CHARGE DEPENDENCE OF THE ENERGY LOSS

OF HEAVY IONS IN MATTER

JOHN MARK ANTHONY

1981

i

Abstract

Charge Dependence o f the Energy Loss

o f Heavy Ions in M atter

John Mark Anthony

Yale U n ive rs ity 1981

Although the energy loss o f charged p a r t ic le s in m atter has long been

thought to be p ro p o rtio n a l to the square o f the p ro je c t i le charge,

recent measurements w ith l ig h t ions suggest th a t higher order charge

dependent corrections are necessary in order to describe experim ental

re s u lts . Several ca lcu la tio n s have been advanced to p re d ic t the

magnitude o f these higher order terms. Any attempt to in ve stig a te these

e ffe c ts fo r heavy ions is complicated, however, by in s u ff ic ie n t

knowledge o f the charge s ta te o f the ions as they in te ra c t w ith the

ta rg e t m a te r ia l. Equ ilibrium charge states o f the ions a f te r

penetra tion have been measured, but a lack of understanding o f the

in fluence o f surface e ffe c ts (such as Auger d eex c ita tio n ) on the

p r o je c t i le charge has precluded any d ire c t c o rre la tio n of the charge

states inside and outside the m a te r ia l.

In order to 1) explore the importance of these higher order terms in

heavy ion stopping powers, and 2) understand the "e ffe c tiv e charge" o f

these heavy ions during p en e tra tio n , we have made a study o f the energy

loss of heavy ions in both th ic k and th in ta rg e ts . The th ic k ta rg e t

measurements, which involve the energy loss of S i, N i and Au ions in

th ic k (5-10 mg/cm2) Cu, Ag and Pb targ ets a t energies E < 2 .5 MeV/amu,

are not w e ll described by current standard tab u la tio n s . This suggests

errors in the Zxz stopping power expression used to produce these

tab u la tio n s . Independent dE/dx measurements have been c a rrie d out w ith

th in ta rg e ts , in which the stopping powers o f C, S i, C l, T i , Fe, N i, Ge,

Br, Nb and I ions in C, A l, Cu, Ag and Au targ ets a t energies near the

peak in the stopping power vs. energy curve were determined. These

measurements confirm the th ic k -ta rg e t re s u lts . The standard tabu la tions

do poorly in p re d ic tin g the magnitude and energy dependence of the

stopping power maximum in most cases.

In an attempt to f in d some simple expression which is v a lid over a

large range of p r o je c t i le , ta rg e t and energy values, we have

param eterized both the higher order corrections and the e ffe c t iv e charge

and f i t them to our data. The large experim ental data base guarantees

th a t random errors in any p a r t ic u la r p ro je c t i le - ta rg e t combination are

not im portant to the f in a l conclusions. The re su lts o f these

ca lcu la tio n s show th a t the h igher-order corrections are indeed important

in describ ing heavy ion energy loss, and the best f i t s to our data are

provided by the terms o f Lindhard. These corrections allow heavy ion

e ffe c t iv e charges fo r a l l ions in a given ta rg e t to be described by a

simple two parameter expression. This e ffe c t iv e charge expression, when

coupled w ith the Lindhard co rrections, provides a much b e tte r

d escrip tio n o f experim ental resu lts than current standard tab u la tio n s .

Also, the smooth behavior o f these parameters allows in te rp o la tio n to

combinations not ye t measured, and th is produces accurate dE/dx values

over a broad range o f p r o je c t i le , ta rg e t, and energy values.

The two parameters generate heavy ion e ffe c t iv e charge values which

agree w e ll both in magnitude and in ta rg e t dependence w ith eq u ilib riu m

charge s ta te measurements in gases, suggesting th a t charge sta tes ins ide

solids and gases (o f approximately the same atomic number) are almost

the same, and th a t the high charge states o f ions when leaving so lids

may be due to processes such as loss o f Auger electrons a t the e x it

surface of the s o lid . Comparison o f average equilibrium charge states

w ith our e ffe c t iv e charge expression may thus give a measure o f the

number o f Auger electrons em itted by the p ro je c t i le upon leaving the

so lid surface.

CHARGE DEPENDENCE OF THE ENERGY LOSS

OF HEAVY IONS IN MATTER

A D iss erta tio n

Presented to the Faculty o f the Graduate School

o f

Yale U n ive rs ity

in Candidacy fo r the Degree of

Doctor o f Philosophy

by

John Mark Anthony

December 1981

TABLE OF CONTENTS

I . In trodu ctio n ................................................................................................................ 1

I I . Stopping Power and Charge State Models ..................................................... 9

A. E lec tro n ic Stopping ..................................................................................... 12

A l. F irs t Order C alcu lations .............................................................. 12

A2. Higher Order Corrections .............................................................. 18

A3. Low V e lo c ity Stopping Powers ..................................................... 21

B. Nuclear Stopping ......................................................................................... 23

C. Average E qu ilibrium Charge States ...................................................... 27

I I I . Experimental Techniques .................................................................................... 35

A. Range and Exploratory dE/dx Measurements .................................... 39

A l. Experimental Geometry .................................................................. 39

A2. Target Fabricatio n ........................................................................... 41

B. Extensive dE/dx Measurements ............................................................... 45

B1. Experimental Geometry .................................................................. 45

B2. Target Fabrication and Thickness Determinations . . . 47

IV . Experimental Results .............................................................................................. 66

A. Thick ta rg e t measurements ........................................................................ 68

A l. Range measurements ........................................................................... 68

A2. In te g ra ted Energy Loss Measurements .................................... 69

B. I n i t i a l Thin Target Measurements ...................................................... 72

C. Extended dE/dx Measurements .................................................................... 77

A b s t r a c t ................................................................ ii

V. Higher Order Corrections and E ffe c tiv e Charge .................................... 110

A. D e r i v a t i o n ...................................................................................................... 114

B. R e s u lts ............................................................................................................... 119

V I. C o n c lu s io n ................................................................................................................... 169

Appendix A ............................................................................................................................ 174

Appendix B ............................................................................................................................ 178

R e fe r e n c e s ............................................................................................................................ 199

CHAPTER I

INTRODUCTION

The study o f the penetra tion o f heavy charged p a r t ic le s in m atter has

been a subject o f continuing in te re s t in physics fo r over e igh ty years.

The importance o f understanding th is process was demonstrated by the

e a rly experiments o f Rutherford (Ru06) and Geiger and Marsden (Ge09), on

the passage o f alpha p a rt ic le s through th in f o i ls , which led d ire c t ly to

the discovery o f the atomic nucleus (R u l l) . This generated both

th e o re tic a l and experim ental in te re s t in the in te ra c tio n o f l ig h t

p ro je c t i le s , such as protons and alpha p a r t ic le s , w ith m atter. The

discovery o f nuclear fis s io n in 1938 (Ha39) made a v a ila b le fo r the f i r s t

time energetic p a rt ic le s o f high charge and mass to be used in

experiments on p enetra tion processes. More rec e n tly , the advent of

heavy ion accelerato rs in nuclear physics has allowed the study of

penetra tion phenomena w ith inc iden t p a rt ic le s varying over a broad range

of atomic numbers and p a r t ic le v e lo c it ie s . Each step in th is process

has led to the discovery o f new techniques and a fu rth e r understanding

of the in te ra c tio n of charged p a rt ic le s w ith m atter.

One important and fundamental aspect o f the penetra tion process is

the method by which charged p a rt ic le s lose energy as they traverse

m a teria ls . The energy loss per u n it pathlength, dE/dx (a lso c a lle d

"stopping power"), can be divided in to two p a rts . Nuclear energy loss

denotes the energy lo s t to e la s t ic c o llis io n s between the p r o je c t i le and

the screened ta rg e t n u c le i; e le c tro n ic energy loss re fe rs to the process

by which the e le c tr ic f ie ld generated by the charge o f the in c id en t

p a r t ic le causes e x c ita tio n and io n iz a tio n o f the electrons in the ta rg e t

atoms. The nuclear energy loss is dominant p r im a rily a t low v e lo c it ie s

(v < 108cm/sec); fo r v 2 108cm/sec i t qu ick ly f a l ls to less than 1% o f

the to ta l energy loss. Thus i t is possible to is o la te the e le c tro n ic

energy loss component by making measurements a t high v e lo c it ie s .

One important reason to study the e le c tro n ic component o f the energy

loss is the wealth o f inform ation on atomic c o llis io n s th a t i t makes

a v a ila b le . The process depends d ire c t ly on the energy tra n s itio n s o f

the ta rg e t electrons caused by the p ro je c t i le charge, and thus i t gives

in form ation on the in te ra c tio n between the two atomic species. In

p r in c ip le any tra n s it io n can be exc ited by the inc ident p a r t ic le , and

the s ta t is t ic a l nature o f th is e x c ita tio n process requires knowledge o f

a l l possible energy le v e ls and tra n s it io n strengths o f the ta rg e t atom

in order to p red ic t the average energy loss per c o llis io n . The ce n tra l

parameter in these stopping power p red ic tions is the logarithm ic mean

e x c ita tio n p o te n tia l, I , defined by ln l = Z ^ ln E ^ , where f^ and

E^ are the o s c il la to r strength and tra n s it io n energy fo r a p a r t ic u la r

atomic t ra n s it io n . This qu an tity can be ca lcu la ted in c e rta in cases w ith

moderate success, but a d e ta ile d d escrip tio n comes only from experiment.

Although ln l w i l l obviously vary w ith the ta rg e t m a te r ia l, i t is not

expected to depend on e ith e r the inc ident p a r t ic le or i t s v e lo c ity . In

cases in which the ta rg e t electrons have o rb ita l v e lo c it ie s much greater

than the p ro je c t i le v e lo c ity ,however, these electrons w i l l tend to

respond a d ia b a tic a lly to the e le c t r ic f ie ld o f the inc iden t p a r t ic le ,

and w i l l not p a rt ic ip a te f u l ly in the e x c ita tio n process. Consequently

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fo r low v e lo c it ie s some co rrec tion fa c to r must be included in any model

o f the energy loss in order to account fo r the absence o f tra n s itio n s

in vo lv in g electrons w ith much higher o rb ita l v e lo c it ie s than th a t o f the

in c id en t p a r t ic le . These "s h e ll corrections" (Bo67,Wa52,Wa55) can also

be determined experim entally from dE/dx measurements. Thus a p r o f i le o f

the e le c tro n ic component o f the stopping power a t medium and high

v e lo c it ie s gives not only a measure o f the average e x c ita tio n energy of

a given ta rg e t atom, but also provides in form ation on the v e lo c ity

dependence of the in te ra c tio n between the bound ta rg e t e lectrons and the

in c id en t p r o je c t i le .

A second important aspect o f the penetra tion process is the study o f

the charge states o f various p a rt ic le s both during and a f te r th e ir

passage through gaseous and s o lid m a te ria ls . The e lectrons of the

in c id en t p a r t ic le w i l l be excited and ion ized during the p a r t ic le 's

p enetra tion o f the ta rg e t, through c o llis io n s w ith ta rg e t atoms. The

re s u lta n t io n iz a tio n s ta te o f the p ro je c t i le increases w ith the

p ro je c t i le v e lo c ity , and many measurements have been made on the charge

states o f these p a r t ic le s a f te r p enetra ting various m ateria ls (Be72).

For l ig h t ions, such as protons and alphas, the charge s ta te o f the ion

can be ca lcu la ted w ith some confidence, but fo r heavier ions the large

number o f e lec tro n capture and loss cross sections complicates any

attempt to p re d ic t ion charge s ta te s , which are usually determined

experim entally a f te r passage through the ta rg e t. The average charge

sta tes o f heavy ions passing through gaseous targ ets are be lieved to be

the same both inside and outside the gas. However, there is s t i l l some

debate about the importance o f charge changing e ffe c ts (such as Auger

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d e ex c ita tio n ) as the ions e x it the surface o f s o lid m a te ria ls , and the

magnitude o f p ro je c t i le charges ins ide so lids is not w e ll known. Thus

the e le c tro n ic energy losses, which depend d ire c t ly on the p r o je c t i le

charge ins ide the s o lid , can give valuable inform ation about io n iza tio n

states in regions not accessible to normal charge s ta te measurement

techniques. The usefulness o f any charge s ta te values derived from

dE/dx measurements by th is method is lim ite d by the accuracy of the

stopping power theory used to describe the data; th is therefore provides

fu rth e r m otivation fo r c a re fu l dE/dx measurements over a broad range o f

energies and p ro je c t i le - ta rg e t combinations as a te s t o f various models.

Although the study of penetra tion phenomena in general, and energy

loss in p a r t ic u la r , can give valuable inform ation on such fundamental

issues as atomic c o llis io n s and p ro je c t i le charge s ta te s , these

phenomena are also im portant in a v a r ie ty o f ap p lications to research

e ffo r ts in many other f ie ld s . For example, 1) one o f the major e f fo r ts

in energy technology in recent years has been the study of in e r t ia l

confinement fusion using heavy ion beams to bombard and implode

deuterium p e lle ts , thereby promoting fusion (Ba76). However, many

aspects o f the in te ra c tio n between high energy heavy ions and l ig h t

ta rg e t m ateria ls are not w e ll understood, and thus fu rth e r experiments

are necessary before any design optim ization can be attempted. 2) The

f ie ld o f ion beam analysis has become extrem ely important in

understanding the surface and bulk properties o f m a te ria ls . However,

the a b i l i t y o f many o f these ion beam techniques to provide q u a n tita tiv e

inform ation depends d ire c t ly on the q u a lity o f the dE/dx measurements

used by the workers. 3) One o f the methods o f tre a tin g some cancers is

through ra d ia tio n bombardment o f tumors. Although the prime source of

ra d ia tio n to date has been electrom agnetic (due to i t s a v a i la b i l i t y ) ,

heavy ions can be a very powerful to o l in ra d ia tio n therapy (Fo65,To66).

Ion beams lo c a liz e the bombarding energy much more e f fe c t iv e ly than

electrom agnetic sources, due to the nature o f the stopping process.

Thus i t is qu ite im portant to understand the energy loss and io n iza tio n

p ro p erties o f heavy ions in compound media, in order to maximize the

q u a lity o f these methods. 4) Many other processes, such as ra d ia tio n

damage stud ies, measurements o f nuclear life t im e s (using the Doppler

S h ift A ttenuation Method), semiconductor doping by ion im plantation ,

e tc . are c o n tro lled by energy loss phenomena.

There is also a tremendous amount o f p ra c tic a l importance in

understanding th is process. For example, many experiments in both

nuclear and p a r t ic le physics require energy loss corrections due to

passage through various th in f o i l windows or ta rg e t m ate ria ls .

Knowledge o f range-energy loss re la tio n sh ip s allows the use of absorber

f o i ls ( in some experiments) in order to remove unwanted charged

p a rt ic le s from the beam of in te re s t . A lso, many charged p a r t ic le

detectors are based on energy loss and io n iz a tio n processes, and in fa c t

some detectors u t i l i z e these methods not only to measure energies but

also to make p a r t ic le id e n t if ic a t io n . Thus any advance in the

understanding o f the energy loss process has d ire c t b e n e fits in

ap p lica tio n s such as these.

Although there is a strong need fo r accurate energy loss

measurements, only a small fra c tio n o f the to ta l region of in te re s t has

been explored experim enta lly . The large number o f p r o je c t i le - ta rg e t

combinations makes any systematic descrip tio n very d i f f i c u l t . With

roughly 100 possible ion beams and 100 possible elem ental ta rg e t

m ate ria ls , there are 104 d if fe re n t combinations, a l l o f which can be

measured a t a wide v a r ie ty o f energies. Even among those measurements

th a t have been made, there are o ften large discrepancies between the

measurements o f d if fe re n t groups. One im portant source o f these

discrepancies is probably improper ta rg e t p rep ara tio n . Target

preparation is a c r i t i c a l p a rt o f a l l measurements, since pinholes,

unwanted oxides, m iso rien ta tio n o f c ry s ta l ta rg e ts , e tc . , can a l l

produce large erro rs in dE/dx and to ta l range measurements. These

problems have led to much confusion in attempts to compare stopping

power models w ith experiment.

T h eo re tica l in ve s tig a tio n s in to the energy loss process began w ith

the work o f Thomson (Th03) and the problem has since been examined by

many authors. One common featu re th a t a l l these in vestig a tio n s share,

however, is a p re d ic tio n th a t the energy loss o f charged p a rt ic le s

should be p rop o rtio n a l to the square o f the p ro je c t i le charge. The bulk

of a l l a v a ila b le dE/dx and range measurements involves l ig h t ions, due

to a c c e s s ib il ity , and th e ir behavior can be f a i r l y w e ll described by

th is Zx2 re la tio n s h ip . However, recent measurements

(An69,An77a,An81,Ba63,He69) suggest th a t higher order charge dependent

corrections to th is scaling are necessary, and these charge dependent

e ffe c ts are expected to be much more important fo r heavy ions than fo r

l ig h t ions. Thus a strong m otivation fo r heavy ion dE/dx measurements

is th e ir a b i l i t y to in ves tig a te the importance of these co rrec tio n s . At

6

lower v e lo c it ie s , several p red ic tions (F i5 9 ,L i6 3 ) o f the energy loss

in d ica te th a t the e le c tro n ic stopping power is d ire c t ly p ro p o rtio n a l to

v e lo c ity . However, recent measurements (Be75,Br72,Mo66,Mo76,Na77,Pi68)

suggest th a t th is re la tio n s h ip may also be in e rro r , and fu rth e r

in ve s tig a tio n would be u s e fu l. Thus there is a current need fo r more

precise measurements and a b e tte r understanding o f the energy loss o f

heavy ions a t a l l energies.

In an e f fo r t to in ve s tig a te various aspects o f th is process, we have

undertaken a series o f measurements on the energy loss o f heavy ions in

both th ic k and th in targ ets a t a v a r ie ty o f energies. Thick ta rg e t

energy loss data were measured fo r S i, N i and Au ions (E < 2 .5 MeV/amu)

in Cu, Ag and Pb ta rg e ts . These targ ets had thicknesses o f 5-10 mg/cm2.

The th in ta rg e t measurements involved ten p ro je c tile s ^C, S i, C l, T i ,

Fe, N i, Ge, Br, Nb and I ) in f iv e s o lid ta rg e t m ateria ls (C. A l, Cu, Ag,

Au) at energies o f 0 .5 < E < 3 .5 MeV/amu. The th ic k ta rg e t transmission

data give values o f heavy ion ranges, as w e ll as some inform ation

concerning the stopping power curves. The th in ta rg e t measurements

allow dE/dx values to be determined d ire c t ly , and these measurements can

then be used to te s t a v a r ie ty o f energy loss models. Any attempt to

examine the charge dependent e ffe c ts fo r heavy ions is complicated,

however, by in s u ff ic ie n t knowledge o f the charge s ta te of the ion as i t

in te ra c ts w ith the ta rg e t m a te ria l. Thus we have examined various

"e ffe c tiv e charge" param eterizations in conjunction w ith the higher

order co rrec tions , in an attempt to fin d some simple expression v a lid

over a broad range of p r o je c t i le , ta rg e t and energy values. Our large

data base guarantees th a t random errors in any p a r t ic u la r p r o je c t i le -

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ta rg e t combination w i l l not be im portant to the f in a l conclusions.

The resu lts o f these ca lcu la tio n s suggest th a t indeed the h igher-

order corrections are im portant in describ ing heavy ion energy loss, and

th a t current standard tabu la tions do poorly in p re d ic tin g the e lec tro n ic

energy loss a t high v e lo c it ie s (v £ 109cm /sec). The best f i t s to our

data are provided by the terms o f Lindhard (L i7 6 ) , which also allow

heavy ion e ffe c t iv e charges fo r a l l ions in a given ta rg e t to be

described by a simple two parameter expression. Use o f th is charge

param eterizatio n , when coupled w ith the Lindhard co rrections, allows

p re d ic tio n o f dE/dx values w ith much b e tte r success than the standard

tabu la tions (N o70,Z i80), which assume a Z j2 stopping power dependence.

The e ffe c t iv e charge param eterization we have used is given by*

^1 . -X v , / . , , * 1 - A exp ( — ) ( 1 •1 )Z 1 v Z ’

1 0 1*

where is the ion e ffe c t iv e charge and v 0 = e2/h . The two

parameters A and X in th is expression vary smoothly w ith the ta rg e t

atomic number, and can be expressed as

A = 1.16 - 1.91x10" 3Z2 + 1. 26x10"5Z 22 ( i . 2 )

and

X= 1.18 - 7.5xlo‘ 3Z2 + 4.53xlo"5Z22 (1 .3 )

This generates heavy ion e ffe c tiv e charge values which agree w e ll both

in magnitude and in ta rg e t dependence w ith eq u ilib rium charge s ta te

measurements in gases. This suggests th a t charge states inside so lids

and gases (o f approximately the same atomic number) are almost the same,

8

and th a t the high charge states observed fo r ions leaving so lids may be

due to processes such as loss o f Auger e lectrons a t the e x it surface o f

the s o lid . Comparison o f average eq u ilib riu m charge states w ith our

e ffe c t iv e charge expression may thus give a measure o f the number o f

Auger e lectrons em itted by the p ro je c t i le upon leaving the s o lid

su rface .

At low v e lo c it ie s the stopping power o f N i ions is found to be very

nonlinear w ith v e lo c ity , a t variance w ith some p red ic tio n s . However,

th is n o n lin e a rity is reasonably w e ll described by the semiempirical

ca lcu la tio n s o f Nesbet and Z ie g le r (Ne77). These re s u lts , as w e ll as

the higher order e ffe c ts discussed above, are consistent w ith our th ic k

ta rg e t data in a l l cases.

In the fo llo w in g pages we w i l l present our measurements and a

discussion o f our re s u lts . The th e o re tic a l basis fo r both the standard

energy loss p red ic tio n s , as w e ll as the higher order co rrections , w i l l

be examined in Chapter I I , along w ith some ju s t i f ic a t io n fo r the

e ffe c t iv e charge re la tio n s h ip assumed here. Chapter I I I discusses the

experim ental techniques involved in both the ta rg e t preparation and the

th ic k and th in ta rg e t energy loss measurements, as w e ll as the data

reduction methods used to derive energy loss measurements from the raw

in fo rm ation . In Chapter IV we present the resu lts o f our th ic k and th in

ta rg e t measurements. Chapter V discusses the a n a ly t ic a l techniques used

to derive higher order corrections and e ffe c t iv e charge va lues, and

examines the a b i l i t y o f these expressions to reproduce our data . We

conclude (Chapter V I) w ith a summary o f our re s u lts .

CHAPTER II

STOPPING POWER AND CHARGE STATE MODELS

As we mentioned in Chapter I , the stopping power o f charged p a rt ic le s

in m atter can be d ivided in to two p a rts , nuclear stopping and

e le c tro n ic stopping. The nuclear stopping power, which qu ickly ris es

to a maximum and then f a l ls asym potically to zero fo r large v e lo c it ie s ,

ty p ic a lly accounts fo r less than 1% of the to ta l stopping power a t

energies above 0 .2 MeV/amu. The whole range of nuclear stopping can be

described by a general formula appropriate fo r a l l values of Z :

(p ro je c t ile atomic number), Z2 (ta rg e t atomic number), and E. The

e le c tro n ic stopping power component also ris es to a maximum (a t energies

of a few MeV/amu) and decreases, but r e la t iv is t ic contraction of the

Coulomb f ie ld o f the in c id en t p ro je c t i le causes a s lig h t increase again

a t high v e lo c it ie s (v i 0 .9 5 c ). No un iversa l theory is a v a ilab le fo r

e le c tro n ic stopping, however, and in fa c t the region below the stopping

power maximum is often trea te d by a d iffe re n t approximation than the

re s t of the curve.

In th is chapter we discuss the th e o re tic a l basis fo r both nuclear and

e le c tro n ic stopping power, includ ing the various approximations used to

describe d if fe re n t regions o f the e le c tro n ic dE/dx curve. The higher

order charge dependent corrections to the e le c tro n ic stopping power are

also examined and compared. One c e n tra l parameter o f importance in a l l

these studies is the charge of the p ro je c t i le inside the ta rg e t, and

therefo re a short d escrip tio n o f eq u ilib rium charge state models and

10

th e ir a b i l i t y to describe experiments is presented. F in a lly the

controversy concerning p ro je c t i le charge sta tes inside and ouside s o lid

targ ets is examined, and possible explanations fo r these re s u lts are

explored.

11

12

As heavy charged p a rt ic le s penetrate m atter, the primary method o f

energy loss is through e x c ita tio n and io n iza tio n o f the atomic

e lectro n s. The e le c tro n ic stopping power is important a t a l l

v e lo c it ie s , and many ca lcu la tio n s have been made in an attempt to

describe th is process. One major obstacle to any general stopping power

expression, however, is the large v a r ie ty o f p r o je c t i le , ta rg e t and

v e lo c ity combinations which must be described. The behavior o f a low

v e lo c ity , h igh ly charged heavy ion is qu ite d if fe re n t from th a t o f an

energetic proton, and d if fe re n t approximations are necessary. Several

c la s s ic a l and quantum mechanical ca lcu la tio n s have been performed, each

w ith a d iffe re n t region o f v a l id i t y , and these models a l l provide

im portant inform ation on the character o f e le c tro n ic energy loss.

A. Electronic Stopping

A l . F irs t Order C alcu lations

One o f the f i r s t comprehensive treatm ents o f energy loss was given by

Bohr (B o l5 ), who described the in te ra c tio n between the p ro je c t i le (o f

v e lo c ity v ) and the ta rg e t e lectrons in terms of a c la s s ic a l impact

parameter, b (see F ig .2 .1 ) . Bohr proposed th a t the maximum value o f b

is the distance fo r which the c o llis io n time b /v is comparable to the

e le c tro n ic o rb ita l period , 1 /v , and th a t fo r la rg e r values o f b the

electrons w i l l respond a d ia b a tic a lly w ith no energy tra n s fe r . Also,

there ex is ts an interm ediate value b a which divides the c o llis io n s in to

"close" and "d is tan t" regions. Bohr argued th a t close c o llis io n s , w ith

b<b1( can be tre a te d as free electrons scattered by the in c id en t

p a r t ic le , w hile d is ta n t c o llis io n s (b>bl ) involve e le c tro n ic e x c ita tio n

o f harm onically bound ta rg e t electrons by the f ie ld o f the p r o je c t i le .

The force equation fo r a harm onically bound e lectron can be given by

( Ja75)

— 2 — e —x + T x + O i x = E (2.1)0 D 3

where E ( t ) is the e le c tr ic f ie ld a t the o r ig in o f the binding force due

to the inc ident p r o je c t i le charge Z: e, w0 is the binding frequency,

and r represents a small damping constant. The ra te o f energy

tra n s fe r to the e lectro n is given by

= J E*. 1 d3x ' (2 .2 )

and thus the to ta l energy loss in the c o llis io n is

& = d t J d 3x 'E - J* (2 -3 )

^The current density is J = -e v 6 (x '-x ) fo r the e le c tro n , and we have

AE = -e r V • E d t (2 .4 )J-00

where v=x and E is taken to be the f ie ld o f the inc ident p a r t ic le a t the

o r ig in (This is c a lle d the dipole approxim ation). Use of the Fourier

transforms o f 5T(t) and E ( t ) , as w e ll as some algebra, allows us to w rite

4 e - - 5 T - f f l + l 2 <*•*>

Evaluating the electrom agnetic f ie ld s (and th e ir transforms) a t the

13

14o r ig in and in s e rtin g them gives us the energy tran s fe r e x p l ic i t ly , i . e .

2Z2e4AE(b) - ------ ^ ( - L - ) U 21L<5) + - !j - 5 2K0<?)] (2 .6 )

mv b ywhere K0(£ ) and Ka(£ ) are Bessel functions, and

y - d - v V f i t . - $ Lyv

I f there are N atoms per u n it volume w ith Z2 electrons per atom, the

number o f electrons w ith impact parameters between b and b+db in a

thickness o f m atter dx is given by

dn = NZ^ffbdbdx (2 .7 )

The energy lo s t per u n it distance can then be determined from

2ffNZ2 y f j £ AEjfbJbdb (2 .8 )

where f ^ , the o s c il la to r strength of the j 1th o s c il la to r , represents

the fra c tio n of e lectrons w ith binding frequency u j . Evaluation of

th is expression gives

dF.1 Zl ej - j „ 4 » N Z 2 ----~• m v

and the "stopping number" L is given by

2 4 le= 41TNZ -------— L (2 .9 )

2 2L = In B - v /2 c 4 g(bp (2 . 1 0 )

where B=1.123y2niv3/Z 1e2<w>, g(t>j) is a correction term dependent on

b j , and <w> is defined by ln < w > = Ifjln u ^ . A s im ila r

ca lc u la tio n fo r the close c o llis io n s , i . e . c o llis io n s w ith free

e lec tro n s , y ie ld s the re s u lt

d E ) 2» N z f z 2e4 _ Z ,e2 _d x ) . " I In [i + (-- ----- ) ] (2 .11 )

/< bL mv mv sbjThe to ta l stopping power o f Bohr is thus the sum o f these two terms. To

evaluate these two expressions, we choose b 1« y v /w , beyond which the

in te ra c tio n is ad iabatic w ith no energy tra n s fe r , and also

b 1>>Z1e2/mv2y, a rough estim ate o f the "size" o f the sc a tte rin g

center. In most cases the b x dependent terms represent a very small

fra c tio n of the to ta l energy loss (Ah78) and can be ignored to give the

Bohr formula

4 1 T N Z .z fe4 , , 0 0 3 9 a2S ---------------- § -* — [ In ( • 2 ) . ln ff . * 2, . A - ] ( 2 . 1 2 )

mv Z e co

I t is c le a r th a t a purely c la s s ic a l c a lc u la tio n is not appropriate

fo r a l l aspects o f energy loss. The small energy tra n s fe r pred icted fo r

large impact parameters in the d is ta n t c o llis io n expression is not

v a lid , since the energy tra n s fe r must be quantized. However, the

expression is co rrec t i f AE(b) is re in te rp re te d as a mean energy loss,

summed over a l l possible atomic tra n s itio n s . Also, quantum mechanics

p ro h ib its the form ation of an in f in i t e ly lo c a liz e d wave packet fo r a

p a r t ic le w ith w e ll-d e fin e d momentum. Thus the c la s s ic a l treatm ent fo r

close c o llis io n s , which presupposes such a wave packet, w i l l break down.

However, fo r slow, heavy p a rt ic le s the fra c tio n of the to ta l energy loss

due to close c o llis io n s is qu ite sm all, and the c a lc u la tio n o f Bohr w i l l

s t i l l be ap pro pria te .

15

The f i r s t co rrect quantum mechanical c a lc u la tio n was performed by

Bethe (Be30), using the Born approximation (Bo26). In th is

approximation the e lec tro n wave functions are assumed to be plane waves,

and momentum tra n s fe r ra th er than impact parameter is used to

ch aracterize the in te ra c tio n . For close c o llis io n s Bethe assumed the

electrons are fre e , w hile the d is ta n t c o llis io n s are trea te d as f i r s t

order d ipole e x c ita tio n s . Since

- s r - i y ^ i 2 ( 2 - 1 3 )

and

J e ik r v i ( r ' ) e i k r dT' (2 .14)2ltti

—* *in the Born approximation (Me70) (K and K are the wave vectors o f the

fre e p a r t ic le before and a f te r s c a tte rin g , re s p e c tiv e ly ), knowledge of

the p o te n tia l Va allows us to ca lcu la te the d i f fe r e n t ia l cross section

(and therefo re the stopping cross section) d ire c t ly . represents the

instantaneous Coulomb in te ra c tio n s o f the system (p ro je c t ile plus

ta rg e t) plus the in te ra c tio n o f the p a r t ic le currents w ith the vector

p o te n tia l A (Ah78). In s e rtin g the p o te n tia l Vj in to Eq.2.14 and using

the re la t io n S=NI/E do (do is the cross section fo rn n n

e x c ita tio n to the atomic s ta te |n>) g ives, to lowest order in Z1# the

Bethe stopping formula (Be30,Fa63), i . e .2 44ffNZ.e 2L 2 „s ----------------------------------- - In (1 - P2) - /s2 ] (2-15>

mvwhere I is the logarithm ic mean e x c ita tio n p o te n tia l per e le c tro n ,

16

In I = I f nlnE n (2 .16 )

17and f is the d ipole o s c il la to r strength fo r the n 'th energy le v e l.

Although values o f I can in p r in c ip le be ca lcu la ted d ire c t ly , the number

of possible tra n s itio n s u sually makes th is im p ra c tica l, and most resu lts

are due to experiment. Bloch has had some success, however, in

p re d ic tin g mean e x c ita tio n energies using a Thomas-Fermi descrip tio n o f

the e le c tro n ic energy le v e ls (B133b).

Since the Born approximation is most appropriate fo r weak p o ten tia ls

and high inc iden t energies, the re s u lt o f Bethe is seen to be p r im a rily

applicab le to l ig h t , energetic p a r t ic le s . Thus, due to the d if fe r in g

regions o f v a l id i ty o f the Bohr and Bethe re s u lts , there is a broad

range of p ro je c t i le -v e lo c ity combinations not accurate ly described by

e ith e r theory. This m otivated Bloch (B133a) to examine the connection

between these two models. He showed th a t the d is ta n t c o llis io n formula

of Bohr was v a lid when viewed as an average energy loss, but he

recognized the necessity o f a quantum mechanical descrip tion of the

close c o llis io n s . Bloch argued, however, th a t the f ie ld of the inc iden t

p a r t ic le produced perturbations o f the wave functions o f the ta rg e t

electron s, and th erefo re the plane wave states of Bethe are not

appropria te . Of course, fo r weak p o te n tia ls the pertu rb atio n is small

and the Bethe expression re s u lts , w hile fo r strong p o te n tia ls the

e ffe c t iv e size o f the p o te n tia l should be large enough to produce

c la s s ic a l sc a tte rin g of wave packets, which is the Bohr approximation.

Thus the Bloch formula was an attem pt to bridge the gap between the

c la s s ic a l and quantum mechanical form ulas. Bloch's re s u lt is

S « -------- — [ I n — j - 1 - + ^ ( l J - R e ^ l + i4ffNZ?e4 Z2 _ 21 r, 2mv ) - £ l n ( l - £ 2) - ] (2 .17 )vmv

where ’J'(z) is the digamma function (Ab70) and v 0=e2/ft . I f we ignore

r e la t iv is t ic co rrec tions , we f in d th a t fo r Z1v 0/ v « l the Bethe formula

re s u lts , w hile fo r Z 1v 0/v > > l,

Z.v vR e * ( l + i ) — l n ( Z . — ) (2 .18a)v ' v 1 v

* ( 1, _ (2 .1 8 b )2

which gives the Bohr form ula. The r a t io y=Z1v 0/v is seen as a

convenient parameter fo r d e lin e a tin g the c la s s ic a l (y> > l) and quantum

( y<<1) regions, and in fa c t the Bloch formula affo rds the only model fo r

experiments w ith y ~ l .

18

A2. Higher Order Corrections

An important featu re o f the Bethe ca lc u la tio n is th a t the stopping

power is expected to be p roportion a l to the square o f the p ro je c t i le

charge. This Z j2 dependence has been the basis fo r almost a l l

comparisons w ith experiment, and i t is assumed in a l l current stopping

power and range com pilations. However, recent measurements on pion

ranges (H e69), as w e ll as p rec is ion data on proton and alpha p a r t ic le

stopping powers (An69), are not consistent w ith th is sca lin g . Moreover,

Andersen e t . a l . have made prec is ion measurements w ith proton, alpha and

L i p ro je c tile s (An77a) which allow them to separate out higher order

contributions to the stopping power, and they have found both Z j3 and

Z1‘* corrections in th e ir re s u lts . For l ig h t , energetic p a r t ic le s , such

as 1 MeV protons, the Bloch co rrec tion reduces to - 1 .123(Z 1v 0/ v ) 2 , which

produces a Zt 4 co rrec tion to the to ta l stopping power. However, there

is no o r ig in fo r a Z j3 co rrec tion in the models discussed above, and

fu rth e r ca lcu la tio n s are necessary.

Recently several attempts have been made to exp la in these

co n trib u tio n s . Ashley, e t . a l . (As72) have extended the ca lcu la tio n s of

Bohr beyond the dipole approximation by includ ing the motion of the

bound e lectro n during the c o ll is io n . They argue th a t th is e ffe c t is

only important fo r d is ta n t c o llis io n s , since fo r close c o llis io n s the

in te ra c tio n w ith free e lectrons w i l l approach the Rutherford sc a tte rin g

law. They therefo re introduce a lower l im i t o f impact parameters,

a^, which is approximately the radius o f the e lectron o rb it . This

generates a Z73 co rrec tion which they have evaluated fo r the Lenz-Jensen

s t a t is t ic a l model o f the atom. This expression can Ho w ritte n as4itZ2Z e4 N

S - ------5 + <2 -»)mv

where L is the Bethe-Bloch stopping number. Here B

Lj = F ( b / x V z ^ x 3/2 (2*20>

w ith x=v2/Z 2v 02 , b is a free parameter and F is evaluated num erically .

H i l l and Merzbacher (H i74) have performed a quantum mechanical

c a lc u la tio n on the quadrapole ex c ita tio n s of a harm onically bound

e le c tro n . This generates a Z 73 term id e n tic a l w ith th a t o f Ashley,

e t . a l . , as expected fo r a harmonic o s c il la to r . Jackson and McCarthy

(Ja72) have made a c a lc u la tio n s im ila r to th a t of Ashley e t . a l . , fo r the

d is ta n t c o llis io n s , but they choose a d if fe re n t value fo r the impact

19

parameter c u to ff . They use the value a =(h/2mw)1^2 , i . e . the(l)

quantal radius o f the o s c i l la to r , and th is allows them to w rite the

f ra c t io n a l co rrection to the energy loss as

20

ZjF(V)

1/2where V=137fJ/Z2 and F(V) is evaluated num erically . F in a lly

Lindhard (L i76 ) has used a fre e e lectron gas w ith plasma frequency u

to derive a Za3 co rrec tion approximately twice th a t o f Jackson and

McCarthy.

Comparison o f these c a lcu la tio n s w ith the experiments mentioned above

suggests th a t the resu lts o f Lindhard may provide the best f i t to

e x is tin g measurements. For l ig h t , s w ift p a r t ic le s such as photons,

however, these corrections are qu ite sm all, and therefore no re a l

conclusion can be drawn from the previously a v a ila b le data.

Experimental measurements w ith h igh ly charged heavy ions would

presumably show much la rg e r e ffe c ts , but precis ion heavy ion stopping

powers have not in general been a v a ila b le .

The corrections discussed here are a c tu a lly the c la s s ic a l equivalents

of the second Born approximation. In p r in c ip le we expect th a t higher

order corrections (such as Z j 5 ^ 6 , e tc . ) w i l l also e x is t , but these may

be d i f f i c u l t to determine experim enta lly . There are problems in

attem pting to separate out even the Zt3 co rrec tio n , since examination o f

th is e f fe c t in current heavy ion stopping power measurements is

complicated by in s u ff ic ie n t knowledge o f the charge state o f the ion as

i t penetrates m a te ria ls . Thus before any d e f in it iv e statement about the

charge dependence o f heavy ion energy loss can be made, some method o f

determ ining the p ro je c t i le charge v a r ia tio n is necessary.

21

A3. Low V e lo c ity Stopping Powers

One im p lic it assumption in a l l models discussed above is th a t the

in c id en t p ro je c t i le is moving a t a v e lo c ity large compared to the

o r b ita l v e lo c ity o f the ta rg e t e lectron s. This condition is not always

f u l f i l l e d , however, and any e lectrons w ith o rb ita l v e lo c it ie s much

g reater than the p ro je c t i le v e lo c ity w i l l respond a d ia b a tic a lly to the

in c id en t f ie ld , w ith no energy tra n s fe r to them. Consequently a t

v e lo c it ie s comparable to the ta rg e t e lectron v e lo c it ie s , "s h e ll

corrections" must be ca lcu la ted to account fo r the change in the

stopping power when various e le c tro n ic sh e lls do not p a r t ic ip a te in the

energy loss process. These corrections have been ca lcu la ted by several

workers, w ith some success (Bo67,Wa52,Wa55).

As the p r o je c t i le v e lo c ity approaches zero, however, almost a l l

ta rg e t e lectrons w i l l need these "s h e ll co rrec tio n s ," and thus the

models discussed above are no longer u s e fu l. D iffe re n t approximations

are therefo re necessary, and Lindhard, e t . a l . (L i63 ) have modelled an

e lectron gas o f constant density to show th a t low v e lo c ity stopping (v <

Z j^ ^ V q) is p roportion a l to v e lo c ity . The v a r ia tio n o f the stopping

power w ith Z 1(Z2 is examined using a Thomas-Fermi p ic tu re o f the atom to

f in d

S = %%ltc2& f L p 2/Z) ( v /v 0) (2 .2 2 )

where ^~Zl 1/6 and Z = (Z j2 /3 + Z22 /3 ) 3 /2 . F irsov (F i5 9 ) has

examined low v e lo c ity stopping w ith a c la s s ic a l model in which the

energy loss is assumed to a ris e from e lectro n exchange between the

in c id en t and ta rg e t p a rt ic le s as th e ir e le c tro n ic sh e lls overlap . He

also reports an energy loss proportion al to the p a r t ic le v e lo c ity over a

broad energy range.

This lin e a r dependence o f the stopping power w ith v e lo c ity has been

accepted by many workers, es p e c ia lly in ion im plantation and low energy

range experiments. I t is obviously very important to understand th is

process, since the range o f most p a rt ic le s is dominated by th e ir low

v e lo c ity behavior. Recent experiments suggest th a t th is v e lo c ity

scaling may be in e rro r , however. One possible explanation may be the

charge changing pro p erties o f low energy p ro je c t i le s , but more study,

both th e o re tic a l and experim ental, is necessary before th is question can

be resolved.

22

B. Nuclear Stopping

The major mechanism fo r energy loss o f very low v e lo c ity heavy ions

is through d ire c t c o llis io n between the in c id en t ions and the screened

nu c le i o f the ta rg e t atoms. This is b a s ic a lly a s c a tte rin g process, and

can be described in terms o f c la s s ic a l mechanics i f the associated

wavelength of the in c id en t p a r t ic le is much less than an atomic

dimension, i . e .

i t s r < 2 - 2 3 )

where h is P lanck's constant, M,E are the mass and energy o f the

inc iden t p a r t ic le , and a is the atomic radius o f the ta rg e t atom. I f we

choose a=a0 , the Bohr radius o f hydrogen, th is condition can be

re w ritte n as

h2 0.3E » --------- « —— e V (2 .24 )2MaQ 1

where A1 is the p r o je c t i le mass in amu. In p rac tice the lowest energies

of in te re s t are in the keV range, and a c la s s ic a l descrip tion is

ju s t i f ie d . Thus we can describe the in te ra c tio n in terms of an impact

parameter formalism.

In F ig . 2 .2 we show the c o llis io n geometry in the center of mass

frame. The in c id en t p a r t ic le has mass, energy and v e lo c ity M1( E and

Uj in the lab frame, and r is the to ta l p a r t ic le separation.

Conservation o f energy and angular momentum in the c o ll is io n gives

23

M2 i M1M2 r « dr 12 . 2, dj/x 2( M j + M 2 )“ E 1 = V ( r ) + * ( M 1 + M 2 ) dt

24andMi Mo rM„

7 ^ % V ■ " , < T v % -> 2 i f * " , < / / / (2 .26 )

Use o f the re la tio n s h ip

(-£->2 + r 2 ( - f f ^ . {2.21)

allows the time dependence o f these equations to be removed. By making

the change o f va ria b le s y = l / r and re a liz in g th a t f va ries from 0/2

to n/2 as r goes from -« to r„ (the distance o f closest approach) we

have

6 = f f - 2 b J U ° [ l - 2> - b V ] ^ d** (2 .28)0 1 2

where y0= l / r 0 .

In p r in c ip le the p o te n tia l V (y) can be introduced and th is equation

evaluated to y ie ld the re la t io n db/d0. This then allows the cross

section do th a t a p a r t ic le w ith impact parameter between b and b+db

w i l l be scattered in to an angle between 0 and 0+d0 (there is no

azim uthal dependence) to be determined by

d o ( e , 0 ) =--- — ------

The to ta l energy tra n s fe r in the c o llis io n is given by4M M 2

E * -------------- jT E 1sin 6 /2 (2 .30 )(m x+ m 2)

and thus the nuclear energy loss is

Sn = NJ Et T F dEt <2' 31)

The choice o f p o te n tia l is obviously o f major importance in

attem pting to evaluate th is expression. Although many forms fo r th is

function have been examined, the most r e a l is t ic expressions are seldom

in teg rab le in eq. 2 .2 8 , and numerical methods are necessary. The most

widely used expression is probably

W 2V(r) = ----- <p / — ) (2 .32 )r ^TF' a 'where ^ ^ ( r / a ) is a Thomas-Fermi screening function and

a o 0 .885aA (zf + z f )"^ (2.33)0 1 2

Lindhard, e t . a l . (L i6 3 ) (LSS) suggest an approximate a n a ly tic form fo r

«fTF, given by

2 1<PTFUj-> » + 3 ] ' (2 .34 )

and th is has been the basis fo r most nuclear stopping power expressions.

Values fo r the nuclear energy loss ca lcu la ted w ith th is p o te n tia l are

given g rap h ic a lly in several papers. The Thomas-Fermi p o te n tia l is

expected to be too repuls ive a t large distances, however, which w i l l

make the corresponding nuclear stopping power too la rg e .

Recently Z ie g le r (Z i77b) proposed a series o f sem iem pirical nuclear

stopping expressions which attempt to correct th is problem. These can

be expressed in terms of the reduced ion energy, e , given by

32. 53 M Ej« = ” I T ~ T (2 .3 5 )z 1z 2 (M 1 + m 2) (Zj + Z ^

where E: is in keV and M: , M2 are in amu. We then have

25

ASn ■ 1.593 e £ e < 0. 01

26(2 .36a)

o _ i „ § In Le + e x p ( l) 1n ! • 7 g ---------- x 0 .0 1 < e < 1 0 (2 .36b)

1 + 6 .8 e + 3 .4 e '

Sn * l n (2(k 4 7 c ) e > 1 0 (2 .36c)

M u ltip ly in g by [ 8 . 4 6 2 3 ^ ^ / (Mj + M gK Zj273 + Z22 /3 ) ] 1/2 gives

Sn in un its o f e V /(1 0 15atoms/cm2) . These re la tio n sh ip s are v a lid fo r

a l l Z j , Z2 and Ej combinations, and they w i l l be used in preference to

the LSS resu lts in the current study.

Any attempt to describe the energy loss o f heavy ions in m atter

depends on an accurate value fo r the p r o je c t i le charge ins ide the

m a te ria l. Although the " e ffe c tiv e charge" of heavy ions can be

ca lcu la ted from experim ental dE/dx measurements by using a p a r t ic u la r

stopping power theory, the re s u lts w i l l obviously depend on the accuracy

of th a t theory. Thus a study o f eq u ilib rium charge states , which are

measured a f te r the ions have passed through so lid and gaseous ta rg e ts ,

is qu ite important as a check o f the consistency o f these e ffe c t iv e

charge values. Although the e ffe c t iv e charge is not id e n tic a l w ith the

average charge, eq u ilib rium charge s tate measurements provide c e rta in

constra in ts which are im portant in any understanding of e ffe c t iv e charge

va lu es .

Consider a beam of ions passing through a gas w ith a density low

enough th a t between c o llis io n s almost a l l the ions w i l l have returned to

th e ir ground s ta te . I f we denote by N(x) the number o f ions carrying

t e lectron s, then the ra te o f change o f N(x) over a pathlength dx of

constant v e lo c ity is given by (see Bo48)

~~dbT^ " P { N (T _ 1 )a c *T ‘ 1 ) + N ( T + l ) o ( T + l ) - N ( T ) [ o c ( T ) + CTi ( T ) ] ^ <2 - 3 7 >

where p is the number o f gas atoms per u n it volume, and oc and

are the cross sections fo r capture and loss of an e lectro n by an

ion carry ing t e lectrons before the c o ll is io n . (We ignore a l l capture

and loss processes invo lv ing more than one e le c tro n ). The ra te o f

change o f the average number o f e lectro n s, "t = t ( x ) , is given by

27

C. Average Equilibrium Charge States

28(2 .38)

where N is the to ta l number o f ions in the beam. I f we make the

where x (x 0) is the average e lectron number a t the p o in t x 0. Although

th is ca lc u la tio n in p r in c ip le allows values o f the average charge s ta te

to be determined, in p ra c tic e such a simple p ic tu re is seldom u sefu l.

We have ignored here a l l capture and loss cross sections invo lv ing more

than one e lec tro n , as w e ll as the e ffe c ts o f p ro je c t i le e x c ita tio n on

the re levant cross sections. Also, the capture and loss p ro b a b ilit ie s

are seldom as w e ll behaved as was suggested in e q s .2 .3 9 ,2 .4 0 . Thus a

rigorous c a lc u la tio n o f average eq u ilib rium charge states from f i r s t

p rin c ip le s is very complicated, and few authors have attempted i t .

However, much work has been done w ith p red ic tin g average charge states

s im p lify in g assumption th a t both oc and cty vary slowly and

lin e a r ly w ith t , we can w rite

O c ( T ) = n [ l + 0£,(T - W) ] (2 .39)

(2 .40 )

where ac and are constants small compared to 1, and w is

the value o f t fo r which oc and have equal magnitude, ft.

This then allows us to w rite

dT (2 .41 )

and, by in te g ra tio n

t (x) = u>+ [ t (x 0 ) - w ] exp [-p - a ^ ) (x - x Q) ] (2 .42)

on more general grounds, w ith vary ing degrees o f success.

One o f the f i r s t o f these ca lcu la tio n s was the work o f Bohr (Bo54),

based on a simple o r b ita l p ic tu re o f the atom. He argued th a t any

e lectro n w ith an o rb ita l v e lo c ity less than the v e lo c ity o f the inc ident

p ro je c t i le would be stripped from the p r o je c t i le . The problem of

c a lc u la tin g average eq u ilib rium charge states was thus reduced to a

v e lo c ity descrip tio n of the electrons in the p r o je c t i le . For an atom or

ion w ith nuclear charge Zx, we can w rite

• - v r " ' v ' vo ? r <2-43>

where a and v are the o r b ita l radius and v e lo c ity o f the p ro je c t ile

e le c tro n , (Z 7-n ) is the number o f e lectrons w ith v e lo c it ie s la rg e r than

v, (o r b ita l radius sm aller than a) and v is the e ffe c t iv e quantum

number o f the binding s ta te . Bohr suggests th a t fo r heavy atoms, the

most t ig h t ly bound electrons in the sh e lls K, L, e tc . , w i l l move in an

approximately Coulomb f ie ld and have values o f v = 1, 2, e tc . Due to

e le c tro n ic screening of the f ie ld of the nucleus, the most loosely bound

electrons are also expected to have values o f v on the order o f u n ity .

Over a large interm ediate region, however, Bohr argues, v is expected

to reach a maximum o f Z j ^ 3 and th is approximation is good fo r values

2/3o f v such th a t v 0 < v < Zx v 0. This maximum is not a tta in e d u n t i l

(Z 1-n )> Z 1/2 , however, and th is v e lo c ity d is tr ib u tio n is most appropriate

fo r ions w ith charge somewhat less than Zx/2. Since, fo r an ion w ith

v e lo c ity v, n represents the number o f e lectrons th a t w i l l be removed

from the ion , we can su b stitu te Z (average charge s ta te ) fo r n to fin d

29

30

Thus Bohr finds

Z, = ( - + V «< Z* < - * - ) (2 .44 )n v o 1 To

z > Z 1 1 v f o r ^ « - + <2-«>

Z 11 1V0

These methods have been used by other workers, as w e ll. Knipp and

T e lle r (Kn41) also assume th a t ~Zl depends p r im a rily on the r a t io of

e lec tro n to ion v e lo c it ie s , but they use a Thomas-Fermi s t a t is t ic a l

model o f the ion to p re d ic t o r b ita l v e lo c it ie s . For large Z J( they fin d

* <( V i > (2 .46 )1 voz i

where v is the root mean square v e lo c ity o f the most loosely bound

electron and f is evaluated num erically . Lamb (La40) assumed th a t

inc iden t p ro je c tile s w i l l be stripped u n t i l the io n iza tio n p o te n tia l of

the next stage o f io n iza tio n is g reater than the k in e tic energy o f the

electrons which, re la t iv e to the ion , bombard i t w ith v e lo c ity v . These

resu lts show an almost l in e a r dependence on the reduced v e lo c ity v^ =

v /V jZj2^3 . B e ll (Be58) attempted to ca lcu la te Z d ire c t ly fo r some

f is s io n fragments using numerical estim ates fo r the capture and loss

cross sections. His work also suggests a un iversal dependence on v^.

The a b i l i t y o f any o f these models to p red ic t average experim ental

charge states va ries strongly w ith the degree of io n iz a tio n . No

ca lc u la tio n is a v a ila b le which describes charge s tate experiments over a

broad range o f p r o je c t i le , ta rg e t and v e lo c ity combinations. In fa c t ,

a l l models discussed here ignore any e x c ita tio n e ffe c ts in the inc ident

p r o je c t i le , and they can only be compared w ith measurements in d ilu te

gases. I t appears from these re s u lts , however, th a t the io n iza tio n

energy is more im portant than the o r b ita l v e lo c ity in describing

experiments, and many charge s ta te d is tr ib u tio n s do show s h e ll e ffe c ts

(Mo67) due to large changes in io n iza tio n energy as new sh ells are

reached.

In view o f the complexity o f the s itu a tio n , i t is not su rp ris in g th a t

most attempts to p re d ic t charge s ta te d is tr ib u tio n s have been based on

em p irica l or sem iem pirical considerations. At low and high v e lo c it ie s ,

the io n iz a tio n w i l l approach values o f 0 and 1, re sp e c tiv e ly . The

dependence of many current models on the reduced v e lo c ity parameter v

2 /3= v / vqZj and the v e lo c ity l im its mentioned above suggest th a t a

form such as

5

31

= 1 - exp(-vr ) (2 .47 )

may be appropriate fo r average charge s ta te s . In fa c t , a s im ila r

expression has been used by Betz, e t . a l . (Be72) w ith much success, i . e .

Z—- — = 1 - A exp-(--------- ) (2 .4 8 )z, „ y1 v Z0 1

where A, y are free parameters to be determined from experiment.

This expression w i l l usually provide good f i t s to a v a ila b le data, w ith

values o f A and y found to be approximately 1 and 2 /3 , re s p e c tiv e ly .

One o f the major resu lts o f experim ental average charge s ta te

measurements has been the fa c t th a t p ro je c t i le charges are much higher

when leaving s o lid targ ets than when in gases, by as much as a fac to r of

two in some cases. This re s u lt has generated much in te re s t in whether

these high io n iz a tio n sta tes occur during the p a r t ic le 's passage through

s o lid ta rg e ts , or a f te r e x it in g the s o lid surface. Betz, e t . a l . (Be76)

suggest th a t the charge o f heavy ions in so lids and gases (o f roughly

the same atomic number) is approximately equal, but th a t in so lids these

p ro je c tile s remain in h ig h ly exc ited atomic states during passage, since

the high density o f the s o lid causes the time between c o llis io n s to be

much less than the l ife t im e o f the excited s ta te s . Thus immediately

a f te r e x it in g the s o lid surface, the p ro je c t i le w i l l re tu rn to the

ground s ta te , and one dominant mode o f d eexcita tio n is through emission

of Auger e lec tro n s . Betz thus argues th a t th is is the o r ig in o f the

high io n iz a tio n states when leaving so lid s , and he ca lcu la tes capture

and loss cross sections fo r the s p e c ific case o f Br ions in C and 02

targ ets which agree w ith th is conjecture.

The e ffe c t iv e charge o f heavy ions, as ca lcu la ted from experim ental

energy loss measurements, also seems to agree w ith th is idea . In most

cases energy loss measurements in so lids and gases o f s im ila r atomic

number produce no re a l d iffe ren ce in e ffe c tiv e charge expressions.

However, eq u ilib riu m charge s ta te measurements outside o f s o lid and

gaseous targ ets show a d e f in ite ta rg e t dependence, in th a t high Z2

targ ets generate lower charge states than low Z2 ta rg e ts , and th is

e ffe c t has not been seen in e ffe c t iv e charge values deduced from dE/dx

measurements. Thus fu rth e r experim entation is necessary to understand

both the ta rg e t dependence of e ffe c t iv e charge and the importance o f

e ffe c t iv e charge values in exploring high p ro je c t i le io n iza tio n s when

leaving so lid s .

32

33

F ig . 2.1 In te ra c tio n between a heavy ion o f charge Z xe and mass M w ith

an e lectron harm onically bound to a s ta tio n ary o r ig in , 0.

34

C D

N

O

35

F ig . 2 .2 C o llis io n geometry fo r two p a r t ic le s c a tte rin g in the center

of mass frame.

36

Chapter III

Almost a l l experim ental in vestig a tio n s in to the stopping power o f

heavy ions involve a measurement o f the energy loss of an ion beam in a

ta rg e t m a te r ia l. Although the resu lts o f energy loss experiments are

c le a r ly re la te d to the stopping power, the dependence can be qu ite

complicated. For example, i t was shown in Chapter I I th a t the stopping

power depends d ire c t ly on the p r o je c t i le charge. I f the energy loss

ta rg e t is very th in , the heavy ion beam w i l l not have time to reach an

eq u ilib rium charge d is tr ib u tio n ,a n d thus the energy loss in the f o i l

w i l l be more dependent on the inc iden t charge than on the p a r t ic u la r ion

being studied. Conversely, i f the ta rg e t m ateria l is too th ic k , the

resu ltan t energy loss spans too large a p ortion of the dE/dx curve, and

only an average value o f the stopping power can be determined. The

importance o f these re s tr ic tio n s w i l l vary as d iffe re n t portions of the

dE/dx curve are explored, and some care must be taken when making these

measurements.

With these conditions in mind, we have made a series of energy loss

measurements using both th ic k and th in ta rg e ts , in order to explore the

importance of higher order charge dependent corrections to the stopping

power. The i n i t i a l experiments involved a study of the energy losses o f

heavy ions (Z 1=14 ,28,79 and £ ^ 2 .5 MeV/amu) in th ick targ ets (Z2=

2 9 ,4 7 ,8 2 ). The ta rg e t thicknesses va rie d from 5-10 mg/cm2 in a rea l

density . These measurements can be analyzed to give values o f heavy ion

EXPERIMENTAL TECHNIQUES

37

ranges, as w e ll as some inform ation concerning the stopping power

curves. The re s u lts suggest m odifications o f current stopping power

com pilations, and therefore conventional th in ta rg e t dE/dx measurements

were made in two cases (N i beams in Cu and Ag) as a check. These

re su lts were consistent w ith the th ic k ta rg e t re s u lts , and are not

reproduced by current stopping power curves. Inc lus ion o f Z j3 and Z a4

e ffe c ts , however, provides good f i t s to the data. To te s t the

g e n e ra lity of these co rrec tio n s , we have measured the stopping power of

several heavy ions (Z j= 6 ,1 4 ,1 7 ,2 2 ,2 6 ,2 8 ,3 2 ,3 5 ,4 1 , and 53) in elemental

ta rg ets (Z2=6,1 3 ,2 9 ,4 7 , and 79) a t energies near the maximum in the

stopping power vs. energy curve.

38

A. Range and Exploratory dE/dx Measurements

39

A l. Experimental Geometry

The apparatus used fo r measuring energy loss both in our th ic k

ta rg e ts , and in the i n i t i a l th in ta rg e t experiments, is shown in

F ig .3 .1 . The heavy ion beams fo r our measurements were generated by a

20 kV Cs sputter ion source and in je c te d in to the Yale MP tandem Van de

G raaff ac ce le ra to r. A fte r ac ce le ra tio n , the ion beam was momentum

analyzed by a 90° bending magnet and then d irec ted by a sw itching magnet

to the experim ental area. The beam passed through three co llim atin g

s l i t s and in to the targ ets a t the center o f the s c a tte rin g chamber.

Emerging p a r t ic le s were then detected a t forward angles (3 ° -5 ° ) to the

beam d ire c tio n and were energy analyzed. Thus these targ ets were used

both to 1) generate energy loss, and 2) sc a tte r the in c id en t p a rt ic le s

to forward angles. S ilic o n surface b a rr ie r detectors were used fo r

measurements w ith S i and Ni beams. The S i detector showed very poor

reso lu tio n fo r the Au beams, however, due to the large nuclear stopping

power o f Au in S i, es p e c ia lly a t low energies. Thus a gas io n iza tio n

chamber was used fo r the Au beams.

The detectors were c a lib ra te d using Rutherford sc a tte rin g o f the

heavy ion beam from th in carbon (5 yg/cm2) and gold (100 yg/cm2)

f o i ls , w ith small corrections made fo r energy loss in these f o i ls .

These c a lib ra tio n f o i ls , w ith thickness u n c e rta in tie s o f ~20%, were

obtained commercially from the Arizona Carbon F o il Company. One

advantage o f th is method is th a t i t makes a v a ila b le c a lib ra tio n energies

much lower than the beam energy, simply by c o lle c tin g a t large

s c a tte rin g angles. A l l measurements discussed here were performed a t

energies w e ll below the Coulomb b a r r ie r .

In both the th ic k and th in ta rg e t measurements, the detector output

was am p lified and shaped and sent d ire c t ly to a M ulti-Channel Analyzer

(MCA). The centroids o f the re s u ltan t peaks were a l l determined by

hand, w ith an assigned uncerta in ty o f T /4 , where T is the f u l l width

a t h a lf maximum. This allowed an em pirica l c a lib ra tio n curve o f channel

number vs. energy to be determined, using the energies produced by the

s c a tte rin g technique mentioned above. Thus the e x it energy fo r a given

energy loss measurement was determined by comparing the channel number

of the e x it in g p a r t ic le s w ith the c a lib ra tio n curve. Since the incident

energy is w e ll known, the d iffe ren ce gives the to ta l energy loss.

A maximum of e igh t targ ets could be mounted on the ta rg e t ladder a t

the center o f the s c a tte rin g chamber. The ta rg e t ladder and the beam

stop were both e le c t r ic a l ly insu la ted from the sc a tte rin g chamber and

were connected in p a ra lle l and p o s it iv e ly biased to act as a Faraday

cup.

Data were taken fo r several targ ets o f each m ate ria l in an attempt to

avoid possible systematic e ffe c ts re f le c t in g inaccuracies in ta rg e t

thickness determ inations. In the th in ta rg e t stopping power

measurements (N i in Cu and in Ag) a kinematic co rrection was made by

40

assuming th a t the sc a tte rin g occurred halfway through the ta rg e t. The

co rrection was always less than 1%. For the th ic k ta rg e t data, the

emerging p a r t ic le s in the forward cone were a l l monoenergetic, because

of small angle m u ltip le s c a tte rin g (Mo48,Mo55,Se77). Thus a l l p a r t ic le s

w ith in a small angular range around 0° had suffered many small angle

c o llis io n s during th e ir passage through the ta rg e t, and had experienced

roughly the same energy loss. The angle a t which p a r t ic le s were

detected was therefore chosen to be w ith in th is small angle

d is tr ib u tio n , and no kinematic correction was necessary.

41

A2. Target Fabricatio n

The th ic k ta rg e t measurements involved sending the beam through the

ta rg e t a t high in c id en t energy and m onitoring both the inc ident and the

e x it energies. The inc iden t energy was then decreased in small steps

u n t i l e x it energies o f only a few MeV were reached. At the lowest

measured energy the res idu al range o f the ions is a small fra c tio n o f

the ta rg e t thickness and can be approximated using low energy range

theory. The to ta l range is given by the sum of th is res idu al range and

the ta rg e t th ickness. Thus these measurements furn ish both range and

in teg ra ted energy loss in form ation .

An obvious requirement fo r these experiments is the production of

th ic k , uniform elem ental fo i ls w ith accurate ly measured thicknesses.

One procedure a v a ila b le fo r making these fo i ls is vacuum evaporation by

re s is tiv e heating . The main requirement is a geometry th a t w i l l produce

a uniform thickness throughout the ta rg e t. A s ing le evaporation boat a t

a large distance from the substrate w i l l provide s u ff ic ie n t u n ifo rm ity ,

but th is method uses more evaporation m ate ria l than is necessary.

We have used a m u ltip le boat arrangement (Ar67) to produce high

un ifo rm ity and also decrease the amount o f m ateria l required fo r these

f o i ls . The geometry chosen involves four boats arranged in a square

w ith sides o f length S. The source m ate ria l is placed at the midpoints

of the sides of the square, and the ta rg e t is centered on the square a t

some distance Z above i t . The advantage o f the 4-boat system is th a t

nonuniform ities due to any boat are compensated fo r by the others. The

un ifo rm ity o f a p a r t ic u la r ta rg e t increases w ith ta rg e t he ig h t, but the

amount o f m ate ria l required also increases, so the height chosen depends

on the av a ila b le m a te ria l as w e ll as the desired u n ifo rm ity .

The targ ets discussed here were evaporated onto 20 yg/cm2 C

backings, w ith a ta rg e t diameter o f 1.25 cm and a b o a t-to -su b stra te

height o f Z = 2S = 15 cm. Id e a lly , the nonuniform ity fo r th is geometry

should be ~0.15%. However, some m ateria ls upon m elting w i l l wet the

boat containing them, and since the spreading o f the m ateria l follow s no

p a rt ic u la r p a tte rn , some u n ifo rm ity is lo s t upon evaporation. This

problem was compensated fo r by mounting a small e le c tr ic motor (10 RPM)

in vacuum and ro ta tin g the ta rg e t during the evaporation, in order to

reduce any lo c a l nonunifo rm ities.

42

H e a t i n g o f a l l f o u r b o a t s s i m u l t a n e o u s l y i s a c c o m p l i s h e d b y p l a c i n g

t h e e l e c t r o d e s o n d i a g o n a l l y o p p o s i t e c o r n e r s o f t h e s q u a r e a n d h e a t i n g

i n p a r a l l e l . T h i s m e t h o d m a y n o t p r o d u c e t h e s a m e e v a p o r a t i o n r a t e f r o m

e a c h b o a t . H o w e v e r , i f e q u a l a m o u n t s ( b y w e i g h t ) o f m a t e r i a l a r e p l a c e d

i n e a c h b o a t , a n d h e a t i n g i s c o n t i n u e d u n t i l a l l b o a t s a r e e m p t y , e q u a l

e v a p o r a t i o n r a t e s a r e n o t n e c e s s a r y .

T h e f o i l t h i c k n e s s e s r e q u i r e d w e r e 5 - 1 0 m g / c m 2 , w h i c h a l l o w e d a n

a c c u r a t e t h i c k n e s s d e t e r m i n a t i o n t o b e m a d e b y w e i g h i n g t h e t a r g e t . T h e

C b a c k i n g s w e r e m o u n t e d o n a l u m i n u m t a r g e t f r a m e s , 2 . 3 cm s q u a r e , w i t h a

1 . 2 5 cm d i a m e t e r h o l e i n t h e c e n t e r . T h e f r a m e w a s t h e n m a s k e d , w i t h

e v a p o r a t i o n a l l o w e d o n l y o n t h e C c o v e r e d h o l e . T h e t a r g e t f r a m e w a s

w e i g h e d b o t h b e f o r e a n d a f t e r e v a p o r a t i o n , a n d t h i s i n f o r m a t i o n p l u s t h e

m a s k a r e a g a v e t h e a v e r a g e t h i c k n e s s .

U n i f o r m i t y w a s d e t e r m i n e d b y m e a s u r i n g t h e e n e r g y l o s s o f a l p h a

p a r t i c l e s t h r o u g h t h e f o i l . 2 4 l Am w a s u s e d a s a s o u r c e o f 5 . 5 M e V a l p h a

p a r t i c l e s , w h i c h w e r e c o l l i m a t e d t o a n a r e a o f ~ 2mm2 a n d u s e d t o

e x a m i n e v a r i o u s l o c a t i o n s o n t h e t a r g e t . T h e p a r t i c l e s w e r e c o l l e c t e d

i n a S i l i c o n d e t e c t o r , a n d t h e e n e r g y s p e c t r u m d i s p l a y e d o n a m u l t i

c h a n n e l a n a l y z e r .

T h e e n e r g y l o s s o f a l p h a p a r t i c l e s i n t h e s e t h i c k t a r g e t s w a s

t y p i c a l l y 2 - 3 M e V , w h i c h w a s l a r g e e n o u g h t o g i v e a c c u r a t e m e a s u r e m e n t s

o f t h e e x i t e n e r g y ( E o u t ) < a n d t h e s p r e a d i n e x i t e n e r g y ( A E o u t )

p r o d u c e d b y t a r g e t n o n u n i f o r m i t y ( a s o p p o s e d t o t h e w i d t h d u e t o e n e r g y

s t r a g g l i n g ) . T h e s t o p p i n g p o w e r S ( E ) o f a l p h a p a r t i c l e s i n e l e m e n t a l

m a t t e r i s w e l l k n o w n ( Z i 7 7 a ) , a n d t h u s E o u t / S ^E o u t * w a s u s e d t o

43

f i n d v e r y a c c u r a t e v a l u e s f o r t h e t h i c k n e s s v a r i a t i o n s i n t h e s e t a r g e t s .

T h i s t h i c k n e s s v a r i a t i o n d i v i d e d b y t h e a v e r a g e t h i c k n e s s g i v e s a v a l u e

f o r t h e t a r g e t u n i f o r m i t y . R e s u l t s f o r t h e t h i c k t a r g e t s u s e d i n t h i s

s t u d y a r e s h o w n i n T a b l e 3 . 1 .

T h e t h i n t a r g e t s u s e d h e r e f o r d i r e c t d E / d x m e a s u r e m e n t s w e r e a l s o

p r o d u c e d w i t h t h i s m u l t i p l e b o a t a r r a n g e m e n t , b y e v a p o r a t i o n o n t o 5

y g / c m 2 C b a c k i n g s . T h e t a r g e t s w e r e a l s o m a s k e d a n d w e i g h e d , b u t t h i s

p r o c e d u r e w a s m u c h l e s s a c c u r a t e b e c a u s e o f t h e s m a l l a m o u n t o f m a t e r i a l

d e p o s i t e d d u r i n g e v a p o r a t i o n ; t h e r e f o r e , t h e t h i c k n e s s e s f o r t h e s e

t a r g e t s w e r e o b t a i n e d f r o m t h e e n e r g y l o s s o f 2 4 1 Am a l p h a p a r t i c l e s ,

w h i c h i s v e r y a c c u r a t e l y k n o w n f o r t h e t a r g e t m a t e r i a l s C u a n d A g ( A n 7 7 ,

Z i 7 7 a ) . T h e t h i c k n e s s e s d e t e r m i n e d b y t h e w e i g h t m e a s u r e m e n t s w e r e

a l w a y s c o n s i s t e n t w i t h t h e e n e r g y l o s s m e a s u r e m e n t s . T h e r e s u l t s f o r

t h e s e t a r g e t s a r e l i s t e d i n T a b l e 3 . 2 .

44

B. Extensive dE/dx Measurements

45

B l . E x p e r i m e n t a l G e o m e t r y

S i n c e w e r e q u i r e d a l a r g e n u m b e r o f p r o j e c t i l e t a r g e t c o m b i n a t i o n s t o

s t u d y t h e g e n e r a l e f f e c t s o f t h e h i g h e r o r d e r s t o p p i n g p o w e r

c o r r e c t i o n s , a m o r e e f f i c i e n t e x p e r i m e n t a l a r r a n g e m e n t w a s d e s i g n e d .

F i g u r e s 3 . 2 a n d 3 . 3 s h o w t h e a p p a r a t u s u s e d f o r t h e s e s t o p p i n g p o w e r

m e a s u r e m e n t s . T h e i n c i d e n t h e a v y i o n b e a m , a f t e r b e i n g a c c e l e r a t e d a n d

m o m e n t u m a n a l y z e d a s b e f o r e , p a s s e d t h r o u g h t h r e e a p e r t u r e s a n d t h e n

t h r o u g h a t h i n g o l d f o i l a t t h e c e n t e r o f t h e s c a t t e r i n g c h a m b e r . M o s t

o f t h e i n c i d e n t b e a m w a s u n d e f l e c t e d , a n d i t p a s s e d t h r o u g h t h e a n n u l u s

c o n t a i n i n g t h e d e t e c t o r s ( F i g . 3 . 3 ) a n d d i r e c t l y i n t o t h e F a r a d a y C u p .

T e n s i l i c o n s u r f a c e b a r r i e r d e t e c t o r s w e r e m o u n t e d i n t h e a n n u l u s , w h i c h

w a s p l a c e d a p p r o x i m a t e l y 2 8 cm d o w n s t r e a m f r o m t h e g o l d s c a t t e r i n g f o i l .

T a r g e t m a t e r i a l a n d a p e r t u r e s w e r e m o u n t e d i n f r o n t o f e a c h d e t e c t o r ,

a n d b e a m p a r t i c l e s s c a t t e r e d b y t h e g o l d f o i l t o 1 0 ° w e r e t h e n c o l l e c t e d

i n a l l t e n d e t e c t o r s s i m u l t a n e o u s l y . T h e t a r g e t f r a m e s a n d c o l l i m a t o r s

w e r e d e s i g n e d s o t h a t e a c h o f t h e s e t e n d e t e c t o r s s i m u l t a n e o u s l y

c o l l e c t e d b o t h 1 ) p a r t i c l e s p a s s i n g t h r o u g h t h e t a r g e t m a t e r i a l a n d

i n t o t h e d e t e c t o r , a n d 2 ) p a r t i c l e s p a s s i n g d i r e c t l y i n t o t h e d e t e c t o r

w i t h o u t g o i n g t h r o u g h t h e t a r g e t m a t e r i a l . T h i s p r o v i d e d a n e n e r g y

c a l i b r a t i o n f o r e a c h b e a m a n d e a c h d e t e c t o r a t t h e s a m e t i m e t h a t t h e

e n e r g y l o s s e s w e r e b e i n g m e a s u r e d . T h e a d v a n t a g e o f t h i s p a r t i c u l a r

a r r a n g e m e n t i s t h a t i t a l l o w s s t o p p i n g p o w e r m e a s u r e m e n t s t o b e m a d e

m u c h f a s t e r a n d m o r e a c c u r a t e l y t h a n t h e p r e v i o u s s e t u p .

I n o r d e r t o m a k e a p r e c i s e d e t e r m i n a t i o n o f t h e e n e r g y l o s s i n o u r

t a r g e t s , a n a c c u r a t e c a l i b r a t i o n c u r v e i s n e c e s s a r y . O n e f u n d a m e n t a l

r e q u i r e m e n t o f o u r c a l i b r a t i o n p r o c e d u r e i s a n a c c u r a t e d e t e r m i n a t i o n o f

t h e b e a m e n e r g y a f t e r 1 0 ° s c a t t e r i n g f r o m t h e A u f o i l . T h e t h i c k n e s s o f

t h e A u s c a t t e r i n g f o i l ( 2 0 . 8 ± 0 . 6 y g / c m 2 ) w a s d e t e r m i n e d b y m e a s u r i n g

t h e b a c k s c a t t e r i n g y i e l d o f 2 M e V a l p h a p a r t i c l e s . S i n c e A u w a s a l s o

o n e o f t h e t a r g e t m a t e r i a l s , a r o u g h e s t i m a t e o f t h e s t o p p i n g p o w e r o f

t h e i n c i d e n t b e a m i n A u w a s a v a i l a b l e , a n d t h u s a n i t e r a t i v e p r o c e d u r e

a l l o w e d a s m a l l c o r r e c t i o n f o r e n e r g y l o s s i n t h e A u s c a t t e r i n g f o i l t o

b e m a d e . S i n c e t h e k i n e m a t i c s o f 1 0 ° e l a s t i c s c a t t e r i n g i s w e l l

u n d e r s t o o d a n d t h e e n e r g y d i s p e r s i o n o f t h e b e a m d u e t o t h e f i n i t e s i z e

o f t h e c o l l i m a t i n g a p e r t u r e w a s a l w a y s < 0 . 1% , t h e c a l i b r a t i o n e n e r g i e s

w e r e v e r y a c c u r a t e l y k n o w n .

D u e t o t h e l a r g e m u l t i p l i c i t y o f d a t a b e i n g c o l l e c t e d , r o u t i n g o f

e a c h s i g n a l t o a p h y s i c a l l y s e p a r a t e M C A w a s n o t p r a c t i c a l . T h u s t h e

o u t p u t p u l s e s f r o m t h e t e n d e t e c t o r s w e r e a m p l i f i e d a n d s h a p e d a n d t h e n

f e d i n t o A n a l o g - t o - D i g i t a l C o n v e r t e r s ( A D C ' s ) o n a " f r o n t e n d "

i n t e r f a c e d t o t h e l a b ' s I B M 4 3 4 1 c o m p u t e r , a s s h o w n i n F i g . 3 . 4 . F i v e

A D C ' s w e r e u s e d , e a c h o f w h i c h w a s s h a r e d b y t w o d e t e c t o r s . E a c h

d e t e c t o r s i g n a l w a s u s e d t o g e n e r a t e b o t h a l i n e a r ( A D C ) a n d a l o g i c

( e v e n t ) p u l s e . U s i n g t h e l a b e l i n g d e t e r m i n e d b y t h e e v e n t p u l s e s , t h e

A D C o u t p u t s i g n a l s w e r e t h e n s e n t i n t o t e n c o m p u t e r - g e n e r a t e d M C A 1s .

46

F o r e v e r y b e a m e n e r g y , t h e t e n M CA s p e c t r a e a c h h a d t w o p e a k s ,

c o r r e s p o n d i n g t o t h e i o n s w h i c h d i d a n d d i d n o t p a s s t h r o u g h t h e t a r g e t .

T h e p e a k l o c a t i o n s w e r e d e t e r m i n e d u s i n g a c o m p u t e r p e a k f i t t i n g r o u t i n e

b a s e d o n c r o s s c o r r e l a t i o n t e c h n i q u e s ( B 1 6 9 ) . B a s i c a l l y , t h e c r o s s

c o r r e l a t i o n b e t w e e n a G a u s s i a n o f w i d t h x a n d t h e d a t a p e a k w a s u s e d t o

g i v e a n e s t i m a t e o f t h e p e a k c h a n n e l n u m b e r . T h i s a l l o w e d a

d e t e r m i n a t i o n o f t h e f u l l w i d t h a t h a l f m a x i m u m , I * . A n e w c r o s s

c o r r e l a t i o n b e t w e e n a G a u s s i a n o f w i d t h T a n d t h e d a t a p e a k w a s t h e n

m a d e , a n d t h e c e n t r o i d o f t h i s c r o s s c o r r e l a t e d p e a k w a s u s e d a s t h e

p e a k l o c a t i o n i n a l l s u b s e q u e n t e n e r g y l o s s d e t e r m i n a t i o n s . T h e

a d v a n t a g e o f t h i s m e t h o d o v e r a s i m p l e G a u s s i a n f i t i s t h a t i t a l l o w s

p e a k s w i t h l o w s i g n a l - t o - n o i s e r a t i o s t o b e m o r e e a s i l y i d e n t i f i e d .

47

B 2 . T a r g e t F a b r i c a t i o n a n d T h i c k n e s s D e t e r m i n a t i o n s

T h e t e n s e l f - s u p p o r t i n g t a r g e t f o i l s ( 2 o f e a c h m a t e r i a l ) w e r e a l l

p r e p a r e d c o m m e r c i a l l y b y M i c r o m a t t e r C o . T h e t e c h n i q u e i n v o l v e d v a c u u m

e v a p o r a t i o n w i t h a l a r g e b o a t - t o - s u b s t r a t e d i s t a n c e , i n o r d e r t o p r o d u c e

v e r y u n i f o r m f o i l s . D a t a w e r e t a k e n f o r t w o s a m p l e s o f e a c h m a t e r i a l i n

a n a t t e m p t t o a v o i d p o s s i b l e s y s t e m a t i c e r r o r s i n o u r d E / d x m e a s u r e m e n t s

r e f l e c t i n g i n a c c u r a c i e s i n t a r g e t t h i c k n e s s d e t e r m i n a t i o n s .

T h e t h i c k n e s s e s o f t h e s e t a r g e t s w e r e m e a s u r e d u s i n g s e v e r a l m e t h o d s ,

i n c l u d i n g 1 ) w e i g h i n g , 2 ) R u t h e r f o r d b a c k s c a t t e r i n g o f a l p h a p a r t i c l e s

a t 2 M e V , a n d 3 ) e n e r g y l o s s o f 2 2 8 T h a l p h a p a r t i c l e s a t 5 - 9 M e V . T h e

i n i t i a l w e i g h t m e a s u r e m e n t s , p e r f o r m e d b y M i c r o m a t t e r , w e r e o n l y

a c c u r a t e t o ~ 1 0 % . T h e b a c k s c a t t e r i n g m e t h o d a c t u a l l y g i v e s t w o

m e a s u r e m e n t s f o r t h e t a r g e t t h i c k n e s s . B o t h o f t h e s e t e c h n i q u e s d e p e n d

o n t h e s t o p p i n g p o w e r o f a l p h a p a r t i c l e s i n t h e t a r g e t , w h i c h i s k n o w n

t o ~ 2 - 3 % i n t h i s e n e r g y r a n g e . T h e f i r s t v a l u e c o m e s f r o m m e a s u r i n g

t h e l o w e s t e n e r g y b a c k s c a t t e r e d p a r t i c l e s , w h i c h a r e s c a t t e r e d f r o m t h e

r e a r o f t h e f o i l ( F i g . 3 . 5 ) . T h e t h i c k n e s s t c a n b e d e t e r m i n e d

i t e r a t i v e l y f r o m

E o u t “ K ( E i n ‘ 4 * S ( E ) d x ) ‘ Ji,1 7 ' 0 0 8 9 ' S < E ) d x < 3 - 1 >

w h e r e a n d E Q U t a r e t h e i n c i d e n t a n d e x i t e n e r g i e s , r e s p e c t i v e l y ,

K i s t h e b a c k s c a t t e r i n g k i n e m a t i c f a c t o r , 8 i s t h e a n g l e o f d e t e c t i o n ,

a n d S ( E ) i s t h e ( e n e r g y d e p e n d e n t ) s t o p p i n g p o w e r . T h e f i r s t t e r m i n

p a r e n t h e s e s i s j u s t t h e e n e r g y o f t h e a l p h a p a r t i c l e s a s t h e y r e a c h t h e

r e a r o f t h e f o i l . T h e s e a l p h a p a r t i c l e s a r e t h e n b a c k s c a t t e r e d t h r o u g h

s o m e a n g l e 0 , a n d t h e e n e r g y i s m u l t i p l i e d b y a k i n e m a t i c f a c t o r K .

T h e p a r t i c l e s a l s o l o s e e n e r g y o n t h e i r r e t u r n t r i p t h r o u g h t h e f o i l .

T h e p a t h l e n g t h h e r e i s s l i g h t l y l o n g e r , d u e t o t h e b a c k s c a t t e r i n g a n g l e

0 , a n d t h i s i s t h e o r i g i n o f t h e t / | c o s 0 | l i m i t o n t h e s e c o n d

i n t e g r a l .

T h e s e c o n d t a r g e t t h i c k n e s s d e t e r m i n a t i o n u s i n g b a c k s c a t t e r i n g c o m e s

f r o m a k n o w l e d g e o f t h e t o t a l n u m b e r o f b a c k s c a t t e r e d p a r t i c l e s . I f w e

c o n s i d e r p a r t i c l e s o f e n e r g y E 0 i n c i d e n t o n a t a r g e t w i t h i n f i n i t e s i m a l

t h i c k n e s s d t , w e c a n r e w r i t e 3 . 1 a s

48

49Eout K(EQ - S (EQ)dt) - S (KE0)dt/|cose | (3.1a)

R e a r r a n g i n g t h i s e x p r e s s i o n g i v e s

dt (3.1b)KS(Eq) + S(KEq)/ | cos 0 |

I f t h e p a r t i c l e s a r e d i s p l a y e d i n a n M CA w i t h e n e r g y w i d t h p e r c h a n n e l

g i v e n b y A E , t h e e n e r g i e s o f t h e p a r t i c l e s c o l l e c t e d i n t h e h i g h e s t

e n e r g y c h a n n e l w i l l r a n g e f r o m a m a x i m u m o f K E 0 d o w n t o a m i n i m u m e n e r g y

o f E Q U t = K E q - A E . T h u s t h e n u m b e r o f p a r t i c l e s H s c a t t e r e d f r o m

t h e f r o n t e d g e o f t h e f o i l i n t o t h e h i g h e s t e n e r g y c h a n n e l i s g i v e n b y

w h e r e N „ i s t h e n u m b e r o f i n c i d e n t p a r t i c l e s , £2 i s t h e d e t e c t o r s o l i d

a n g l e , n i s t h e n u m b e r o f p a r t i c l e s p e r u n i t v o l u m e i n t h e t a r g e t , a n d

o ( E d ) i s t h e b a c k s c a t t e r i n g c r o s s s e c t i o n . T h e t o t a l n u m b e r o f

p a r t i c l e s s c a t t e r e d i n t o t h e d e t e c t o r , N , i s g i v e n b y

w h e r e t h e e n e r g y E a t a g i v e n d e p t h b e l o w t h e s u r f a c e d e p e n d s o n t h e

a l p h a p a r t i c l e s t o p p i n g p o w e r . T h u s t h e r a t i o o f t h e s e t w o q u a n t i t i e s

g i v e s

(3.2)

(3.3)

NH (3.4)

w h i c h d e t e r m i n e s t h e t h i c k n e s s t .

I n t h e t h i r d t e c h n i q u e , t h e 2 2 8 T h a l p h a p a r t i c l e m e a s u r e m e n t s w e r e

m a d e u s i n g e x a c t l y t h e s a m e g e o m e t r y a n d a n a l y s i s t e c h n i q u e s a s t h e

h e a v y i o n m e a s u r e m e n t s . T h u s a t h o r i u m s o u r c e w a s p l a c e d i n t h e

s c a t t e r i n g c h a m b e r i n t h e p o s i t i o n o f t h e A u s c a t t e r i n g f o i l ( F i g 3 . 2 ) ,

a n d t h e c r o s s c o r r e l a t i o n p e a k f i t t i n g r o u t i n e w a s u s e d t o d e t e r m i n e t h e

c a l i b r a t i o n c u r v e s a n d a l p h a p a r t i c l e e n e r g y l o s s e s i n o u r t a r g e t s . T h e

s t o p p i n g p o w e r o f a l p h a p a r t i c l e s i n t h i s e n e r g y r a n g e h a s b e e n v e r y

a c c u r a t e l y m e a s u r e d f o r A l , C u , A g a n d A u t a r g e t s ( A n 7 7 ) , a n d t h i s

p r o c e d u r e t h u s y i e l d e d v e r y p r e c i s e v a l u e s f o r o u r t a r g e t t h i c k n e s s e s .

T h e a d v a n t a g e o f t h i s m e t h o d o v e r t h e o t h e r s i s t h a t i t a l l o w s t h i c k n e s s

m e a s u r e m e n t s o v e r t h e s a m e a r e a u s e d i n t h e d E / d x m e a s u r e m e n t s , a n d t h e

o v e r a l l u n i f o r m i t y o f t h e f o i l i s t h e r e f o r e n o t c r u c i a l . ( H o w e v e r , t h e

a l p h a p a r t i c l e b a c k s c a t t e r i n g w a s u s e d t o t e s t t h e u n i f o r m i t y o f t h e s e

t a r g e t s , w h i c h w a s b e t t e r t h a n 1% i n a l l c a s e s ) . T h e r e s u l t s f o r a l l

t h r e e t h i c k n e s s d e t e r m i n a t i o n t e c h n i q u e s ( w e i g h i n g , a l p h a p a r t i c l e

b a c k s c a t t e r i n g , a n d a l p h a p a r t i c l e e n e r g y l o s s ) w e r e c o n s i s t e n t w i t h o n e

a n o t h e r i n e a c h c a s e . T h e r e s u l t s a r e l i s t e d i n T a b l e 3 . 3 .

51

T a b l e 3 . 1 T a r g e t t h i c k n e s s e s u s e d i n o u r t h i c k t a r g e t e n e r g y l o s s

m e a s u r e m e n t s . A l s o l i s t e d i s t h e m e a s u r e d t h i c k n e s s u n i f o r m i t y f o r e a c h

t a r g e t .

52

T H I C K T A R G E T S

E L E M E N T A V E R A G E T H I C K N E S S U N I F O R M I T Y

( m g / c m 2 ) ( % )

C u 7 . 0 8 ± 0 . 0 3 0 . 3

C u 4 . 8 8 ± 0 . 0 2 0 . 3

A g 8 . 3 9 ± 0 . 0 4 0 . 4

A g 6 . 9 2 ± 0 . 0 3 0 . 4

A g 4 . 2 4 ± 0 . 0 2 0 . 5

P b 1 0 . 2 5 ± 0 . 0 4 1 . 2

P b 7 . 4 3 ± 0 . 0 3 2 . 5

53

T a b l e 3 . 2 T a r g e t t h i c k n e s s e s u s e d i n o u r e x p l o r a t o r y d E / d x

m e a s u r e m e n t s .

54

T H I N T A R G E T S

( E x p l o r a t o r y d E / d x M e a s u r e m e n t s )

T a r g e t M a t e r i a l T h i c k n e s s ( p g / c m 2 )

C u 7 5 9 ± 2 4

C u 5 1 6 ± 1 8

C u 4 5 8 ± 1 6

A g 8 6 5 ± 2 2

A g 6 9 4 ± 1 7

A g 5 3 6 ± 1 6

55

T a b l e 3 . 3 T a r g e t t h i c k n e s s e s u s e d i n o u r e x t e n s i v e d E / d x m e a s u r e m e n t s .

56

T H I N T A R G E T S

( E x t e n s i v e d E / d x M e a s u r e m e n t s )

T a r g e t M a t e r i a l T h i c k n e s s ( y g / c m 2 )

1 0 5 ± 2 . 4

9 8 ± 3

A l 2 5 2 ± 4 . 5

A l 2 4 7 ± 4

C u 3 8 2 ± 5

C u 3 8 7 ± 6

A g 3 6 5 ± 6

A g 3 9 6 ± 5

A u 6 1 4 ± 1 0

Au 623 ± 11

57

F i g . 3 . 1 S c h e m a t i c r e p r e s e n t a t i o n o f t h e e x p e r i m e n t a l a p p a r a t u s f o r o u r

t h i c k t a r g e t e n e r g y l o s s a n d e x p l o r a t o r y d E / d x m e a s u r e m e n t s . T h e h e a v y

i o n b e a m , i n c i d e n t f r o m t h e r i g h t , p a s s e s t h r o u g h t h r e e c o l l i m a t i n g

s l i t s a n d i n t o t h e t a r g e t . S c a t t e r e d p a r t i c l e s a r e d e t e c t e d a t f o r w a r d

a n g l e s i n t h e s i l i c o n s u r f a c e b a r r i e r d e t e c t o r a n d t h e g a s i o n i z a t i o n

c h a m b e r . T h e t a r g e t l a d d e r a n d t h e b e a m d u m p a r e c o n n e c t e d i n p a r a l l e l

t o a c t a s a F a r a d a y c u p .

G a sI o n i z a t i o nC h a m b e r

I n s u l a t i n gC o u p l i n g

B e a mS t o p S i

S u r f a c eB a r r i e rD e t e c t o r

B e a mD e f i n i n g

A p e r t u r e s

I n c i d e n tB e a m

3 0 " O r t e c S c a t t e r i n g C h a m b e r01oo

59

F i g . 3 . 2 S c h e m a t i c r e p r e s e n t a t i o n o f t h e e x p e r i m e n t a l a r r a n g e m e n t f o r

o u r e x t e n s i v e d E / d x m e a s u r e m e n t s . T h e h e a v y i o n b e a m , i n c i d e n t f r o m t h e

r i g h t , p a s s e s t h r o u g h t h r e e c o l l i m a t i n g s l i t s a n d i n t o t h e A u s c a t t e r i n g

f o i l . P a r t o f t h e b e a m p a r t i c l e s a r e e l a s t i c a l l y s c a t t e r e d t o 1 0 ° a n d

i n t o d e t e c t o r s m o u n t e d o n t h e a n n u l u s - - t h e m a i n p o r t i o n o f t h e b e a m i s

u n d e f l e c t e d a n d p a s s e s t h r o u g h t h e h o l e i n t h e a n n u l u s a n d i n t o t h e b e a m

d u m p .

SCATTERING FOIL

BEAM DEFINING

APERTURESINSULATINGCOUPLING

BEAM STOP

ANNULAR TARGET-DETECTOR

ARRAY

INCIDENTBEAM

3 0 n0RTEC SCATTERING CHAMBERoo

61

F i g . 3 . 3 T a r g e t - d e t e c t o r a n n u l u s . P a r t i c l e s s c a t t e r e d t o 1 0 ° b y t h e A u

f o i l ( F i g . 3 . 2 ) p a s s d i r e c t l y i n t o t e n t a r g e t - c o l l i m a t o r - d e t e c t o r

a r r a n g e m e n t s m o u n t e d o n a n a n n u l a r r i n g . T h e g e o m e t r y s h o w n a l l o w s t h e

d e t e c t o r t o c o l l e c t b o t h 1 ) p a r t i c l e s p a s s i n g d i r e c t l y t h r o u g h t h e

t a r g e t m a t e r i a l ( l a r g e h o l e i n c o l l i m a t o r ) a n d , 2 ) p a r t i c l e s

e x p e r i e n c i n g n o i n t e r a c t i o n w i t h t h e t a r g e t m a t e r i a l ( s m a l l h o l e i n

c o l l i m a t o r ) .

S i S u r f a c e B a r r i e r D e t e c t o r

T a r g e t / M a t e r i a l

C o l l i m a t i n gA p e r t u r e s

Target-Detector Annulus

to

63

F i g 3 . 4 E x p e r i m e n t a l e l e c t r o n i c s f o r t h e e x t e n s i v e d E / d x m e a s u r e m e n t s .

T h i s a r r a n g e m e n t w a s r e p e a t e d f i v e t i m e s , t o a c c o m o d a t e a l l t e n

d e t e c t o r s . S i g n a l s f r o m e a c h d e t e c t o r w e r e a m p l i f i e d a n d s h a p e d t o

p r o d u c e l o g i c a n d l i n e a r p u l s e s . T h e l o g i c p u l s e s w e r e b o t h c o u n t e d

( s c a l e r ) a n d u s e d a s " e v e n t " p u l s e s t o t e l l t h e c o m p u t e r w h e n t o r e a d

t h e c o n t e n t s o f t h e a n a l o g - t o - d i g i t a l c o n v e r t e r ( A D C ) . E a c h A D C w a s

s h a r e d b y t w o d e t e c t o r s , a n d t h e r e f o r e t h e s u m a m p l i f i e r ( S U M ) r e c e i v e d

t w o l i n e a r p u l s e s , o n e f r o m e a c h d e t e c t o r . ( T S C A - T i m i n g S i n g l e

C h a n n e l A n a l y z e r , L G S - L i n e a r G a t e S t r e t c h e r , GDG - G a t e a n d D e l a y

G e n e r a t o r )

C O M P U T E R

65

F i g . 3 . 5 T y p i c a l b a c k s c a t t e r i n g s p e c t r u m f o r a l p h a p a r t i c l e s s c a t t e r e d

f r o m a t h i n f o i l . E a a n d E ^ a r e t h e e n e r g i e s o f p a r t i c l e s

b a c k s c a t t e r e d b y a n a n g l e 8 f r o m t h e f r o n t a n d r e a r o f t h e f o i l ,

r e s p e c t i v e l y . T h e e n e r g y d i f f e r e n c e , E ^ E ^ , g i v e s a d i r e c t m e a s u r e

o f t h e t a r g e t t h i c k n e s s i f t h e a l p h a p a r t i c l e e n e r g y l o s s i s k n o w n . T h e

r a t i o o f t h e t o t a l n u m b e r o f b a c k s c a t t e r e d p a r t i c l e s t o t h e n u m b e r a t

t h e f r o n t e d g e o f t h e s p e c t r u m ( i . e . , w i t h e n e r g y E ) a l s o g i v e s adv a l u e f o r t h e t a r g e t t h i c k n e s s , a s d e s c r i b e d i n t h e t e x t .

mo

ENER

GY

N u m b e r o f C o u n t s

0 50 5

CHAPTER IV

EXPERIMENTAL RESULTS

I n C h a p t e r I I I , w e d i s c u s s e d t h e e x p e r i m e n t a l t e c h n i q u e s f o r m a k i n g

b o t h o u r t h i c k a n d t h i n t a r g e t e n e r g y l o s s m e a s u r e m e n t s . H e r e w e

p r e s e n t t h e r e s u l t s o f t h e s e m e a s u r e m e n t s , a n d u s e t h e m t o e x a m i n e t h e

a c c u r a c y o f c u r r e n t h e a v y i o n s t o p p i n g p o w e r a n d r a n g e c o m p i l a t i o n s ,

p r i m a r i l y t h o s e o f N o r t h c l i f f e & S c h i l l i n g ( N o 7 0 ) a n d Z i e g l e r ( Z i

8 0 a , b ) . T h e i n a b i l i t y o f t h e s e s t a n d a r d t a b u l a t i o n s t o r e p r o d u c e o u r

r a n g e a n d t h i c k t a r g e t e n e r g y l o s s d a t a ( S e c . A ) s u g g e s t e d a n e e d f o r

b e t t e r s t o p p i n g p o w e r c u r v e s , a n d t h u s t h e e x p l o r a t o r y d E / d x

m e a s u r e m e n t s w e r e m a d e ( S e c . B ) . T h e a b i l i t y o f t h e h i g h e r o r d e r c h a r g e

d e p e n d e n t c o r r e c t i o n s t o f i t t h e s e s t o p p i n g p o w e r m e a s u r e m e n t s t h e n

s e r v e d a s m o t i v a t i o n f o r o u r e x t e n s i v e d E / d x m e a s u r e m e n t s ( S e c . C ) .

T h e s e m e a s u r e m e n t s w e r e m a d e i n o r d e r t o e x a m i n e t h e g e n e r a l i t y o f t h e

h i g h e r o r d e r c o r r e c t i o n s o v e r a w i d e v a r i e t y o f p r o j e c t i l e , t a r g e t , a n d

e n e r g y v a l u e s , a n d a l s o t o e x p l o r e t h e t a r g e t d e p e n d e n c e o f v a r i o u s

e f f e c t i v e c h a r g e p a r a m e t e r i z a t i o n s . T h u s , t h e d a t a a r e p r e s e n t e d i n

b o t h a l o g i c a l a n d h i s t o r i c a l o r d e r . T h e a n a l y s i s t e c h n i q u e s f o r u s i n g

d E / d x m e a s u r e m e n t s t o d e r i v e b o t h 1 ) h i g h e r o r d e r c h a r g e d e p e n d e n t

c o r r e c t i o n s t o t h e s t o p p i n g p o w e r , a n d 2 ) e f f e c t i v e c h a r g e s t a t e v a l u e s

a r e i n t r o d u c e d i n S e c . B a n d w i l l b e m o r e f u l l y d i s c u s s e d i n C h a p t e r V .

67

68

A. Thick target measurements

A l . R a n g e m e a s u r e m e n t s

A s w e m e n t i o n e d i n C h a p t e r I I I , t h e t e c h n i q u e f o r d e t e r m i n i n g h e a v y

i o n r a n g e s i n v o l v e d m o n i t o r i n g b o t h t h e i n c i d e n t a n d t h e e x i t e n e r g i e s

i n t h e t h i c k t a r g e t e n e r g y l o s s m e a s u r e m e n t s . A t t h e l o w e s t m e a s u r e d

e n e r g y t h e r e s i d u a l r a n g e o f t h e i o n s c a n b e a p p r o x i m a t e d u s i n g l o w

e n e r g y r a n g e t h e o r y , s i n c e e v e n a l a r g e e r r o r i n t h e s e c a l c u l a t i o n s i s

s m a l l c o m p a r e d t o o u r t a r g e t t h i c k n e s s e s . T h i s p r o c e d u r e y i e l d s a t o t a l

r a n g e g i v e n b y t h e s u m o f t h i s r e s i d u a l r a n g e a n d t h e t a r g e t t h i c k n e s s .

We h a v e u s e d t h e l o w e n e r g y r a n g e p r e d i c t i o n s o f L i t t m a r k a n d Z i e g l e r

( L i 8 0 ) t o e s t i m a t e t h e r e s i d u a l r a n g e o f o u r p a r t i c l e s . T h e e m e r g e n t

b e a m h a s a n a n g u l a r d i s t r i b u t i o n , h o w e v e r , a n d w i l l h a v e a s l i g h t l y

s m a l l e r r a n g e t h a n a m o n o d i r e c t i o n a l b e a m a t t h a t e n e r g y . T o a c c o u n t

f o r t h i s a p p r o x i m a t i o n a n d a n y u n c e r t a i n t i e s i n t h e c a l c u l a t i o n s , w e

h a v e a s s u m e d a 2 5 % u n c e r t a i n t y i n t h e r e s i d u a l r a n g e v a l u e s p r e d i c t e d b y

Z i e g l e r . T h i s u n c e r t a i n t y w a s t h e n a d d e d i n q u a d r a t u r e t o t h e e r r o r s i n

b e a m e n e r g y a n d t a r g e t t h i c k n e s s . R e s u l t s a r e l i s t e d i n T a b l e 4 . 1 ,

t o g e t h e r w i t h t h e p r e d i c t i o n s o f Z i e g l e r a n d N o r t h c l i f f e a n d S c h i l l i n g

( N S ) i n e a c h c a s e . ( N S v a l u e s f o r C u a n d P b w e r e f o u n d b y i n t e r p o l a t i n g

b e t w e e n n e a r b y t a r g e t s . )

69

T h e r a n g e p r e d i c t i o n s o f Z i e g l e r a r e i n m u c h b e t t e r a g r e e m e n t w i t h

o u r m e a s u r e d r a n g e s t h a n a r e t h o s e o f N S , a n d i n f a c t a r e c o n s i s t e n t

w i t h a l l o u r d a t a a t t h e 5 - 1 0 % l e v e l . H o w e v e r , a s s h o w n b e l o w , t h i s

d o e s n o t i m p l y a 5% a c c u r a c y i n t h e s t o p p i n g p o w e r a t a l l v e l o c i t i e s .

A 2 . I n t e g r a t e d E n e r g y L o s s M e a s u r e m e n t s

E a c h t h i c k t a r g e t e n e r g y l o s s m e a s u r e m e n t g i v e s i n f o r m a t i o n a b o u t t h e

i n t e g r a t e d s t o p p i n g p o w e r i n t h e t a r g e t , w h i c h c a n b e u s e d t o t e s t t h e

c o m p i l a t i o n s o f Z i e g l e r a n d N S . F i g s . 4 . 1 - 4 .6 s h o w p l o t s o f e x i t e n e r g y

( E 1 v s . i n c i d e n t e n e r g y ( E . ) f o r S i , N i , a n d A u b e a m s i n A g o u t ^ i n

t a r g e t s , t o g e t h e r w i t h t h e p r e d i c t i o n s o f Z i e g l e r a n d N S f o r e a c h t a r g e t

( A c o m p l e t e l i s t o f o u r m e a s u r e d t h i c k t a r g e t e n e r g y l o s s e s i s g i v e n i n

A p p e n d i x A ) . E o u t i s g i v e n b y

E = E. - f (-r^) dx (4.1)out in J0 v dx '

w h e r e t i s t h e t a r g e t t h i c k n e s s a n d d E / d x i s t h e t o t a l s t o p p i n g p o w e r

( e l e c t r o n i c p l u s n u c l e a r ) . B e c a u s e o f n u c l e a r s c a t t e r i n g , t h e t o t a l

p a t h l e n g t h t h r o u g h t h e s e t h i c k t a r g e t s i s a c t u a l l y s o m e w h a t l o n g e r t h a n

t h e t h i c k n e s s . T h e c o r r e c t i o n t o t h e i n t e g r a t e d e n e r g y l o s s i s

n e g l i g i b l e , h o w e v e r , s i n c e t h i s e f f e c t i s o n l y i m p o r t a n t n e a r t h e e n d o f

t h e p a t h , w h e r e t h e s t o p p i n g p o w e r i s s m a l l c o m p a r e d t o t h a t a t h i g h

e n e r g i e s .

70

F o r t h e S i i n A g d a t a , F i g s . 4 . 1 , 4 . 2 , t h e i n t e g r a t e d e n e r g y l o s s i s

g r e a t e r t h a n t h e Z i e g l e r a n d N S p r e d i c t i o n s f o r a l l e n e r g i e s . T h i s

s u g g e s t s t h a t d E / d x b e t w e e n t h e e n e r g i e s o f 1 6 M e V ( h i g h e s t e x i t e n e r g y

m e a s u r e d ) a n d 3 8 M e V ( l o w e s t i n c i d e n t e n e r g y m e a s u r e d ) i s g r e a t e r t h a n

t h e Z i e g l e r a n d N S v a l u e s . T h e s t o p p i n g p o w e r m e a s u r e m e n t s o f F o r s t e r ,

e t a l . ( F o 7 6 ) s h o w t h i s e f f e c t , w i t h a m a x i m u m e n e r g y l o s s o f 8 . 4 M e V -

c m 2 / m g a t a n i n c i d e n t e n e r g y o f 2 5 M e V , w h i c h i s s i g n i f i c a n t l y d i f f e r e n t

f r o m t h e t a b u l a t i o n s o f b o t h Z i e g l e r a n d N S . A n u m e r i c a l i n t e g r a t i o n o f

F o r s t e r ' s e t a l . e x p e r i m e n t a l d E / d x m e a s u r e m e n t s i s a l s o s h o w n i n F i g s .

4 . 1 , 4 . 2 , a n d i s c o n s i s t e n t w i t h t h e p r e s e n t t h i c k t a r g e t d a t a .

T h e t h i c k t a r g e t e n e r g y l o s s m e a s u r e m e n t s f o r N i i n A g a r e s h o w n i n

F i g s . 4 . 3 , 4 . 4 . T h e r e i s a d i s c r e p a n c y b e t w e e n t h e p r e d i c t i o n s a n d t h e

d a t a f o r e x i t e n e r g i e s > 3 M e V . T h e a v e r a g e e x p e r i m e n t a l s t o p p i n g

p o w e r s b e t w e e n ~ 3 0 - 1 2 0 M e V a r e t h u s e x p e c t e d t o b e g r e a t e r t h a n t h e

v a l u e s o f Z i e g l e r a n d N S . T h e f a c t t h a t t h e c u r v e s c r o s s a t l o w e x i t

e n e r g i e s , h o w e v e r , s u g g e s t s t h a t d E / d x m u s t f a l l b e l o w t h e p r e d i c t e d

v a l u e s a t l o w v e l o c i t i e s . A s d i s c u s s e d b e l o w , o u r t h i n t a r g e t d E / d x

m e a s u r e m e n t s f o r N i i n A g d e m o n s t r a t e t h e s e e f f e c t s , a n d a n u m e r i c a l

i n t e g r a t i o n o f o u r i n d e p e n d e n t , t h i n t a r g e t d E / d x m e a s u r e m e n t s

( d i s c u s s e d i n S e c . 4 B ) i s c o n s i s t e n t w i t h t h e t h i c k t a r g e t d a t a .

F i g s . 4 . 5 , 4 . 6 s h o w t h e t h i c k t a r g e t e n e r g y l o s s e s f o r A u i n A g . T h e

h i g h e s t e x p e r i m e n t a l A u e n e r g y w a s 2 0 0 M e V , w h i c h i s w e l l b e l o w t h e

m a x i m u m i n t h e d E / d x v s . E c u r v e . T h e A u d a t a i s t h u s r e s t r i c t e d t o t h e

l o w v e l o c i t y r e g i o n . T h e Z i e g l e r c a l c u l a t i o n s a r e s h o w n t o b e i n f a i r l y

g o o d a g r e e m e n t w i t h t h e s e m e a s u r e m e n t s o v e r t h i s v e l o c i t y r a n g e .

71

S u m m a r i z i n g , t h e l o w v e l o c i t y s t o p p i n g p o w e r c a l c u l a t i o n s o f Z i e g l e r

a n d N S a r e f a i r l y a c c u r a t e f o r S i i n A g , b u t a r e t o o h i g h f o r N i i n A g .

A t h i g h e r e n e r g i e s , n e a r t h e m a x i m u m i n t h e d E / d x v s . e n e r g y c u r v e , t h e

s t o p p i n g p o w e r p r e d i c t i o n s f o r S i a n d N i b e a m s i n A g a r e t o o l o w . T h e s e

r e s u l t s w e r e i n g e n e r a l t r u e f o r C u a n d P b t a r g e t s a s w e l l . F o r A u i o n s

a t e n e r g i e s b e l o w t h e m a x i m u m i n t h e s t o p p i n g p o w e r v s . e n e r g y c u r v e ,

t h e Z i e g l e r v a l u e s a p p e a r r e a s o n a b l y a c c u r a t e i n a l l t a r g e t s a t e n e r g i e s

u p t o - 2 0 0 M e V .

T h e Z i e g l e r c a l c u l a t i o n s a r e b a s e d o n a n o n l i n e a r s t o p p i n g p o w e r v s .

v e l o c i t y r e l a t i o n s h i p ( N e 7 7 ) ; o u r d a t a a l s o s u g g e s t t h i s b e h a v i o r . I n

o r d e r t o f u r t h e r i n v e s t i g a t e t h i s e f f e c t , a s w e l l a s t h e c h a n g e s i n t h e

m a x i m a o f t h e d E / d x c u r v e s , c o n v e n t i o n a l t h i n t a r g e t s t o p p i n g p o w e r

m e a s u r e m e n t s w e r e m a d e , a s d i s c u s s e d b e l o w .

72

B. Initial Thin Target Measurements

B e c a u s e o f t h e d i s c r e p a n c y b e t w e e n p r e d i c t i o n a n d e x p e r i m e n t f o r t h e

i n t e g r a t e d ( t h i c k t a r g e t ) e n e r g y l o s s m e a s u r e m e n t s d i s c u s s e d a b o v e ,

s t a n d a r d d E / d x v s . E m e a s u r e m e n t s w e r e m a d e f o r N i b e a m s i n t h i n C u a n d

A g t a r g e t s . F i g s . 4 . 7 a n d 4 . 8 s h o w t h e r e s u l t s ( a f t e r s u b t r a c t i n g

n u c l e a r s t o p p i n g , ( Z i 7 7 b ) ) o f t h e l o w v e l o c i t y d E / d x m e a s u r e m e n t s

t o g e t h e r w i t h t h e p r e d i c t i o n s o f Z i e g l e r , N o r t h c l i f f e a n d S c h i l i n g , a n d

t h o s e o f L i n d h a r d , S c h a r f f a n d S c h i o t t ( L S S ) ( L i 6 3 ) . T h e L S S t h e o r y ,

w h i c h i s b a s e d o n a T h o m a s - F e r m i d e s c r i p t i o n o f t h e a t o m , i s u s e d

e x t e n s i v e l y i n a n a l y s e s o f l o w e n e r g y s t o p p i n g p o w e r a p p l i c a t i o n s , s u c h

a s i o n i m p l a n t a t i o n ( S e e C h a p t e r I I ) . T h i s t h e o r y p r e d i c t s a l i n e a r

s t o p p i n g p o w e r v s . v e l o c i t y r e l a t i o n s h i p . T h e N S c a l c u l a t i o n s a l s o

r e d u c e t o v e l o c i t y p r o p o r t i o n a l s t o p p i n g p o w e r s a t l o w e n e r g i e s .

T h e Z i e g l e r c a l c u l a t i o n s b e l o w 6 X 1 0 8c m / s a r e a p p r o x i m a t e d u s i n g t h e

r e l a t i o n :

z i 2 <- f r > P ■ z i2 <s>(- f - ’p ( 4 -2 >* .

w h e r e Z x i s t h e p r o j e c t i l e e f f e c t i v e c h a r g e , Z x i s t h e p r o j e c t i l e

a t o m i c n u m b e r , < s > r e l a t e s t h e h e a v y i o n a n d p r o t o n e f f e c t i v e c h a r g e s

( Z i 7 7 b ) , a n d ( d E / d x ) i s t h e e x p e r i m e n t a l p r o t o n s t o p p i n g p o w e r .P

( A n 7 7 b ) . E x p e r i m e n t a l d a t a b e l o w 4 X 1 0 8c m / s a r e e x t r a c t e d f r o m o u r

t h i c k t a r g e t e n e r g y l o s s m e a s u r e m e n t s . B y m a k i n g a s m a l l c h a n g e i n t h e

i n c i d e n t e n e r g y a n d m e a s u r i n g t h e c o r r e s p o n d i n g c h a n g e i n t h e e x i t

e n e r g y , t h e r a t i o o f t h e s t o p p i n g p o w e r s a t t h e e n t r a n c e a n d e x i t

73

e n e r g i e s i s d e t e r m i n e d . C o m b i n i n g t h i s w i t h a n i n d e p e n d e n t t h i n t a r g e t

m e a s u r e m e n t o f t h e e n t r a n c e e n e r g y d E / d x g i v e s d E / d x a t t h e e x i t e n e r g y .

T h e m e a s u r e m e n t s e x h i b i t a s t r o n g n o n l i n e a r b e h a v i o r a n d a r e q u i t e

i n c o n s i s t e n t w i t h b o t h t h e N S a n d L S S c a l c u l a t i o n s . T h e y f a l l s l i g h t l y

b e l o w t h e p r e d i c t i o n s o f Z i e g l e r , b u t a r e g e n e r a l l y i n g o o d a g r e e m e n t

w i t h t h e v e l o c i t y d e p e n d e n c e o f t h e Z i e g l e r c u r v e s . T h u s a v e l o c i t y

p r o p o r t i o n a l s t o p p i n g p o w e r d o e s n o t d e s c r i b e t h e s e d a t a a c c u r a t e l y .

( S e e F i g s . 4 . 7 a n d 4 . 8 )

A s t h e p r o j e c t i l e e n e r g y i n c r e a s e s , t h e s t o p p i n g p o w e r r e a c h e s a

m a x i m u m a n d t h e n s l o w l y d e c r e a s e s . T h e t h i c k t a r g e t d a t a d i s c u s s e d

e a r l i e r s u g g e s t i n a c c u r a c i e s i n t h e s t o p p i n g p o w e r p r e d i c t i o n s o f

Z i e g l e r a n d N S . F o r S i a n d N i p r o j e c t i l e s i n C u , A g , a n d P b t a r g e t s ,

t h e e x p e r i m e n t a l m a x i m a i n t h e d E / d x v s . E c u r v e s a r e f o u n d ( 1 ) t o b e

l a r g e r i n m a g n i t u d e , a n d ( 2 ) t o o c c u r a t s u b s t a n t i a l l y d i f f e r e n t

e n e r g i e s f r o m t h e p r e d i c t e d v a l u e s . F i g s . 4 . 9 a n d 4 . 1 0 s h o w p l o t s o f

o u r d E / d x v s . E m e a s u r e m e n t s f o r N i b e a m s i n C u a n d A g t a r g e t s . T h e

d i s c r e p a n c i e s s e e n i n F i g s . 4 . 9 , 4 . 1 0 c a l l i n t o q u e s t i o n t h e c h a r g e

d e p e n d e n c e a s s u m e d i n t h e Z i e g l e r a n d N S c a l c u l a t i o n s .

A s w e d e s c r i b e d i n C h a p t e r I I , t h e e n e r g y l o s s i n t h i s v e l o c i t y

r e g i o n i s u s u a l l y w r i t t e n a s

- 4w h e r e C = 3 . 0 7 X 1 0 Z 2 / A 2 M e V - c m 2 / m g , a n d t h e s t o p p i n g n u m b e r L

d e p e n d s o n t h e p a r t i c u l a r t h e o r y u s e d t o d e s c r i b e t h e e n e r g y l o s s . B o t h

Z i e g l e r a n d N S a s s u m e L = L 0 ( v , Z 2 ) , i . e . i t d e p e n d s o n l y o n t a r g e t

74

m a t e r i a l a n d p r o j e c t i l e v e l o c i t y , w h i c h r e s u l t s i n a s i m p l e Z t 2 s t o p p i n g

p o w e r d e p e n d e n c e . R e c e n t m e a s u r e m e n t s o n p i o n r a n g e s ( H e 6 9 ) , a s w e l l

a s p r e c i s i o n d a t a o n p r o t o n a n d a l p h a p a r t i c l e s t o p p i n g p o w e r s ( A n 6 9 ) ,

a r e n o t c o n s i s t e n t w i t h t h i s s c a l i n g , h o w e v e r . A s h l e y , e t a l . ( A s 7 2 )

a n d J a c k s o n a n d M c C a r t h y ( J a 7 2 ) h a v e c a l c u l a t e d Z x3 c o r r e c t i o n s t o t h e

s t o p p i n g p o w e r , a s s u m i n g

L = L ^ v .Z ^ + Z ^ f v . Z ^ (4.4)

w h i l e L i n d h a r d ( L i 7 6 ) s u g g e s t e d a Z x3 t e r m a p p r o x i m a t e l y t w i c e t h a t o f

J a c k s o n a n d M c C a r t h y , a s w e l l a s t h e c o r r e c t i o n o f B l o c h ( B 1 3 3 a ) , i . e .

L = L0(v .Z2) + Z 1L1(v ,Z2) +<p(v,Z1) (4.5)

T h e L i n d h a r d c o r r e c t i o n s p r o v i d e t h e b e s t f i t t o t h e p i o n a n d p r o t o n -

a l p h a p a r t i c l e d a t a . M o r e o v e r , A n d e r s e n , e t a l . ( A n 7 7 a ) h a v e m a d e

p r e c i s i o n m e a s u r e m e n t s w i t h p r o t o n , a l p h a a n d L i p r o j e c t i l e s i n A l , C u ,

A g a n d A u w h i c h a l l o w t h e m t o s e p a r a t e o u t h i g h e r o r d e r c o n t r i b u t i o n s t o

t h e s t o p p i n g p o w e r . T h e y h a v e f o u n d Z23 a n d Z x4 c o r r e c t i o n s w h i c h a r e

w e l l d e s c r i b e d b y t h e L i n d h a r d p r e d i c t i o n s . T h e s e c o r r e c t i o n s a l l

v a n i s h a t h i g h v e l o c i t i e s .

We h a v e i n v e s t i g a t e d t h e e f f e c t o f h i g h e r o r d e r c o r r e c t i o n s f o r N i

p r o j e c t i l e s i n C u a n d A g t a r g e t s , u s i n g t h e t e r m s p r o p o s e d b y L i n d h a r d .

V a l u e s f o r L 0 a r e t a k e n f r o m t h e e x p e r i m e n t s o f A n d e r s e n , e t a l . , w i t h a

q u o t e d a c c u r a c y o f 0 . 5 % . T h e J a c k s o n a n d M c C a r t h y Z *3 c o r r e c t i o n s a r e

t a b u l a t e d , a n d w e h a v e a s s u m e d t w i c e t h e s e v a l u e s f o r L a . T h e B l o c h

c o r r e c t i o n i s g i v e n b y

<p (v,Z ) = -y2 s — — — i y = — — with v„ = ~ <4-6>n(n + y ) v 0 *

Z .v„ 2 e_0 = ft

75

w h i c h f o r s m a l l y r e d u c e s t o

22 v n 2

- 1 . 1 2 3 Z 1 - y - = - Z : L 2 ( v ) ( 4 . 6 a )v

a s d e s c r i b e d b y L i n d h a r d ( L i 7 6 ) . F o r h e a v y i o n s a t t h e s e v e l o c i t i e s , y

i s n o t s m a l l , h o w e v e r , a n d t h e f u l l e x p a n s i o n m u s t b e u s e d .

T h i s e n t i r e f o r m a l i s m a s s u m e s a d e f i n i t e p r o j e c t i l e c h a r g e . A t

e n e r g i e s o f 1 - 3 M e V / a m u l o w Z t p a r t i c l e s , s u c h a s p r o t o n s a n d a l p h a s ,

a r e d e s c r i b e d b y t h e n u c l e a r c h a r g e . H e a v y i o n s a t t h e s e v e l o c i t i e s a r e

n o t f u l l y s t r i p p e d o f e l e c t r o n s , h o w e v e r , a n d s o m e a s s u m p t i o n a b o u t t h e

p r o j e c t i l e c h a r g e m u s t b e m a d e . ( S e e S e c . 2 . C ) F o l l o w i n g P o r t e r ( P o 7 7 ) ,

w e h a v e c h o s e n :

*i <4 '7 >

Z 1 v Z1 0 1kw h e r e Z j = i o n e f f e c t i v e c h a r g e a n d X i s a f r e e p a r a m e t e r . T h i s

f u n c t i o n a l f o r m w a s o r i g i n a l l y p r o p o s e d b y N o r t h c l i f f e ( N o 6 1 ) a n d i s

b a s e d o n a T h o m a s - F e r m i d e s c r i p t i o n o f t h e p r o j e c t i l e . T h e r e i s n o

e x p l i c i t t a r g e t d e p e n d e n c e i n t h i s e x p r e s s i o n .

W e h a v e t h u s a t t e m p t e d a o n e - p a r a m e t e r f i t t o t h e s t o p p i n g p o w e r

m e a s u r e m e n t s r e l e v a n t t o o u r t h i c k t a r g e t d a t a . T h e r e s u l t s a r e s h o w n

i n F i g s . 4 . 9 , 4 . 1 0 . S i n c e o u r f i t s a r e b a s e d o n A n d e r s e n , e t a l . ' s

e x p e r i m e n t a l v a l u e s f o r L 0 , t h e c u r v e s g o n o l o w e r i n v e l o c i t y t h a n

t h e i r m e a s u r e m e n t s ( w i t h s o m e s l i g h t e x t r a p o l a t i o n ) , i . e . E ~ 1 . 0

H e V / a m u .

76

V a l u e s o f t h e c h a r g e s t a t e p a r a m e t e r X w e r e f o u n d t o b e 0 . 8 2 f o r N i

i n b o t h C u a n d A g , w h i c h s u g g e s t s t h a t t h e p r o j e c t i l e c h a r g e i s

i n d e p e n d e n t o f t h e t a r g e t m a t e r i a l . W i t h e x a c t l y t h e s a m e c h a r g e s t a t eIe x p r e s s i o n , w e a r e t h u s a b l e t o f i t b o t h t h e C u a n d A g d a t a , w h i c h

d i s p l a y r a d i c a l l y d i f f e r e n t b e h a v i o r w i t h r e s p e c t t o t h e c o n v e n t i o n a l

c u r v e s .

77

A l t h o u g h t h e c o r r e c t i o n s o f L i n d h a r d p r o v i d e g o o d f i t s t o o u r i n i t i a l

d E / d x m e a s u r e m e n t s f o r N i i n C u a n d i n A g , t h e t a r g e t i n d e p e n d e n c e o f

t h e r e s u l t i n g e f f e c t i v e c h a r g e e x p r e s s i o n i s n o t c o n s i s t e n t w i t h t h e

r e s u l t o f a v e r a g e c h a r g e s t a t e m e a s u r e m e n t s m a d e a f t e r p a s s a g e t h r o u g h

s o l i d a n d g a s e o u s t a r g e t m a t e r i a l s . B o t h s o l i d a n d g a s t a r g e t s g e n e r a t e

t h e s a m e t a r g e t d e p e n d e n c e i n h e a v y i o n c h a r g e s t a t e s , i n t h a t l o w Z 2

t a r g e t s p r o d u c e h i g h e r p r o j e c t i l e c h a r g e s t h a n h i g h Z 2 t a r g e t s . T h i s

t a r g e t d e p e n d e n c e o f h e a v y i o n s a f t e r e x i t i n g m a t e r i a l s i s a p o s s i b l e

c o n s t r a i n t o n t h e e f f e c t i v e c h a r g e v a l u e s c a l c u l a t e d f r o m e n e r g y l o s s

m e a s u r e m e n t s , w h i c h a p p l y t o t h e i o n s i n s i d e t h e m a t e r i a l . H o w e v e r ,

t h e r e i s n o w a y t o u n i q u e l y d e t e r m i n e t h e e f f e c t i v e c h a r g e v a l u e s f r o m

h e a v y i o n d E / d x m e a s u r e m e n t s , s i n c e t h e c h a r g e d e p e n d e n c e o f t h e

s t o p p i n g p o w e r i s n o t k n o w n . T h e b e s t w a y t o s t u d y t h e s e e f f e c t s w o u l d

b e t h r o u g h a n a n a l y s i s o f p r e c i s i o n d E / d x d a t a o v e r a b r o a d r a n g e o f

p r o j e c t i l e - t a r g e t - e n e r g y c o m b i n a t i o n s . T h i s l a r g e d a t a b a s e w o u l d a l l o w

t h e g e n e r a l f e a t u r e s o f b o t h t h e h i g h e r o r d e r c o r r e c t i o n s a n d t h e

e f f e c t i v e c h a r g e v a l u e s t o b e e x a m i n e d ; h o w e v e r , s u c h a d a t a b a s e h a s

n o t e x i s t e d .

I n o r d e r t o r e s o l v e t h i s p r o b l e m , w e h a v e c h o s e n t o s t u d y t h e e n e r g y

l o s s o f f i v e d i f f e r e n t e l e m e n t a l t a r g e t s f o r a v a r i e t y o f d i f f e r e n t

b e a m s . R e c e n t l y A n d e r s e n , e t . a l . m a d e a p r e c i s i o n s t u d y o f t h e

s t o p p i n g p o w e r o f p r o t o n , a l p h a a n d L i p a r t i c l e s i n A l , C u , A g a n d A u

t a r g e t s w h i c h a l l o w e d t h e m t o s e p a r a t e o u t v a l u e s f o r t h e B e t h e s t o p p i n g

C. Extended dE/dx Measurements

78

n u m b e r ( L „ i n e q . 4 . 5 ) t o a n a c c u r a c y o f 0 . 5 % . T h e s e m e a s u r e m e n t s c o v e r

m o s t o f t h e v e l o c i t y r e g i o n o f i n t e r e s t t o u s , s o w e h a v e u s e d t h e s a m e

m a t e r i a l s i n o r d e r t o u t i l i z e t h e i r r e s u l t s f o r L 0 . W e h a v e a l s o c h o s e n

t o s t u d y C , d u e t o i t s l o w a t o m i c n u m b e r a n d i t s i m p o r t a n c e a s a t a r g e t

m a t e r i a l i n m a n y n u c l e a r p h y s i c s e x p e r i m e n t s .

T h e r e s u l t s o f o u r e n e r g y l o s s m a s u r e m e n t s ( a f t e r a c o r r e c t i o n f o r

t h e n u c l e a r s t o p p i n g c o n t r i b u t i o n ( Z i 7 7 b ) ) a r e s h o w n i n F i g s . 4 . 1 1 - 4 . 1 5 ,

a l o n g w i t h t h e p r e d i c t i o n s o f Z i e g l e r a n d N S . D a t a w e r e t a k e n f o r t w o

t a r g e t s o f e a c h m a t e r i a l , i n o r d e r t o a v o i d p o s s i b l e s y s t e m a t i c

u n c e r t a i n t i e s i n o u r d E / d x m e a s u r e m e n t s d u e t o a n y e r r o r s i n t h e t a r g e t

t h i c k n e s s d e t e r m i n a t i o n s . O u r r e s u l t s f o r S i a n d N i r e p r o d u c e t h e

e f f e c t s d i s c u s s e d i n S e c . B , a s e x p e c t e d , b u t i n m o s t o f t h e o t h e r c a s e s

t h e r e i s n o p r e e x i s t i n g d a c a w i t h w h i c h t o c o m p a r e . T h e m a i n s o u r c e s o f

u n c e r t a i n t y i n t h e s e m e a s u r e m e n t s w e r e 1 ) s t a t i s t i c a l f l u c t u a t i o n s i n

d e t e r m i n i n g t h e p e a k l o c a t i o n s , a n d 2 ) u n c e r t a i n t y i n t h e t a r g e t

t h i c k n e s s e s . A l i s t i n g o f t h e s e d a t a i s a v a i l a b l e i n A p p e n d i x B .

I n g e n e r a l t h e s t a n d a r d t a b u l a t i o n s d o p o o r l y i n p r e d i c t i n g b o t h t h e

m a g n i t u d e a n d t h e l o c a t i o n o f t h e s t o p p i n g p o w e r m a x i m u m f o r n e a r l y a l l

t h e p r o j e c t i l e - t a r g e t c o m b i n a t i o n s w h i c h w e s t u d i e d . T h e b e h a v i o r o f

a l l o u r h e a v y i o n s i n a g i v e n t a r g e t m a t e r i a l i s f a i r l y c o n s i s t e n t ; i n

A g t a r g e t s , f o r e x a m p l e , t h e p e a k i s a l m o s t a l w a y s 1 ) l a r g e r i n

m a g n i t u d e , a n d 2 ) l o w e r i n e n e r g y t h a n t h e s t a n d a r d p r e d i c t i o n s . T h e s e

f a c t s s u g g e s t t h a t t h e Z xz d e p e n d e n c e o f t h e s t a n d a r d t a b u l a t i o n s i s n o t

s u f f i c i e n t t o d e s c r i b e t h e s e h e a v y i o n m e a s u r e m e n t s , a n d t h a t t h e

t e c h n i q u e s u s e d i n S e c . B s h o u l d b e a p p l i c a b l e h e r e a s w e l l . T h e r e f o r e

79

w e h a v e e x a m i n e d s e v e r a l f o r m s f o r t h e h i g h e r o r d e r c o r r e c t i o n s , i n

c o n j u n c t i o n w i t h v a r i o u s e f f e c t i v e c h a r g e p a r a m e t e r i z a t i o n s , i n a n

a t t e m p t t o p r o v i d e b e t t e r a g r e e m e n t b e t w e e n e x p e r i m e n t a n d t h e o r y .

T h e s e m e t h o d s , a n d t h e r e s u l t s , a r e d i s c u s s e d i n C h a p t e r V .

80

T a b l e 4 . 1 R a n g e m e a s u r e m e n t s . A l s o l i s t e d a r e t h e p r e d i c t e d r a n g e s o f

Z i e g l e r ( R z ) a n d N o r t h c l i f f e a n d S c h i l l i n g ( R N S ) i n e a c h c a s e .

81H E A V Y I O N R A N G E S

I n c i d e n t B e a m T a r g e t T o t a l R a n g e P r e d i c t e d R a n g eB e a m E n e r g y M a t e r i a l ( m g / c m 2 ) ( m g / c m 2 )

( M e V )r n s r z

S i 6 3 . 8 C u 7 . 8 6 + 0 . 2 0 8 . 0 7 . 8S i 4 1 . 7 C u 5 . 5 8 ± 0 . 1 8 5 . 6 5 . 6S i 5 1 . 8 A g 7 . 4 2 ± 0 . 1 3 8 . 4 7 . 9S i 3 7 . 7 A g 5 . 3 9 ± 0 . 3 1 6 . 6 6 . 0S i 5 1 . 8 P b 1 1 . 1 5 ± 0 . 2 6 1 4 . 1 1 2 . 0S i 4 1 . 7 P b 8 . 4 3 ± 0 . 3 2 1 2 . 1 9 . 5

N i 1 0 2 . 4 C u 7 . 4 1 ± 0 . 1 2 7 . 0 7 . 0N i 5 9 . 2 C u 5 . 2 8 ± 0 . 1 3 4 . 9 5 . 0N i 8 5 . 4 A g 7 . 2 9 ± 0 . 1 1 7 . 8 7 . 3N i 3 7 . 4 A g 4 . 6 1 ± 0 . 1 1 4 . 7 5 4 . 4N i 1 1 5 . 3 A g 8 . 8 4 ± 0 . 1 5 9 . 6 9 . 3N i 8 5 . 4 P b 1 0 . 7 5 ± 0 . 1 9 1 3 . 3 1 1 . 0N i 5 3 . 4 P b 7 . 9 0 ± 0 . 2 3 1 0 . 3 7 . 9

A u 1 6 5 . 5 C u 7 . 6 8 ± 0 . 2 0 6 . 8 7 . 5A u 8 5 . 4 C u 5 . 3 7 ± 0 . 1 9 4 . 7 5 5 . 4A u 1 1 4 . 3 A g 7 . 4 8 ± 0 . 2 2 7 . 0 7 . 1 5A u 1 6 5 . 5 A g 9 . 0 7 ± 0 . 2 3 8 . 5 8 . 5A u 1 1 4 . 3 P b 1 1 . 1 5 ± 0 . 3 3 1 1 . 9 1 0 . 8A u 7 8 . 8 P b 8 . 5 1 ± 0 . 3 9 9 . 7 5 8 . 3

82

F i g . 4 . 1 I n t e g r a t e d e n e r g y l o s s o f S i i n A g ( 4 . 2 4 m g / c m 2 ) . A l s o s h o w n

a r e t h e p r e d i c t i o n s o f Z i e g l e r ( -----------) a n d N S ( - • - ) , a s w e l l a s a

n u m e r i c a l i n t e g r a t i o n o f t h e e x p e r i m e n t a l d E / d x m e a s u r e m e n t s o f F o r s t e r ,

e t a l .

EXIT

ENER

GY(M

eV)

83

3 9 41 4 3 4 5INCIDENT ENERGY (MeV)

4 7

84

F i g . 4 . 2 I n t e g r a t e d e n e r g y l o s s o f S i i n A g ( 6 . 9 2 m g / c m 2 ) . A l s o s h o w n

a r e t h e p r e d i c t i o n s o f Z i e g l e r ( ----------- ) a n d N S ( - • - ) , a s w e l l a s a

n u m e r i c a l i n t e g r a t i o n o f t h e e x p e r i m e n t a l d E / d x m e a s u r e m e n t s o f F o r s t e r ,

e t a l .

EXIT

ENER

GY

(MeV

)

85

86

F i g . 4 . 3 I n t e g r a t e d e n e r g y l o s s o f N i i n A g ( 4 . 2 4 m g / c m 2 ) . A l s o s h o w n

a r e t h e p r e d i c t i o n s o f Z i e g l e r ( ----------- ) a n d N S ( - • - ) , a s w e l l a s a

n u m e r i c a l i n t e g r a t i o n o f o u r t h i n t a r g e t d E / d x m e a s u r e m e n t s .

EXIT

ENER

GY

(MeV

)

1I

87

iiIi

88

iii

F i g . | 4 . 4 I n t e g r a t e d e n e r g y l o s s o f N i i n A g ( 6 . 9 2 m g / c m 2 ) ,

a r e t i h e p r e d i c t i o n s o f Z i e g l e r ( ----------- ) a n d N S ( - • - ) , a sI

n u m e r i c a l i n t e g r a t i o n o f o u r t h i n t a r g e t d E / d x m e a s u r e m e n t siiiiiiiii

iiiiiIiiii

A l s o s h o w n

w e l l a s a

EXIT

ENER

GY

(MeV

)

II

89

II

INCIDENT ENERGY (MeV)

90

F i g . 4 . 5 I n t e g r a t e d e n e r g y l o s s o f A u i n A g ( 6 . 9 2 m g / c m 2 ) . A l s o s h o w n

a r e t h e p r e d i c t i o n s o f Z i e g l e r ( -----------) a n d N S ( - • - ) .

II

EXIT

ENER

GY

(MeV

)

18

15

12

9

6

3

01 0 5 1 2 0 1 3 5 1 5 0 1 6 5

INCIDENT ENERGY (MeV)

1

A u in A g ( 6 . 9 2 m g / c m 2 )--------------ZIEC--------------N S

j L E KS'*s 's '

*s 's's ' " + /

s ''

s Ts* ’/X

s'^ '

s 'sSS'sSsS

SS 'X ‘

‘/

92

F i g . 4 . 6 I n t e g r a t e d e n e r g y l o s s o f A u i n A g ( 8 . 3 9 m g / c m 2 ) . A l s o s h o w n

a r e t h e p r e d i c t i o n s o f Z i e g l e r ( ----------- ) a n d N S ( - • - ) .

EXIT

ENER

GY

(MeV

)

93

INCIDENT ENERGY (MeV)

94

F i g . 4 . 7 E l e c t r o n i c s t o p p i n g p o w e r o f N i i n C u , v s . v e l o c i t y . A l s o

s h o w n a r e t h e p r e d i c t i o n s o f Z i e g l e r ( -------------------) , N S ( - • - ) a n d L S S

( )•

STOP

PING

(M

eV/m

g/cm

2)

95

96

F i g . 4 . 8 E l e c t r o n i c s t o p p i n g p o w e r o f N i i n A g , v s . v e l o c i t y . A l s o

s h o w n a r e t h e p r e d i c t i o n s o f Z i e g l e r ( ----------------- ) , N S ( - • - ) a n d L S S

( )•

STOP

PING

(M

eV/m

gAm

2)

97

98

F i g . 4 . 9 E l e c t r o n i c s t o p p i n g p o w e r o f N i i n C u , v s . e n e r g y . A l s o s h o w n

a r e t h e p r e d i c t i o n s o f Z i e g l e r ( --------------) a n d N S ( - • - ) a s w e l l a s t h e

c a l c u l a t i o n s o f t h e p r e s e n t s t u d y ( S e c . 4 B ) ( ----------- ) . T h e c h a r g e s t a t e

e x p r e s s i o n u s e d i n t h e Z 3 + Z 4 f i t ( L i n d h a r d p l u s B l o c h ) i s e x a c t l y t h e

s a m e a s f o r t h e N i i n A g d a t a , F i g . 4 . 1 0 . T h i s p r o d u c e s r e a s o n a b l e f i t s

f o r b o t h d a t a s e t s ( N i i n C u a n d i n A g ) , e v e n t h o u g h t h e y b e h a v e q u i t e

d i f f e r e n t l y w i t h r e s p e c t t o t h e c o n v e n t i o n a l c u r v e s .

STO

PPIN

G

(MeV

/mg/

cm2)

99

E N E R G Y ( M e V )

100

F i g . 4 . 1 0 E l e c t r o n i c s t o p p i n g p o w e r o f N i i n A g , v s . e n e r g y . A l s o

s h o w n a r e t h e p r e d i c t i o n s o f Z i e g l e r ( ------------- ) a n d N S ( - • - ) a s w e l l a s

t h e c a l c u l a t i o n s o f t h e p r e s e n t s t u d y ( S e c . 4 B ) ( -----------) .

STOP

PING

(M

eV/m

g /cm

2)

E N E R G Y ( M e V )

102

F i g 4 . 1 1 E l e c t r o n i c s t o p p i n g p o w e r o f a l l o u r h e a v y i o n s i n C ,

e n e r g y . A l s o s h o w n a r e t h e p r e d i c t i o n s o f Z i e g l e r ( -----------------) a n d

( - • - ) f o r e a c h i o n .

v s .

N S

43;

2 018161412

242 22 0183329254036322824

103

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104

F i g 4 . 1 2 E l e c t r o n i c s t o p p i n g p o w e r o f a l l o u r h e a v y i o n s i n A l ,

e n e r g y . A l s o s h o w n a r e t h e p r e d i c t i o n s o f Z i e g l e r ( --------------) a n d

( - • - ) f o r e a c h i o n .

v s .N S

105

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106

F i g 4 . 1 3 E l e c t r o n i c s t o p p i n g p o w e r o f a l l o u r h e a v y i o n s i n Cu,

e n e r g y . A l s o shown a r e t h e p r e d i c t i o n s o f Z i e g l e r ( --------- ) a n d

( - • - ) f o r e a c h i o n .

vs .

NS

STOPPI

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eV/mg

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107

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108

F i g 4 . 1 4 E l e c t r o n i c s t o p p i n g p o w e r o f a l l o u r h e a v y i o n s i n Ag,

e n e r g y . A l s o shown a r e t h e p r e d i c t i o n s o f Z i e g l e r ( --------- ) a n d

( - • - ) f o r e a c h i o n .

vs.NS

2.52.01.54

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F i g 4 . 1 5 E l e c t r o n i c s t o p p i n g p o w e r o f a l l o u r h e a v y i o n s i n Au,

e n e r g y . A l s o shown a r e t h e p r e d i c t i o n s o f Z i e g l e r ( --------- ) a n d

( - • - ) f o r e a c h i o n .

vs.NS

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in

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EN ER G Y (MeV/amu)

CHAPTER V

HIGHER ORDER CORRECTIONS AND EFFECTIVE CHARGE

As we sh o w ed i n C h a p t e r IV, t h e r e i s a n e e d t o u n d e r s t a n d h i g h e r

o r d e r c o n t r i b u t i o n s t o t h e s t o p p i n g p o w e r i n o r d e r t o d e s c r i b e o u r

e x p e r i m e n t a l r e s u l t s . I f t h e c h a r g e s t a t e o f t h e p r o j e c t i l e a s i t p a s s e s t h r o u g h t h e t a r g e t m a t e r i a l i s w e l l known, t h e h i g h e r o r d e r

c o r r e c t i o n s c a n b e d e d u c e d d i r e c t l y f ro m p r e c i s i o n d E / d x m e a s u r e m e n t s .

H o w e v e r , s i n c e t h e B o r n a p p r o x i m a t i o n , w h i c h i s t h e b a s i s o f t h e B e t h e Z j 2 s t o p p i n g p o w e r e x p r e s s i o n , b e c o m e s i n c r e a s i n g l y v a l i d a s t h e

i n c i d e n t e n e r g y i n c r e a s e s , t h e h i g h e r o r d e r c o r r e c t i o n s a l l v a n i s h a t

h i g h v e l o c i t i e s . At l o w e r e n e r g i e s ( ~ l - 3 MeV/amu) o n l y l i g h t

p r o j e c t i l e s a r e f u l l y s t r i p p e d o f t h e i r e l e c t r o n s , a n d t h e c o r r e c t i o n s a r e d i f f i c u l t t o d e t e r m i n e f ro m l i g h t i o n d E / d x d a t a du e t o t h e low

c h a r g e . T h i s s e v e r e l y l i m i t s t h e a c c u r a c y o f a n y d e t e r m i n a t i o n o f

h i g h e r o r d e r e f f e c t s . T h e r e f o r e , a n a l y s i s o f a l a r g e n u m ber o f a c c u r a t e d E / d x m e a s u r e m e n t s f o r h e a v y i o n s u s i n g t h e t e c h n i q u e s i n t r o d u c e d i n

C h a p t e r IV may p r o v i d e t h e o n l y m e c h a n i s m f o r d e t e r m i n i n g t h e h i g h e r o r d e r c o r r e c t i o n s a n d t h e e f f e c t i v e c h a r g e o f t h e s e i o n s i n a s e l f -

c o n s i s t e n t way .

I n t h i s c h a p t e r we e x t e n d t h e a n a l y s i s t e c h n i q u e s i n t r o d u c e d i n

C h a p t e r IV t o c o v e r o u r w h o l e r a n g e o f s t o p p i n g p o w e r m e a s u r e m e n t s . By

e x a m i n i n g t h e h i g h e r o r d e r e f f e c t s i n s u c h a l a r g e d a t a b a s e , we h a v e

r e d u c e d t h e i n f l u e n c e o f random e r r o r s f ro m a n y p a r t i c u l a r p r o j e c t i l e -

112

t a r g e t c o m b i n a t i o n . S e v e r a l e x p r e s s i o n s f o r t h e e n e r g y - l o s s p r o c e s s ,

b o t h w i t h a n d w i t h o u t t h e h i g h e r o r d e r c o r r e c t i o n s , h a v e b e e n e x p l o r e d

a l o n g w i t h v a r i o u s e f f e c t i v e c h a r g e p a r a m e t e r i z a t i o n s . The r e s u l t s a r e

c o m p a r e d w i t h t h e r e s u l t s o f p r e v i o u s i n v e s t i g a t i o n s o f t h e h i g h e r o r d e r

d E / d x c o r r e c t i o n s , a s w e l l a s a v e r a g e e q u i l i b r i u m c h a r g e s t a t e

m e a s u r e m e n t s o f i o n s a f t e r p a s s i n g t h r o u g h s o l i d s a n d g a s e s . A l s o , t h e

a b i l i t y o f t h e s e e x p r e s s i o n s t o r e p r o d u c e o u r d a t a i s c o m p a r e d t o t h e s t a n d a r d t a b u l a t i o n s o f Z i e g l e r a n d NS.

11 3

1 1 4

A. D e r i v a t i o n

I n t h e p r e v i o u s c h a p t e r we p l o t t e d ( F i g s . 4 . 1 1 - 4 . 1 5 ) t h e r e s u l t s o f

a l l o u r s t o p p i n g p o w e r m e a s u r e m e n t s v s . t h e p r e d i c t i o n s o f Z i e g l e r a n d

NS. I n m o s t c a s e s t h e s e s t a n d a r d t a b u l a t i o n s do p o o r l y i n p r e d i c t i n g

b o t h t h e m a g n i t u d e a n d t h e l o c a t i o n o f t h e s t o p p i n g p o w e r maximum, a n d

t h e r e i s a s y s t e m a t i c d i s c r e p a n c y b e t w e e n p r e d i c t i o n a n d e x p e r i m e n t f o r a l l i o n s i n a g i v e n t a r g e t m a t e r i a l . I n F i g s . 5 . 1 - 5 . 1 0 we show t h e

r a t i o o f o u r d E / d x m e a s u r e m e n t s t o t h e t a b u l a t i o n s o f Z i e g l e r a n d NS,

p l o t t e d v s . t h e i o n e n e r g y , w h i c h d e m o n s t r a t e c l e a r l y t h e s y s t e m a t i c

b e h a v i o r o f a l l o u r m e a s u r e m e n t s i n a g i v e n t a r g e t . The f a c t t h a t t h e

Z i e g l e r a n d NS c u r v e s f o r a p a r t i c u l a r t a r g e t m a t e r i a l b e h a v e so

s i m i l a r l y i s f u r t h e r e v i d e n c e t h a t t h e Z j 2 d e p e n d e n c e u s e d t o p r o d u c e

t h e s e v a l u e s may n o t b e s u f f i c i e n t .

I f a s i m p l e Z : 2 s t o p p i n g p o w e r d e p e n d e n c e i s a s s u m e d , t h e n a r a t i o o f e x p e r i m e n t a l d E / d x v a l u e s f o r two d i f f e r e n t h e a v y i o n s , m e a s u r e d i n t h e

same t a r g e t m a t e r i a l a n d a t t h e same v e l o c i t y , w i l l g i v e a v a l u e f o r t h e

s q u a r e o f t h e r a t i o o f t h e two h e a v y i o n e f f e c t i v e c h a r g e s , i . e .*2( d E / d x ) Z

-------------- A 1A ( 5 . 1 )( d E / d x ) 2B ZIB

w h e r e Z j i s t h e e f f e c t i v e c h a r g e o f a p r o j e c t i l e o f a t o m i c nu m ber Z 1#

a n d A a n d B r e p r e s e n t t h e two i o n s u n d e r c o n s i d e r a t i o n . I f on e o f t h e

i o n s i s c h o s e n t o b e h y d r o g e n , a n d t h e m e a s u r e m e n t s a r e made a t

v e l o c i t i e s l a r g e e n o u g h t h a t t h e p r o t o n i s s t r i p p e d o f i t s e l e c t r o n , t h e n t h e h y d r o g e n e f f e c t i v e c h a r g e i s e q u a l t o on e a n d we h a v e a d i r e c t

m e a s u r e o f t h e h e a v y i o n e f f e c t i v e c h a r g e . T h i s t e c h n i q u e h a s b e e n u s e d

b y many w o r k e r s ( N o 7 0 , S a 7 2 , Z i 8 0 ) , b u t t h e r e s u l t a n t e f f e c t i v e c h a r g e

e x p r e s s i o n s show no t a r g e t d e p e n d e n c e . H o w e v e r , b o t h s o l i d a n d g a s

t a r g e t s g e n e r a t e a t a r g e t d e p e n d e n c e i n h e a v y i o n c h a r g e s t a t e s m e a s u r e d

a f t e r t h e i o n e x i t s t h e t a r g e t ; low Z2 t a r g e t s p r o d u c e h i g h e r p r o j e c t i l e

c h a r g e s t h a n h i g h Z2 t a r g e t s , a n d t h i s t a r g e t d e p e n d e n c e i s a p o s s i b l e

c o n s t r a i n t on t h e e f f e c t i v e c h a r g e v a l u e s c a l c u l a t e d f ro m e n e r g y l o s s

m e a s u r e m e n t s . T h i s s u g g e s t s t h a t t h e Z xz s t o p p i n g p o w e r e x p r e s s i o n may n o t b e c o m p l e t e .

We h a v e r e e x a m i n e d t h i s Z j 2 r e l a t i o n s h i p i n l i g h t o f t h e f a c t t h a t

o u r m e a s u r e m e n t s 1) a r e more a c c u r a t e t h a n m o s t p r e v i o u s l y a v a i l a b l e

d a t a a n d 2 ) e x p l o r e p r o j e c t i l e - t a r g e t c o m b i n a t i o n s n o t p r e v i o u s l y

i n v e s t i g a t e d . I n F i g s . 5.1*1 a n d 5 . 1 2 we h a v e a n a l y z e d o u r e n e r g y l o s s

m e a s u r e m e n t s f o r S i a n d Br i o n s a s s u m i n g o n l y a Z a 2 s t o p p i n g p o w e r d e p e n d e n c e . V a l u e s o f t h e i o n e f f e c t i v e c h a r g e w e r e c a l c u l a t e d f ro m t h e

r a t i o o f o u r d a t a t o p r o t o n s t o p p i n g p o w e r s , w h i c h h a v e b e e n v e r y a c c u r a t e l y m e a s u r e d f o r t h e f o u r m a t e r i a l s shown ( A n 7 7 ) . I n e a c h c a s e

t h e v e l o c i t i e s a r e h i g h e n o u g h t o g u a r a n t e e t h a t t h e p r o t o n i s s t r i p p e d .

The r e s u l t a n t e f f e c t i v e c h a r g e v a l u e s ( d i v i d e d b y t h e i o n a t o m i c nu m b er ) a r e p l o t t e d , v s . t h e i o n v e l o c i t y . ( V a l u e s f o r t h e C d a t a h a v e n o t b e e n

c a l c u l a t e d , s i n c e t h e d E / d x m e a s u r e m e n t s f o r p r o t o n s i n C a r e n o t a sw e l l known a s i n t h e o t h e r m a t e r i a l s . ) T h e r e i s no s i m p l e t a r g e t

d e p e n d e n c e e v i d e n t i n t h e s e v a l u e s , a n d t h u s t h e Z xz s t o p p i n g p o w e r

d e p e n d e n c e d o e s n o t a l l o w o u r d a t a t o r e p r o d u c e t h e t a r g e t d e p e n d e n c e o f

e q u i l i b r i u m c h a r g e s t a t e s . I t i s o b v i o u s , h o w e v e r , t h a t i f t h e

e x p e r i m e n t a l u n c e r t a i n t i e s i n o u r d a t a w e r e l a r g e r ( a s i s t h e c a s e w i t h

1 1 5

many d E / d x m e a s u r e m e n t s f o r h e a v y i o n s ) a u n i v e r s a l c u r v e c o u l d b e d r a w n

t h r o u g h a l l t h e d a t a . T h i s i s e x a c t l y t h e r e s u l t o f Z i e g l e r a n d NS, a n d

i t d e m o n s t r a t e s t h e n e e d f o r a c c u r a t e h e a v y i o n d E / d x d a t a i n o r d e r t o r e c o g n i z e t h e l i m i t a t i o n s o f t h e Z x2 e x p r e s s i o n .

We h a v e a l s o a n a l y z e d t h e same d a t a b y i n c l u d i n g h i g h e r o r d e r c h a r g e

d e p e n d e n t c o r r e c t i o n s t o t h e s t o p p i n g p o w e r . I n t h i s c a s e n o s i m p l e

r a t i o w i l l p r o v i d e v a l u e s f o r t h e e f f e c t i v e c h a r g e , a n d we m u s t a s su m e

some p a r t i c u l a r m o d e l . F o l l o w i n g t h e m e t h o d s o f C h a p t e r IV, we h a v e

u t i l i z e d t h e c o r r e c t i o n s o f L i n d h a r d ( L i 7 6 ) , i n w h i c h t h e s t o p p i n g num be r L c a n b e w r i t t e n a s

L = L0(v,Z2) + Z2 LjCv.Zg) + <p (v,Z1) (5.2)

L i n d h a r d s u g g e s t s a Z x3 t e r m ( L x ( v , Z2 ) ; a p p r o x i m a t e l y t w i c e t h a t o f

J a c k s o n a n d M c C a r t h y , a n d we h a v e c a l c u l a t e d L x a s s u m i n g t w i c e t h e i r

t a b u l a t e d v a l u e s , w h i l e $ ( v , Z a ) i s j u s t t h e B l o c h c o r r e c t i o n ( s e e e q . 4 . 6 ) . V a l u e s f o r L „ ( v , Z2 ) h a v e b e e n d e t e r m i n e d e x p e r i m e n t a l l y by

A n d e r s e n , e t a l . (A n7 7a) t o h i g h a c c u r a c y f o r t h e t a r g e t s shown

( a c c u r a t e d a t a f o r C t a r g e t s a r e n o t a v a i l a b l e ) , a n d c o n s e q u e n t l y we h a v e u s e d t h e i r v a l u e s f o r L „ . ( S i n c e o u r c a l c u l a t i o n s a r e b a s e d on t h e

e x p e r i m e n t a l w o rk o f A n d e r s e n , e t a l . , we go no l o w e r i n v e l o c i t y t h a n

t h e i r m e a s u r e m e n t s , i . e . E ~ 1 . 0 MeV/amu.) As b e f o r e , we m u s tfts u b s t i t u t e Z x f o r Z x , s i n c e t h e h e a v y i o n s a r e n o t f u l l y s t r i p p e d .

T h u s , i n t h e a b o v e e x p r e s s i o n f o r L a l l t e r m s e x c e p t t h e i o n e f f e c t i v e

c h a r g e a r e known, a n d t h e i o n e f f e c t i v e c h a r g e c a n b e c o m p u t e d

i t e r a t i v e l y f ro m o u r d E / d x m e a s u r e m e n t s . The r e s u l t s o f t h e s e

c a l c u l a t i o n s a r e shown i n F i g s . 5 . 1 3 a n d 5 . 1 4 , w h e r e we h a v e a g a i n

116

p l o t t e d t h e e f f e c t i v e c h a r g e v a l u e s v s . t h e i o n v e l o c i t y . I n e a c h c a s e ,

u s e o f t h e h i g h e r o r d e r c o r r e c t i o n s a l l o w s o u r c a l c u l a t e d e f f e c t i v e

c h a r g e v a l u e s t o r e p r o d u c e o v e r t h e w h o l e v e l o c i t y r a n g e t h e t a r g e t

d e p e n d e n c e o f h e a v y i o n c h a r g e s t a t e m e a s u r e m e n t s made a f t e r t h e i o n s

e x i t m a t e r i a l s . T h i s e f f e c t i s t r u e o f a l l o u r h e a v y i o n d a t a , a n d i sn o t p e c u l i a r t o S i a n d B r .

The s u c c e s s o f t h i s p r o c e d u r e s u g g e s t s t h a t a g e n e r a l e x p r e s s i o n f o r

h e a v y i o n e f f e c t i v e c h a r g e may b e p o s s i b l e . H o w e v e r , some a s s u m p t i o n s

a b o u t t h e f o r m o f t h i s e f f e c t i v e c h a r g e e x p r e s s i o n a r e n e c e s s a r y . P r e v i o u s l y ( B e 6 6 a ) , t h e a v e r a g e c h a r g e s t a t e s o f i o n s e x i t i n g s o l i d a n d

g a s e o u s t a r g e t s , Z x, h a v e b e e n f i t w i t h a s e m i e m p i r i c a l f o r m u l a g i v e n b y

- X v— — = 1 - A e x p -(--------- - ) ( 5 . 3 )1 v Z y0 1

w h e r e v a r i o u s c o m b i n a t i o n s o f A, X a n d % a r e u s e d a s s e a r c hk —p a r a m e t e r s ( s e e C h a p t e r I I ) . We h a v e s u b s t i t u t e d Z x f o r Z x i n t h e

a b o v e e x p r e s s i o n a n d u s e d i t , i n c o n j u n c t i o n w i t h t h e v a r i o u s h i g h e r o r d e r s t o p p i n g p o w e r c o r r e c t i o n s , i n an a t t e m p t t o p r o v i d e a f i t t o o u r

h e a v y i o n e n e r g y l o s s m e a s u r e m e n t s . A T h o m a s - F e r m i s t a t i s t i c a l

d e s c r i p t i o n o f t h e t a r g e t a to m s u g g e s t s y = 2 / 3 ( B o 4 8 , B o 5 4 ) , a n d we h a v e a d o p t e d t h i s v a l u e i n o u r f i t s , l e a v i n g A a n d X a s f r e e p a r a m e t e r s .

( O t h e r c o m b i n a t i o n s o f A, X a n d y a s f r e e p a r a m e t e r s h a v e a l s o b e e n

t r i e d , b u t w i t h l e s s s u c c e s s . ) The t a r g e t d e p e n d e n c e o f t h e e f f e c t i v e

c h a r g e i m p l i e s t h a t a s e p a r a t e f i t i s n e c e s s a r y f o r e a c h t a r g e t

m a t e r i a l . H o w e v e r , due t o t h e c o n s i s t e n t b e h a v i o r o f a l l o u r h e a v y i o n s

i n a g i v e n t a r g e t , a s i l l u s t r a t e d i n F i g s . 5 . 1 - 5 . 1 0 , we h a v e a t t e m p t e d

t o f i t o u r e n e r g y l o s s m e a s u r e m e n t s i n e a c h t a r g e t , r a n g i n g f ro m C i o n s

1 17

t o I i o n s , w i t h o n e s e t o f v a l u e s f o r A a n d X. T h r e e d i f f e r e n t f o r m s

f o r t h e h i g h e r o r d e r c o r r e c t i o n s h a v e b e e n e x a m i n e d , i n c l u d i n g 1)

L i n d h a r d ( L i 7 6 ) , 2 ) A n d e r s e n , e t a l . ( A n 7 7 a ) , a n d 3 ) A s h l e y , R i t c h i e

a n d B r a n d t ( A s 7 2 , R i 7 8 ) , w h e r e t h e c o r r e c t i o n s o f A s h l e y e t a l . h a v e on e

a d d i t i o n a l a d j u s t a b l e p a r a m e t e r . ( B o t h t h e L i n d h a r d a n d A s h l e y , e t . a l .

e x p r e s s i o n s u t i l i z e t h e B l o c h ( B 1 3 3 a ) c o r r e c t i o n . ) Thu s we h a v e u s e d a t

m o s t t h r e e p a r a m e t e r s f o r e a c h t a r g e t , two f o r t h e e f f e c t i v e c h a r g e

e x p r e s s i o n a n d on e f o r t h e h i g h e r o r d e r c o r r e c t i o n s o f A s h l e y , e t a l . ,t o f i t a l l o u r h e a v y i o n e n e r g y l o s s m e a s u r e m e n t s o v e r a w i d e r a n g e o f

e n e r g i e s ( 0 . 5 t o 4 . 0 MeV/amu) a n d p r o j e c t i l e a t o m i c n u m b e r s ( Z x = 6 t o

5 3 ) .

118

1 19

The s t o p p i n g p o w e r c o r r e c t i o n s t h a t p r o v i d e t h e b e s t f i t t o o u r d a t a

a r e t h o s e o f L i n d h a r d . T h e s e h i g h e r o r d e r t e r m s , i n c o n j u n c t i o n w i t h

t h e two p a r a m e t e r e f f e c t i v e c h a r g e e x p r e s s i o n d i s c u s s e d a b o v e , a l l o w

a c c u r a t e f i t s t o a l l o u r d a t a i n A l , Cu, Ag a n d Au t a r g e t s . T h i s i s

c o n s i s t e n t w i t h t h e r e s u l t s o f o t h e r w o r k e r s u s i n g v e r y l i g h t

p r o j e c t i l e s , who f i n d t h a t t h e L i n d h a r d c o r r e c t i o n s p r o v i d e g o o d f i t s t o

t h e i r d a t a . A n d e r s e n , e t . a l . (An 7 7 ) u s e d t h e i r m e a s u r e m e n t s f o r p ,

a a n d L i p r o j e c t i l e s i n A l , Cu, Ag a n d Au t o s e p a r a t e o u t Z a3 a n d Z x4 e f f e c t s c o n s i s t e n t w i t h t h e c o r r e c t i o n s o f L i n d h a r d a n d B l o c h , w h i l e t h e

r e s u l t s o f Heckman a n d L i n d s t r o m ( H e 6 9 ) , on t h e s t o p p i n g p o w er

d i f f e r e n c e b e t w e e n JI+ a n d II p a r t i c l e s , a r e w e l l d e s c r i b e d b y t h e L i n d h a r d r e s u l t s .

I n F i g s . 5 . 1 5 - 5 . 1 8 we h a v e c a l c u l a t e d p r o j e c t i l e e f f e c t i v e c h a r g e s f o r a l l o u r i o n s , u s i n g t h e s t o p p i n g p o w e r t e r m s o f L i n d h a r d , a n d

2 / 3p l o t t e d th em v s . t h e r e d u c e d v e l o c i t y , g i v e n b y v ^ = v / v j Z j

A l s o shown i s t h e two p a r a m e t e r e f f e c t i v e c h a r g e e x p r e s s i o n f o r e a c hkt a r g e t , a n d t h e s e f i t s a r e s e e n t o r e p r o d u c e t h e e x t r a c t e d Z x v a l u e s

o v e r a b r o a d r a n g e . Thus t h e u s e o f t h e h i g h e r o r d e r c o r r e c t i o n s n o t

o n l y a l l o w s u s t o r e p r o d u c e t h e t a r g e t d e p e n d e n c e o f a v e r a g e e q u i l i b r i u m*c h a r g e s t a t e m e a s u r e m e n t s , b u t i t a l s o p r o v i d e s Z x v a l u e s w h i c h c a n

b e d e s c r i b e d o v e r a b r o a d r a n g e o f e n e r g i e s a n d p r o j e c t i l e a t o m i c

n u m b e r s b y a s i m p l e two p a r a m e t e r e x p r e s s i o n f o r e a c h t a r g e t m a t e r i a l .

B . R e s u l t s

The s u c c e s s o f o u r f i t s c a n a l s o b e c o m p a r e d d i r e c t l y w i t h t h e

t a b u l a t i o n s o f Z i e g l e r a n d NS. U s i n g t h e v a l u e s f o r A a n d X a s

d e t e r m i n e d b y o u r f i t t i n g t e c h n i q u e s , we c a n g e n e r a t e a n e f f e c t i v e

c h a r g e e x p r e s s i o n w h i c h , when c o u p l e d w i t h t h e L i n d h a r d h i g h e r o r d e r

c o r r e c t i o n s , a l l o w s u s t o p r e d i c t t h e e n e r g y l o s s f o r a l l o u r h e a v y i o n

m e a s u r e m e n t s . To i l l u s t r a t e t h i s , we h a v e t a k e n t h e r a t i o o f o u r

e x p e r i m e n t a l m e a s u r e m e n t s t o t h e p r e d i c t i o n s o f o u r two p a r a m e t e r f i t ,

a n d p l o t t e d t h e s e v s . t h e i o n e n e r g y ( F i g s . 5 . 1 9 - 5 . 2 2 ) (Once a g a i n , we

go no l o w e r i n e n e r g y t h a n t h e m e a s u r e m e n t s o f A n d e r s e n , e t . a l . ) . T h i s

t e c h n i q u e p r o d u c e s s u b s t a n t i a l l y b e t t e r r e s u l t s t h a n t h e s t a n d a r d

t a b u l a t i o n s , a n d c a n r e p r o d u c e e s s e n t i a l l y a l l o u r d a t a a t t h e 5% l e v e l .

The s y s t e m a t i c d i s c r e p a n c i e s b e t w e e n p r e d i c t i o n a n d e x p e r i m e n t , a s shown

i n F i g s . 5 . 1 - 5 . 1 0 , h a v e b e e n e s s e n t i a l l y e l i m i n a t e d . F u r t h e r r e f i n e m e n t s o f t h e s e s t o p p i n g p o w er c o r r e c t i o n s may p r o d u c e e v e n b e t t e r a g r e e m e n t .

A l t h o u g h t h e r e i s no s t r o n g i n i t i a l m o t i v a t i o n f o r t h e v a l u e s o f t h e

two p a r a m e t e r s A a n d X i n t h e e f f e c t i v e c h a r g e p a r a m e t e r i z a t i o n u s e dh e r e , t h e a b i l i t y o f t h i s e x p r e s s i o n t o f i t o u r d a t a s u g g e s t s t h a t t h e r e

may b e some p h y s i c a l j u s t i f i c a t i o n f o r t h em . F i g . 5 . 2 3 shows t h e v a l u e s o f t h e s e p a r a m e t e r s a s d e t e r m i n e d b y o u r f i t s , p l o t t e d v s . t h e a t o m i c

num be r o f t h e t a r g e t , w h i l e T a b l e 5 . 1 l i s t s t h e s e p a r a m e t e r s d i r e c t l y .

The c u r v e s i n F i g . 5 . 2 3 a r e g i v e n by

- 3 - 5 2A = 1.16 - 1.91x10" Z 2 + 1 .26x10 Z g ( 5 . 4 )

a n d

120

T h e s e e x p r e s s i o n s a l l o w i n t e r p o l a t i o n b e t w e e n o u r m e a s u r e d v a l u e s , a n d

t h e r e s u l t a n t e f f e c t i v e c h a r g e e x p r e s s i o n c a n t h e n b e c o u p l e d w i t h t h e

L i n d h a r d c o r r e c t i o n s t o a c c u r a t e l y p r e d i c t e n e r g y l o s s c u r v e s f o r

p r o j e c t i l e - t a r g e t c o m b i n a t i o n s n o t s p e c i f i c a l l y m e a s u r e d i n t h i s w o r k .

The v a l u e s o f A a n d X shown h e r e d e m o n s t r a t e t h a t t h e e f f e c t i v e

c h a r g e e x p r e s s i o n h a s a s t r o n g t a r g e t d e p e n d e n c e f o r a l l o u r m e a s u r e d

v e l o c i t i e s , c o n s i s t e n t w i t h c h a r g e s t a t e m e a s u r e m e n t s o f h e a v y i o n s a f t e r e x i t i n g m a t e r i a l s . F u r t h e r m o r e , t h e m a g n i t u d e o f t h e e f f e c t i v e

c h a r g e s c a l c u l a t e d u s i n g t h i s e x p r e s s i o n a r e v e r y c l o s e t o a v e r a g e

e q u i l i b r i u m c h a r g e s t a t e m e a s u r e m e n t s i n g a s e s ( B e 7 2 ) , s u p p o r t i n g t h e i d e a t h a t c h a r g e s t a t e s i n s i d e s o l i d s a n d g a s e s a r e a p p r o x i m a t e l y t h e

s am e , b u t t h a t n e a r s u r f a c e e f f e c t s , s u c h a s t h e e m i s s i o n o f A u g e r

e l e c t r o n s b y p r o j e c t i l e s a f t e r l e a v i n g s o l i d s u r f a c e s , r e s u l t i n h i g h v a l u e s o f i o n i z a t i o n ( s e e S e c . 2 . C ) . I n d e p e n d e n t c a l c u l a t i o n s on

e l e c t r o n c a p t u r e a n d l o s s c r o s s - s e c t i o n s b y B e t z , e t a l . ( B e76 ) a l s o

s u p p o r t t h i s i d e a . T hu s a c o m p a r i s o n o f o u r e f f e c t i v e c h a r g e v a l u e s c a l c u l a t e d i n s i d e t h e s o l i d w i t h a v e r a g e e q u i l i b r i u m c h a r g e s t a t e

m e a s u r e m e n t s o u t s i d e may g i v e a m e a s u r e o f t h e nu m ber o f A u g e r e l e c t r o n s

e m i t t e d b y t h e p r o j e c t i l e n e a r t h e s o l i d s u r f a c e . B a s e d on c u r r e n t d a t a f o r a v e r a g e h e a v y i o n c h a r g e s t a t e s ( B e 6 6 b , D a 7 1 ) , t h e nu m ber o f

e l e c t r o n s s h o u l d v a r y o v e r a b r o a d r a n g e , f r o m a r o u n d two f o r 10 MeV

S u l f u r i n G o l d , f o r e x a m p l e , t o a b o u t t e n f o r 180 MeV I o d i n e i n Aluminum. U n f o r t u n a t e l y t h e r e i s no e x p e r i m e n t a l d a t a c u r r e n t l y

a v a i l a b l e on t h e n u m ber o f s e c o n d a r y e l e c t r o n s e m i t t e d b y f a s t h e a v y

121

X = 1 .1 8 - 7 . 5 x l o " 3 Z 2 + 4 . 5 3 x 10_ 5 Z 2 2 ( 5 . 5 )

i o n s a f t e r p e n e t r a t i n g s o l i d f o i l s ; p e r h a p s t h i s w o rk w i l l h e l p

s t i m u l a t e s u c h e x p e r i m e n t s .

122

1 2 3

T a b l e 5 . 1 V a l u e s o f t h e p a r a m e t e r s A, X u s e d i n o u r e f f e c t i v e c h a r g e e x p r e s s i o n .

1 2 4

E f f e c t i v e C h a r g e P a r a m e t e r s

T a r g e t A t o m ic Number

13 1 . 1 3 9 1 . 0 9 9

29 1 . 1 1 7 0 . 9 8 4

47 1 . 1 0 0 . 9 4 2

79 1 . 0 9 0 . 8 6 9

1 2 5

F i g . 5 . 1 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n C

t a r g e t s t o t h e p r e d i c t i o n s o f N o r t h c l i f f e a n d S c h i l l i n g , v s . e n e r g y .

dE/dx

(ex

p)/dE

/dx (

NS)

1.25

1.20

1.15

1.10

1.05

1.00

0.95

0.90

0.85

0 .80

0.75

1 rC TARGET

■ BAA O ♦A

*

.o* 2 *, *> & * 'X A a" X A X A o: * *

f A■ OA o O

OA

AOXO▲

cSiClTiFeNi

♦ Ge Z Br □ Nb■ I

J____ I0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00

ENERGY (MeV/amu)

1 2 7

F i g . 5 . 2 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n C

t a r g e t s t o t h e p r e d i c t i o n s o f Z i e g l e r , v s . e n e r g y .

dE /d

x (ex

p) /dE

/dx

(Zieg

ler)

1.25

1.20

1.15

1.10

1.05

1.00

0.95

0.90

0.85

0 .8 0

0.75 -----------------------------0 0.40 0.80 1.20 1.60 2 .00 2.40 2.80 3.20 3.60 4.00

ENERGY (M eV/am u)

■<♦

d .0 O■■

1

o — o

• CA SiO ClX TiO Fe▲ Ni♦ Gez Br□ Nb■ I

12 9

F i g . 5 . 3 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n A l

t a r g e t s t o t h e p r e d i c t i o n s o f N o r t h c l i f f e a n d S c h i l l i n g , v s . e n e r g y .

dE/dx

(ex

p)/d

E/dx

(NS

)1.20

1.15

1.10

1.05

1.00

0.95

0.90

0.85

1.25

0.75

1— A1

1TARGCTAl

At 1

*J k - AZ$ . j o

OA01

0 $s * t□

o ° <

Ao cdA

(* .1 ■ 1►• •« «

o•

O •■ I S 2® " A

■-la4 "a■ a

t A A * • • A • A ••

•■ A

' A • CA Si O Clx TI O Fe . A Ni♦ Ge z Br □ Nb■ I1

0 0.40 0.80 1.20 1.60 2 0 0 2.40 2.80 3.20 3.60 4.00ENERGY (MeV/amu)

1 31

F i g . 5 . 4 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Al

t a r g e t s t o t h e p r e d i c t i o n s o f Z i e g l e r , v s . e n e r g y .

dE/dx

(ex

p) /d

E/dx

(Z

iegler

)

1.20

1.15

1.10

1.05

1.00

0 .95

0 .90

1.25

0.75

-----"TAl

-------1TARGIET

aw a*

' - ± Ktk* <•PO 1* <

O* •

x l f □ c*o <

• oA ••O

■ ■

■■■ ■

L i f i AA

A A•

x » V♦ ** • CA S O C x T O F< A N♦ G z B □ N ■ I

i1OA ee -rb

E N E R G Y ( M e V /a m u )

1 3 3

F i g . 5 . 5 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Cu

t a r g e t s t o t h e p r e d i c t i o n s o f N o r t h c l i f f e a n d S c h i l l i n g , v s . e n e r g y .

dE/dx

(e

xp)/d

E/dx

(NS

)

1.25

1.20

1.15

1.10

1.05

1.00

0.95

0.90

0.85

0 .80

0.75

C u 1'ARGE.T «1

> cSi —

) Cl Ti

> Fe — Ni

> Ge Br NbT

2 2Z

£C><A4□_ z 2

O2o

zz —

7 a z ° ca i *■xfi2!

aa . 8 ° ^ 2* ♦- a L °

ZcI

:° 90 o■*■i■■ B> ■■

■ ■ o • •<>• Ao• o o6 OA•

1 1

:•

■■■■■ ■ ■ ■ A •

■O

0 0.40 0.80 1.20 1.60 2.00 240 2.80 3.20 3.60 4.00

ENERGY (M eV/am u)

1 3 5

F i g . 5 . 6 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Cu

t a r g e t s t o t h e p r e d i c t i o n s o f Z i e g l e r , v s . e n e r g y .

d E /dx

(ex

p) /d

E /dx

(Z ieg

ler)

1.25

1.20

1.15

1.10

1.05

1.00

0 .9 5

0 .9 0

0 .8 5

0 .8 0

0 .7 5

i i Cu TARGET

□7 z yz9 O-

o•

Zl y .

L ■■ n $i

<yf A| * *A ^ ^ o

•• •

■Z 1

A OA « '. 4ko0 C

A OA O

* •' •

a r i► cc AO c *OOa

o* r

a Si o Cl x TiO F e a Ni♦ G e z B r □ Nbi1 1

0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00ENERGY (M eV/am u)

1 37

F i g . 5 . 7 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Ag

t a r g e t s t o t h e p r e d i c t i o n s o f N o r t h c l i f f e a n d S c h i l l i n g , v s . e n e r g y .

dE/d

x (e

xp)/d

E/d

x(N

S)1.20

1.15

1.10

1.05

1.00

0 .9 5

0 .9 0

0 .8 5

0 .8 0

0 .75

1.25A A i ^

i iTADCCTT

1

i #h

. ■ i I PI * ! ? * * ! V'0

< 4

AO\ A A .. . 4

Ba0 °□ iB ^ O " * 1 1

“ 0 0*1r x ct * ♦ -

>

' cX 0

• •

■&

A AS A° 3 ooX

«« •

o•° * 0o

■

• Ca Si o Cl

i — o<J

• AA

• 4

■ x Ti O Fe a Ni ♦ Ge z Br□ N b ■ II 1

0 .0 0 .40 0 .8 0 1.20 1.60 2 .0 0 2 .40 2 .8 0 3 .20 3 .60 4 .0 0

ENERGY (MeV/amu)

1 3 9

F i g . 5 . 8 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Ag

t a r g e t s t o t h e p r e d i c t i o n s o f Z i e g l e r , v s . e n e r g y .

dE/d

x (e

xp)/

dE/d

x(Zi

egle

r)

1.25

1.20

1.15

1.10

105

1.00

0 .9 5

0 .9 0

0 .8 5

0 .8 0

0 .7 5 .

O/v u i

A g T A R G E T

* s1 g * -

f> o tz <8A Z X5§> fo o

• CA Si O Cl X Ti O Fe ▲ Ni♦ G e Z Br □ Nb ■ I

8 ooAA

0 .4 0 0 .8 0 1.20 1.60 2 .0 0 2 .40 2 .8 0 3 .2 0 3 .60 4 .0 0

E N E R G Y ( M e V / a m u )

141

F i g . 5 . 9 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Au

t a r g e t s t o t h e p r e d i c t i o n s o f N o r t h c l i f f e a n d S c h i l l i n g , v s . e n e r g y .

dE/dx

(ex

p)/dE

/dx (

NS)

1.25

1.20

1.15

1.10

1.05

1.00

0.95

0 .90

0 .85

0 .8 0

0.75

................ . 1 1- *T'♦

1 1 Au TARGET

w a r #

■■ * » x * & . * . A o O1'V % 8 » &* z fi b

A - 0 o

k A£ <* 4 y,A

0"■ - ■ ,-ga o o □ . o• V . *

O 0* q <

o o «

•• •• o C l4+ • • A

•A CSiA

Vy 1X Ti O Fe▲ Ni ♦ Ge z Br

•

□■NbI0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00

ENERGY (M eV/am u)

1 4 3

F i g . 5 . 1 0 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Au

t a r g e t s t o t h e p r e d i c t i o n s o f Z i e g l e r , v s . e n e r g y .

dE /dx

(e

xp)

/dE

/dx

(Zie

gler

)

1 .2 5

1 .2 0

1.15

1 .1 0

1.05

1 .0 0

0 .9 5

0 .9 0

0 .8 5

0 .8 0

0 .7 5

1 1 A u T A R G E T

a• ■

■

s f e n4 a •

•O•

* DX °

DZAq AO f O

♦ T .% a I O a ft

• "•

A ^ o '2 4 7 Z A

o ® oOA o A O

• CA S i

Z& a '

A

<O Cl X T i 0 F e A N i ♦ G e Z B r □ N b ■ I

•

0 .4 0 0 .8 0 1.20 1.60 2 .0 0 2 .4 0 2 .8 0 3 .2 0 3 .60 4 .0 0E N E R G Y ( M e V / a m u )

14 5

F i g . 5 . 1 1 V a l u e s o f t h e e f f e c t i v e c h a r g e o f S i i o n s ( d i v i d e d b y t h e

a t o m i c n u m b e r o f S i ) , c a l c u l a t e d f r o m e x p e r i m e n t a l d E / d x m e a s u r e m e n t s b y

a s s u m i n g a Z x z s t o p p i n g p o w e r d e p e n d e n c e , v s . v e l o c i t y .

146

0.85

0.82

0.79

0.76

0.73

r T 0 .70 \* - 0.67 M

0 .64

0.61

0 .58

0.55

1 1 1 1 1a: « —!—a:i— /--

1 1 ■4 1wi ■ rujvwiiiCd \i v l <

• Al'

O Ull ' ■a Ag a Au #

i *

4. ! ♦

*A•

$ -- 4 i -

¥

1 *

i i i i2.00 14.60 17.20 19.80 22.40 25.00

V elocity (10® c m /s e c )

14 7

F i g . 5 . 1 2 V a l u e s o f t h e e f f e c t i v e c h a r g e o f B r i o n s ( d i v i d e d b y t h e

a t o m i c n u m b e r o f B r ) , c a l c u l a t e d f r o m e x p e r i m e n t a l d E / d x m e a s u r e m e n t s b y

a s s u m i n g a Z j 2 s t o p p i n g p o w e r d e p e n d e n c e , v s . v e l o c i t y .

148

0.60

0.57

0.54

0.51

0 .48

N 0 4 5* - 0 4 2 N

0.39

0 .36

0 .33

0 .3 0

1 | 1 " T I Br Projectiles (no Z ? )

1 " 1 , j.

• A__ o C

A Aa A

1u

t igu

* ♦ *A A

*T 4

Jt* 4&

%

i 1 1 1 _ 113.00 14.40 15.80 17.20 18.60

V e lo c ity (IO8 c m /s e c )2 000

1 4 9

F i g . 5 . 1 3 V a l u e s o f t h e e f f e c t i v e c h a r g e o f S i i o n s ( d i v i d e d b y t h e

a t o m i c n u m b e r o f S i ) , c a l c u l a t e d f r o m e x p e r i m e n t a l d E / d x m e a s u r e m e n t s b y

i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s o f L i n d h a r d , v s . v e l o c i t y .

150

0.82

0.76

0.73

M -0 7 0 \

★ — 0.67 NQ 64

Q6I0.58

0 .55

1 1 1 1 1 1 1 ♦i i

• Al “ o Cu ▲ Ag

CV.IIIC9 ' |I \i

** * ta Aui * *

tt 9|

T1

* ’ ♦i

♦ * H*

T 1

♦. a*

i 1 1 • i1200 14.60 17.20 19.80 2 2 .4 0

Velocity (I08 cm/sec)25 .00

1 51

F i g . 5 . 1 4 V a l u e s o f t h e e f f e c t i v e c h a r g e o f B r i o n s ( d i v i d e d b y t h e

a t o m i c number o f B r ) , c a l c u l a t e d f ro m e x p e r i m e n t a l d E / d x m e a s u r e m e n t s by

i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s o f L i n d h a r d , v s . v e l o c i t y .

152

0 .60

0.57

“I 1-------1------- I------- 1-----Br Projectiles (Z? included)

0.54

0.51

0 .48

0.45

t 7 0 - 4 20 .39

0 .3 6

0 .3 3

0 .3 0

• Al o Cu a Ag a Au

001ft-

1 1

*o

ft

1

t V

$ 4

13.00 14.40 15.80 17.20Velocity (IO 8 cm /se c )

I860 20 .00

1 5 3

F i g . 5 . 1 5 V a l u e s o f t h e e f f e c t i v e c h a r g e o f a l l o u r h e a v y i o n s ( d i v i d e d

b y t h e i o n a t o m i c n u m b e r , Z x ) i n A l , c a l c u l a t e d f ro m e x p e r i m e n t a l d E / d x

m e a s u r e m e n t s b y i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s o f L i n d h a r d , v s .

2 / 3t h e r e d u c e d v e l o c i t y v/viZ l . A l s o shown i s o u r two p a r a m e t e r

e f f e c t i v e c h a r g e f i t f o r A l .

154

1 5 5

F i g . 5 . 1 6 V a l u e s o f t h e e f f e c t i v e c h a r g e o f a l l o u r h e a v y i o n s ( d i v i d e d

b y t h e i o n a t o m i c n u m b e r , Z : ) i n Cu, c a l c u l a t e d f ro m e x p e r i m e n t a l d E / d x

m e a s u r e m e n t s b y i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s o f L i n d h a r d , v s .

2 / 3t h e r e d u c e d v e l o c i t y v / V qZ j . A l s o shown i s o u r two p a r a m e t e r

e f f e c t i v e c h a r g e f i t f o r Cu.

156

1 57

F i g . 5 . 1 7 V a l u e s o f t h e e f f e c t i v e c h a r g e o f a l l o u r h e a v y i o n s ( d i v i d e d

by t h e i o n a t o m i c n u m b e r , Z x) i n Ag, c a l c u l a t e d f ro m e x p e r i m e n t a l d E / d x

m e a s u r e m e n t s by i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s o f L i n d h a r d , v s .

2 / 3t h e r e d u c e d v e l o c i t y v / v 0Z 1 . A l s o shown i s o u r two p a r a m e t e r e f f e c t i v e c h a r g e f i t f o r Ag.

158

1 5 9

F i g . 5 . 1 8 V a l u e s o f t h e e f f e c t i v e c h a r g e o f a l l o u r h e a v y i o n s ( d i v i d e d

b y t h e i o n a t o m i c n u m b e r , Z j ) i n Au, c a l c u l a t e d f r o m e x p e r i m e n t a l d E / d x

m e a s u r e m e n t s b y i n c l u d i n g t h e h i g h e r o r d e r c o r r e c t i o n s o f L i n d h a r d , v s .

2 / 3t h e r e d u c e d v e l o c i t y v / v 0Z : . A l s o shown i s o u r two p a r a m e t e re f f e c t i v e c h a r g e f i t f o r Au.

zf

/z

160

1 6 1

F i g . 5 . 1 9 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n A l

t a r g e t s t o t h e p r e d i c t i o n s o f t h e p r e s e n t s t u d y .

dE/dx

(exp

) /dE

/dx

(th)

1.20

1.15

1.10

1.05

1.00

0.95

0.90

0.85

0.80

0.75

1.25 ...... “ T--------Al TARGE T

■. ■ z2A

■A• i

. V-IDX ft*<£* V <& o

i&jfin A n m 91©o* l±- o -

•••

♦ cAA©© A

AO •

A "• CA Si O Cl x Ti0 Fe ▲ Ni♦ Ge z Br □ Nb ■ I1

A

10.40 0.80 1.20 1.60 2 .00 2.40 2 .80 3.20 3.60 4 .00

E N E R G Y (M e V /a m u )

1 6 3

F i g . 5 . 2 0 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Cu

t a r g e t s t o t h e p r e d i c t i o n s o f t h e p r e s e n t s t u d y .

dE/dx

(exp

) /dE

/dx

(th)

1.20

1.15

1.10

1.05

1.00

0.95

1.25

0.85

0.80

0.75

— r_ _ n Cu TARGET

□°N*1 Z Z2 •

<f a b *

* •0• Of* oo <XA

•* o'▲ A ° 0 6

•b •

0 1 •• •AO

O •A ^ A V 4 A° 0 r ° a o

•AO

C A Ni Si ♦ Ge Cl z BrX

O11 Fe

□ ND■ I

0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00ENERGY (M eV /am u)

1 6 5

F i g . 5 . 2 1 R a t i o o f a l l o u r e x p e r i m e n t a l e n e r g y l o s s m e a s u r e m e n t s i n Ag

t a r g e t s t o t h e p r e d i c t i o n s o f t h e p r e s e n t s t u d y .

dE/d

x (e

xp)

/dE

/dx(

th)

1.25

1.20

1.15

1.10

1.05

1.00

0 .95

0 .9 0

0 .8 5

0 .8 0

0 .75

A g ‘rARGE:t

■■ ■4 4 Hi• •

Pa

•A

•. O • •

Z 2t t

• a oz D l " * '

O ----1

ftS

• C.° o Z Z a AA

- o A

a S i o Cl

— x T i O F e a N i♦ G e □ N b z B r , ■ I

0 .40 0 .8 0 1.20 1.60 2 .0 0 2 .40 2 .8 0 3 .2 0 3 .60 4 .0 0

E N E R G Y ( M e V / a m u )

167

Fig. 5.22 Ratio of a l l our experimental energy loss measurements in Au

targets to the predictions of the present study.

dE

/d

x (e

xp)/

dE

/d

x(t

h)

1.25

1.20

1 . 1 5

1.10

1 . 0 5

1.00

0 . 9 5

0 . 9 0

0 . 8 5

■T..........1

Au T A R G E T

4■ O

J

&L □ ^ f * ° J

• 9 •

*••

••

•A

CC2

* r£ °

ko * A ©• OA O

OX

_________ OA♦

. Z

ClTiFe -

■ o oa

* 2 i

A

A

NiGeB r

■ □ Nb

I•

0 . 7 50 . 4 0 0 . 8 0 1 . 2 0 1 . 6 0 2 . 0 0 2 . 4 0 2 . 8 0 3 . 2 0 3 . 6 0 4 . 0 0

ENERGY (MeV/amu)

169

Fig. 5.23 Values of the two parameters A and X in our e f fe c t iv e

charge expression, vs. ta rg et atomic number. The so lid curves represent

a quadratic f i t in Z2 (See Sec 5 .B ). The uncerta in ties in these

parameters are ~2-3%.

Z2

CHAPTER V I

CONCLUSION

In th is work, we have presented the resu lts o f energy loss

measurements fo r heavy ions in both th ick and th in targ ets , over a broad

range of p r o je c t i le , ta rg e t , and energy values, in an e f f o r t to

understand the energy loss of heavy ions in matter. The th ick target

measurements involve S i, N i, and Au ions in th ick (5 to 10 mg/cm2) Cu,

Ag, and Pb targets at energies o f E<2.5 HeV/amu. The th in target data

give d ire c t dE/dx values fo r C, S i, C l, T i , Fe, N i, Ge, Br, Nb, and I

ions in C, A l, Cu, Ag, and Au targets at energies E < 3 .5 MeV/amu.

Our resu lts fo r both th ick and th in targets can not be explained

using current stopping power and range tabu la tions . The low v e lo c ity

data are not w e ll described by the ca lcu la tions of N o r th c l i f fe and

S c h i l l in g (No70) or Lindhard, Scharff and Schiott (L i6 3 ) , but can be

f a i r l y w e ll reproduced by the ca lcu la tions of Z ieg le r (Z i8 0 ) , which

include the v e lo c ity dependence of the p r o je c t i le charge at low

v e lo c i t ie s . This produces a nonlinear stopping power vs. v e lo c ity

re la t io n s h ip . At high v e lo c i t ie s the tabulations of Z ieg le r and

N o r th c l i f fe and S c h i l l in g , which assume that the stopping power is

proportional to the square of the p r o je c t i le charge, do poorly in

p red ic tin g both the magnitude and the energy dependence of the stopping

power peak. However, recent range and dE/dx measurements show that the

addition of higher order charge dependent corrections to the stopping

power produces much b e tte r agreement w ith experimental data fo r pions

171

and l ig h t ions. Thus we have extended these methods to our heavy ion

data.

We have examined several energy loss formulae, both w ith and withoutI

the higher order charge dependent terms, in conjunction w ith various

e f fe c t iv e charge expressions, in an attempt to f in d some combination

which w i l l describe our re s u lts . The best f i t s to our data are provided

by the terms of Lindhard, which allow heavy ion e f fe c t iv e charges fo r

a l l ions in a given targ et to be described by a simple two parameter

e f fe c t iv e charge expression. Thus through the use o f only two free

parameters we are able to f i t a l l our dE/dx measurements in a given

ta rg e t , which includes ions from C to I ranging in energy from 0.1 - 4 .0

MeV/amu. Use of th is charge param eterization, when coupled w ith the

Lindhard corrections, allows prea ic tio n o f dE/dx values w ith much b e tte r

success than the standard tabula tions , and the systematic discrepancies

between experiment and pred ic tion are la rg e ly removed. The smooth

behavior of the two parameters A and X allows in te rp o la t io n and

extrapo la tion of th is expression to a wide v a r ie ty of target atomic

numbers. These e f fe c t iv e charge values can then be coupled w ith the

Lindhard corrections to provide accurate stopping powers fo r a broad

spectrum of p r o je c t i le , targ et and energy values. The a b i l i t y of the

Lindhard corrections to reproduce our data serves as motivation for

th e o re t ic a l investigations in to even higher order terms (Z j S.Zj6 . . . ) ,

which could become increasingly important as the energy losses of very

highly charged heavy ions are explored.

Our e f fe c t iv e charge expression shows a strong ta rg et dependence

172

which can not be reproduced by a simple Zxz stopping power re la t io n s h ip .

We have shown th at the use o f higher order stopping power terms produces

heavy ion e f fe c t iv e charges in solids which ex h ib it the same ta rg e t

dependence as equilibrium charge state measurements made on heavy ions

e x it in g so lid and gaseous ta rg ets .

This e f fe c t iv e charge expression generates values which agree w e ll

with equilibrium charge s ta te measurements in gases. This suggests that

charge states inside solids and gases (o f approximately the same atomic

number) are almost the same, and that the high charge states of ions

when leaving solids may be due to processes such as loss o f Auger

electrons at the e x i t surface of the s o lid . Comparison of average

equilibrium charge states w ith our e f fe c t iv e charge expression may thus

give a measure of the number o f Auger electrons emitted by the

p ro je c t i le upon leaving the so lid surface. Perhaps these re la tionsh ips

w i l l stimulate fu rth e r experimental investigations which can determine

more d i re c t ly the magnitude of p r o je c t i le charges inside so lids .

173

174

Appendix A

Incident Beam Target Target E x itBeam Energy* M a te r ia l Thickness Energy

(MeV) (mg/cm2)

T h ic k T a rg e t Energy Lo sse s

Si 69.7 Ag 6.92 ± 0.03 17.05 + 0.3157.7 4.34 + 0.1455.7 3.12 + 0.1253.7 2.31 + 0.1151.7 1.62 + 0.09

Si 45.7 Ag 4.24 ± 0.02 10.99 + 0.2043.7 9.01 + 0.1841.6 7.26 + 0.1537.6 4.37 ± 0.12

68.7 Pb 10.25 + 0.04 12.88 + 0.2759.7 5.05 + 0.1557.7 4.01 + 0.1455.7 3.06 + 0.1253.7 2.29 ± 0.1151.7 1.71 + 0.10

69.7 Pb 7.43 + 0.03 30.24 + 0.8053.7 10.48 + 0.2645.7 4.68 + 0.1543.7 3.59 + 0.1341.6 2.83 + 0.1137.6 1.47 + 0.09

Si 69.7 Cu 7.08 ± 0.03 5.37 ± 0.1668.7 4.74 ± 0.1567.7 4.10 ± 0.1465.7 3.24 ± 0.1363.7 2.37 ± 0.11

Si 53.7 Cu 4.88 ± 0.03 8.15 ± 0.1845.7 3.40 ± 0.1243.7 2.63 ± 0.1041.6 1.99 ± 0.09

U n c e r ta in t ie s of 0.1 MeV ( S i ) , 0.25 MeV (N i) and 0.5 MeV (Au) inthe beam energy are due to passage through the C backings on our ta rg e ts .

175

IncidentBeam

Ni

Ni

Ni

Ni

Beam Target Target E x itEnergy M ate r ia l Thickness Energy

(MeV) (mg/cm2)

118.7 Ag 6.92 + 0.03 7.49 + 0108.0 4 .24 ± 0

94.2 2.46 ± 093.6 2.05 + 092.2 1.97 + 090.2 1.83 ± 088.3 1.64 + 087.2 1.41 + 085.2 1.28 + 0

71.5 Ag 4.24 + 0.02 7.80 + 059.0 4.20 ± 052.3 3.14 + 047.6 2.30 + 043.1 1.77 ± 041.2 1.56 ± 039.2 1.44 + 037.2 1.23 + 0

118.8 Ag 8.39 ± 0.04 1.71 + 0118.7 1.71 ± 0116.7 1.49 + 0115.1 1.45 + 0

118.7 Pb 10.25 + 0.04 6.22 + 0108.0 3.96 ± 0

98.0 2.45 + 093.6 1.93 ± 091.2 1.77 + 089.2 1.62 + 087.2 1.47 + 085.2 1.34 + 0

93.6 Pb 7.43 + 0.03 8.33 + 085.2 5.55 + 071.5 3.23 + 065.1 2.20 + 063.2 2.06 + 059.2 1.73 + 059.0 1.64 + 057.2 1.48 + 055.2 1.33 + 053.2 1.20 + 0

.19

.15

.11

.10

.10

.10

.09

.09

.08

.15

.11

.09

.08

.07

.07

.06

.06

.11

.11

.10

.10

.15

.11

.09

.08

.08

.08

.07

.07

.16

.11

.09

.07

.07

.07

.07

.06

.06

.06

176

118.7 Cu 7.08 ± 0.03 3.15 ± 0100.3 1.87 + 0108.3 1.67 + 0108.1 1.71 ± 0106.2 1.51 ± 0104.2 1.34 + 0102.2 1.24 + 0

93.6 Cu 4.88 + 0.03 9.90 + 085.2 6.40 + 077.2 4.36 + 071.5 2.98 + 065.1 2.15 ± 063.2 1.82 + 061.2 1.61 + 059.0 1.50 + 0

18141313121211

2520161311101009

177

IncidentBeam

Au

Au

Au

Au

Au

BeamEnergy

(MeV)

160.2147.6 141.8130.6124.5114.5

200.4184.7179.8165.5

179.8 160.2147.6141.8130.6124.5 114.0

124.5 108.4

99.8 93.285.9 79.4

200.4184.7179.8165.5

124.5 108.4 100.2

93.285.9

Target Target E x itM ate r ia l Thickness Energy

(mg/cm2)

Ag 6.92 ± 0.03

Ag 8.29 ± 0.04

17.08 + 2.7613.99 ± 2.4112.55 ± 2.2910.25 ± 2.07

9.10 ± 1.956.78 ± 1.73

13.07 ± 2.7810.70 ± 2.489.53 ± 2.378.29 + 2.16

Pb 10.25 ± 0.04

Pb 7.43 ± 0.03

25.10 + 2.3513.67 + 1.9313.18 + 1.7811.90 + 1.6610.38 + 1.56

8.25 + 1.477.15 ± 1.39

25.62 + 1.8917 ..59 ± 1.7212.23 ± 1.5111.16 + 1.45

9.06 + 1.347.98 + 1.29

Cu 7.08 ± 0.03

Cu 4.88 ± 0.03

18.96 + 4.8513.15 + 3.9912.59 ± 3.86

9.90 ± 3.34

23.28 + 3.8315.02 + 3.1011.40 ± 2.7110.10 + 2.53

7.61 ± 2.20

178

Appendix B Heavy Ion E lectronic Stopping Powers*

Inc ident Beam Stopping Power (MeV-cm2/mg)Beam Energy C Al Cu Ag Au

(HeV/amu)

c 1.328 5.09±0.15

4.04±0.09

3.00±0.04

2.29±0.03

1.53±0.03

---- 4.05±0.10

2.96±0.06

2.32±0.03

1.54±0.03

c 1.659 4.73±0.14

3.73±0.08

2.81±0.04

2.12±0.03

1.45±0.03

---- 3.77±0.10

2.74±0.05

2.16±0.03

1.48±0.03

c 2.075 4.27±0.13

3.35±0.07

2.56±0.04

1.97±0.03

1.35±0.03

---- 3.41±0.09

2.48±0.05

2.02±0.03

1.40±0.03

c 2.366 3.91±0.12

3.11±0.07

2.39±0.04

1.87±0.03

1.29±0.03

---- 3.16±0.08

2.33±0.05

1.93±0.03

1.34±0.03

c 2.491 3.80±0.11

3.03±0.06

2.33±0.03

1.83±0.03

1.27±0.03

---- 3.08±0.08

2.28±0.04

1.92±0.03

1.32±0.03

c 2.907 3.47±0.10

2.75±0.06

2.13±0.03

1.72±0.03

1.26±0.03

---- 2.82±0.07

2.09±0.04

1.85±0.03

1.24±0.03

*Data fo r two d i f fe r e n t targets of each m a te r ia l , as w e ll as theresu lts fo r three separate experiments, are tabulated. The nuclearstopping contribution (Zi77b) has been subtracted from the raw data. The l is te d energy is the energy of the beam incident on the target(a f t e r 10° scatte ring from the Au f o i l - - s e e Sec 3B).

179

3.324

4.013

3.19± 0 . 1 0

2.69±0.08

2.53 1.97 1.59 1.19±0.05 ±0.03 ±0.02 ±0.02

2.56 1.93 1.73 1.18±0.07 ±0.04 ±0.03 ±0.02

2.14 1.71 1.41 ----±0.05 ±0.03 ±0.02

2.27 1.67 ---- 1.05±0.06 ±0.03 ±0.02

180

In c id e n t Beam Sto p p in g Power (MeV-cm2/mg)Beam Energy

(MeV/amu)C Al Cu Ag Au

Si 0.698 19.87±0.59

12.78±0.27

9.01±0.13

8.32±0.12

5.28±0.11

---- 12.59±0.32

9.06±0.18

8.15±0.12

5.16±0.11

Si 0.882 19.31±0.57

13.35±0.28

9.59±0.14

8.49±0.12

5.48±0.11

---- 13.20±0.34

9.67±0.19

8.18±0.12

5.28±0.11

Si 1.094 18.87±0.56

13.59±0.29

9.92±0.15

8.43±0.12

5.50±0.11

---- 13.45±0.34

10.07±0.20

8.21±0.12

5.41±0.11

Si 1.238 18.70±0.55

13.61±0.29

9.97±0.15

8.23±0.12

5.48±0.11

---- 13.72±0.35

10.16±0.20

7.94±0.12

5.41±0.11

Si 1.416 18.23±0.54

13.54±0.29

9.96±0.15

8.05±0.12

5.40±0.11

---- 13.14±0.34

10.14±0.20

7.73±0.11

5.34±0.11

Si 1.579 17.72±0.52

13.28±0.28

9.90±0.15

7.82±0.11

5.31±0.11

---- 13.19±0.34

9.98±0.19

7.60±0.11

5.27±0.11

Si 2.152 16.23±0.48

12.29±0.26

9.33±0.14

7.13±0.11

5.00±0.10

---- 12.19±0.31

9.28±0.18

6.93±0.19

5.06±0.10

Si 2.813 14.60±0.43

11.00 ±0.23

8.60±0.13

6.57±0.10

4.71±0.10

---- ---- 8.36±0.16

6.52±0.10

4.71±0.10

181

S i 3.197 13.61±0.41

10.43 8.03 6.25 4.75±0.22 ±0.12 ±0.09 ±0.15

---- 7.83 6.39 4.47±0.15 ±0.09 ±0.11

182

In c id e n t Beam Sto p p in g Power (MeV-cm2/mg)Seam Energy

(MeV/amu)C Al Cu Ag Au

Cl 0.701 23.85±0.57

16.49±0.32

11.20±0.17

10.27±0.18

6.38±0.11

---- ---- 11.11±0.19

9.61±0.16

6.29±0.12

Cl 0.900 23.52±0.57

17.20±0.33

12.02±0.18

10.75±0.19

6.77±0.12

---- ---- 11.90±0.20

10.11±0.15

6.67±0.13

Cl 1.128 23.08±0.56

17.87±0.34

12.59±0.19

10.71±0.19

6.94±0.12

---- ---- 12.43±0.21

10.61±0.16

6.87±0.13

Cl 1.270 22.88±0.55

17.83±0.34

12.78±0.19

10.84±0.19

6.96±0.12

---- 18.40±0.33

12.58±0.22

10.58±0.16

6.95±0.13

Cl 1.412 22.56±0.54

18.57±0.36

12.86±0.19

10.69±0.19

6.96±0.12

---- 18.38±0.33

12.63±0.22

10.45±0.17

6.98±0.13

Cl 1.555 22.28±0.54

17.91±0.35

12.89±0.19

10.57±0.19

6.94±0.12

---- 18.46±0.33

12.61±0.22

10.39±0.15

7.11±0.13

Cl 1.697 21.90±0.53

17.88 , ±0.35

12.83±0.19

10.39±0.19

6.92±0.12

---- 18.03±0.32

12.52±0.21

10.27±0.15

6.95±0.13

Cl 1.982 21.27±0.51

17.70±0.34

12.64±0.19

9.99±0.18

6.81±0.12

---- 17.71±0.31

12.22±0.21

10.01±0.15

6.89±0.13

183

C l 2.266

Cl 2.551

Cl 2.836

Cl 3.120

Cl 3.405

20.54±0.50

20.00±0.49

19.28±0.47

18.60±0.45

17.84±0.44

16.74 12.36 9.70 6.72±0.33 ±0.18 ±0.17 ±0.12

16.97 11.85 9.80 6.91±0.30 ±0.20 ±0.15 ±0.13

------------ 12.00 9.42 6.57±0.18 ±0.17 ±0.12

16.40 11.43 9.55 6.75±0.29 ±0.20 ±0.14 ±0.13

------------ 11.59 9.17 6.42±0.17 ±0.17 ±0.12

15.70 11.03 9.22 6.62±0.28 ±0.19 ±0.14 ±0.13

. - _____ 11.15 8.95 6.32±0.17 ±0.16 ±0.11

14.82 10.63 8.86 6.39±0.27 ±0.18 ±0.13 ±0.12

------------ 10.68 ------------ 6.17±0.16 ±0.11

14.04 10.18 8.47 ------------

±0.25 ±0.17 ±0.23

184

IncidentBeam

Ti

T i

T i

T i

T i

T i

T ii

Beam Sto p p in g Power (MeV-cm2/mg)Energy C Al

(MeV/amu)

0.406 29.03±0.70

17.01±0.33

28.66±0.91

----

0.509 31.18±0.75

18.57±0.36

30.68±0.97

18.19±0.32

0.612 32.09±0.77

20.12±0.39

31.83±1.01

19.67±0.36

0.818 32.65±0.79

22.27±0.43

32.16±1.01

21.58±0.38

1.025 32.45±0.78

23.65±0.46

32.00±1.01

23.07±0.41

1.232 32.11±0.77

24.53±0.47

31.65±1.00

24.00±0.43

1.439 31.72±0.77

24.98±0.48

30.99±0.98

24.52±0.44

1.543 31.59±0.76

25.13±0.48

31.11±1.15

24.56±0.44

Cu Ag Au

10.71 10.03 6.09±0.16 ±0.18 ±0.11

10.75 9.52 6.26±0.19 ±0.14 ±0.12

11.96 11.22 7.19±0.18 ±0.20 ±0.13

12.01 10.70 7.09±0.21 ±0.16 ±0.14

13.04 12.18 7.82±0.20 ±0.22 ±0.14

13.14 11.66 7.75±0.23 ±0.17 ±0.15

14.73 13.34 8.70±0.22 ±0.24 ±0.16

14.66 12.96 8.63±0.25 ±0.19 ±0.17

15.82 13.82 9.13±0.24 ±0.25 ±0.16

15.68 13.68 9.15±0.27 ±0.20 ±0.18

16.54 14.23 9.34±0.25 ±0.26 ±0.17

16.41 13.95 9.40±0.28 ±0.21 ±0.18

16.97 14.26 9.42±0.25 ±0.26 ±0.17

16.68 14.00 9.57±0.29 ±0.21 ±0.18

17.16 14.19 9.43±0.26 ±0.26 ±0.17

16.78 18.97 9.62±0.29 ±0.21 ±0.18

T i 1.132 33.23±0.99

23.65±0.46

13.99±0.27

T i 1.753 31.82±0.95

24.63±0.48

Ti 2.168 31.00±1.02

13.50±0.27

185

T i 1.646 31.32±0.76

25.10± 0.4 8

17.24±0.26

14.07±0.25

9.43±0.17

30.76±0.97

24.58±0.44

16.84±0.29

13.98±0.21

9.62±0.19

23.67±0.60

16.33±0.32

14.11±0.31

9.27±0.21

24.03±0.61

14.13±0.31

9.61±0.22

T i 1.964 23.72±0.63

16.78±0.34

14.06±0.32

9.38±0.21

23.41±0.61

16.39±0.34

14.04±0.31

9.59±0.22

186In c id e n t Beam Sto p p in g Power (MeV-cm2/mg)

Seam Energy(MeV/amu)

C Al Cu Ag Au

Fe 0.611 37.51±0.90

22.78±0.44

14.60±0.22

14.37±0.26

9.23±0.16

---- 21.96±0.39

14.55±0.25

13.92±0.21

9.02±0.17

Fe 0.788 38.98±0.94

25.26±0.49

16.43±0.25

15.82±0.28

10.24±0.18

---- 24.38±0.43

16.31±0.28

15.38±0.23

10.09±0.19

Fe 0.876 39.20±0.95

26.21±0.50

17.19±0.26

16.37±0.29

10.66±0.19

---- ---- 16.95±0.29

---- ----

Fe 0.965 39.27±0.95

27.06±0.52

17.81±0.27

16.82±0.30

10.89±0.19

---- 26.25±0.47

17.54±0.30

16.4340.24

10.79±0.21

Fe 1.053 39.49±0.96

27.80±0.54

18.41±0.28

17.25±0.31

11.15±0.20

---- 27.06±0.49

18.12±0.31

16.63±0.25

11.05±0.21

Fe 1.230 39.64±0.96

28.91±0.56

19.28±0.29

17.51±0.32

11.43±0.20

---- 28.24±0.51

18.83±0.34

17.14±0.26

11.34±0.22

Fe 1.407 39.31±0.95

29.81±0.57

19.97±0.30

17.65±0.32

11.59±0.21

---- 28.85±0.51

19.55±0.34

17.36±0.26

11.57±0.22

Fe 1.496 39.33±0.95

30.02±0.58

20.16±0.30

17.64±0.32

11.61±0.21

---- 29.24±0.52

19.61±0.34

17.35±0.26

11.61±0.22

187

Fe 1.797

Fe

Fe

2.294

1.411

Fe 1.943

Fe 2.439

Fe 2.652

Fe 2.829

Fe 3.010

38.96 30.42 20.74 17.28 11.65±0.94 ±0.59 ±0.31 ±0.31 ±0.21

---- 29.44 20.07 17.35 11.84±0.53 ±0.35 ±0.26 ±0.23

37.48 ---- 20.94 16.62 11.59±0.91 ±0.31 ±0.30 ±0.21

40.59 29.20 20.31 17.56 _-±1.21 ±0.56 ±0.40 ±0.34

---- 28.31 19.66 17.52 11.65±0.71 ±0.39 ±0.38 ±0.26

40.10 29.84 20.97 16.98 ----±1.19 ±0.58 ±0.41 ±0.33

---- 28.88 19.91 17.50 11.70±0.73 ±0.40 ±0.38 ±0.26

39.24 28.29 20.41 17.36 ----±1.22 ±0.55 ±0.41 ±0.35

---- 27.13 19.51 16.96 11.87• ±0.71 ±0.39 ±0.38 ±0.27

39.86 28.14 19.65 15.98 ----±1.52 ±0.56 ±0.39 ±1.02

---- 26.20 19.21 17.16 11.69±0.72 ±0.40 ±0.41 ±0.29

37.87 28.18 19.68 17.08 ----±1.20 ±0.55 ±0.39 ±0.35

— _ _ 26.09 18.84 16.21 11.74±0.73 ±0.38 ±0.36 ±0.27

39.88 27.88 18.84 16.70 ----±1.33 ±1.26 ±0.40 ±0.51

188

In c id e n t Beam Sto p pin g Power (MeV-cm2/mg)Beam Energy

(MeV/amu)C Al Cu Ag Au

Ni 0.590 39.95±0.94

23.73±0.46

15.26±0.23

15.44±0.28

9.66±0.17

----- 23.19±0.41

15.28±0.26

14.72±0.22

9.56±0.18

Ni 0.709 40.41±0.97

25.62±0.49

16.72±0.25

16.65±0.30

10.54±0.19

---- 24.89±0.44

16.81±0.29

16.02±0.24

10.49±0.20

Ni 0.845 41.41±1.00

27.52±0.53

18.09±0.27

17.56±0.32

11.24±0.20

---- 26.56±0.47

18.05±0.31

17.08±0.25

11.18±0.21

Ni 0.982 42.14±1.01

29.00±0.56

19.22±0.29

18.18±0.33

11.77±0.21

----- 28.12±0.50

19.03±0.33

17.86±0.26

11.75±0.22

Ni 1.118 42.51±1.02

30.21±0.58

20.14±0.30

18.65±0.33

12.11±0.22

----- 29.31±0.52

19.87±0.34

18.40±0.27

12.18±0.23

Ni 1.255 42.76±1.03

31.17±0.60

20.87±0.31

19.02±0.34

12.36±0.22

---- 30.12±0.53

20.52±0.35

18.72±0.27

12.41±0.24

Ni 1.358 42.84±1.03

31.67±0.61

21.35±0.32

19.43±0.35

12.49±0.22

---- 30.77±0.55

20.87±0.36

18.89±0.28

12.55±0.24

Ni 1.443 42.82±1.03

32.03±0.62

21.65±0.32

19.36±0.35

12.54±0.22

----- 31.03±0.55

21.20±0.36

18.84±0.28

12.66±0.24

189

Ni 1.700 42.75±1.03

32.66±0.63

22.30±0.33

19.08±0.34

12.68±0.23

----- 31.73±0.57

21.57±0.37

18.95±0.28

12.83±0.25

Ni 2.042 42.21±1.02

---- 22.66±0.34

18.65±0.34

12.56±0.23

----- 32.00±0.58

21.60±0.37

18.86±0.28

13.07±0.25

Ni 2.470 41.50±1.02

---- 22.42±0.34

18.09±0.33

12.64±0.23

---- 30.89±0.56

21.75±0.38

18.61±0.28

12.98±0.25

Ni 1.533 44.01±1.31

31.50±0.61

21.94±0.43

18.61±0.36

----

---- 30.62±0.77

21.58±0.43

19.19±0.42

12.72±0.28

Ni 1.789 43.81±1.30

32.05±0.62

22.43±0.44

18.70±0.37

----

---- 31.22±0.79

21.55±0.43

19.17±0.42

12.80±0.28

Ni 2.131 43.61±1.34

31.67±0.61

22.57±0.45

19.20±0.38

----

---- 31.00±0.81

21.70±0.44

18.96±0.42

12.64±0.29

Ni 2.559 42.81±1.29

31.21±0.60

22.09±0.44

18.75±0.38

----

---- 30.12±0.77

21.29±0.42

18.57±0.41

13.03±0.29

Ni 2.901 41.21±1.30

30.01±0.59

22.01±0.44

18.28±0.40

----

---- 28.52±0.75

20.81±0.43

18.15±0.41

12.42±0.29

190

IncidentBeam

Ge

Ge

Ge

Ge

Ge

Ge

Ge

Beam Sto p p in g Power (MeV-cm2/mg)Energy C Al

(MeV/amu)

0.527 41.24±0.99

25.55±0.49

40.45±1.28

24.87±0.44

0.660 43.87±1.06

28.07±0.54

43.28±1.37

27.18±0.48

0.794 45.78±1.10

30.18±0.58

44.86±1.42

29.23±0.52

0.927 46.84±1.13

31.88±0.61

45.53±1.44

31.03±0.55

1.061 47.41±1.14

33.52±0.65

46.37±1.47

32.44±0.58

1.461 49.24±1.19

36.19±0.71

47.30±1.50

35.03±0.63

1.617 49.32±1.19

----

47.76±1.51

35.88±0.65

1.796 49.57±1.21

----

46.89±1.49

35.96±0.66

Cu Ag Au

16.67 16.31 10.19±0.25 ±0.29 ±0.18

16.60 15.67 10.04±0.29 ±0.23 ±0.19

18.62 18.04 11.37±0.28 ±0.33 ±0.20

18.55 17.52 11.26±0.32 ±0.26 ±0.22

20.36 19.42 12.28±0.30 ±0.35 ±0.22

20.26 19.02 12.20±0.35 ±0.28 ±0.23

21.80 20.23 12.99±0.33 ±0.36 ±0.23

21.48 20.07 12.96±0.37 ±0.30 ±0.25

23.03 21.25 13.48±0.34 ±0.38 ±0.24

22.56 20.93 13.48±0.39 ±0.31 ±0.26

25.47 22.29 14.17±0.38 ±0.40 ±0.25

24.66 21.94 14.26±0.43 ±0.27 ±0.32

26.10 22.00 14.28±0.39 ±0.40 ±0.26

25.06 22.16 14.61±0.44 ±0.33 ±0.28

26.39 21.95 14.21±0.40 ±0.40 ±0.26

25.20 22.02 14.49±0.47 ±0.33 ±0.28

191

Ge 1.996 49.25 — 26.57 21.49 14.38±1.20 ±0.40 ±0.57 ±0.26

47.35 36.00 25.81 22.03 14.60±1.52 ±0.67 ±0.54 ±0.33 ±0.30

192

Beam Energy C Al Cu Ag Au(MeV/amu)

In c id e n t Beam Sto p p in g Power (MeV-cm2/mg)

Br 0.495 43.74±1.30

26.37±0.56

17.24±0.26

15.61±0.23

9.95±0.21

---- 26.48±0.68

18.07±0.36

15.34±0.23

10.03±0.21

Br 0.620 47.50±1.41

29.16±0.62

29.53±0.29

17.62±0.26

11.20±0.23

---- 29.36±0.76

20.60±0.40

17.23±0.25

11.37±0.23

Br 0.744 49.00±1.45

31.38±0.66

21.44±0.32

19.17±0.28

12.82±0.26

---- 31.39±0.81

22.76±0.45

18.75±0.27

12.33±0.25

Br 0.994 51.23±1.52

34.98±0.74

24.50±0.37

21.27±0.31

13.55±0.28

---- 35.45±0.91

26.10±0.51

20.74±0.28

13.67±0.30

Br 1.119 51.61±1.53

36.50±0.77

25.74±0.38

22.00±0.32

13.93±0.29

---- 36.22±0.93

27.17±0.53

21.25±0.31

14.16±0.29

Br 1.307 52.63±1.56

38.06±0.81

27.35±0.41

22.70±0.33

14.30±0.30

---- 38.20±0.98

28.96±0.57

21.84±0.32

14.53±0.30

Br 1.729 53.19±1.59

40.23±0.85

29.13±0.44

22.90±0.34

14.62±0.31

---- ---- 30.87±0.75

22.08±0.33

14.83±0.30

Br 1.991 52.29±2.27

40.58±0.87

29.47±0.45

22.60±0.35

15.13±0.42

---- 42.46±1.38

31.20±0.87

---- 14.93±0.41

193

Br 2.058 52.61 40.86 28.85 22.42 14.77±1.58 ±0.87 40.44 ±0.34 ±0.41

42.76 31.16 21.98 14.80±1.38 ±0.75 ±0.34 ±0.31

194

Inc ident Beam Stopping Power (MeV-cm2/mg)Beam Energy C Al Cu Ag

(MeV/amu)

Nb 0.525 52.83±1.57

30.72±0.65

19.86±0.30

17.60±0.26

---- 30.80±0.80

20.87±0.41

17.25±0.26

Nb 0.736 57.99±1.72

36.13±0.77

24.07±0.36

20.81±0.31

---- 35.75±0.92

25.24±0.50

20.69±0.30

Nb 0.948 59.72±1.78

40.27±0.85

27.44±0.41

23.50±0.35

---- 40.11±1.04

28.93±0.57

22.96±0.34

Nb 1.160 60.93±1.83

43.57±0.93

30.36±0.46

24.89±0.37

---- 43.77±1.14

32.04±0.69

24.45±0.37

Nb 1.372 60.02±1.81

45.97±0.98

32.47±0.49

25.96±0.39

---- 46.36±1.30

33.99±0.73

25.03±0.38

Nb 1.478 61.76±1.86

47.04±1.00

33.28±0.50

26.32±0.39

---- 47.76±1.33

35.06±0.74

25.31±0.38

Nb 1.055 61.17±1.81

41.98±0.81

29.40±0.58

24.83±0.49

---- 40.67±1.03

29.09±0.58

25.56±0.56

Nb 1.267 62.95±1.87

44.79±0.87

31.84±0.63

26.55±0.52

---- 42.96±1.09

31.11±0.62

26.98±0.59

Au

11.58 40.24

11.46±0.24

13.58 ±0.29

13.91±0.29

15.15±0.32

15.37±0.32

16.22±0.34

16.30±0.34

16.63±0.35

16.98±0.35

16.84±0.35

17.05±0.35

16.61±0.37

17.36±0.39

195

Nb 1.585

Nb 1.797

Nb 2.012

64.34±1.95

65.24±2.01

65.08±2.04

47.24 33.85 27.84 ----±0.92 ±0.68 ±0.55

45.90 33.02 28.19 17.95±1.18 ±0.66 ±0.62 ±0.41

47.83 34.96 27.79 ----±0.94 ±0.70 ±0.56

45.25 33.38 28.27 18.39±1.18 ±0.72 ±0.63 ±0.53

47.35 34.61 28.59 ----±0.94 ±0.69 ±0.66

196

Beam Energy C Al Cu Ag Au(MeV/amu)

In c id e n t Beam Sto p p in g Power (MeV-cm2/mg)

1 0.366 30.84±0.66

18.69±0.29

17.50±0.26

11.25±0.24

---- 31.25±0.83

20.14±0.41

17.26±0.27

11.33±0.24

I 0.536 63.91±1.90

37.19±0.79

23.45±0.35

22.11±0.33

14.21±0.30

---- 38.03±1.00

25.25±0.52

21.81±0.34

14.41±0.30

I 0.690 71.12±2.12

42.56±0.91

27.56±0.42

25.78±0.38

16.52±0.35

---- 42.35±1.12

29.58±0.82

25.02±0.39

16.78±0.35

I 0.844 75.04±2.24

46.99±1.00

31.11±0.47

18.63±0.43

18.13±0.38

---- 47.72±1.47

33.15±0.87

27.84±0.43

18.49±0.39

I 0.998 77.84±2.33

50.99±1.09

33.88±0.51

30.76±0.46

19.43±0.41

---- 50.99±1.72

34.77±0.90

30.26±0.72

19.85±0.54

I 0.112 18.69±0.52

---- 7.24±0.13

---- ----

18.41±0.66

---- 7.40±0.15

---- 2.60±0.07

I 0.160 26.60±0.68

17.90±0.37

9.84±0.16

9.00±0.18

5.23±0.34

26.14±0.88

---- 10.14±10.19

8.25±0.14

5.29±0.11

I 0.219 34.73±0.86

22.56±0.45

12.56±0.20

11.98±0.23

7.27±0.14

34.31±1.11

---- 13.03±0.24

11.33±0.18

7.19±0.15

197

I 0.285 43.04 26.98±1.06 ±0.53

42.34 26.36±1.37 ±0.49

I 0.361 51.21 30.93±1.25 ±0.61

50.15 30.24±1.61 ±0.55

I 0.447 57.73 34.72±1.41 ±0.68

56.91 33.49±1.82 ±0.61

I 0.541 63.22 38.67±1.54 ±0.75

62.88 37.12±2.01 ±0.67

I 0.757 72.07 46.10±1.75 ±0.90

70.74 43.63±2.39 ±0.79

I 0.765 72.64 46.37±1.76 ±0.90

71.11 44.22±2.25 ±0.80

I 1.009 77.70 52.96±1.90 ±1.03

74.76 49.79±3.69 ±0.90

I 1.309 81.41 ----±2.69

76.52 54.69±5.77 ±1.29

I 0.832 75.27 46.44±2.26 ±0.91

__ 45.13±1.16

15.42 15.00 9.10±0.24 ±0.28 ±0.17

15.88 14.35 9.20±0.28 ±0.22 ±0.18

18.29 17.97 11.14±0.28 ±0.33 ±0.21

18.94 17.36 11.22±0.34 ±0.27 ±0.22

21.22 20.83 13.08±0.32 ±0.38 ±0.24

21.68 20.23 13.34±0.38 ±0.31 ±0.26

23.93 23.37 14.90±0.36 ±0.43 ±0.27

24.73 23.10 15.10±0.47 ±0.35 ±0.29

29.59 28.39 18.10±0.45 ±0.52 ±0.33

29.72 28.00 18.39±0.62 ±0.42 ±0.36

29.78 28.10 18.25±0.45 ±0.51 ±0.33

29.93 28.36 18.48±0.62 ±0.42 ±0.36

34.79 31.53 20.57±0.53 ±0.67 ±0.39

34.51 32.36 21.12±0.88 ±0.49 ±0.56

39.08 35.30 22.23±0.63 ±0.91 ±0.45

38.42 35.24 23.00±1.49 ±0.83 ±0.63

31.65±0.63

29.25±0.58

----

---- 29.59±0.66

----

1.079 79.78±2.41

51.78± 1 . 0 1

35.43±0.71

32.92±0.65

1.220

51.32±1.32

56.05±1.59

3 .03 ±0.92

33.53±0.75

1.507 56.76±1.18

40.61 ±1.02

35.89±1.28

1.388 83.53±2.53

56.83±1.11

39.90±0.79

35.66±0.71

36.43±0.84

1.469 83.36±2.50

57.10±1.11

40.45±0.81

35.38±0.75

54.08±1.44

36.60±0.81

36.60±1.14

198

199

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