fuse depends on the direction α of its motion. This arises from thenon-uniform density distribution in the plume which implies, that the deeper thelaser penetrates the plume, the denser the plasma becomes, too. Furthermore, if theion’s path is mainly within the hot filament the probability is higher that it collideswith another hot ion, such that in some cases their velocities partially sum up leadingto substantially increased cross sections.

The overall angular distribution Φ(β) can now be obtained by evaluating

the integral over the differential cross section times the probability for such a fusion

event to happen. Since the total angular distribution is a superposition of all angular

probability distributions of a single fusion event, we have to integrate over all ions

(i.e. over all initial positions r0) as well as over all possible directions of motion3:3Our model does not account for the dependence of the anisotropy of the differential cross section

on the energy (s. Figure 2.8). In other words, for all neutron emissions form nuclear reactions anaverage anisotropy is assumed, which is justified as long the reaction mechanism is the same for

52

Φ(β) =

2π∫

0

∫

x0, y0

P (ro, α) · dσ

dΩ(β − α) d2r0 dα (5.1)

where P (ro, α) is the probability that a particle which initially was situated at

r0 = (x0, y0) is going to fuse on its path at a random angle α (with respect to the

laser’s axis of propagation). Once a suitable parametrization for all quantities is

found the probability for a fusion event can easily be evaluated

P0(r0, α) =

T∞∫

0

|r(α)|σ(E(r(t, α))) · ni(r(t, α), t)

dt (5.2)

where the path integral was parameterized by the time it takes the ion to leave

the plume, T∞. Here we assume that the trajectory of the particle, r(t, α), is

determined by α, its initial position and its constant velocity. The velocity r can

then be obtained from the temperature at the ion’s initial position, which provides

us with its energy and hence its velocity, too4. Moreover the cross section σ depends

on the total energy of the collision E, this is the energy of the observed particle plus

a variable fraction of the energy of the target ion (which is give by the temperature

at its position), depending on the random direction of its motion5 . Finally the

ion density ni(r, t) depends on the position within the plume and furthermore on

the expansion of the filament (which can be assumed to happen at the average

ion velocity). This means that (5.1) can be evaluated by means of a Monte Carlo

simulation that randomly varies the starting position of the ion, the angle of its

motion, and the degree of parallelism of the motion of the studied ion and the

motion of a potential target ion.

ions of all energies. Nevertheless the strong dependence of the total cross section on the energy istaken into account, when the probability for a fusion event is computed.

4Of course the energy is given by a Maxwellian distribution which may not be disregarded sincethis is crucial for the fact that more events happen in the filament than in the beam target.

5More rigorously, if both ions move along the same axis but in opposite direction their energiessum up whereas for different directions the energy is determined by the velocity of the ions withrespect to each other.

53

This simulation would be the most logical continuation of the work presented

in this thesis, especially since it is the only way to determine whether we in fact

understand the data previously discussed.

5.1.2 High Energy Neutrons

Furthermore we report a few detector events, that supposedly are caused by high

energy neutrons arising from DT fusion. We failed so far to explain these events

exclusively from secondary reactions. Although the fusion cross section6 for a single

DT event is larger than for a single DD event by four too five orders of magnitude7,

the abundance is significantly smaller. Based on the data gathered so far we have

to assume therefore, that our deuterium source may be contaminated with small

amounts of tritium. However after the completion of the full capabilities of the

THOR laser we can expect an increased overall yield to several 105 neutrons pe

shot, such that the observation of high energy neutrons (from secondary reactions)

may become feasible.

5.2 Outlook - Future Experiments

For a qualitative and quantitative improvement of angular measurements, as well

as for any kind of neutron measurement, an enhancement of the neutron yield is

crucial. For this purpose a list of suggestions is provided in the following.

5.2.1 Machining Beam

A very interesting technique to increase the amount of energy depleted in the high-

density region of the gas jet was lately developed by D. Symes, [22], in the context

of x-ray generation and shock-wave experiments. In order to avoid the absorbtion6This is the cross section for one particle at rest.7For the initial energy of the triton the cross section is larger by four orders of magnitude whereas

it grows by another factor of ten if it gets slowed down to ≈ 200keV .

54

of the heating beam in the low-density wings of the gas jet, this is before the fusion

relevant high-density region, they destroy these cluster using a moderate intensity

(1015 W/cm2) beam propagating transverse to the heating beam. They have shown

that the spatial energy deposition in the plasma channel produced by the heating

beam can be modify by the machining beam by as much as a factor of 15.

It is hard to assess by how much this may improve the neutron yield, sine we

do not know, where exactly the energy in our experiment is depleted. Based upon

the rayleigh scattering measurement in [16] it can be infered, that on the wings

of a jet of hydrogen clusters the density changes by as much as 400%/mm. Since

the fusion yield scales quadratically with the density this is a technique that hence

should be explored for fusion experiments, too. Even if the enhancement of the yield

should turn out to be not substantial, it would help to be in a more homogenous

and hence more reproducible regime.

5.2.2 Magnetic Electron Confinement

It is subject to an ongoing discussion whether the ion energies observed by means of

Faraday cups far apart from the hot plasma filament are meaningful for the energies

of the ions acquired by coulomb explosions. These questions arise from the fact, that

the energetic electrons created by the laser field are confined by space charge forces

to the hot ion plasma. Therefore it may very well be, that the ions detected on

Faraday cups have acquired their energy from interactions with these hot electrons.

Therefore it would be an interesting experiment to apply a magnetic field

along the axis of the filament to see how this modifies the number and energy of

the ions. More specifically, if the applied magnetic field is strong enough, it can

be expected that electrons expelled from clusters in a direction perpendicular to

the field undergo a circular motion. If the magnetic field is strong enough, i.e.

55

B ≈ 5T for typical electron energies of W < 50keV , the radius of their circular

motion would be r =√

2Wme/eB ≤ 150µm which can be considered to be within

the filament8. In other words this would imply a confinement for the vast majority

of the electrons which may lead to a confinement of the ions due to space charge

forces, too. This is particulary interesting since not only the effect of the electrons

on the final ion energies can be studied for different spatial confinements (realized

by different magnetic fields), but also an increase of the plasma disassembly time

can be expected to lead to an enhanced fusion yield.

5.2.3 Solid Density Beam Target

For a typical shot at a laser energy of 1J , a total ion yield into the full angle of

1013 − 1014 ions is measured by means of Faraday cups , [16]. Although it may be

argued that energetic ions create an unknown number of secondary electrons on the

plate of a faraday cup and that therefore faraday cup signals may tend to overes-

timate the ion yield, there is doubtlessly a large number of ions emerging from the

plasma without any interaction. Thus it sounds natural to suggest to encircle the

gas jet by a solid density target. However the mean free path of a deuterium ion,

even with an energy of 20keV , is short in a dense target (on the order of micrometers

or less). This is due to the fact, that the velocity of such an ion is slow compared to

the typical electron motion in an atom (which can be assumed to have an energy of

13.6eV ). For this reason most thermal deuterium ions would lose their energy before

they had a chance to fuse. Nevertheless for tritium ions at an energy of 1.01MeV

(from DD fusion) the penetration depth is much higher and hence the probability

for fusion, too.

Nevertheless the range of deuterons or tritons grows approximately quadrat-8Such a magnetic field may be feasible since the dimensions of the field do not need to exceed

2mm neither does it have to last for longer than at most a few microseconds.

56

ically with the energy and hence does not exceed a millimeter for energies less than

10MeV . This fact underlines the need for higher intensities to strip larger clusters

which may facilitate multi-MeV ion energies in single cluster explosions. In other

words, for a significant increase of the fusion yield it may be necessary to produce

lesser ions at higher energies. With the advent of femtosecond, peta-watt laser

facilities with peak intensities of I0 = 1023 W/cm2 it soon will become feasible to

observe coulomb explosions of single micrometer size droplets, leading to several 1010

ions of hundreds of MeV energy and a range of tens of centimeters in deuterated,

solid-density targets.

57

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60

Vita

Federico Francisco Bursgens was born in Herdecke (Westphalia/Germany) on Au-

gust 30th in 1979. After finishing his military service in spring 2000, he went to

college at the Westfalische Wilhelms Universitat Munster, where he was admitted

to the German Academic foundation in the fall of 2000. After takeing his interme-

diate examination for the dipoloma program in 2001, he changed to the Bayerische

Julius Maximillians Unviersitat Wurzburg. Since the fall of 2002, he has been a

graduate student in the physics program of the University of Texas at Austin work-

ing for Prof. Todd Ditmire.

Permanent Address: Federico F. Buersgens

Am Born 9

59821 Arnsberg/Germany

e-mail: federico@bursgens.de

This thesis was typeset with LATEX2ε9 by the author.

9LATEX2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademarkof the American Mathematical Society. The macros used in formatting this thesis were written byDinesh Das, Department of Computer Sciences, The University of Texas at Austin, and extendedby Bert Kay and James A. Bednar.

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