K- and L-shell x-ray production cross sections for alpha particles in the 1 to 4 MeV range on selected thin targets

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Title:
K- and L-shell x-ray production cross sections for alpha particles in the 1 to 4 MeV range on selected thin targets
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viii, 95 leaves : ill. ; 28 cm.
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English
Creator:
Soares, Christopher Graham
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Subjects / Keywords:
Alpha rays   ( lcsh )
Ionization   ( lcsh )
Atomic transition probabilities   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 90-94).
Statement of Responsibility:
by Christopher Graham Soares.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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Full Text











K- AND L-SHELL X-RAY PRODUCTION CRUSS SECTIONS FOR
ALPHA PARTICLES IN THE 1 TO 4 MeV RANGE .ON
SELECTED THIN TARGETS







By

CHPISTOPHER GRAHAM SCARES


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA


1976















ACKNOWL EDGEMENTS


There have been a great many good people who have helped

me along the long and sometimes tortuous path through grad-

uate school. My primary movers have been the members of. the

X-Ray Physics group at the University of Florida, Dr. Henri

A. Van Rinsvelt, Dr. Richard Lear, and Mr. John Sanders,

who offered invaluable help in every step of the data ac-

quisition and analysis. I would also like to thank Gary

Rochau, whose expert assistance allowed me to get everything

possible out of the available equipment. Impetus for this

work was provided 7by Dr. Louis Brown, who loaned us target

materials for analysis, and Dr. Tom Gray, who has done so

much important work in this field and who provided us with

data prior to publication. Dr. George Pepper generously

provided us with a copy of his computer program XCODE and

this made possible the theoretical calculations shown in

this work. Finally, I wish to acknowledge the hard work of

my parents in typing and proofreading the manuscript and the

patience and serenity of my wife during even the most try-

ing of times.


















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . .

LIST OF TABLES . .

LIST OF ILLUSTRATIONS . .

ABSTRACT . .

CHAPTER


Page

. .. ii

iv

v


vii


INTRODUCTION . .

EXPERIMENTAL PROCEDURES .

EXPERIMENTAL PRODUCTION CROSS SECTIONS

THEORETICAL IONIZATION CROSS SECTIONS

RADIATIVE AND NON-RADIATIVE RATES .

COMPARISON OF EXPERIMENT TO THEORY .

CONCLUSIONS . .


REFERENCES . .

BIOGRAPHICAL SKETCH . .


. 1

. 8

19

. 35

. 41

S. 57

. 87


. 90

. 95


iii


I.

II.

III.

IV.

V.

VI.

VII.


I *


* ,















L15T OF T/ALE5


Table Page

I. Recent K-shell x-ray production experiments 5

II. Recent L-shell x-ray production experiments 6

III. Target thicknesses . 9

IV. K-shell x-ray production cross sections .28

V. Measured mid Z K-shell x-ray production cross
sections . . 30

VI. measured L-shell x-ray production cross
sections . . 31

















LIST OF ILLUSTRATIONS


Figure

1. X-ray detector efficiency ....

2. Block diagram of experimental setup .

3. K-shell x-ray spectrum of Fe .

4. L-shell x-ray spectrum of Au .

5. Elastic scattering spectrum .

6. Major L-shell x-ray transitions .

7. Computer fitted L spectrum .

8. K-shell x-ray production cross sections

Fe, Ge, Co, and Se . .

9. K-shell x-ray production cross sections

Ti, Ni, V, Cu, fin, and Zn .

10. K-shell x-ray production cross sections

Y, Ag, and Sn . .

11. L production cross sections for Hf, Pb,

12. L- production cross sections for Hf, Pb,

13. Lg production cross sections for Hf, Pb,


Page



* 15

15

S 17

. 18

S 47

51


for



for



for

* .

and

and


and U


14. Lyl production cross sections for Hf, Pb, and

15. L2,3,6 production cross sections for

Hf, Pb, and U . .

16. Lg production cross sections for Au .

17. L and L production cross sections for Au

18. Lyl and Ly2,3,6 production cross sections for


U


Au.


* *









LIST OF ILLUSTRATIONS continued


Figure Page

19. Theoretical L-subsnell ionization cross
sections for Au . . 72

20. Production cross section ratios L /Lyl for
Hf, Au, Pb, and U . 74

21. Production cross section ratios Ly2,3,6/Lyl for
Hf, Au, Pb, and U . 75

22. L1 subshell ionization cross sections for
Hf, Au, and U . . 78

23. L2 subshell ionization cross sections for
Hf, Au, and U. .. . 79

24. L3 subshell ionization cross sections for
Hf, Au, and U . .. 80

25. Subshell ionization cross sections for Pb 81

26. Subshell ionization cross section ratios for
L3/L2 for Hf, Au, Pb, and U .... 84

27. Subshell ionization cross section ratios for
L1/L2 for Hf, Au, Pb, and U . 85















Abstract of Dissertation Presented to the
Graduate Council of the University of Florida
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



K- AND L-SHELL X-RAY PRODUCTION CROSS SECTIONS FOR
ALPHA PARTICLES IN THE 1 TO 4 [feV RANGE ON
SELECTED THIN TARGETS

By

Christopher Graham Soares




Chairman: Henri A. Van Rinsvelt
major Department: Physics

K- and L-shell x-ray production cross sections have

been measured for nlpha particle bombardment of a wide range

of thin targets. Total K-shell x-ray production cross sec-

tions were measured for 1.0 to 4.4 MeV alpha particles on

Ti, V, in, Fe, Co, Ni, CL, Zn, Ge, and Se. Also, K and K

production cross sections were measured for 1.5 to 4.25 MeV

alpha particles on Y, Ag, and Sn, while 1.0 to 4.0 meV alpha

particles were used to measure L, L LP, LY1, and LY2,3,6

production cross sections for Hf, Au, Pb, and U. All data

were taken with thin targets evaporated onto thin carbon

backings. The measured cross sections were compared to

the predictions of the binary encounter approximation (BEA),

the plane-wave Born approximation (PWBA), the PWBA with


vii









binding energy corrections (.'.iBAB), the Pb'BA uith Coulomb

trajectory corrections (PWBAC), and the PWJBA with both the

above corrections (PWBABC). It was found that the PLBABC

fits the K-shell data best, while it consistently underpre-

dicts the L-shell data. The latter is best described by

the PWBA with no corrections. In addition, L-subshell

ionization cross sections ipere inferred from the measured

production cross sections and the appropriate experimental

radiative and non-radiative transition probabilities. These

too were compared to the various theoretical models con-

sidered, and,again, the best results were obtained with the

PWBA.


V i 1















CHAPTER 1

INTRODUCTION

The observation of energetic x-ray photons following

inner-shell ionization was first reported over sixty years

ago in radioactive source experiments (Chadwick, 1912, 1913;

Russell and Chadwick, 1914; Thomson, 1914). Since that

time, these experiments and their interpretations have grown

much more refined due to many experimental and theoretical

improvements. The advent of higl energy particle accelera-

tors in the 1930's allowed the use of controllable amounts

of light ions of specific energy for the study of the energy

dependence of x-ray and electron yields from particle bom-

bardment in the meV energy range. The field up to about

1950 was thoroughly reviewed by Massey and Burhop (1952).

Between that time and the mid 1960's, the field of inner-

shell ionization phenomena was relatively dormant, since the

only high resolution experiments performable were quite dif-

ficult and lengthy. However, with the advent of relatively

high efficiency and good resolution solid state x-ray detec-

tors, the study of x-ray emission following charged particle

bombardment has enjoyed a renaissance in the past ten years.

Garcia et al. (1973) have published a very good review of

the subject and Crasemann (1975) has edited a series which

summarizes the field up to essentially the present.









One might ask why positive-ion-induced x-ray cross sec-

tions should be measured at all. From a theoretical physics

standpoint, the Experimentally determined cross sections

serve as important tests of theoretical assumptions and

models. For example, the hydrogenic character of the inner

shells of atoms is verified oy cross section measurements

which show agreement to theoretical predictions which as-

sume hydrogenic wave functions to describe inner-shell

atomic states. Another theoretical assumption which has

been verifieo is that the interaction between a charged par-

ticle and an atomic electron can be described in terms of a

classical collision between two particles. As time passes,

more detailed information is gained from cross section mea-

surements which test the theories to a greater extent. Theo-

retical refinements have reached a point, at least for pro-

ton-induced emission, where theory and experiment are in ex-

cellent agreement. The effect of projectile charge is cur-

rently of much interest, with alpha particle measurements

being the next step beyond protons for comparison between ex-

periment and theory. Again as in the case of protons, there

has been excellent agreement for the K-shell, but there are

discrepancies in the L-shell results which must be explored.

Projectiles with higher charge show still more discrepancies,

and more effects in the ionization process have to be consid-

ered to describe these results. Measurement of positive-ion-

induced x-ray cross sections is thus an important testing

ground for theoretical modeling of inner-shell atomic

processes.









ValLes for x-ray production cross sections are poten-

tially very important from an applied physics standpoint

also. The field of trace element analysis using x-ray emis-

sion spectroscopy, which until recently included only fluo-

rescence-induced emission, has expanded to include charged-

particle-induced emission (Johansson et al., 1970). With

accurate x-ray production cross sections, and a knowledge of

target matrix effects on x-ray absorption, it is possible to

quantify this extremely sensitive method of trace element

analysis. Proton cross sections have been used in the anal-,

ysis of aerosol samples using proton-induced emission

(Johansson et al., 1975) and alpha particle cross sections

could prove important in this field since they too are often

used in trace element studies (Van Rinsvelt et. al., 1974).

In this work, alpha particles from 1.0 to 4.4 mev from

the University of Florida Van de Graaff accelerator were

used to measure K-shell x-ray production cross sections for

the low Z elements Ti, V, In, Fe, Co, Ni, Cu, Zn, Ge, and

Se. Also, 1.0 to 4.25 MeV alphas were used to measure K-

shell x-ray production cross sections for the medium Z tar-

gets Y, Sn, and Ag,while 1.0 to 4.0 MeV alphas were used

to measure L-shell production cross sections for the high Z

elements Hf, Au, Pb, and U. The K-shell results are com-

pared to the theoretical predictions of the uncorrected

plane-wave Born approximation (PlIBA), the PIBA corrected for

increased binding energy and projectile trajectory (PlUBABC)

and the binary encounter approximation (BEA). The L-shell









spectra were resolved into several groups of characteristic

peaks, and production cross sections Lwere determined for

each of them and then compared to the theories mentioned

above. In addition, L-suashell ionization cross sections

were inferred from the data, and these are also compared to

theory. Where possible, the results of this work are com-

pared to the results of previous experiments.

Experimental effort in this field prior to 1973 has

been the subject df two excellent reviews by Rutledge and

Watson (1973), and Garcia et al. (1973). Work which has

appeared since then is listed by source and date in Table I

for K-shell data and in Table II for L-shEll data. These

tables are by no means completE, and include only results

which are readily accessible in the literature. Further,

the tables are restricted to experiments done with thin,

transmission targets, denoted by either 'thin' or 'foil'

(i.e., less than or greater than 100ug/cm2 respectively).

The remainder of this work is organized as follows.

Chapter II contains a description of the experimental pro-

cedures used to obtain the x-ray and particle spectra used

in the analysis, while Chapter III describes the analysis of

the spectra, which results in the measured x-ray production

cross sections. Chapter IV describes in brief the various

theoretical approaches which have been taken to describe

charged particle induced x-ray emission. In Chapter V, the

various atomic parameters which describe radiative and non-

radiative emission rates are discussed, and procedures for


















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converting production to ionization cross sections, and vice

versa, using these parameters are examined. Finally, in

Chapter VI, the results of this work are compared with the

theories described in Chapter IV using the methods discussed

in Chapter V, and conclusions are drawn as to the relative

success of each.















CHAPTER II

EXPERIMfENTAL PROCEDURES


The University of Florida 4 MV Van de Graaffaccelerator

wJas used to obtain bear.s of singly ionized 4He, which were

collimated to a 2.5 mm diameter beam spot on the target by

a carbon collimator placed 2 cm from the target. Up to ten

targets could be analyzed without opening the system to air,
-6
and the vacuum uas generally kept at 1 x 10- Torr or lower.

The normal to the targets was oriented at 22.5 with re-

spect to the incident beam and the targets were thin enough

to alloij the beam to bL? accurately monitored using a Faraday

cup. All targets were from 3 to 60 pg/cm2 thick and were

made by evaporating the element to be investigated onto 10

to 50 pg/cm2 thick carbon backings. The target thicknesses

were determined using Rutherford elastic scattering of pro-

jectiles at 135 relative to the beam axis in the vertical

plane. Thicknesses were measured by 2.0 MeV proton bombard-

ment both before and after the experiment, and were

monitored by scattered alpha particle counting during the

experiment. The range of measured values was within the ex-

perimental uncertainties in almost all cases, with the ex-

ceptions to be described later on. The average measured

target thicknesses and their uncertainties are shown in

Table III.
















TABLE III


Target thicknesses


.6

+ .4

_ .5

.3
+ ,3
3

1.0

+ 3

+ 1

+ 3


Ti

V

Mn

Fe

Co

Ni

C u

Zn

Ge


a


5.8

3.8

4.5

2.6

25

9.8

30

12

28


All target thicknesses are in ag/cm2


6.9

2.9

6.9

32

4.8

57

52

17


.7

.3

.7

3

.5

6

5

2









The scattered projectiles L:ere ceLected by a silicon

surface barrier detector which was collimated to 1.55 x
-3
0 sr. This value wiEs measured by ,lacing a calibrated
741
Am radioactive alpha source at Lhe target posit r_- and

comparing the number of alpha particles detected to the num-

ber which were expected to be emitted. The above value is

the average of measurements taken on three different occa-

sions over the course of four months. The standard devia-

tion of these measurements is 0.02 x 10 sr.

For some of the data, Rutherford scattering of projec-

tiles was observed, using a particle detector of the same

type as described above placed at 450 relative to the beam
direcion. ~iis-4
direction. This detector was collimated to 1.77 x 104 sr

as determined in the same manner as above. Target thick-

nesses measured with this (front) detector agreed well with

thicknesses measured with the other (back) detector for pro-

tons only. For alpha particles, it was necessary to use the

back detector only since passage of the scattered alpha par-

ticles through the carbon backing made it impossible to

measure target thicknesses accurately due to further scat-

tering by the backing. There were also problems encountered

in the collimation of the front detector since it was nec-

essary to keep it from 'seeing' the component of the main

beam which was scattered from the carbon collimators in

front of the target. This necessitated using parallel col-

limators for the front detector which introduced further

scattering problems from the outer collimator aperture.









Again this was only a problem for alpha particles due to

their much higher Rutherford elastic scatLerinq cross sec-

tions relative to protons at the same energy. The carbon

collimator scattering u~as not a problem for the c-ck par-

ticle detector since it was located behind the plane of the

carbon collimator.

The.x-rays produced were detected by a KEVEX 5i'l.i)

detector with an 80 mm active area and a resolution of 230

eV at 5.9
to the beam direction, in the horizontal plane. he culli-

mated beam of x-rays passed through the 0.025 mm-thick fmlylar

wiii dow of the scattering chamber, a short (le-s than 5 -mm)

-ir path, and into the detector through a 0.025 mmi-thick-

beryllium window. The absolute detector efficiency was de-

termined in the standard manner of Gehrke and Lnkken (1971)

and Hansen et al. (1973), using calibrated radioactive
51 57 65 241
sources of Cr, Co, Zn, and Am located at the actual

beam spot position. The size of the radioactive source ap-

proximated the beam spot size. The number, N(E), of de-

tected characteristic x-rays of each ericrgy E, was corrected

for absorption in the Mylar source cover and compared to the

number of x-rays which were expected to be emitted by the

source, considering its initial calibration, half-life, and

specific x-ray activity relative to its calibrated gamma-ray

activity. The efficiency e is thus

e(E) = N(E)/AIft} exp (-0.693T/T1) (11.1)
2
wh-here A is the calibrated gamma-ray activity, T i u the








relative intensity of x-rays of energy E compared to refer-

ence source gamma-rays, f is the correction For absorption

by the mylar source cover, t is the counting time period, T

is the time since source calibrauion, and Tj is the source
2
half-life. The values for the quantities I were taken from

Gallagher and Cipolla (1974), who averaged earlier measured

values, mylar absorption coefficients were calculated from

data given by Veigele (1973). The resulting values for e(E)

were fitted to a function of the form

e(E) = A{-exp(-BE) (11.2)

since it is known that efficiency at low energies is expo-

nentia' in character, being mainly determined by absorbers

in the path of the x-rays, while at energies above about 15

keV the efficiency is relatively flat. The efficiency again

falls off sharply above 30 keV, but since this was above the

range of calibration, it was not considered in the fitting

function. The results of the fit give values for the param-

eters of A = 4.84 x 10- and B = 0.198. The measured points,

their uncertainties, and this fit, are shown in Figure 3.

Pulses from both detectors were amplified and then

stored and analyzed by an 1830 General Automation on-line

computing facility. In order to reduce pulse pileup of the

amplifier and keep dead time corrections negligible, count-

ing rates in both the x-ray and particle channels were kept
-1
below 1000 sec This was accomplished by limiting beam

currents when necessary. Beam currents were generally from

10-200 nA. A block diagram of the components of the

















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C\J








C\JC







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14



experimental set-up is shown in Figure 2. The collected

spectra were stored on magnetic tape for off-line analysis,

and displayed on-line for monitoring the results of the ex-

periment. A typical K-shell x-ray specLrum is shown in Fig-

ure 3, while a typical L-shell spectrum is shown in Figure

4. A typical scattered particle spectrum is shown in

Figure 5.







X-ray Det.
Bias Power
Supply



X-ray
Detector


X-ray Det.

Preamplifier


X-ray
Pulse
Amplifier


X-ray
Pulse
ADC



Display


Van de Graaff
Positive Ion
Accelerator



Target
Chamber

^1


Beam Current
Integrator


X-ray Clock Part.
Live Time Live
Timer Timer



Computer


Particle Del.
Bias Power
Supply



Particle
Detector


Particle Det.

Preamplifier
- --T

Particle
Pulse
Amplifier


Particle
Pulse
ADC


Figure 2. Block diagram of experimental setup.


Faraday Cup


Magnetic
Tape




















I0

4-



0







10




I0O


Figure 3. K-shell x-ray spectrum of Fe showing
characteristic peaks and peaks from contaminants in the
carbon target backing.





17


S I i i I I a I a I I I -.. T I


Au

L




O
10








102







IO'






10 I I n I I d I 0

10.0 20.0
Photon Energy (keV)

Figure 4. L-shell x-ray spectrum of Au showing
characteristic L- and M-shell pueas.
























I0 -
o







SII
SI I







1.60 1.80
Particle Energy (MeV)
Figure 5. Elastic scattering particle spectrum of 2.0
[Yle' protons on Ag, detected at lab angle 9= 1450 and shou-
ing target, backing, and low Z contaminants.















CHAPTER [II

EXPERIMENTAL PRODUCTION CROSS SECTIONS


In the preceding chapter, the methods used for obtaining

x-ray and scattered particle spectra were examined. In this

chapter, methods for obtaining x-ray production cross sec-

tions from these spectra are outlined. The.production cross
x
section o. for a particular peak i is related to the number

of detected x-rays characteristic of that peak, Ni, by

N. = N Nt e. .x (11.1)
i pN i

where N is the number of projectiles which were incident on
p
the target during the time the spectrum was collected, Nt is

the area target density in atoms/cm2, and .i is the x-ray

detector efficiency for the photon energy of the peak i.

Also, the areal target density is related to the number of

elastically scattered projectiles detected by the particle

detector, NR, by

NR = N pNtCotR() (III.2)

where again N is the number of incident projectiles, and

0 is the measured particle detector solid angle. The term

OR(O) is the Rutherford elastic scattering cross section

(Rutherford, 1911),









kZlZZe2 2 (l+x)2(l+x2+2xcos) 3/2
R()1 = ----- 1 1-s--sin-4( /2)
RM 4 E l+xcos sn (/)
(III.3)

Ljwhro 0 is the scattering angle as measu ret in the l.bura-

tory frame of reference, k is the constant in the Coulomb

force law (9 x 10 nt-m2/coul2 in 51 units), Zle is the pro-

jectile nuclear charge, Z2e is the target nuclear charge, E

is the projectile energy as measured in the laboratory frame,

x is the ratio of projectile mass to target mass, and X. is

the scattering angle as measured in the center of mass frame.

The angles 0 and X are related by

2 s2o(, 2 2 ( .
cosX = -xsin 2 cos (l-x sin20) (III.4)

In applying this set of equations, there are certain

assumptions one must make. For the first equation (III.1)

they are that (1) the x-rays jere emitted isotropically and

(2) the number of x-rays detected, corrected for detector

efficiency, represents the number emitted by the target,

i.e., that absorption of x-rays in the target, and loss of

counts due to electronic daadtimes and pulse pileup in the

amplifiers,are both negligible. The first assumption is

quite valid for K-shell x-rays as would be expected from the

zero angular momentum characteristic of Is wave functions.

For the L-shell, it is not quite so straightforward an as-

sumption due to the different angular momentum states of the

subshells. It has been suggested (merzbacher, private com-

munication) that any anisotropy in the L-shell x-rays would

be quite small and probably well within the other









experimental uncertainties. Further experiments in this

area have been planned (Van Rinsvelt and I.ear, private com-

munication). As for the second assumntiCnr underlying use of

Eq. (III.1), target thicknesses were sm-ll enough so that

self-absorption could be neglected, and count rates were

kept low to minimize electronics losses, as was mentioned

previously.

In applying Eq. (III.2), the primary assumption is that

elastic scattering is the only mechanism at work causing

particles to be detected at the laboratory angle and pre-

dicted elastic scattering energy. The energies considered

in this work, less than 4.5 meV, are much less than the

Coulomb barrier height of the nucleus (about 10 meV for

alphas incident on Ti) at which nuclear potential scatter-

ing could be expected to come into play (mcDaniel et al.,

1975b). In addition it is assumed that all the particles

scattered into solid angle 0 at laboratory angle 0 were de-

tected. This assumption is valid since the particle de-

tector is 100% efficient if the incident particles are com-

pletely stopped in the active volume of the detector, as

they were in these experiments, and if counts were not

lost in the electronics. Again, low count rates were used

to minimize such losses.

In applying the Rutherford formula some further assump-

tions must be made. They are that (1) this equation holds,

that is, the projectile energies are non-relativistic, (2)

the energy of the projectile is the same for all scattering









events and (3) the effective charge of the projectile is

2.0. Assumption (1) is obviously true for the alpha par-

ticle energies considered. Assumption (2) is met best for

thin targets and high energies. It is assumed that projec-

tile energy loss in the target is negligible up to the time

of the elastic scattering event, and further that only a

single scattering event occurs for each projectile scattered

into solid angle ) at laboratory angle 0. Thus projectiles

scattered by the target are not scattered again either be-

fore or after the original scattering. Obviously, the

thinner the target, the better this assumption. Energy loss

of projectiles in thin targets is evidenced by low energy

tailing in the scattered particle spectra. This was present,

tn some extent, in all targets for low bombarding energies

and in the thicker targets at medium energies also. In some

cases of low projectile energy, it was not possible to mea-

sure target thicknesses because the tailing was so severe.

However, in the majority of cases, thicknesses agreed quite

well between energies, indicating that the tailing, although

nearly always present, was not bad enough to cause loss of

scattered particle information.

Finally, the third assumption rests on the belief that

the scattering can be described as a nucleus-nucleus inter-

action with no screening effect from the projectile electron.

This is equivalent to saying that the projectile electron is

lost before the scattering event takes place, through inter-

actions with the outer target electrons. If this were not









so in all cases, then an equilibrium charge state of between

1 and 2, which would no doubt be energy dependent, would be

needed to determine the scattering cross section. However,

from the measured target thicknesses, this would not seem to

be the case, since a charge state of 2 describes the target

thicknesses so well over a wide range of energies. Also,

the degree with which the thicknesses as measured with pro--

tons (where this is not a problem) agreed with the alpha

scattered thicknesses allows one to make the third assump-

tion, at least for the energies where particle counts could

be used to measure target thickness.

Again considering Eqs. (III.1) and (III.2), it is seen

that if one is divided by the other, the resulting equation

can be rearranged to give
N. 0
x I
i e N R (III.5)

Thus one can determine the production cross section simply

from a knowledge of the ratio of detected x-rays to scat-

tered particles, and without the necessity of measuring the

number of incident projectiles, Np, and of knowing the tar-

get thickness Nt. Of course, with a knowledge of these

latter quantities, one can also get the production cross

section from Eq. (III.l) without the scattered

particle spectrum. Thus there are two methods which can be

used, depending upon what information one has available.

The first method, Eq. (III.5), is considered more accurate,

because measurements of the incident beam can be quite








difficult. This is due to the possibility of scattered

electrons from the collimators and parts of the scattering

chamber, including the target, being read on the Faraday cup

along with the main transmitted beam, thus giving a net cur-

rent less than what is actually incident. In practice, this

is circumvented in a number of ways, such as electrically

isolating the Faraday cup and grounding the scattering cham-

ber, and by placing the end of the Faraday cup uell back of

the scattering chamber. Since the scattered Plectrcns will

have such low energies relative to the beam, it is also

possible to electrically bias the collimator relative to the

chamber by a small amount, so as to attract electrons scat-

tered from the collimator towards the walls of the chamber,

where they will be lost to ground. Probably the best solu-

tion is to keep a well focused beam and thus minimize the

scattering problems in the first place. In this work, val-

ues of the integrated charge were used to calculate target

thicknesses from Eq. (II1.2), again assuming a charge per

ion of 2.0. The degree with which the values agreed, both

between different energies and different projectiles, allows

one to assume that values obtained from integrated beam

currents were not seriously in error.

Thus, wherever possible, production cross sections were

determined from Eq. (III.5) using the ratio of x-rays to

scattered particles. When this was not possible due to lack

of resolution in the particle detector or severe low energy

tailing, Eq. (III.1) was used, calculating N from the
P





25


integrated beam current, and Nt from the target thicknesses

as averaged over the oroton and other alpha particle deter-

minations. It is recognized that uncertainties in the

latter method are greater. SpeciFic uncertainties will be

dealt with later in this chapter.

The measured quantities N. and N are determined from
1 R
the x-ray and particle spectra respectively. The particle

spectra consist of carbon, target element, and lo!j Z con-

taminant (such as oxygen) elastic scattering peaks on a flat

background, and in most cases the target peak is easily sum-

med to get NR. Only in the cases where tailing was so

severe as to cause interference between the carbon and the

target peaks was it not possible to measure a realistic

value for NR. Backgrounds were generally so low as to have

a negligible effect on the peak sums. For the low Z K-shell

spectra (Z = 22 to 34) the intensities N. were determined
1
for the K and the Kg components combined. Again as in the

case of the particle spectra, backgrounds were low and the

peak sums easily determined. For the high Z L-shell x-ray

spectra however, this was not the case. Referring to Figure

4, it is seen that the L-shell spectra can be resolved into

four separate lines or groups of lines. They are labeled

L2, La, L, and Ly. The L and L components result from

transitions to the L3 subshell while the L group consists

of lines resulting from transitions to the L1 and _2 sub-

shells. The Lg group contains transitions to all three sub-

shells. The detailed relationships of these transitions to








the subshells will be dealt with in a later chapter. Suf-

fice it to say that it is important to get production cross

sections for the L L and L lines or groups of lines and

Lo separate the main two components nf the L group: the L
Y1
and L2,3, To accomplish this, various computer programs

are commonly used to fit the x-ray spectra. Some such as

REX (Kauffman, 1976) and the least squares code of Trompka

and Schmadebeck (1968) use matrix inversion techniques to

unfold the spectra. Others such as SAMPO (Routti and

Prussin, 1969) and XSPEC (Henke and Elgin, 1970) use itera-

tive procedures to achieve the best fit. In order to ana-

lyze the data presented here, an iterative type program was

developed to fit unresolved or barely resolved groups of

peaks using sums of Gaussian error functions resting on

either linear or exponential backgrounds. It was necessary

for this program to be relatively small in order for it to

be used on available facilities, yet general enough to be

used on both K- and L-shell data. Such a program was de-

veloped and used successfully on the L-shell data, as well

as the medium Z K-shell data in this work. In handling the

L-spectra the following fits were made: the L line was

fitted separately with a single Gaussian, while the La and L

components were fitted as the sum of two Gaussians. The L

group was fitted as the sum of three Gaussians, while the L
Y
group was fitted as the sum of four. It was thus possible, in

the case of the latter group, to almost completely separate

lines due to transitions to the L1 subshell from lines due








to L2 transitions. In this way it is possible to deduce

subshell cross sections from experiment, as will be dis-

cussed later.

The medium Z K-shell data (Z = 37 to 50) iere also fitted

using one or two Gaussians for the K group and two Gaussians

for the K, group. The results of all K and L fits gave nor-

malized chi-squared values of between 0.8 and 1.9 for the

spectral regions fitted. The fitted areas thus obtained

were used to calculate production cross sections using

either Eq. (III.5) or Eq. (II.1). The low Z K-shell pro-

duction cross sections were obtained prior to the develop-

ment of this fitting routine and have been reported pre-

viously (Soares et al., 1976). They are shown in Table IV.

Table V contains the measured medium Z K-shell production

cross sections as well as the ratios K /K while L-shell

production cross sections for the various major components

are shown in Table VI.

The total uncertainty in the production cross sections

reported here is due to two types of uncertainty. The rela-

tive uncertainty is that uncertainty between values measured

at different energies on the same target, while the normali-

zation uncertainty affects all values measured with the same

target. The latter includes uncertainties in the efficiency

calibration and the particle detector solid angle measure-

ment, as well as angular uncertainties in the position of

the particle detector. The efficiency determination is

affected by uncertainties in the source strengths used (8%),















TABLE IV


K-shell x-ray production cross sections


Ti
Z=22


2.63
3.37
5.43
7.80
11.5
18.3
22.8
30.6
34.9
40.2
45.1
46.7
56.7
63.8
70.1
77.5
84.6
90.8
100.
111.
119.
123.
134.
146.
158
169.
175.
190.
204.
203.


V
Z=23

1.21






19.7
22.6
25.9
29.6
29.6
37.0
41.5
45.4L
50.0
55.1
59.3
64.6
68.6
77.1
83.7
90.1
97.0
105.
112.
111.
125.
136.
138.


Mn
Z=25

.814
1.02
1.66
2.50
3.54
6.27
--H
10.8
12.3
13.5
16.1
15.9
19.8
22.2
24.7
28.8
30.1
31.9
35.3
38.4
42.5
46.0
49.4
54.3
55.7
62.0
67.4
71.9
77.6
83.6


Z=26

.453
m--





7.48
8.66
9.86
11.0
11.7
14.1
35.9
17.9
19.3
21.5
23.4
26.5
29.9
33.3
35.9
38.9
41.4
47.7
49.4
52.1
56.6
60.0
61.2


E
(meV))


1.0
1.1
1.3
1.5
1.7
1.9
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4


Z=27

.399
.619
.975
1.47
2.12
3.39

5.31
6.00
6.80
7.83
9.69
9.86
11.1
12.3
13.6
14.8
16.2
17.9
19.7
21.6
23.1
25.2
27.2
29.5
32.2
34.0
36.9
39.2
43.5















TABLE IV continued


E Ni Cu Zn Ge Se
(meV) Z=28 Z=29 Z=30 Z=32 Z=34

1.0 .207 .201 .104 .0676 .0303
1.1 -- .270 -- .0908 .0482
1.3 -- .464 -- .161 .0736
1.5 -- .709 -- .262 .121
1.7 -- 1.02 -- .389 .186
1.9 -- 1.85 -- .630 .295
2.1 -- 2.81 -- 1.05 .386
2.2 3.95 3.02 2.11 1.11 .665
2.3 4.52 3.38 2.38 1.30 .801
2.4 4.97 3.74 2.68 1.48 .887
2.5 5.80 4.22 3.05 1.66 .943
2.6 6.66 4.70 3.39 1.83 1.12
2.7 7.32 5.45 3.94 2.17 1.30
2.8 8.21 6.12 4.44 2.42 1.48
2.9 9.08 6.84 5.01 2.73 1.62
3.0 10.2 7.67 5.58 2.96 1.76
3.1 11.0 8.32 6.24 3.28 1.89
3.2 12.0 8.99 6.42 3.68 2.11
3.3 13.4 9.80 7.11 4.08 2.62
3.4 14.6 10.8 7.74 4.54 3.15
3.5 16.1 12.1 8.44 5.17 2.84
3.6 17.6 13.5 9.42 5.56 3.41
3.7 19.2 14.6 10.4 5.91 3.56
3.8 20.9 15.9 11.3 6.49 3.87
3.9 21.7 16.7 11.4 6.92 4.26
4.0 24.3 18.5 12.3 7.59 4.92
4.1 26.1 19.7 14.3 8.04 5.08
4.2 28.1 21.5 15.4 8.72 5.90
4.3 29.7 22.7 16.5 9.17 5.76
4.4 32.4 25.7 18.1 10.0 5.81


Cross sections are in


barns











t L N L TIN Nr\i CC L 0 C tI L Ln [N,
y 0C C\ r-H r -4 r i C[ r-i rN NJ C '-J -
0 ca C- o. ,- .-
(E 020. o o-






C
z O Cm
0-- C C7 D CO 0 C- CD C3 C7 C
C ccn rc .- c- L r c7 o
ro .-i c3 -r i- iD cn 0 E E COD D C D D- CD 3 CD E --4 O



c
C-
0

,-

ELI
0n


ci
U)



'--4 i Lo cOc D [N, r-i ( -I
u N V7 N N N N N l N N Il N
c C; ; .
002
c o o d d d d d d o o o o
o II


U

D E

-a N Lq cl :t
d '-^ m-A NM '-4 Ln ED -t ED
IJ .*- O C- -1 '- -1 0> Ln CO --t -) 0. I T -
.j >, i n o D i d Ln c o C' ED 0 < 0


x



Q(;

-CC






II I1





"0 0

00
) ED
N 2 (0 C- C- E -D CO O CO N N
>- c4 TC CO N O ED- ED co 3 ED 'i r-
D [\1 N ED E -I C)







> J cc r c C 0 -
S *H






a) '-'-.4
E 0 D D cc
a L 4'1 4
01 cc C- tED N c C- E N 10 1o ED N







-N-N


a In c^- o (M Ln r- o EN LD to O IN n

LbJ v- -iA rsi c M rMH N N N rn n ct















TABLE VI

measured L-shell x-ray production cross sectionsa


Hafnium Z = 72


0.0148
0.0338
0.0564
0.0984
0.225
0.200
0.439
0.417
0.831
0.918
1.19
1.51
1.56


0.356
0.867
1,79
2.77
5.67
5.98
9.93
11.5
17.2
20.9
25.8
30.9
35.2


0.285
0.603
1.11
1.66
3.15
3.28
5.29
6.06
8.84
11.0
13.6
16.2
18.7


0.0210
0.0394
0.0892
0.121
0.295
0.237
0.464
0.496
0.781
1.04
1.24
1.50
1.77


0.0165
0.0362
0.0515
0.0442
0.103
0.098
0.158
0.165
0.207
0.181
0.240
0.282
0.358


Gold Z = 79


0.0129
0.0162
0,0348
0.0558
0.109
0.143
0.118
0.265
0.240
0.404
0.545
0.649
0.784
0.842


L

0.171
0.315
0.716
1.10
1.87
2.57
2.21
3.86
4.27
6.25
8.22
9.95
11.8
12.9


0.113
0.218
0.454
0.696
1.13
1.59
1.34
2.34
2.63
3.88
4.89
5.91
7.10
7.65


0.0103
0.0203
0.0403
0.0532
0.116
0.116
0.139
0.250
0.282
0.419
0.489
0.605
0.697
0.755


LY 2.3,6

0.00722
0.0135
0. 0335
0.0319
0.0588
0.101
0.0770
0.122
0.161
0.242
0.270
0.342
0.400
0.419


E b
E

1.00o
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00


b
E
ex
1.00
1.25
1.50
1.75
2.00
2.25
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00















- continued


Lead Z = 82


0.135
0.707
1.06
1.74
2.57
2.97
4.22
5.57
6.98
7.98
9.23


0.0939
0.427
0.632
1.02
1.52
1.83
2.63
3.22
4.06
4.80
5.51


Uranium Z = 92


0.00508
0.0166
0.0258
0.0376
0.0371
0.0610
0.0868
0.106
0.132
0.189
0.220
0.255


0.0882
0.195
0.352
0.518
0.510
0.792
1.16
1.45
1.80
2.43
2.95
3.56


All cross sections are in
All energies are in MeV


b
E b

1.25
1.75
2. 00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00


0.00784
0.0403
0.0601
0. G2
0.165
0.175
0.265
0.339
0.428
0.478
0.581


Lyl

0.00529
0.0369
0,0497
0.0783
0.138
0.176
0.257
0.317
0.399
0.481
0.569


LY23.6

0.00562
0.0332
0.0443
0.0663
0.106
0.125
0.167
0.205
0.252
0.296
0.321


Eh

1. 50
1.75
2.00
2.25
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00


0.0550
0.120
0.199
0.301
0.292
0.434
0.638
0.817
0.984
1.29
1.56
1.88


barns


0.00348
0.00789
0.0142
0.0258
0.0284
0.0448
0.0559
0.0874
0.0909
0.125
0.155
0.197


0.00290
0.0115
0.0185
0.0216
0.0175
0.0318
0.0469
0.0421
0.0427
0.0710
0.0820
0.0931


TABLE VI









values taken for the relative number of x-rays emitted per

decay (2%), statistical and background subtraction uncer-

tainties in the x-ray peak integration (2O), as well as

fitting uncertainties (5%). These combine to give an over-

all uncertainty of 10% for the value assigned as the effi-

ciency of the detector for the photon energy considered.

This uncertainty is taken as a constant for all analyses

considered in this work. Also, the calibrated intensity of

the particle source used to measure the detector solid angle

had an 8% uncertainty and this was the primary source of

error in this determination. The error in the Rutherford

cross section due to uncertainty in the laboratory angle was

estimated to be 5% or less. Combining these three as inde-

pendent sources of error yields an overall normalization

uncertainty of 14%,

The relative uncertainties are different depending on

which method is used to measure the cross section, either

Eq. (Ill.1) or Eq. (111.5). Use of the latter introduces

only uncertainties in the x-ray and particle counts. The

uncertainty in the number of scattered particles counted

varies between 1 and 5% depending on the counting statistics

and the degree of peak tailing present. The x-ray counting

uncertainties on the other hand have a much wider variation

owing to the fact that additional uncertainty arises when

spectra must be unfolded. Thus the uncertainty in the num-

ber of x-rays detected is from 1 to 5% where individual

lines are concerned and from 5 to 15% when multiple peak








fitting is necessary. The above numbers are estimated from

observed variations between repeated data points and differ-

ent fits of the same spectrum. If these errors are com-

bined, a relative uncertainty of up to 16% results when

Eq. (III.5) is used. On the other hand, the use of Eq.

(III.l) introduces uncertainties due to beam integration

(3z), and target thickness due to nonuniformities in both

target and incident beam (10%). The latter number was ob-

served from the various thickness determinations and is the

uncertainty given in Table III. Thus use of Eq. (11I.1)

results in a relative uncertainty of up to 20%. Combining

both relative and normalization uncertainties, we have a

range of uncertainty, depending mostly on counting statis-

tics in the x-ray peaks, of 15-20% when Eq. (I11.5) is

used and 20-25% when Eq. (III.1) is used.















CHAPTER IV

THEORETICAL IONIZATION CROSS SECTIONS


Until now, only production cross sections have been

discussed. These are proportional to the probability per

incident projectile of seeing an x-ray due to a certain

transition between energy levels. The various theories, on

the oLher hand, predict ionization cross sections, which are

proportional to the probability per incident projectile of

creating an inner-shell vacancy in a certain level. The two

quantities are not the same for any given level due to the

possibility of radiationless transitions. The factors which

describe radiationless transition probabilities will be dis-

cussed in the following chapter. In this chapter, the var-

ious theories which predict ionization cross sections will

be briefly discussed.

One may describe a charged-particle-induced ionization

event in the following terms: the incident particle col-

lides with an atom which is at rest. The collision imparts

energy to the atom via the Coulomb interaction and leaves it

in some final state in which there is an inner-shell vacancy

and either an electron with some kinetic energy in the con-

tinuum, or an electron still bound to the atom, but promoted

to a higher energy level. The total Hamiltonian describing









this system is the sum of an unperturbed Hamiltonian, Ho,

and an interaction potential, V. The transition amplitude

from the initial state to the final state is then the matrix

element between these two states and the interaction poten-

tial. All quantum mechanical treatments of the ionization

process start from this point. In practice H is chosen

such that the interaction term is small enough to be consid-

ered as a perturbation. The unperturbed Hamiltonian is

usually chosen as the sum of the atomic and free particle

Hamiltonians. The latter has plane-wave eigenfunctions, and

the eigenfunctions of the former are known, at least in

principle. Ideally one would use the properly anti-sym-

metrized self-consistent Hartree-Fock-Slater-Dirac wave

functions which describe atoms with single (or multiple) in-

ner-shell vacancies. However, calculations have as yet been

confined to simpler schemes, such as product wave functions.

The total wave function is then approximated to be an ini-

tial state eigenfunction of H and thus projectiles are

considered to be plane waves both initially and finally.

This is the plane-wave Born approximation, which has been

the subject of much attention in the literature (fMerzbacher

and Lewis, 1958; Khandelwal and Merzbacher, 1966; Choi and

rflerzbacher, 1969). Treating the projectiles as plane waves

is most realistic for high incident energies, where the de

Broglie wavelengths are large relative to the classical dis-

tances of closest approach. For lower energies this approx-

imation breaks down, among other reasons, because of









Coulcmo revulsion of the projectile by the target nucleus.

A better aporoxination for low energies involves the use of

distorted plane-viaves (DWBA) (Miadison and Shelton, 1973).

In any event, in the spirit of First order perturbation

theory, ionization cross sections are proportional to the

square of the transition amplitude, where V is taken as the

Coulomb interaction. The basic theory behind the PWBA has

been reviewed recently in detail (fladison and merzbacher,

1975). Tables of data which allow one to calculate ioniza-

tion cross sections for arbitrary projectiles, energies, and

targets have been available for several years (Khandelwal,

Choi, and merzbacher, 1969). The most recent and comprehen-

sive of these Lables (Choi, merzhacher, and Khandelwal,

1973) was used in tnis work to compute the uncorrected PWBA

ionization cross sections. The calculations were performed

using the computer code XCCDE (Pepper, 1974).

Although the PWBA is well applicable for larger colli-

sion energies, problems arise for lower energies. Essen-

tially there are two low energy effects which must be con-

sidered. As was mentioned previously, straight line projec-

tile trajectories are not applicable for low energy and hence

close (in the classical sense) collisions. Account must be

taken of the effect of the Coulomb repulsion between projec-

tile and target nucleus upon the trajectory. The resultant

loss of projectile energy can be quite crucial to the cross-

section magnitudes. This effect has been analyzed using the

semiclassical approach (Bang and Hansteen, 1959; Hansteen









and fosebekk, 1970). Using this approach, correction fac-

tors have been derived Luhich can be applied to the PUBA cal-

culations. The result, denoted PWBAC, was again performed

using XCODE for the targets and energies considered.

Related to the trajectory effect is the problem caused

by the perturbation of the target atom by a slowly moving

projectile. This has the effect of increasing the electro-

static binding energies due to the temporary presence of the

projectile within the atom. This effect has been investi-

gated at length (Brandt et al., 1966; Rasbas et al.,

1973a,b), and correction factors to the PWBA can be calcu-

lated from data supplied by Brandt and Lapicki (1974).

These calculations, denoted PUBAB, are also included in the

program XCODE. When both the above corrections are applied

to the PLUBA, the result is denoted PWIBABC and this too was

performed by XCODE.

An alternative approach to the ionization problem is to

consider the event es a classical collision between the pro-

jectile and the atomic electron. This is not an unreason-

able assumption since the Rutherford elastic scattering

formula is derivable from both classical and quantum mechan-

ical approaches. In this scheme, the target atom and its

remaining electrons are passive spectators, and serve only

to determine the velocity distribution of the electron to be

ionized. The cross section is then an average of the free-

particle ionization cross sections over all possible energy

exchanges and all possible electron velocities. This









approach is known as the binary encounter approximation

(BEA). Calculations can be broken into two groups according

to the method of determining the velocity distribution of

the orbital electron. In quantal BEA calculations, tne ve-

locity distribution is the square of the momentum-space wave

function of the atomic electron being ejected, while in clas-

sical BEA calculations, it is obtained from a microcanonical

ensemble. The two approaches give identical distributions

for hydrogenic states. A thorough review of BEA calcula-

tions has been published by Garcia et al. (1973). Data

which may be used to calculate ionization cross sections in

the BEA have been published by Hansen (1973), icGuire and

Richard (1973), and PcGuire and Omidvar (1974). These

tables yield results which are essentially identical.(Pepper,

1974) and are included in XCODE.

It should be noted that the above theories and the cal-

culations which result from their use are for single vacancy

production only. There is, however, much experimental evi-

dence that, to some extent, multiple ionization takes place

in all positive ion experiments, especially at higher ener-

gies and for higher Z projectiles. This evidence, which has

been reviewed by McGuire (1974), is the presence of higher

energy 'satellite' lines in the x-ray spectra measured with

high resolution spectrometers. These satellites are higher

in energy since they represent transitions from outer levels

which have been shifted up in energy due to the reduced nu-

clear screening in atoms which have been multiply ionized.









In low resolution experiments such as those reported here,

these satellites show up as centraid shifts in the peak

energies relative to the single Jcancry transition energies.

within the framework of the EEA (icGuire and Richard, 1973),

it is possible to estimate the degree of multiple ionization

as a function of projectile, energy, and target. McGuire

(1974) has published data in the form of graphs showing mul-

tiole ionization cross sections for alpha particle bombard-

ment up to 100 PeV for several targets. These graphs show

relative probabilities of ionizing a single K electron

alone, as well as a single K and 1,2,3,4,5, or 6 L elec-

trons. it is interesting to note that for the energies and

elements in this work, these graphs indicate that there is

no significant multiple ionization occurring, at least so

far as the K-shell data are co-cerned. This will have im-

portant implications later in this work when experimental

and theoretical results are compared. At this time, there

is no experimental information available for estimating the

effect of multiple ionization in L-shell studies (Richard,

1975). This information is crucial for realistic compari-

son of theory and experiment, which will be discussed in the

next two chapters.















CHAPTER V

RADIATIVE AND NON-RADIATIVE RATES


If vacancy production in inner atomic shells was always

followed by photon emission characteristic of that shell,

then ionization cross sections would be identically equal to

production cross sections. However, a radiative transition

is not the only mechanism for atomic de-excitation following

vacancy production. Auger (1925), in cloud chamber experi-

ments, showed direct proof that vacancy production is some-

times followed by the ejection of an electron (rather than

a photon) of energy characteristic of the atomic shell, thus

leaving the atom twice ionized. Such processes are now

known as Auger transitions. Thus, for every K-shell vacancy

produced, there is a probability OK that the vacancy will

be filled with a radiative transition, and a probability AK

that the vacancy will be filled by an Auger or non-radiative

transition. Since these are the only possibilities in the

K-shell, the sum of these two probabilities is unity. The

quantity wK is known as the K-shell fluorescence yield, and

is thus the ratio of characteristic photons produced to

initial vacancies and as such


x I
e K s te K- l (s s 1)
where aKx is the K-shell production cross section and aK I is









the i(-shell ionization cross section. blith the above rela-

tion and values for wK, one can easily convert theoretically

predicted ionization cross sections to production cross sec-

tions for comparison to experimental results.

A great deal of work has gone into both the experimen-

tal and theoretical determination of fluorescence yields for

the K-shell. Experimental determination of wK hinges on be-

ing able to produce known vacancy rates to compare to ob-

served x-ray emission rates. The most common means of pro-

ducing known K-shell vacancy rates is through the use of

radioactive sources which decay by K-electron capture. If

one knows the branching ratio for this process, one can de-

termine wK as being proportional to the ratio of the number

of observed K-shell x-rays to the number of expected nuclear

decays. This is only one of a wide number of possible

methods. The field of K-shell fluorescence yield measure-

ment has been thoroughly reviewed by Bambynek et al. (1972),

who have selected a set of 'best' experimental values of wK

Fluorescence yields may be calculated theoretically by

considering the Auger process as an electrostatic inter-

action between two electrons in an atom with an inner-shell

vacancy. The interaction is thus Coulombic and the transi-

tion probability is proportional to the matrix element be-

tween the proper initial and final states. In the most

recent calculations (rcGuire, 1970; Kostroun et al., 1971;

Walters and Bhalla, 1971),Hartree-Fock-Slater wave functions

were used and the results for all authors are essentially








the same over a very wide range of Z. Theory and experiment

also agree very well for the K-shell fluorescence yield.

In the case of the L-shell, the problem of non-radia-

tive transitions is considerably more complicated. To be-

gin with, there are tree subshells of different energies to

which transitions can take place, and hence three fluores-

cence yields and Auger rates are needed. Further, there is

the possibility of transitions between the subshells them-

selves. Such transitions, which are almost completely non-

radiative, are known as Coster-Kroniq transitions after

their discoverers (Coster and Kronig, 1935). Coster-Kronig

transitions result in the transference of a vacancy from a

lower subshell to a higher one, and in the emission of an

electron from a higher shell. For the L-shell, with its

three subshells, the Coster-Kronig transition probabilities

are labeled f12, f23, and f13, denoting vacancy promotion

between the L1 d L and [L3 L and L, and Lnd L subshells

respectively. Thus for the L-shell there are six non-

radiative probabilities one must consider, rather than just

one, as in the K-shell case.

It is possible to correct the fluorescence yield for

each subshell (ui) for possible Coster-Kronig transitions by

considering the ratio of characteristic x-rays from the L-

shell (without regard to subshell origin) to initial vacan-

cies in the ith subshell. Thus for the L1 subshell, the

corrected fluorescence yield (vl) is the sum of the L1 sub-

shell fluorescence yield (w1), the fraction of vacancies








originally in the L1 subshell Luhich has shifted up to the

L2 suhshell (f12) times that subshell's fluorescence yield,
and the fraction of vacancies originally in the LI subshell

whlcn has shifted to the L subsnell, through either di-

rect transfer between subshells 1 and 3 (f13) or via the L2

subshell (fl2f23), times the L3 subshell fluorescence yield.

A similar relation holds for the L2 subshell, but its fluo-

rescence yield need only be corrected for Coster-Kronic

transitions to the L3 subshell, u!hile the L3 subshell fluo-

rescence yield needs no correction at all, since its va-

cancies have nowhere they can go within the L-shell. Thus

the Lhree equations for Coster-Kronig-corrected fluorescence

yields are


1 1 + f12 2 + (f13 + f12f23)03

v2 =2 f23 3

3 = "3 (V.2)

It is possible to define an average fluorescence yield ZL as

the sum of the corrected fluorescence yields, which

obeys the relation

WL = LX/aLI (V.3)

x
where aL is the total L-shell x-ray production cross sec-

tion and aL is the total L-shell ionization cross section.

By total is meant

CL = C + a2 + 03 (V.4)

where the quantities a. are the individual subshell cross
sections, either production or ionization.









Because of the more complicated transition possibili-

ties in the L-shell, the quantities w. and f.. are very much
1 13
more difficult to determine experimentally than wK. To do

so accurately, one needs both a controlled production of

primary L-subshell vacancies and high resolution spectrom-

etry to detect the electrons and photons which are emitted

(Bambynek et al., 1972). The available experimental data

are, as a result, quite scarce, especially for the innermost

subshell, the L1, where experimental data exist for wl and

f12 only for elements of Z greater than 80. Available ex-

perimental data are summarized and evaluated by Bambynek

et ai. (1972).

The situation is much better from a theoretical stand-

point. Calculations for the L-shell non-radiative rates

exist for a wide range of 7 and compare well with each other

(McGuire, 1971; Chen et al., 1971; Crasemann et al., 1971).

However, the scarce experimental data are sometimes at wide

variance with these calculations, particularly for the

quantities fl2 and f23 where measurements generally exceed

calculated values by as much as 30%. This discrepancy can

have a large effect upon cross sections which are inferred

from the use of these parameters, as will be discussed

later. At this point, further theoretical refinements are

being considered by several authors to remedy these dis-

crepancies.

The complexity of the typical L-shell spectrum (Fig-

ure 4) relative to the typical K-shell spectrum (Figure 3)









is due to the fact that L-shell spectra show transitions to

three different L-subshell levels whose energy differences

are resolvable using solid state detectors. For the lou and

medium Z targets considered in this work, K-shell spectra

were resolvable as two peaks, the K and K, peaks. The K

peak is actually the sum of the Ka1 and K 2 lines which

represent radiative transitions from the L3 and L_2 subshells

respectively to the K-shell. The K peak is made up of the

sum of possible transitions from the ilT and higher shells to

the K-shell. For the low Z targets (Z = 22-34), the L-shell

x-rays fall below the low energy cutoff of the detector.

Fhe three mid Z targets (Z = 37-50) have L-shell x-rays

which show up as a single structured peak. As one goes up

in Z, the binding energies of the various shells rise also,

and differences in subshell energies become correspondingly

greater. Thus, for high Z targets, the L-shell x-ray struc-

ture becomes resolvable into the groups of peaks discussed

in Chapter III. It is useful to consider an energy level

diagram to show houi this spectrum is related to transitions

between subshell levels. Referring to Figure 6, it is seen

that vacancy, say in the L2 subshell, can be filled by

radiative transitions from the Mi, V4, N1, N4, or 04 sub-

shells,resulting in L, Ll L LY5, Lyl, or LyG characteristic

x-rays. There are numerous other possible transitions to

the L2 level from other subshells, but they are generally

not observed due to their relatively low probabilities of

occurrence. Transition probabilities, or radiative widths








05
04
03
02
01


N7
N6
N5
N4
N,
N2
NI


Ms
M4
M3
M2
M,


Figure 6. Major L-shell radiative transitions.


I









as they are often called, are determinable both experimen-

tally and theoretically. The best current theoretical val-

ues are from Scofield (1974), who uses relativistic Hartree-

Fock-Slater wave Functions to determine the transition am-

plitudes. There has been a great deal of experimental work

using high resolution x-ray spectrometry to measure ratios

of K- and L-shell emission rates from relative peak areas.

This work has been recently summarized by Salem et al.

(1974) who have included tables of 'best' values obtained

by a weighted least squares fit of the available data. Ex-

periment and theory generally compare quite well, especially

for ratios of transitions between the same two shells, such

as L,]/L 2. However, the theory predicts consistently low

values for ratios of transitions between different shells,

such as L.5/LQ,. This discrepancy is particularly apparent

in the L-shell ratios, and further theoretical work is

needed to resolve it.

All the various parameters needed to convert production

cross sections to ionization cross sections have now been

discussed. Suppose one wishes to relate the production of

a particular L-shell x-ray to the various subshell ionization

probabilities. Consider,for example, the transition from

the M1 level to the L2 level which results in an L, x-ray.

First of all, one must consider how vacancies can arise in

the L2 subshell. They can either be created outright in an

L2 ionization, or they can be shifted from vacancies in the

L1 subshell through a Coster-Kronig transition. These L2








JE:Ecies rLill now be filled by either radiative or non-

ri
ra-iative transitions is just the L fluorescence yield w,.

rather, of the possible radiative transitions, only a cer-

tain fraction will come from the rl subshell and hence re-

sult in Lr7 x-rays being emitted. Since the number of

r-_served L77 x-rays emitted is proportional to the L' x-ray

production cross section, and the number of initial L2 and

L, subshell vacancies have the same proportionality to 92
I
and G3 respectively, we can summarize all the above infor-

mation as


x = (a2 1+ ) 2 ( /'r 2) (v.5)

i-here r-.and 2 are the radiative widths of the Lr7 transi-

tion, and the sum of all possible L2 subshell transitions

respectively. In a similar manner, one can write for all

the major L-shell components

aL = {3 f 2 12f23 + f13)a01w3 F3 (U.6)

S= 03 + f232 + (f12f23 + f13) l} 3 F3 (V.7)

L = C1 F1 + (f12o1 + 02w2 F2


+ {^3 3"2+ 22 13 f 23)al}3 F3 (V.8)

"L = 1 W1F1 + ( 1f12 + a2)w2 2 (2v.)
Y Y Y
here Fiy = F /Fi and y refers to the sum of all component

transitions characteristic to group y which occur to subshell








, and F. is the sum of all transitions to the ith subshell.

The suoer-scripts x and I have been omitted since from now on

all icnization cross sections will be subscripted with num-

C:-, d.-d all production cruss sections will be subscripted

*ith the transition in Siegbahn notation.

The above equations can be used as they stand to com-

pure experimental cross sections to theory, and this is the

mosL common procedure for doing so. The subshell ionization

cross sections o. are taken from the various theories to-
1
gather with appropriate values for fij, w. and F. to pre-

dict production cross sections, which are then compared to

experimental values. However, closer inspection of the data

indicates that it is possible to resolve the L group into
-Y
transitions which are characteristic of the L2 subshell, and

those characteristic of the L1 subshell. Such an unfoldjnq,

performed by the program Lritten for this experiment, is

shown in Figure 7. The transitions Ly and LY. are charac-

teristic of the L2 subshell, while transitions LY2,3 and

LY4,4' are characteristic of the LI subshell. There is a

problem since the component Ly. (which is an L2 transition)

is not resolved from Ly2, However, the above suggests a

procedure for inverting Eqs. (V.6-9) to yield subshell ioni-

zation cross sections in terms of measured production cross

sections (Chang et al., 1975). The L transitions equation
Y
can be broken into two parts


S= (f12al + a2)w2 F2 (V.10)














1000


U)
-4-

0

0I
0


500


13.0 14.0
Photon Energy (keV)
Figure 7. Computer-fitted AL L spectrum obtained with
4.0 MeV a-particles. Data points are shown as well as fitted
Gaussian peaks on an exponential background.








7L = C IF 1 (f12 a 2) 2 F2 (V.11)
Y2,3,6 Y2,3 Y6

Comparing Eq. (V.10) with Eq. (V.9) it is seen that they can

be solved for the two unknowns oa and a2 in terms of the
measured quantities aLy and aLy These results may then be

put in a third equation such as Eq. (V.7) to solve for a3.

An alternative approach due to Datz et al. (1974) is

nearly equivalent except that it is considerably more round-

about. In this method, one begins by relating the number of

characteristic x-rays produced, A to the number of ini-
th
tially present vacancies in the i subshell, Vi. (The

quantity A is related to the number of observed character-
y
istic x-rays, N through the detector efficiency e .) The

suhshell vacancies may then be related to the x-rays pro-

duced through the set of equations

V = A2 (r 6/r )A, r 1Y2,3)(1/i) (v.12)


2 = AY ( 2/ l)(l/w2) f12V1 (V.13)

S= A 2 r3/ra,2)(1/w3) f23v2

(f13 + f12f23)v1 (V.14)

The latter two equations are entirely equivalent to Eqs.

(V.10) and (V.7) respectively since both A and V. are pro-
y i
portional to ay and a. respectively, with the same propor-

tionality factors. The difference comes in Eq. (V.12) where

the LY, portion of the LY2,3, peak is corrected out using

the radiative width ratio rF6/ 1 and the production number








AyI. WLith the transformations V. -- a. and A -- these
1 1 y y
three equations may be solved sequentially as they stand to

yield the ionization cross sections j:. However, in their

original paper, Datz et al. chose to complicate matters by

introducing a vacancy production cross section for the L3

subshell C3v and related it to the La,,2 production cross

section and the ionizatijn cross sections through

S=L (r3/r )(I/A3) (V.15)
L 1, ,2

and

3 = 0 + f232 + f13 + 12f23)l (V.16)

The vacancy cross section (which is the L3 subshell cross

section corrected for Coster-Kronig transitions) is calcu-

lated -from a measured value of the L, production cross sec-

tions, and then Eq. (V.16) is divided by oa, C2, and 03a

each in turn, which yields equations for each of these in

terms of cross-section ratios. Thus one has


01 = 3" (f13 + f1223) + a/ +23(2/

(V.17)
a2 = a3 {f23 + (13 + f12f23)a/a2 + C3/02
(v.18)

03 = C3 {1 + F23((2/03) +(f13 + f12f 13) 1/3 }
(V.19)

The cross section ratios are then quite correctly equated to

the vacancy ratios, i.e., i./oa = Vi/V .

There are several differences between this approach and

the more direct approach of Chang et al. The most serious





54


one is having to rely on the ratio r,,F./ for the subtrac-

tion of the L2 component from the LY2,3,6 peak. This ratio

rises quite steeply as a function of Z from zero at about

Z = 70, and is not well known experimentally. Further,

deviations between theory and the experimental fit (Salem

et al., 1974) are typically 20%. Thus it would be expected

that the equivalence between the two iiethods would fail as

one increases the Z of the tE.rget. This will be examined

in the next chapter. One further caution in the use of the

method of Datz et al. is that the A's refer to the effi-

ciency-corrected peak areas. This detail was not mentioned

in their paper and will lead to significant error if the un-

corrected areas are used for lower Z targets, where tne

efficiency may vary over the energy range of the L-shell

transitions. ,Both methods were used in this work to convert

production cross sections to suhshell ionization cross sec-

tions, and the results are compared to theory in the follow-

ing chapter.

There has been much concern about the effect of mul-

tiple ionization on single vacancy radiative and non-radia-

tive rates. A statistical procedure has been described

(McGuire, 1969) which corrects single vacancy Auger and x-ray

transition rates for population changes in the outer shells.

Applied to argon (Larkins, 1971) it predicts an upward

change of about 50Q in the K-shell fluorescence yield as

additional vacancies in the L-shell are created. This pro-

cedure essentially agrees with the results of Bhalla and









kialters (1972) who have calculated the fluorescence yield

using wave functions appropriate to multiply ionized atomic

states.

In the case of the L-shell fluorescence yield, the

statistical approach predicts an upward change of up to two

orders of magnitude (Larkins, 1971) as the number of M-shell

vacancies is increased. Argon is an extreme case, since an-

other calculation of the same type for copper (Fortner

et al., 1972) yields results for wL Luhich are relatively in-

sensitive to ff-shell vacancies. Argon is such an extreme

case because the dominant L-shell x-ray transitions for it

are 3s to 2p (L2 and L ). Thus as one removes the less

tightly bound 3p electrons, the Auger rates are strongly

reduced, with the x-ray rates relatively unchanged. Copper,

on the other hand, has dominant x-ray transitions which are

more characteristic of higher 7 atoms, with the principal

transitions being 3d to 2p (L ,2). Hence, as the most

weakly bound M-shell electrons (3d) are removed, Auger and

x-ray rates are about equally affected. As in the case of

the K-shell, the L-shell fluorescence yields are relatively

unaffected by additional ionizations from the levels higher

than the r-shell, since the dominant transitions are from

the M-shell for all higher Z atoms. From this we may con-

clude that, granting there is no additional L- and M-shell

ionization occurring with single vacancies in the L-shell,

then single vacancy values For the L-shell fluorescence

yields may be used Lwith relative confidence. For alpha









particles at the energies considered in this work, this is

probably a very good assumption, since the L-shell binding

energies of the elements considered in the high Z range are

about the same as the K-shell energies of the elements con-

sidered in the low Z work. The degree of multiple ioniza-

tion in the latter is estimated to be negligible (McGuire,

1974). Thus, on the basis of this very tenuous comparison

between K- and L-shells, the single vacancy values in

Bambynpk et al. (1972) will be used for comparison between

experiment and theory For the L-shell data of this work.














CHAPTER VI

COMPARISON OF EXPERIMENT TO THEORY


The K-shell x-ray production cross sections which were

measured in this work are tabulated in Tables IV and V. The

latter table includes values For the ratio K /K The total

K-shell x-ray production cross sections for all the elements

considered are shown in Figures 8-10. Also shown in these

figures are the predictions of the BEA, the PIBA, and the

P1LBA with binding energy and trajectory corrections (PWFABC).

The ionization cross sections predicted by these theories

were converted to production cross sections using the single

vacancy fluorescence yields of McGuire (1969) as tabulated

by Bambynek et al. (1972). In addition, results of avail-

able earlier measurements are shown for comparison. These

include a partial use of measurements from McDaniel et al.

(1975a) which coversenergies up to 2.5 mYeV on a wide range

of targets. Also shown are the results of measurements by

Carlton et al. (1972) for Ni, Cu, Fe, Co, and Ag, and those

of Lin et al. (1973) for Ti, Mn, Zn, Se, and Sn. Finally,

thin target ionization cross sections for 1-5 MeV alpha

particle bombardment of Fe and Cu given by McKnight et al.

(1974) have been converted to x-ray production cross sec-

tions using the K-shell fluorescence yields cited previously.












_o


0
0


aL)


uJ


0 0 T 0
(S U) o Q

(Suoq)o


0
0


1T~ im ~Tn-rrnT


Ct


0


-0 I





aCLCL
<1<

Lt. nr


I


S0 0 0


(suJDq) o


III I l4 I I 111I 1 1I


II 11t I I 1111 1 I I


o. '
0.3



0.--


aj
u
-.-I I 0



IO C -

i- El 0
oE E --
n L .CH -
4- r Li 0E


uD -l
Vn ^-/



LO m J 0.


0 C)
0 03) _0



c) C -C
04 0x L
O- BU
u 0
TJ LO LO,--
0 E LN
Li 0 C [-
.D C 1110) L
a .-fdi m0J



X X [U
--4 -] I

CJ 0_ C
(-i cr) cH
UI I.)
[a --. -

.-0 kl Lj I
nl en )u [







1i -
-j Mua+
I > *H



0 0 01
S- I *r




- 7 M C.)

OL OmUf



Sn -o o c
0 0o



.- U C 0- -
w LL- I M I
C



EO0
-0 -1)'
*-i

co E a
CD 3 w -F1
CD 0[ 0 H

0.OB
.A- OE


rrrrrr~-~mrrr-r --1 -rrrn r T-l rm i TT 1 7


,,















Q


O. -



cW


-o
b o
(sujoq)to


0. 3
r a)

a -1


Sbo
(suJDq)!


b


i -
< <


, O < %




oam


( ob o
(suDoq)Jo


0o.








Q


u m
c cd
UCD E-

UCU --

I )
--Li w" C
m cix-
o n ,-
O 3 C X]-
o m -
0- L .- i -





U') .-I M
C "C .
CDL



U CiL




4L
Da c

0 0


oU C
-0 01 : a

3 *- n (








"u o
- -- q *H
-I Ci > C
0 0 -,q




C 0 .-J
U UC CO


S-o

3 LO 4
O -
C: -o ,j -
M > c- 0_O










.- :3 (u|v
.a -H













oanB
wI c c *a
0c- -M -i -i-










C m Hrm
C) Ci C C *-
3 n 'O 0r


(00 0 l-e
Li- 3
Ea
cr t- ii


: ao o
Ot O


Q o




65



O0 0 '"
10. Y



--o



S/ 0- __Ag
10


i. ---
10 / /









0o Sn
O C-

-31




I I I
1.0 2.0 3.0 4.0
Ea( MeV)
Figure 10. Measured mid Z K-shell x-ray production cross
sections for ct-particle bombardment of Y, Ag, and Sn. Pre-
dictions of the PUJBA, PLUBABC, and BEA are shown as well as
date from previous experiments. Legend: PWBA- -; PWBABC
; BEA-.--. ; 0 this experiment; 1 McDaniel eL al.
(1975); + Lin et al. (1972); x Carlton et al. (i2)-.
-3 / /



-3/ /












(1575); + Lin et ^S_. (1972); X Carlton et al. (lOT?).









As is seen from the figures, all previous results agree

with the results oF the present work (with only a few excep-

tions), within the experimental uncertainty estimated in

Chapter III. While all theories appear to reproduce the

general shape of the data, the PWBA consistently over-pre-

dicts the measured cross sections for all elements and ener-

gies considered, as does thr BEA for the lower Z elements.

For these latter elements, the best results are obtained

using the PLBABC, as is the case with low energy proton bnm-

bardmenrL also. However, there is a trend with increasing

target Z for the experimental cross sections to rise above

the PWJBABC and more closely agree to the PUBA. It has been

suggested (Chui, 1971) that for higher Z atoms, relativistic

effects in the K-shells may be important in determining

cross section magnitudes. It was noted that the inclusion

of relativistic waIe Functions in the PWBA analysis tended

to raise the predicted cross sections. Perhaps a combina-

tion of relativistic corrections and the corrections de-

scribed in Chapter IV will be needed to adequately describe

the K-shell cross sections for all elements.

The major components of the L-shell characteristic

x-ray spectrum were resolved as described in Chapter III.

The production cross sections were then determined for each

component, and these are shown in Table VI. They are also

shown plotted in Figures 11-18. For comparison, the ioni-

zation cross sections predicted by the various theories were

converted to production cross sections using Eqs. (V.6-8)











10





10


0 _-



102





i02
102


Hf
-


0
/
/O
/ /
/0/
A/


lol
U


I I I I


10


2.0 3.0
Ea(MeV)


40


Figure 11. Measured La production cross sections
for a-particle bombardment of Hf, Pb, and U. Legend:
PL.'A- -; FPIBASC ; BEA- ---; 0 this experiment.


- o 0


/ O


/
/

/
//


A


/


T











0 Hf ,o-



10 -" .



/ f
--2o










/I t
/ 0 .
/ /



o / / o H Pb, a L


-- / / 0

0 / /0 Pb

10 0












Ea (MeV)
Figure 12. Freasured Ly production cross sections
for ca-particle bombardment of Hf, Ph, and U. Legend:
PlUHA PEABA ; BEA-- this experiment.
PLUBA --; P'uJ~aABC---; BEA------; 0 this experiment.










Hf 0, 0


0O
O't>
^f>o^


/o

//0
/0


I0'
101




I0

(n
O-


/
/


I I I I


1.0


2.0
Ea


3.0
MeV)


4.0


Figure 13. measured Lp production cross sections
for a-particle bombardment of HF, Pb, and U. Legend:
PW'BA- -; PIBAC --- ; BEA ----; 0 this experiment.


Pb -p
0
0


I02
I0


_ __ _




02



----------..-^ ---.--.------__---

O Hf .-o




10 /
S./ Pb o00-
-2 / /o t-- .
0 -




--- ,./ o
S./ '

3 / 7 'U
10 ,/ o- .
/,' / _

-33---
10 /







I I I I
1.0 2.0 3.0 4.0
Ea(MeV)
Figure 14. Measj-red Ly1 production cross sections
for a-oErticle bomardnent of Hf, Pb, and U. Legend:
PLUBA- -; PLJBABC -- ; BEA-----; 0 this experiment.
/ P












P;UA--- -; PL'JBAEE-- ; 3EA- ---; 0 this exp~erimneuL.











Hf
O C
A/ "'
ADO


-p


o00
Pb o
0 o 0


d10



-2
10

C:


0


/
/
/


/
/
/


I I I I i I

1.0 2.0 3.0 4.0
Ea(MeV)
Figure 15. Fr'easured LY2,3,6 production cross sections
for a.-pirticle bomrb=rdment of Hf, Pb, and U. Legend:
F.'BA- -; PLUBAC ; BEA- ---; 0 this experiment.


o-0o


-3
O


-3
10


_ ___ 7 ___


o/~
o
0 .0
11 0






I I I I


10'


100


0
.0 I

b)
ci

-o


-I
10


PWBAC


PWBAB --- .---..o -.
PWBA ...
,PW-BA .- .. -
S./ ./
/-0 ...



S.: Au
/ 4"
/':
/ :
,,'
//
PI



/ :1

I


1.0


2.0


3.0


4.0


Ea(MeV)
Figure 16. Measured LO production cross
sections for a.-particle bombardment of Au.
Legend: 0 this experiment.


1 I


I I










10








10
I0



..-Q









10'
)ooid


hl* ,ow ....
i


S0.


/o0'
/L
.a//**

/ l. o. ...-
/ 0-


/ I

/ / ..
"/ / j "" L-'







I,.I
- .-: A L



/ .*-'
"1*' o


1.0 2.0 3.0 4.0
Ea(MeV)
Figure 17. measured La and L, production cross sec-
tions for x-particle bombardment of Au. Legend: PLUBA
- -; PUBAB -............. ; PLUBABC--.-- ; 0 this experiment.














-10
10


I02




id3
!


001


<-- L
/2,5,6


o -.
0 ..... ''' *'
--.
-



/ o ..
: .Au

S- /
5/ /
/ 0-- ...
/ 1, *.*.*.
/ 0
!/ /. .-"
/ 0, ."


o/ /

./

//.


1.0 2.0 3.0 4.0
Ea(MeV)
Figure 18. measured Lyl and Ly2,3,6 production cross
sections for a-particle bombardment of Au. Legend: PWBA
S -: PIJBAB .............; P'BAC--.-- ; 0 this experiment.


U)

a-


10'


-1>
10-






102


0
o)


f









and (V.10-11). The non-radiative rates used for this con-

version were those calculated by ,:-cuire (1970) and tabu-

lated by Bambynek et a1. (1972), while the radiative rates

were taken from t-,e latest relativistic calculations of

Scofield (1974). There are no published experimental values

for alpha-induced L-shell production cross sections, so com-

parison to previous experiments is strictly qualitative.

The most important thing to note on these graphs is the

apparent overall agreement of the PWLBA to the results ob-

tained. The only exceptions to this are the results for the

L and L components, which show a tendency to fall below

the predictions of the PUIBA. This tendency has been noted

by Gray et al. (1975) in data taken with protons, alphas,

and lithium ions, and :ho attribute the discrepancy in the

Ly1 and LY2,3,6 relative to the L production cross sections

to the effect of the binding energy correction for the L1

and L2 subshells. They suggest that a fixed average bind-

ing energy increase may nct be sophisticated enough to de-

scribe the differences in the L-subshells. Brandt and

Lapicki (1974) have suggested that similar results in the

subshell data of Datz et al. (1974) might be due to changes

in the radiative and non-radiative atomic parameters due to

multiple vacancy production. Possibilities for this have

been discussed in Chapters IV and V. Evidence of multiple

inner-shell ionization shows up as energy shifts in the peak

positions. As an experimental check, the data reported here

were analyzed for peak position shifts as functions of









energy and target, as well as between proton and alpha bom-

bardment. No such shifts were measurable. While this does

not rule out the possibility of multiple ionization, it does

indicate that the multiple ionizatiorn iias not severe enough

to cause measurable peak shifts.

An interesting feature of the L1 subshell atomic wave

function is apparent in the cross section for the L 2,
12,3,6
group of transitions. The L1 ubshell is described by a 2s

wave function, which for hydrogenic atoms has a node at a

radial distance of approximately 2a /ZL, where a is the

Bohr radius and ZL is the screened nuclear charge (usually

taken as Z-4.15). This node causes an inflection in the

predicted LI ionization cross section, which is shown in

Figure 19 for the PWBA. The energy dependence of this

'kink' in the cross section can be understood in terms of

the semiclassical approach of Bang and Hansteen (1959). This

analysis indicates that the ionization cross section is pro-

portional to an 'optimum penetration distance,' which is a

function of projectile energy. When this distance corre-

sponds to the position of the node in the 2s wave function,

i.e., where the 2s electron has the least chance to be, then

an inflection in the cross section is predicted. For haf-

nium, this inflection occurs between about 3.0 and 3.5 MeV

for incident alpha particles and is observable in the

LY2,3,6 production cross section. The form of the BEA used

(Hansen, 1973) does not predict the inflection at this en-

ergy; however, the position is predicted quite well for both
















102








H
b.





10


I/
I/
I
I
I ,'
I/I/
I/I
II

II


2.0


Figure 19.
cross sections
for the PL;3A.


4.0 6.0 8.0
E (MeV)
Theoretical L-subshell ionization
for a-particle bombardment of Au


L3


/


Au


1 1









the PWBA and PWBABC. The energy range covered here was not

high enough to fully explore the behavior of this inflection

as a function of energy and projectile. Gray et al. (1975)

have done so and determined that the inflection is best de-

scribed by the PWBA, since the binding energy correction

causes an upward shift in its position, which is not ob-

served.

It is often instructive to consider ratios of cross

sections to bring out certain details of structure, since

magnitude discrepancies cancel. The ratios L to Ly and

Ly2,3,6 to Lyl are shown in Figures 20 and 21. For higher

projectile energies the latter ratio shows a minimum and

then a gradual rise. The energy range considered here was

such that this minimum was reached only for hafnium. For

this ratio, the PWBA appears to describe the data best.

However, the PWBA predictions of L /LY1 fall consistently

above the data, again due to the fact that the PLBA over-

estimates the L production cross section, but not the Lyl.

This trend increases with target Z and may be able to be

explained using relativistic corrections, which have yet to

be applied fully to analyses of this type. Chang et al.

(1975) have noted that where relativistic calculations have

been completed, they tend to raise the subshell cross sec-

tions rather more for the L1 and L2 subshells than the L3,

and especially more at lower projectile energy. This would

seem to support the use of relativistic corrections along

with the binding and trajectory corrections to get the best

theoretical modelino.







I I o I I
0

- Hf


_- '



o o o- ooo -Au
j o o o r A


-.




-
-S.
-5
.5


0


a Pb
ooo-I


'I 5
I. -* -
1'ooa


0
0 0


0 00


1.0 2.0 3.0 4.0

Ea(MeV)
Figure 20. measured x-ray production cross section
ratios La/Ly1 for a-particle bombardment of Hf, Au, Pb,
and U. Legend: PWBA -; BEA--- ; 0 this
experiment.


T1


30


20


_30
b

20

40


30


20











1.0








1.0





S 1.0











2.0




1.0


0N
0-
0 *1'


-0 o o .
o- --3


-1,

\



iN
0


\\

N


0 0-o0.


*- -


S'o 0o F
S- .. 0- cooo


\ \

N


O -
NI
0 % N
0 1%,


0 0


-0 0
0000
I I


1.0 2.0 3.0 4.0

Ea(MeV)
Figure 21. Measured x-ray production cross section
ratios Lyl/L2, 3 6 for a-particle bombardment of Hf, Au,
Pb, and U. Legend: PUfIA----; BEA-.-.- ; 0 this
exoerimient.


Hf


-- --


L


\u


I









In Chapter V, two methods of converting experimental

production cross sections to iqoization cross sections were

discussed. The method of Chang et al. (1975) was the more

straightforward, and was used to convert the measured pro-

duction cross sections to ionization cross sections for com-

parison to theory, using Eqs. (V.I1), (V.9) and (V.7). How-

ever, unlike Chang et al. who used theoretical values for

atomic parameters to do this conversion, the available ex-

perimental values were used. This approach was chosen to

give a more realistic experimental value for the ionization

cross sections so inferred, rather than a mixture between

theory and experiment. Another reason was the possibility

of better agreement being reached in the comparison of these

more truly experimental values to the direct predictions of

the theory, free of possible errors introduced through the

use of theoretical atomic parameters. In most cases, there

is little variance between theoretical and experimental

atomic parameters. Ihe exceptions to this are the critical

values of L1 and L2 fluorescence yields and Coster-Kronig

transition rates involving these subshells, where there is

considerable discrepancy as discussed in Chapter V. The

experimental values for the radiative widths were taken from

the 'most probable values' in the review by Salem et al.

(1974). This source did not include some less important

transitions, which are needed to determine the total radia-

tive widths of the subshells. To estimate these undetermined

radiative widths, a scaling procedure was used between









theoretical and known experimental rates which occur between

the same two shells. Thus the unknown experimental rate rY5

L.jas determined from the known experimental rate IFr jy

scaling Fy1 to the theoretical ratio F / 1. This proce-

dure is assumed to be valid since known experimental rates

scale like the theory between the same two shells. Also,

the unknown rates never contributed more than 10% to the

total subshell width for any subshell.

Experimental non-radiative rates were taken from exper-

imental data tabulated by Bambynek et al, (1972). Where

values were not given, they were estimated from the remain-

ing data. A full set of measured w. and f.. were available

only for lead, hence this element will be focused upon for

the comparison between theory and experiment. The subshell

cross sections that mere determined using the method of

Chang et al. and the experimental atomic parameters are

shown in Figures 22-25. For lead, which was typical of the

other elements studied, the effect of using the experimental

atomic parameters was to raise the calculated L3 subshell

cross sections by an average of 17% relative to calculations

made with the theoretical atomic parameters. The L2 and L1

cross sections, on the other hand, were lowered by an aver-

age of 16% and 11 respectively. These changes have the

effect of causing better agreement with the predictions of

the PWBA. The difference in cross sections is due to the

difference between the experimental and theoretical Coster-

Kronig rates f12, and the L1 and L2 subshell fluorescence







IH I I_
Hf .


Io
10 -




I0
10-




C'






-1
10-




6oT


-/0.*


oo
o ---
~-."


Au


'00
/ /


/ /


I I I I


1.0


2.0 3.0
Ea(MeV)


4.0


Figure 22. YPeasured L1 subshell ionization cross
section for a-particle bombardment of Hf, Au, and U.
Legend: PWBA- -; BEA- -. ; PLUBABC- ; 0 this
experiment.


-C


-o
,o *.



































0i2


o
Hf 1-a
o .

/o /
/V"


10,




100




E I0
EO

S1 -

10io


0-
0-0
0.--1
0- ;?
^ ^^


Au


0 -P


0 0


0 _.


0 I
0


0 /


I I I I I
1.0 2.0 3.0 4.0
Ea (MeV)
Figure 23. Measured L2 subshell ionization cross
section for a-particle bombardment of Hf, Au, and U.
Legend: PWBA- -; BEA-----; PVLBABC-
0 this experiment.


/
/


1H-
I0


-g/








102



10'


I0

10

Id'



-!0



10l


Hf ,- S
00'
0 -o e
0*o0


" 0


Au


/


o


/ 0
""0o


i I I I


1.0


2.0


3.0


4.0


Ea(MeV)
Figure 24. Measured L3 subshell ionization cross
section for a-particle bombardment of Hf, Au, and U.
Legend: PUBA- -; BEA-----; PIJBABC ; 0 this
experiment.










10'






0

1-
.0 -


b io0



10



10'


L-
0,


O 0 0


0 o
d -.-


Pb


I I


1.0


2.0


3.0


4.0


Ea(MeV)
Figure 25. Measured subshell ionization cross
sections for a-particle bombardment of Pb. Legend:
PLJBA ; PLUBAB-- ......- ; PLWBAC--------; 0 this
experiment.









yields. The experimental value of the former is larger by

over a factor of two, while the ratio of the two latter is

larger by about 15% relative to tre theory. These larger

values cause decreases in the L1 and L2 and a consequent in-

crease in the L3 subshell cross sections. Since more con-

sistent results relative to the theories are gained, it is

nct unreasonable to assume that the experimentally measured

non-radiative rates are probably more correct than their

theoretical counterparts. The relative difference in the

radiative rates, although present, is not sufficient to have

noticeable effect on the cross sections. Certainly more

work is necessary both theoretically and experimentally in

the determination of fluorescence yields and Coster-Kronig

transition rates.

An interesting exercise is to use the experimentally

determined ionization cross sections in Eq. (V.8) to calcu-

late a value for the L, production cross section. This

value may then be compared to the production cross section

which was measured directly from the L x-ray peak as a con-

sistency check. This was done for all the L-shell data and

agreement was generally quite good. There was a trend for

the measured values to fall below the computed values with

increasing energy. This may possibly be due to inaccurate

fitting of the very complicated L, group of peaks. As the

energy was increased, too much background may have been sub-

tracted in the fitting routine, which should actually have

been included as part of the L complex. However, the









effect of this was not serious, with the difference at 4 r(eV

beinn only about 10% for gold, which was typical of the

other elements considered.

The subshell cross section ratios L3/L2 and L1/L2 are

shown in Figures 26 and 27, compared to the predictions of

the PWBA and the BEA. They show the same features as the

ratios L /L-y and Ly2 ,, /L- respectively, as would be ex-

pected. Considering the ratio L1/L2 first, one sees good

agreement fcr hafnium, gold, and lead, but serious disagree-

ment with uranium. Because the cross sections decrease

sharply with increasing Z, the latter poor agreement may be

due to low counting statistics. There may also be problems

with the atomic parameters which were used in the analysis,

since they are not at all well knonn for uranium. Also,

relativistic effects are bound to be.more important for

higher Z elements, so perhaps this too is a factor.

The other ratio L3/L2 has the same characteristics as

the previously discussed ratio, agreement with the PLBA being

good for hafnium, but successively poorer for the higher Z

elements. The fact that this ratio falls so far below the

theory is due to the overestimation of the measured L3 sub-

shell cross section by the PUBA. Again, as discussed pre-

viously, this may be due to differences in the subshells as

far as the effect of increased binding in slow collisions is

concerned. The low energy dip in the ratio L /L2 for gold,

reported by Datz et al. (1974) and Chang et al. (1975), was

not observed in this work. Again, this may be due to the








5 -- Hf
SO. O.. .- -. -- O r .---0


Jo Au

S 1 ooooo _o




5 ..
k"' ~ o o o o o oo
0 0 0 0 0 0




U
10 -



5
0 8000 0



1.0 2.0 3.0 4.0
Ea(MeV)
Figure 26. measured subshell ionization cross
section ratios L3/L2 for a-particle bombardment of Hf,
Au, Ph, and U. Legend: PUBA- ; BEA-----; o this
experiment.








S1 I I I


\Hf
o* o r-O-0 z- +.--


0
N.
5*-'v
0 ''


N1
a
N


Au


*5-.


Pb
0-
o D- 0-o 0- .C.._
o~ 0- Q._ 2 o-Q "-


\ \
\ \
N N
'N N.
"S
N.,


0


2.0


~*5 .5--.
-S-
-5--
-5.


3.0


Ea(MeV)
Figure 27. Measured subshell ionization cross
section ratios L1/L2 for a-particle bombardment of Hf,
Au, Pb, and U. Legend: PUJBA--- ; BEA----; o this
experiment.


H ~



H-


10




5


1.0


4.0


0 ~-- o-o- .


o0 0 0 0 o


So u









effect of the lo i statistics. The error bars shown are

typical of this, and for the lowest energy data, the uncer-

tainty is as high as 35%.

Finally, the effect of the different methods of infer-

ring ionization cross sections from measured production

cross sections discussed in Chapter V were compared. It was

found that the two methods agreed within experimental error

for nearly all targets and energies. The scattered devia-

tions which vJere observed uere probably due to inaccuracies

in the fitting of the L complex, since both methods rely

(although in different ways) on being able to accurately

separate the Ly peak from the Y2,36 peak. There were ap-

parently no systematic deviations with Z, probably because

the factor /r1 mas not that critical in determining the

cross sections. From this it is concluded that the two

methods are indeed equivalent; however, the method of Chang

et al. is certainly the more straightforward and is less

dependent upon the atomic parameters.















CHAPTER VII

CONCLUSIONS


To summarize, alpha-particle-induced K-shell x-ray

production crcss sections have been measured for 13 elements

of Z Letween 22 and 50. These have been compared to various

theoretical predictions and the results of other experiments.

The comparison indicates that the data are best described

by the plane-wave Born approximation with binding energy and

CouJomb trajectory corrections (PW3ABC). There was a ten-

dency with increasing target Z for the measured cross sec-

tions to fall above the predictions of the PWBABC. It was

suggested that this deviation may be due to relativistic

effects in K-shell wave functions.

In addition, L-shell x-ray production cross sections

were determined for the major components of the alpha-par-

ticle-induced L-shell x-ray spectra of four elements of Z =

72 to 92. These cross sections were compared to the various

theories and the best overall agreement was obtained with

the plane-wave Born approximation with no corrections (PWBA).

Inconsistencies between production cross sections charac-

teristic of different subshells were discussed and mention

was made of the possibility of binding energy correction

variations between the subshells. Ratios of cross sections

were examined, which enhance the trends noticed.

87









Finally, L-subshell ionization cross sections were in-

ferred from measured cross sections using the method of

Chang et al. (1975) and available experimental radiative

and non-radiative transition probabilities. These were com-

pared to the direct predictions of the various theories,

with more consistent results with the PWBA being obtained,

relative to tne same calculations performed using theorati-

cal atomic pErameters. This result points out the need for

Further ref nemerts in the theoretical predictions of the

L-subshell fluorescence yields and Coster-Kronig transition

probabilities.

There were several problems associated with the data

obtained in this experiment. To begin with, all the cross

section values measured were affected by the large uncer-

tainty in the measured x-ray detector efficiency and parti-

cle detector solid angle. This uncertainty was due to the

high uncertainty in the strengths of the sources used for

this determination. A much more precise experiment could

have been performed using sources which were calibrated to

a higher degree of accuracy.

The scatter which was sometimes apparent in the mea-

sured cross sections was mostly due to low counting statis-

tics. The low energy cross sections were often quite small

in magnitude and target thicknesses used were quite small,

thus requiring large amounts of time to accumulate sufficient

counts. A better experiment could have been performed if

more time had been allotted to measuring fewer points more


accurately.









For the L-shell cross sections, additional scatter was

introduced because of fitting inaccuracies. The fitting

program which was developed for use on these spectra was

quite sufficient for well resolved peaks of good statistics,

but the fitting quality decreased when resolution of multi-

ple peaks with low statistics was necessary. This problem

could have been circumvented by using a more accurate and

elaborate fitting routine, such as one of those described in

Chapter III. Circumstances did not permit this, and thus

the present experiment had a rather large uncertainty as-

sociated with it.

For all the problems outlined above, the results ob-

tained in this experiment have.still been able to shed

light on, and raise further.questions concerning,the theo-

retical treatment of K- and L-shpll ionization by alpha par-

ticles. It is expected that future experiments of this

nature will be performed, and further theoretical develop-

ments will arise, which will make charged-particle-induced

inner-shell ionization phenomena still better understood.















REFERENCES


Abrath, F., and T. J. Gray, 1974a, Phys. Rev. A 9, 682.

-- and T. J. Gray, 1974b, Phys. Rev. A 10, 1157.

Auger, P., 1925, J. Phys. Radium 6, 205.

ALwaya, Y. K., K. Izuro, T. Hamado, M. Okano, T. Takahashi,
A. Hashizume, Y. Tendon, and T. Katou, 1976, Phys. Rev.
A 13, 992.

Bambynek, W., B. Crasemann, R. W. Fink, H.-U. Freund, H.
Mark, C. D. Swift, R. E. Price, and P. V. Rao, 1972,
Rev. Mod. Phys. 44, 716.

Bang, J., and J. M. Hansteen, 1959, K. Dan. Vedensk. Selsk.
Mat.-Fys. medd. 31, No. 13.

Basbas, G., W. Brandt, and R. Laubert, 1973a, Phys. Rev.
A 7, 983.

W. Brandt, and R. H. Ritchie, 1973b, Phys. Rev.
A 7, 1971.

Bearse, R. C., D. A. Close, J. J. Malanify, and C. J.
Umbarger, 1973, Phys. Rev. A 7, 1269.

Bhalla, C. P., and D. L. Walters, 1972, in Proceedings of
the International Conference on Inner Shell Ionization
and Future Applications, Atlanta, Georgia, edited by
R. W. Fink, S. T. Manson, J. fM. Palms, and P. V. Rao
(ERDA Report No. CONF-720404, Oak Ridge, Tenn.), p. 1572.

Bissinger, G. A., A. B. Baskin, B.-H Choi, S. M. Shafroth,
J. M. Howard, and A. W. Waltner, 1972, Phys. Rev. A 6,
545.

Brandt, W., and G. Lapicki, 1974, Phys. Rev. A 10, 474.

,R. Laubert, and I. Sellin, 1966, Phys. Rev. 151, 56.

Busch, C. E., A. B. Baskin, P. H. Nettles, and S. M.
Shafroth, 1973, Phys. Rev. A 7, 1601.









CarlLon, R. F., J. L. Duggan, J. Lin, K. Eger, l. T. Lu,
m. J. Kelly, J. R. Dunning, and H. D. Fetzer, 1972, Bull.
Am. Phys. Soc. 17, 89.

Chadwick, J., 1912, Phil. Mag. 24, 594.

1913, Phil. mag. 25, 193.

Chang, C. N., J. F. Morgan, and S. L. Blatt, 1975, Phys.
Rev. A 11, 607.

Chaturvedi, R. P., R. r. Wheeler, R. B. Liebert, D. J.
Miljanic, T. Zabel, and G. C. Phillips, 1975, Phys. Rev.
A 12, 52.

Chen, J. R., J. D. Reber, D. J. Ellis, and T. C. Miller,
1976. Phys. Rev. A 13, 941.

Chien, M. H., B. Crasemann, and V. 0. Kostroun, 1971, Phys.
Rev. A 4, 1.

Choi, B.-H., 1971, Phys. Rev. A 4, 1002.

-- and E. MYerzbacher, 1969, Phys. Rev. 177, 233.

-- and E. rerzbacher, and G. S. Khandelwal, 1973, At.
Data 5, 291.

Coster, D., and R. de L. Kronig, 1935, Physica 2, 13.

Crasemann, B., editor, 1975, Atomic Inner Shell Processes
(Academic Press, New York).

-- H. Chen, and V. O. Kastroun, 1971, Phys. Rev.
A 4, 2161.

CrisLell, T. L., and T. J. Gray, 1974, Phys. Rev. A 10,
1145.

Datz, S., J. L. Duggan, L. C. Feldman, E. Laegsgaard, and
J. U. Anderson, 1974, Phys. Rev. A 9, 192.

Ferree, D. V., 1972, Thesis, University of Tennessee
(unpublished).

Fortner, R. J., R. C. Der, T. m. Kavanagh, and J. D. Garcia,
1972, J. Phys. B 5, L73.

Gallagher, W. J., and S. J. Cipolla, 1974, Nucl. Inst. Meth.
122, 405.

Garcia, J. D., R. J. Fortner, and T. m. Kavanagh, 1973, Rev.
Mod. Phys. 45, 111.









Gehrke, R. J., and R. A. Lokken, 1971, Nucl. Inst. Mrfeth.
97, 219.

Gray, T. J., G. Mi. Light, R. K. Gardner, and F. D. McDaniel,
1975, Phys. Rev. A 12, 2393.

Hansen, J. S., 1973, Phys. Rev. A 8, 822.

SJ. C. McGeorge, D. Nix, W. D. Schmidt-Ott, I. Unus,
and R. W. Fink, 1973, Nucl. Inst. Meth. 106, 365.

Hansteen, J. M., and 0. P. Mosebekk, 1970, Z. Phys. 234,
284.

Henke, B. L., and R. L. Elgin, 1970, in Advances in X-ray
Analysis, edited by B. L. Henke, J. B. Newkirk, and G. R.
Maltett (Plenum Press, New York), Vol. 13, p. 648.

Hopkins, F., R. Brenn, A. Whittemore, J. Karp, and S. K.
Bhattacherjee, 1975, Phys. Rev. A 11, 916.

Johansson, T. B., R. Akselsson, and S. A. E. Johansson,
1970, Nucl. Inst. meth. 84, 141.

-- R. E. Van Grieken, J. W. Nelson, and J. W. Winchester,
1975, Anal. Chem. 47, 855.

Kauffman, H., 1976, Florida StaLe University (unpublished).

Khandelmal, G. S., R.-H. Choi, and E. Merzbacher, 1969, At.
Data 1, 103.

,and E. Merzbacher, 1966, Phys. Rev. 151, 12.

Khelil, N. A., and T. J. Gray, 1975, Phys. Rev. A 11, 893.

Kostroun, V. 0., i. H. Chen, and B. Crasemann, 1971, Phys.
Rev. A 3, 533.

Larkins, F. P., 1971, J. Phys. B 4, L29.

Lear, R., and T. J. Gray, 1973, Phys. Rev. A 8, 2469.

Lennard, WI. N., and J. V. mitchell, 1975, Phys. Rev. A 12,
1723.

Lin, J., J. L. Duggan, and R. F. Carlton, 1973, in Proceed-
ings of the International Conference on Inner Shell
Ionization Phenomena and Future Applications, Atlanta,
Georgia, edited by R. W. Fink, S. T. Manson, J. m. Palms,
and P. V. Rao (ERDA Report No. CONF-720404, Oak Ridge,
Tenn.), p. 998.