Backscattering of light atomic projectiles from thin gold films in the energy range 50-110 keV.

University of Windsor University of Windsor

Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations Theses, Dissertations, and Major Papers

1-1-1970

Backscattering of light atomic projectiles from thin gold films in

Backscattering of light atomic projectiles from thin gold films in

the energy range 50-110 keV.

the energy range 50-110 keV.

Everett Brimmer University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation Recommended Citation

Brimmer, Everett, "Backscattering of light atomic projectiles from thin gold films in the energy range 50-110 keV." (1970). Electronic Theses and Dissertations. 6847.

https://scholar.uwindsor.ca/etd/6847

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BACKSCATTERIM6 OF LIGHT ATOMIC PROJECTILES FROM THIN Au FILMS IN THE ENERGY RANGE 50-110 keV

by

E ve re tt Brimner

A Thesis

Submitted to the Faculty of Graduate Studies through the Department of Physics In P a rtia l F u lfillm e n t of the Requirements fo r

the Degree of Master of Science a t the U n iv e rs ity o f Windsor

Windsor, Ontario 1970

UMI Number: EC52789

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ABSTRACT

Thin gold film s , vacuum deposited onto a b erylliu m s u b strate, have been irra d ia te d w ith ,?Li , and p ro je c tile s in the energy range 50-110 keV. The backscatter in te n s ity a t a laboratory s c a tte rin g angle o f 136.4 degrees was measured. The in te n s ity is not e x a c tly proportional to film thickness but increases more ra p id ly . This e f fe c t has been a ttrib u te d to m u ltip le sm all-angle c o llis io n s .

ACKNOWLEDGEMENTS

I wish to thank D r. A. van Wijngaarden fo r his supervision throughout the course of th is work. I should also lik e to thank D r. W. Bay!is fo r suggesting the s h e ll-s h ie ld e d Coulomb p o te n tia l, and Mr. B. Miremadi fo r his valuable assistance during the measure­ ments.

Acknowledgements are also due to D r. J . P. Marton who measured the thicknesses of the th in gold film s .

I am als o indebted to the National Research Council of Canada fo r its fin a n c ia l support in the form of a Graduate Scholarship.

F in a lly , I am g ra te fu l to my w ife fo r her understanding and encouragement.

11

TABLE OF CONTENTS

ABSTRACT î

ACKNOWLEDGEMENTS ! i

LIST OF FIGURES Iv

CHAPTER 1 - INTRODUCTION I

CHAPTER n - THEORY 3

CHAPTER 111 - EXPERIMENTAL 10

3:1 D etector E ffic ie n c y 10

3 :2 Film Thickness 12

CHAPTER IV - RESULTS 14

4:1 Backscatter of 'h 14

4 :2 Backscatter of ^He, ^Ll and '*B 16 CHAPTER V - ENERGY LOSS IN THE FILM 20 CHAPTER VI - MULTIPLE SCATTERING EFFECTS 25

6:1 Angular Divergence 26

6:2 Correction Factor 27

6:3 Dependence of Backscatter Y ie ld on Film

Thickness 30

CHAPTER V I 1 - DISCUSSION 33

APPENDIX 1 - THE SHELL-SHIELDED COULOMB POTENTIAL 34 APPENDIX I I - THE CLASSICAL CROSS SECTION FOR THE

SHELL-SHIELDED COULOMB POTENTIAL 36 APPENDIX 111 - TRANSFORMATION FORMULAE BETWEEN LAB

AND CM FRAMES 40

111:1 R elationship of the Cross Sections 43 APPENDIX IV - EXPERIMENTAL DIFFERENTIAL SCATTERING

CROSS SECTION 44

BIBLIOGRAPHY 45

VITA AUCTORIS 46

LIST OF FIGURES

1. Various P o ten tials as a Function of In tern u clear

Distance R 7

2 . The Energy Dependence of Various Laboratory Cross Sections fo r S catterin g Through 136,4

Degrees of Various P ro je c tile s From Gold 9 , 33a 3 . Schematic Diagram of the Apparatus 11 4 . Energy Dependence of the Observed Backscatter

Y ie ld of 'h From Various Thin Gold Films 15 5 . Energy Dependence of the Observed Backscatter

Y ie ld of ^ From Various Thin Gold Films 17 6 . Energy Dependence of the Observed Backscatter

Y ie ld o f a i From Various Thin Gold Films 18 7. Energy Dependence of the Observed Backscatter

Y ie ld of *'B From Various Thin Gold Films 19 8 . The Nuclear Stopping Power (d f/d ? )n as a

Function of (from Lindhard e t a l . 1963) 22 9 . Correction Factor fo r M u ltip le S c atterin g

as a Function of Evaluated a t H alf the

Film Thickness 29

10. Observed and Corrected Backscatter Y ie ld as

a Function of Film Thickness 31 11. Asymptotic Views of an E la s tic C o llis io n in

the Lab System 40

12* Asymptotic Views of an E la s tic C o llis io n in

the CM System 4 l

13. A Vector A ddition Diagram o f the V e lo c itie s

of P a rtic le s in the CM and Lab Frames 42

iv

CHAPTER 1 INTRODUCTION

The energy loss of atomic p ro je c tile s in solids can be

a ttrib u te d to two separate processes: nuclear reco il and e le c tro n ic e x c ita tio n . In a nuclear c o llis io n the ta rg e t nucleus together w ith

its o rb ita l electro ns re c o ils , and the p r o je c tile is d e fle c te d . In general slow ions lose energy p rim a rily to ta rg e t atoms and t h e ir tra je c to ry consists of a devious path, w hile fa s t ions d iss ip a te energy mainly through e le c tro n ic e x c ita tio n and t r a v e l, th e re fo re , in nearly s tra ig h t lin e s . Although the major energy loss mechanism fo r lig h t

k

atomic p ro je c tile s above about 10 eV is the e le c tro n ic one, some nuclear c o llis io n s occur and backseatt e ring of in cident ions can r e s u lt. At

6 "7

energies above 10 -1 0 ' eV, such in te ra c tio n s are almost purely Coulomb!c fo r the lig h te s t atomic p r o je c tile s , and the s c a tte r in g ,

th e re fo re ,e x h ib its Rutherford behaviour. At lower energies the Incident p a r tic le a t it s point of maximum p o te n tia l energy in the c o llis io n sees a weakened Coulomb p o te n tia l, due to e le c tro n ic screening between the two n u c le i. One of the aims of th is in v e s tig a tio n is to study the in te ra c tio n p o te n tia l fo r nearly head-on c o llis io n s of lig h t atomic p ro je c tile s w ith gold. In such c o llis io n s the in cident ions can be scattered back through the surface of the ta rg e t sample in to the surrounding vacuum.

Backscatter!ng of lig h t atomic p ro je c tile s is used to in vestig ate the locatio n of im purity atoms in c ry s ta ls (Mayer e t a l . 1968), and to exp lain sp u tterin g (Behrisch 1969; Sigmund 1968 and 1969; van

Wijngaarden e t a l . 1970). McCracken and Freeman (1969) have studied

2

the energy d is tr ib u tio n of hydrogen ions backscattered from th ic k heavy ta rg e ts . They assumed a model fo r the process In which "the incident

ion slows down in the ta rg e t m aterial w ithout undergoing any s c a tte rin g , th a t i t is then scattered [in a pure Coulomb p o te n t!a f| through a large angle and th a t i t returns to the surface w ithout fu rth e r s c a tte rin g ." The energy dependence of t h e ir observed backscatter In te n s ity agrees f a i r l y well w ith predictions based on t h e ir model, but tl% in te n s ity is high by a fa c to r of about 4 , A norm alization by 3.1 is required to f i t Behrisch's angular d is tr ib u tio n of backscattered from copper. They th erefo re concluded th a t a s in g le -c o llis io n model was inadequate. To determine the sig n ific a n c e of m u ltip le sm all-angle c o llis io n s in th is work, an analysis of the dependence of backscatter y ie ld on ta rg e t

thickness has been made.

CHAPTER 11 THEORY

Consider an ion beam in cident on a ta rg e t film of several atomic layers thickness. I f the ion mass m^ is less than the ta rg e t atom mass m^, s in g le -c o llis io n backscattering can occur. In such a v io le n t c o llis io n only the ta rg e t atom and the p r o je c tile need be)considered. This is so because the ta rg e t atom is nearly fre e (Bergstrom and Domeij I966) since bond energy of neighbouring atoms is small fo r keV p r o je c tile s . The p r o je c tile does not see the ta rg e t atom's neighbours since the tr a n s it time across the in te ra c tio n region of the ta rg e t atom ( t y p ic a lly about

10” '^ sec) is much sho rter than normal v ib ra tio n a l periods (lO "*^ -lO ” *^ sec) in the s o lid . M u ltip le s c a tte rin g re su lts whenever the p r o je c tile s u f f ic ie n t ly penetrates the e le c tro n clouds of more than one atom, and occurs th erefo re In a l l th ic k ta rg e t media. The two-body s c a tte rin g problem, however, must be understood before we can proceed to study m u ltip le s c a tte rin g e ff e c t s .

The s c a tte rin g of keV p ro je c tile s from atoms is not s t r i c t l y e la s t ic , since e le c tro n ic e x c ita tio n of both the incident ion and the ta rg e t atom usu ally re s u lts . The energy loss to e le c tro n s , however, is a small fra c tio n of the in cident ion energy and the sing le s c a tte rin g event can be considered e la s t ic .

Because of e le c tro n ic screening, the actual p o te n tia l energy of two in te ra c tin g atoms is unknown. At high energies, when the c o llid in g nuclei in terp e n e tra te each o th e r's e le c tro n clouds, e le c tro n ic screening

V ( R ) r r Z ^ Z ^ e ^ / R ( 1 )

The s c a tte rin g is described by the fa m ilia r Rutherford cross section

(d c r/d n )^ = : (b /4 )^ csc^ (0 /2 ) * (2)

where b is the c o llis io n diam eter, the distance of clo sest approach of two unscreened nuclei in a head-on c o llis io n :

b iz Z iZ g e Z /tiu v ^ ). (3)

Here u is the reduced mass of the p r o je c tile -ta r g e t system, and v is

t h e ir i n i t i a l r e la tiv e v e lo c ity . The Rutherford cross section is accurate only as long as the minimum distance of approach i b ( 1 + esc 6 /2 ) is much less than the e ffe c tiv e screening radius o f the in te ra c tin g p a r tic le s . For 'h backscattered from Au a t 8|_=136.4°, fo r example, we fin d

d eviations from Rutherford behaviour a t energies as high as 100 keV. For more massive p ro je c tile s and or sm aller s c a tte rin g angles, the lower energy bound fo r Rutherford behaviour increases.

Several interatom ic p o te n tia ls have been proposed to allo w fo r the e le c tro n ic screening of the nuclear Coulomb f i e l d . Bohr (1948) suggested th a t the in te ra c tio n between two atomic stru ctures is given by the screened Coulomb p o te n tia l energy

V(R) = (Z^Z2e^/R) e x p (-R /a s ). (4 )

* A ll Center-of-Hass (CM) q u a n titie s w i ll be w r itte n w ithout a s u b scrip t. Laboratory (la b ) q u a n titie s w i ll be denoted by the subscript J l .

5

The range of the screening, and hence a measure of the o v e ra ll size of the two in te ra c tin g p a r tic le s , is given by the Bohr screening length

a

Here ajj = 0.529 the Bohr radius. Firsov (1958) employs a screening fa c to r c a lcu lated from the Thomas-Femi s t a t i s t ic a l model of the atom

a = 0.8853 a^. (6)

We sh all r e fe r to and use th is l a t t e r value e x c lu s iv e ly as the screening parameter in the remainder o f th is th e s is .

For R « a , the Bohr p o te n tia l energy is approximately given as

V (R )= (Z^Z^e^/R) (1 -R /a ) H (l-R /a ) (?)

where H(x) is the Heavi side step function

f 1 I f x > 0 H(x) =

Lo

x < o .

Eq. 7 represents the in te ra c tio n energy between a charge Z^e and a charge Z^e when the l a t t e r is surrounded by a spherical s h e ll of negative charge density

-Zge (4-jra^) ^ 5 (R -a) (See Appendix 1 ).

An important advantage of the s h e ll-s h ie ld e d p o te n tia l is th a t the corresponding cross section has a simple a n a ly tic form:

2

ab (a + b /2)

4a (a -f b) s in ^ (0 /2 ) + b^ É2T

where the c o rrec tio n fa c to r k Is

1 + ^ / a

1 + b/a + b^/(4a^sIn^i© )

(9)

(See Appendix I I ) .

Eq. 8 is re a d ily in teg rated to give the c la s s ic a lly expected f i n i t e to ta l cross section

c r = \ d i i ^ = a ^ T T • (10) ; dxx

4-TT

In the lim it of high in cident energy (small b) and large screening radius, b/a is sm all, and /dG 7\(E q. 8) approaches / (Eq. 2 ) ,

\ d / i j idXLjp;

Smith e t a l . (1967) have used v a ria tio n a l methods to obtain the best em pirical f i t of the function al form

V(R) =Ae^/R exp(-R /C ) (11)

to experimental data fo r s c a tte rin g of He on Ne and Ar in the energy range of 10 eV to 100 keV. In Eq. 11 the v a ria tio n a l parameters are A and C. The em pirical values fo r C are about twice as large as the corresponding Thomas-Fermi screening parameters (Eq. 6 ) . Furthermore Abrahamson (1963) has calc u la te d Thomas-Fermi-Dirac p o te n tia ls fo r the

in te ra c tio n of noble-gas atoms, and has found good agreement w ith experiment a t small R. In Figure 1 Abrahamson's resu lts fo r Ne-Ne, K r-K r, and

Rn-Rn in tera c tio n s are shown, along w ith the a n a ly tic p o te n tia ls of Eqs. 1, 4 and 7, fo r both screening lengths a and 2a. Curve number 3 ,

representing the s h e ll-s h ie ld e d Coulomb p o te n tia l energy

»

Ne-Ne (TFD)

0

Kr - Kr (TFD)

®

Rn —Rn (TFD)

1. V=Z| Zge^ FT

2 .V = Z | Zg ( f FT

exp (-R /2 a )

3 . V = Z | Zge^R"'

(I- R/2o) h(l-R/2o)

4.V= Z|

T-

1

^

IT' exp(-R/a)

5 . V = Z , Zge^ R"'

(l-R /a) hd-R/o)

R/a

Figure I . Various p o te n tia ls as a function of in te rn u c le a r distance R. The points are Abrahamson's (1963) c a lc u la tio n s fo r in te ra c tio n s between

V ( R ) = (Z, Z ^ e ^ /R ) ( l - R / 2 a ) H ( l - R / 2 a ) ( 1 2 )

gives the best f i t fo r R ^ a , The corresponding cross section is given by Eqs. 8 and 9 w ith a replaced by 2a.

Cross sections fo r Bohr's screened p o te n tia l have been calcu lated num erically fo r several values of b/a (Everhart e t a l . 1955). For

7 T /2 ^ 0 < n and b /a ; ^ l, the Rutherford cross section is as much as a fa c to r of 2 .5 la rg e r than E v erh art's values, whereas the s h e ll-s h ie ld e d cross section (Eq. 8) always lie s w ith in 10% o f the numerical re s u lts .

Our experimental s c a tte rin g cross sections ( t o be discussed

1 4 7 11

below) fo r H, He, Li and B on Au f o r a fix e d laboratory s c a tte rin g angle of 136.4° and a t laboratory energies from 50 to 110 keV are

presented as c ir c le s in F ig . 2. The Rutherford cross sections (dash-dot lin e s ) li e w ell above the s h e ll-s h ie ld e d (dashed lin e s ) and E v erh art's exponentially-screened Coulomb values ( O ' s ) . The good agreement between the s h e ll-s h ie ld e d and exponentially-screened Coulomb cross sections is evidence th a t backscattering is governed p rim a rily by the p o te n tia l a t distances R < a . This is so because the former in te ra c tio n is zero fo r R > a w hile the l a t t e r is no t. The s o lid curve, presenting the cross

sections of the s h e ll-s h ie ld e d Coulomb p o te n tia l w ith the screening radius doubled, agrees f a i r l y w ell w ith the reduced experimental data (Chapters V and V I) both in energy dependence and absolute valu e.

•>19

w w

i

Z O

LU C/)

s

o

U

\ \\\

20

50

100

ENERGY (keV)

300

CHAPTER 111 EXPERIMENTAL

Figure 3 is a schematic diagram of the apparatus. Honoenergetic ion beams a t energies below 120 keV are obtained from a magnetic

analyzer (van Wijngaarden e t a l . 1970). The angular divergence of the emerging beam is lim ite d to w ith in 0 .2 ° by a c ir c u la r s l i t system. A fte r c o llim a tio n the ion beam enters the ta rg e t chamber which is

- 7

maintained a t a pressure of the order of 10 T o rr. Regularly spaced th in ta rg e t film s of Au, vacuum deposited onto a th ic k Be p la te , can be moved across the path of the ion beam as indicated by the arrows. A Faraday cup connected to a bellows can be moved in to the path of the

ion beam. The absolute value o f the curren t is measured to w ith in

2% by a K eith le y 410 e le c tro m e te r. An Ortec Model E-013-025-100 surface -4

b a r r ie r d e te c to r, subtending a s o lid angle û, n = ( 5 .2 5 ± 0 .0 3 ) x 10

s r , is positioned a t 136.4° w ith respect to the incident ion beam. The pulses from the d etecto r are fed through a p re a m p lifie r (Ortec Model 109a), a main a m p lifie r (Ortec Model 4 8 5 ), a sin g le channel analyzer (Ortec Model 4o 6a), and are recorded by a d ig it a l ratem eter (Ortec Model 4 3 4 ).

3:1 Detector E ffic ie n c y

Any ion which is detected has passed through the Au electro de (228 % thickness) of the surface b a r r ie r d e te c to r, and then produced s u ffic ie n t io n iz a tio n to create a pulse d is c e rn ib le above the noise of the d e te c to r. To fin d the minimum energy fo r 100% d e te c tio n , the detector was placed d ir e c t ly in the path of monoenergetic *H + and ^He beams. With the lower d isc rim in ato r level of the single channel analyzer

10

11

0)

O'

k_

i

lO

OJ

(A 3 W

2

S.

a.

m j

I

cn

ID

1

u </>

2

12

fix e d ju s t above the detector noise le v e l, the detector e ffic ie n c y

(number of counts per incident p r o je c tile ) was observed to be e s s e n tia lly p e rfe ct fo r protons a t primary energies E % 40 keV and fo r ^He a t E ^ 50 keV. As the primary energy decreased below these values, so did the e ffic ie n c y of the d e te c to r. Below 10 keV no p a rtic le s could be detected.

I t is the counting e ffic ie n c y fo r the backscattered p ro je c tile s which is of in te re s t in the present experiment. The energy loss fo r these p ro je c tile s in the th in ta rg e t film s is q u ite small (Chapter V ), and 100% b a c k s c a tte re d -p ro je c tile counting e ffic ie n c y was expected fo r primary energies down to about the same lim it s . To v e r ify t h is , only incident energies were used fo r each p r o je c tile and each film fo r which the counting rate o f the backscattered ions in to the detector (8^ = 136.4°)

remained constant fo r a small range of the lower d isc rim in ato r s e ttin g s .

3 :2 FiIm Thickness

Five th in Au film s were vacuum deposited simultaneously onto Be and glass substrates using an Edwards Coating U n it (Model I2E A /722). The

thicknesses of the Au film s on the glass substrate were measured using o p tic a l techniques (Marton and Schleslnger 1969). To check th a t

corresponding Au film s on the two substrates were of equal thickness, the fo llo w in g experiment was performed. The glass-backed and Be-backed Au film s were successively irra d ia te d w ith ^He p ro je c tile s and the in te n s itie s of the backscattered p ro je c tile s from the various th in film s and the two substrates were measured. The backscatter y ie ld (number of backscatter counts per in cident p r o je c tile ) from the Be substrate was 0.2% of th a t from the thinnest Be-backed Au f ilm , w hile th a t of the glass was about 15% of the y ie ld from the thinnest glass-backed f ilm . A fte r

13 subtraction o f the substrate backscatter y ie ld s from those of the

th in film s , the re s u ltin g corrected y ie ld s were equal, w ith in experimental e rro r of about 2%, fo r corresponding film s on glass and Be, This agreement was found over the primary energy range 50-110 keV and fo r various film

thicknesses In the range 75-350 X. I t was, th e re fo re , concluded th a t ( I ) corresponding film s on glass and Be had the same thickness, and (2) the observed backscatter yie ld s could be corrected by subtraction of the

substrate y ie ld .

The measured thicknesses of the fiv e Au film s are 75, 120, 170,

CHAPTER IV RESULTS

4:1 Backscatter of

The fiv e Au film s and the Be substrate were bombarded in succession by proton beams of the order of lo ” *® A a t various fix e d primary energies

in the range 50-110 keV, The to ta l backscatter counts during 10 sec in te rv a ls were recorded fo r each film and the substrate a t each primary energy. Just p rio r to and immediately a f t e r each counting in te rv a l the incident ion curren t was measured to an accuracy of b e tte r than 2%. I f these two ion current measurements d iffe r e d by more than 5%, the data were discarded. In order to minimize s t a t i s t ic a l e rro rs a miniminn to ta l of 3000 backscatter events were recorded fo r each film a t each energy. The gross backscatter y ie ld , b' , was obtained by d iv id in g the backscatter counts per second by the p r o je c tile p a r tic le c u rre n t.

For a given film b" includes c o n trib u tio n s from both the Au film and the su b s tra te. At the highest primary energy the B' value of the substrate reached a maximum and equalled about 5% of the B^ value of

the thinnest Au f il m . The corrected backscatter y ie ld , 1^, was calcu lated (Sec 3*2) by subtracting the substrate y ie ld from each recorded film y ie ld . The energy dependence of the observed backscatter y ie ld , B^, fo r the

various Au film s is presented In F ig . 4 by the s o lid curves through

th e p o in ts m arked as c i r c l e s . The dashed c u r v e s , w h ic h r e p r e s e n t th e

same data corrected fo r energy loss in the film s and fo r m u ltip le sm all-angle s c a tte rin g e ff e c t s , w i ll be discussed in Chapter V I.

14

t (Â)

CL

350

265

170

Ü J

120

c/)

75

o

10

30

50

100

ENERGY (keV)

200

16

4 :2 Backscattering o f Ste. ^Li. and

The fiv e Au film s and the Be substrate were a ls o irra d ia te d w ith monoenergetic beams of \ e ^ , ^ L i \ and ions. With these

p ro je c tile s the substrate y ie ld was e ith e r zero , or n e g lig ib le , being three orders o f magnitude sm aller than the thinnest film y ie ld in the

4

case of He. The observed values fo r the various th in film s are presented as c ir c le s In Figs. 5~7 as a function of primary energy. As the p r o je c tile mass increased, the primary energy fo r which 100% detection could be obtained increased ra p id ly , lim itin g the energy range of our

in v e s tig a tio n s .

17

6

He on Au

Q .

350

265

170

120

75

8

5 X

50

ICO

ENERGY (keV)

200

18

2 X 1 0

Q.

265

170

120

75

c/)

7

50

100

200

ENERGY (keV )

Figure 6 , The s o lid curves through the experimental points (c ir c le s ) show the dependence of the observed backscatter y ie ld on primary energy fo r ^Li impinging on various gold film s . The dashed curves, w ithout experimental p o in ts, are plots of the corresponding backscatter y ie ld s corrected fo r m u ltip le s c a tte rin g e ffe c ts versus the average energy.

19

2 X I ( T

on

Q _ j Ü J >

-120

75

3 X

60

100

200

ENERGY (keV)

CHAPTER V

ENERGY LOSS IN THE FILM

The observed Bq value consists o f p ro je c tile s backscattered anywhere Inside the film m a te ria l. Since the ion c o n tin u a lly loses energy as i t traverses the ta rg e t medium, there is a range of energies over which the main s c a tte rin g event can occur. In the absence of m u ltip le s c a tte rin g , the backscatter y ie ld is d ir e c t ly proportional to film thickness, and on the average a l l v io le n t c o llis io n s can be considered to occur a t the centre of the th in f ilm . To each Bq value we assign an average energy of in te ra c tio n E = E q -^ E , where Eq

Is the primary energy and aE is the energy loss in o n e-h alf of the film thickness. The /^E values have been computed using the energy-loss theory of Lindhard and S charff (1961) (Lindhard e t a l . I963) . We sh all now summarize some of the important aspects of th is theory.

The to ta l stopping power o f a ta rg e t m a terial fo r a p r o je c tile is re la te d to the to ta l stopping cross section S by

Æ = NS (13)

dR

where N is the atomic density o f the ta rg e t medium. Although an energetic atomic p r o je c tile is d e fle c te d only by nuclear c o llis io n s

(Bohr 1948), the energy of the p r o je c tile is dissipated to both electro ns

and r e c o i l i n g atom s In th e s to p p in g m edium . Thus th e t o t a l s to p p in g c ro s s

section consists o f the stopping cross section ^ fo r less in energy in nuclear c o llis io n s and the stopping cross s e c t Iw fo r loss In

energy to e le c tro n s :

20

21

S = S ,+ Sq . (14)

From Eqs. 13 and 14 we see th a t

dR IdRln WRie (15)

/dE\ =NSn and m =NS@. (16)

IdRln idRJe

To obtain a universal d e scrip tio n fo r energy loss, which is v a lid fo r a l l atomic p r o je c tile -ta r g e t combinations, Lindhard e t a l . introduced the dimensionless va riab le s

€ = E amg (1 7 )

Z|

and P = R Nrog4iTa^mi

(m, i- m^) (1 8 )

fo r energy and range re s p e c tiv e ly . In terms o f these v a riab le s Eqs. 16 can be w r itte n as

'dE) _ 4naZ, Zz«P (19)

dR/f, (m,+ m j _n

and = 4naZ, Z g ^ . ^ . (20)

IdRL (m, + mz) \d ? /*

Lindhard e t a l . present a universal curve (F ig . 8) fo r / § £ \ versus ( d ? / n

6^ which is based on the Thoroas-Fermi s t a t i s t ic a l model of the c o llid in g atoms. In terms of the new v a riab le s the e le c tro n ic stopping power is given by

22 ro lO

Vü

$

1

§

1

2

23

where K = z}^^Q.0793 z t z f (A, + (22) ( z p T z p p - Â p ^

in which and Ag are the atomic numbers of the p r o je c tile and ta rg e t atoms re s p e c tiv e ly . The values of K fo r ^H, ^He, ^Li and on Au are

15.13, 3*00 , 1.71, and 1,216 re s p e c tiv e ly . Thus using F ig . 8 fo r /dg\ and Eqs. 21 and 22 fo r /d e l, one can compute the dE. values from

Id fin Id We dR

Eqs. 15, 19 and 20.

The energy loss aE In a th in stopping region of thickness <&R (much sm aller than the range of the p r o je c tile ) is then found from the r e la tio n ­ ship

AE e /d & j AR (23)

where + (24)

and

The simple a n a ly tic form (Eq. 21) ap p lies only fo r p r o je c tile 2/3

24

energies of 120 keV, allow ing us to compute fo r protons as w ell as fo r heavier p ro je c tile s over the e n tir e energy range in ve s tig ate d .

2/3

For v e lo c itie s v> Vj= VqZj e le c tro n ic stopping completely dominates, but i t is no longer proportional to v . For v < V j e le c tro n ic stopping s t i l l dominates fo r Z^< Zg. For heavier p ro je c tile s and or

decreasing energy, nuclear stopping gradually overtakes e le c tro n ic stopping to become the major mechanism of energy loss. However even fo r our heaviest p r o je c tile ( * * B ) , a E^ remains a fa c to r of 4 la rg e r than A E ^. For th is reason the model described by McCracken and Freeman (1 9 69 ), where the

incident ion tra v e ls in s tra ig h t lines except fo r v io le n t c o llis io n s , should be co rrec t to a f i r s t approximation fo r lig h t ions.

The average aE values in the thinnest film are 0.7 keV fo r protons,

4 7 I I

/^ l keV fo r He and Li and ^ 1 . 8 keV fo r B. Even large u n c e rta in tie s in these small energy corrections s t i l l allows us to obtain the average energy of in te ra c tio n , E = E ( ,-aE, to a s u f f ic ie n tly high degree of

p recisio n .

CHAPTER VI

MULTIPLE SCATTERING EFFECTS

I f the in cident ions tra v e lle d s t r a ig h t - lin e paths in the ta rg e t m a terial u n til they suffered a sing le v io le n t in te ra c tio n and were

scattered , and then proceeded re c ti I in ea rl y out of the stopping medium, only p a rtic le s scattered in to A jn = 5 ,2 5 x l o ”^ s r about 8 ^ = 1 3 6 ,4 ° would reach the d e te c to r. As a consequence of m u ltip le sm all-angle scatterin g s before the v io le n t c o llis io n , the beam gradually spreads out about the d ire c tio n o f i n i t i a l incidence, in any in fin ite s im a l time in ­ te rv a l

dT,

cr(e,E)dT

scattered back, where 6 is the s c a tte rin g angle w ith respect to the d ire c tio n of motion immediately preceding s c a tte rin g and E Is the p r o je c tile energy a t th a t in s ta n t. The p r o je c tile density in the diverging beam is azim uthally is o tro p ic . Because the Rutherford cross section is h igh ly preferred in the forward d ire c tio n , p a rtic le s

tr a v e llin g a t angles 0<0|_= 136.4° w ith respect to the d etecto r are more l i k e l y to e n te r the detector than those w ith 0 > 0 ^ . Con­

sequently more p a rtic le s are m u ltip ly -s c a tte re d in to the s o lid angle of the detector than out of i t , re s u ltin g in an enhancement of the

backscatter y ie ld . Backscattered p a rtic le s a ls o s u ffe r small d e fle c tio n s , fu rth e r enhancing the backscatter y ie ld . To compare the observed

b a c k s c a t t e r y ie ld w ith the th e o r e tic a lly predicted one fo r a sing le s c a tte rin g e ven t, a c o rrectio n roust be applied to B^.

26

6:1 Angular Divergence

To fin d the co rrec tio n fa c to r by which the observed backscatter y ie ld must be divided in order to c o rrec t fo r m u ltip le sm all-angle d e fle c tio n s , an estim ate of the angular divergence of the beam in t r a v e llin g through a th in stopping region is required. Suppose the p r o je c tile encounters several s c a tte rin g centers on its passage through the stopping medium. Since the in d ivid u al d e fle c tio n s are completely random, the average d e fle c tio n is ze ro . The root-mean-square (rms) d e fle c tio n w i ll not vanish, however, since there e x is ts a random walk

in angle away from the d ire c tio n of i n i t i a l incidence. The to ta l rms d e fle c tio n associated w ith the p r o je c tile penetrating a depth aR in to the medium can be estim ated as fo llo w s* Consider an e la s t ic c o llis io n between two atoms in which the p a rtic le s are d eflected through the common angle of s c a tte rin g 6 in the CM system. From Appendix I I I the p r o je c tile

(m^) tran s fe rs an amount of energy to the re c o ilin g ta rg e t (m^) equal to

T = 4m,m2 E s i n 2 ( 8 / 2 ) , (27) (m |+ mg)^

where E is the impact energy. For lig h t p ro je c tile s (m j^«m ^), the lab and CM systems are almost id e n tic a l, and 6-6^^. Since the vast m a jo rity of c o llis io n s are sm all-angle ones Eq. 27 reduces to

T 2 4 (m ,/m 2 )E (8 /2 )2 . (28)

When a s w ift ion traverses a th in stopping region, i t undergoes several small d e fle c tio n s from it s o rig in a l d ire c tio n of motion, losing an amount of energy

27

*E n = = T, (29)

in atomic r e c o ils , where the summation includes a l l s c a tte rin g events. S u b stitu tin g fo r T; from Eq. 28, the nuclear energy loss becomes

2 2

AEn=i(m|/m2) L E ;8| ^(m |/m 2)E [ 6 ; (30)

where E; is the p r o je c tile energy p rio r to the i^^ c o llis io n . In a th in film the p r o je c tile energy remains n early constant and has, th e re ­ fo re , been taken outside the summation sig n . I f a l l s c a tte rin g angles are sm all, the re s u ltin g angular d is tr ib u tio n is approximately gaussian

(Bohr, 1948) w ith a root-mean-square width

♦ m s - (3 1)

i yE m,

The c a lc u la tio n of a E^, has been described in the previous Chapter. Because a l l s c a tte rin g events are not small angle ones, the actual angular d is tr ib u tio n w i ll d i f f e r from a gaussian. Eq. 31 is , th e re ­ fo re , considered to be only a rough estim ate of the angular divergence.

6 :2 Correction Factor

The c o rrec tio n fa c to r (CF = B<j/B) is the fa c to r by which the

o b s e rv e d b a c k s c a t t e r y i e l d , enhanced by th e e f f e c t s o f m u l t i p l e s m a

28

(1 ) The Rutherford cross section gives the form of the

backscattered angular d is tr ib u tio n . Since fo r large s c a tte rin g angles the shape of the angular d is tr ib u tio n is but l i t t l e a ffe c te d by e le c tro n ic screening, th is assumption introduces only small e rro rs .

(2 ) A ll backscatter events occur a t a film depth AR = t / 2 where t is the f ilm thickness.

(3 ) Just p rio r to backscattering the incident ions are d is trib u te d over an angular d is tr ib u tio n of gaussian form, centered about the incident d ire c tio n , whose rms width is given by Eq. 31 w ith 6 R = t / 2 .

(4) As the backscattered ions tra v e l out of the film they

again spread out in a gaussian angular d is tr ib u tio n about t h e ir o rig in a l s c a tte r d ire c tio n s , w ith an rms width a t the film surface again given by Eq. 31 but w ith ^ R = ( t / 2 ) / T , the average distance the ions tra v e l through the f ilm a f t e r being scattered from the film center towards the d e te c to r.

The results of the computation are presented in F ig . 9 as a p lo t of the c o rrectio n fa c to r versus $ , the predicted rms width of the

angular d is tr ib u tio n fo r AR = t / 2 . The co rrec tio n fa c to r was found to be s lig h t ly sm all, A b e tte r one was obtained by replacing $ , as

calcu lated from Eq* 31, w ith I . 10 ÿ and reading the co rrec tio n fa c to r from the same curve (F ig . 9 ) , The new values were found to be adequate (See Section 6 :3 ) fo r a l l film thicknesses and a l l p ro je c tile s a t a l l energies in ve s tig ate d .

The dependence o f B (the Bq value corrected fo r m u ltip le s c atte rin g e ffe c ts ) on the average energy of in te ra c tio n (the primary energy corrected fo r energy loss In the film ) is represented by the

29

cr

o

<

o

o

ÜJ

cr

cr

o

5

.4

3

$

2

0

6

12

18

<f>

R.M.S.

(Degrees)

24

3 0 dashed curves in Figs. 4 - 7 . Both axes in each of these diagrams have a dual meaning. The abscissa represents the primary energy Eq fo r the s o lid curves, and the average energy E = Eo- a£ fo r the dashed curves. The ordinate represents the observed backscatter y ie ld B© fo r the s o lid curves, and the corrected y ie ld B=Bo/CF fo r the dashed curves.

6:3 Dependence of Backscatter Y ie ld on FIIm Thickness

The dependence of the observed backscatter y ie ld Bo on the measured film thicknesses a t an average In te ra c tio n energy of 100 keV

is shown by the dashed curves through the experimental points marked as squares in F ig . 10. Since smooth curves can be f i t t e d qu ite n ic e ly through these p o ints, the film thickness measurements appear to be accurate w ell w ith in the u n certain ty of ± 5

In the absence of m u ltip le s c a tte rin g the backscatter y ie ld should be proportional to film thickness fo r t values much sm aller than the

range of the p r o je c tile in the f ilm . The c ir c le s in F ig . 10 represent the same data as the squares, but the ordinate now represents the y ie ld corrected fo r m u ltip le d e fle c tio n s (B = Bo/CF). S tra ig h t lin e s (the s o lid curves) passing through the o rig in provide good f i t s to the B values, in d ic a tin g th a t the co rrec tio n fa c to r fo r m u ltip le d e fle c tio n s may be f a i r l y accu rate. I t w i ll be noticed th a t m u ltip le s c a tte rin g e ffe c ts increase ra p id ly w ith increasing p r o je c tile mass. The reason fo r th is is th a t the r a tio AE„/m^ (See Eq. 31) increases w ith p r o je c tile mass.

The e r ro r in the c o rrec tio n fa c to rs is d i f f i c u l t to estim ate, but the deviations from u n ity of these fa cto rs appear to be accurate to , a t le a s t, w ith in 20%, For the thinnest film the average co rrectio n

31

_0> 0

<D 1

<D

a .

00

6 (— Xl(T XIO

5

4

3

2

_1 01^

4 0 0 0 4 0 0

100 200 3 0 0 100 200 3 0 0

- XIO

200 3 0 0

FILM THICKNESS (Â)

3 2 fa c to rs in the energy range in vestigated are 1 .0 2 , 1 .0 3 , 1.05 and 1.12 fo r **He, ^Li and re s p e c tiv e ly . A 20% e rro r in the deviations from u n ity of these numbers resu lts in a change in B of less than 1% fo r the three lig h te s t p ro je c tile s and less than 3% fo r the fo u rth . We conclude, th e re fo re , th a t the u n c e rta in tie s In B introduced by m u ltip le s c a tte rin g are p r a c tic a lly n e g lig ib le fo r the thinnest f ilm .

CHAPTER V il DISCUSSION

Because the data fo r the thinnest film are the most c e rta in , only the 75 % film re su lts w i ll be compared w ith theory. The laboratory backscatter cross section is given by (See Appendix IV)

i s : - _ B (32) dA N ( A /i) t

where a a is the s o lid angle subtended by the d etector and N is the

atomic density of the stopping medium. The cross sections fo r the various p ro je c tile s were c a lcu lated a t each energy in vestigated by s u b s titu tio n of nianerical values in to Eq. 32. The cross sections fo r near 100 keV were found to be 1.03 times the values predicted by the s h e ll-s h ie ld e d Coulomb p o te n tia l w ith a replaced by 2a (Eq. 1 2 ). Since the conversion from B value to cross section involves the measured value of the film thickness and its u n c e rta in ty , and the s t a t is t ic a l e rro r in B (le s s than 2%), no physical s ig n ifican ce can be a ttrib u te d to such a fa c to r . The observed cross sections fo r a l l p ro je c tile s have been divided by 1 .0 3 . This normalizes the th e o re tic a l and experimental values fo r a t 100 keV. The normalized cross sections are presented as c ir c le s in F ig . 2 . The

s o lid curves provide f a i r l y accurate f i t s to the data, both in magnitude and energy dependence. The s h e ll-s h ie ld e d Coulomb p o te n tia l, th e re fo re , q u ite accu rately describes the in te ra c tio n involved in backscattering of

lig h t atomic p ro je c tile s in the energy range in ve s tig ate d .

\ \ 0

m:

LU

33a

20

50

100

ENERG

Y (keV)

300

Figure 2, Energy dependence of lab cross sections a t 0l= 136.1+“ f o r various p r o je c tile s on g o ld, R utherford; ____ s h e ll-s h ie ld e d Coulomb a->2 a ; — - s h e ll-s h ie ld e d Coulomb; O screened Coulomb; o e xp erim en tal.

APPENDIX 1

THE SHELL SHIELDED COULOMB POTENTIAL Consider a p o te n tia l

:(0 /R ) (1 -R /a ) H (l-R /a ) (7)

where H(x) Is the Heavi side step fu n c tio n . Define U(R) = R (|>(R) = 0 ( l- R /a ) H ( l- R /a ) .

The charge d is tr ib u tio n g (R) giving ris e to th is p o te n tia l ^(R) is given by Polsson's equation

-4TTC(R).

For R > a , vanishes since iji does. For R < a , v^(jl= - 4ttQ. £(R)

For R = a , v^dl=ri d l LR ^>(«31 = 1 ^ U(R) R dR^ lR = a R dR^ R = a

Here 6 (R ) Is the Dirac d e lta fu n c tio n .

Therefore the to ta l v^^ (R ) =J. U"(a)-4TrQ. &XR) R

Note; U " (R )= 0 everywhere except a t R = a , To fin d U "(a), le t x = ( 1 -R /a ).

Then d f = 1 d l <jr2 dx^

and i d i U = l _ d iC x H C x )] R dR^ R a^ d x ^

=0. f z 6 ( x ) + X 5 * (x ) l since H '(x ) = <5(x). Ra^

35

2L & (x ) since f ( x ) & '( x ) = - 6 ( x ) f ' ( x ) Ra^

= & (1 -R /a ) Ra^

Q. 6^ (R-a) since S ( x ) = a ^ ( a x ) Ra

and 8 ( x ) = S ( -x )

0 &(R-a) since 8 ( x ) r O unless R = a. a£

Therefore V ^ t(R ) = & 8(R -a ) - I+ttO, <5(R) a&

= - 4 r r 6 ,

and the charge d is tr ib u tio n is

Ç r Q. &(R)-G(4Tra%) * 6 (R-a) ,

corresponding to a point charge Q. a t the centre of a spherical shell (radius a) of surface charge density -Q(4Tra^)

APPENDIX 11

THE CLASSICAL CROSS SECTION FOR THE SHELL-SHIELDED COULOMB POTENTIAL The v a lid it y of the c la s s ic a l c a lc u la tio n has been discussed by Bohr (1948) and by Mott and Massey (1 9 6 5 ). For a c la s s ic a l t r e a t ­ ment to be v a lid , two conditions must be s a tis fie d :

(1 ) A « a , X « b where X is the de B roglie wavelength of the p r o je c t ile , b is the c o llis io n d ia m e te r,

and a is the screening length.

(2) e > X /2 T T a (See Bohr),

For the cases tre a te d In th is thesis ( H, He, L i, and B a t 50-110 keV on A u), the c la s s ic a l c a lc u la tio n is v a lid a t a l l s c a tte rin g angles 8 > 0 . 1 ° ,

The c la s s ic a l d if f e r e n t ia l s c a tte rin g cross sectio n .

= i d ( p 5 l., = i

d-n- 2 n s in 6 d0 d(cosO)

4 C lln! e /2 ) d(p%)

- 1

(33)

fo r a c en tral p o te n tia l, can be calcu lated from the re la tio n s h ip between the Center-of-Mass (CM) s c a tte rin g angle 6 and the impact parameter p. From the conservation of energy Goldstein (1959) finds

Xo

e=Tr-2 j d y / ^ l- V ( p /y ) /E - y 2 (34)

where y = p /R and y^ c p/R^. Here R^ is the turning point of the o r b it , determined by

E = L^/2uR2 + V (R J r Epf/R + V (R ,) (35)

37

where E is the CM energy, u is the reduced mass and L is the angular momentum.

o In terms of the new v a ria b le y and the parameter (l = Zj^Z2e ‘^ the

p o te n tia l (Eq, 7)

V (R )= d /R (1 -R /a ), O ^ R ^ a = 0 , R > a

becomes V ( p /y ) - Q y /p ( l - p / y a ) , y:^p /a = 0 , y < p/a

S u b stitu tio n of th is p o te n tia l in to Eq. 34 y ie ld s

p/a

8 = "rr-Z \ d y / \Jl-y^' -2 \ d y / \/ i -Qy( 1 -p /y a )/ (pE) -y ^ .

With w =Ea/Q., and c = p /a , we obtain

8 = TT-2 ^ d y / ^1-yZ -2 ^ d y / \ f l + ( l - y / c ) l / w -y ^

(7)

= TT-2 Arcsin y -2 Arcsin j^ (2 y + 1 /w c )/\/(w c )^ + 4 t 4 /w j ^

Using Eq. 35 , the upper lim it % becomes

2y„ = -l/w c + \ l i v i c y ^ + 4 + 4/w

On s u b s titu tio n of th is value in to the previous equation, we fin d

(1 + 2wc^)/ ^1 + (2w c)^(l 1/w^ 8 - 2 Arcsin

= 2 (A-B)

-2 Arcsin c

where sîn A = (I-(-2W c^)/ |/l + (2w c)^(l + 1/w)

3 8

cos A = 2wc / \/r+ (2wc)^ (1 + 1/w)

sin B = c

cos B = ^T-c^

From the previous equations i t follow s th a t

sin 0 /2 = sin(A-B) = sin A cos B - cos A s in B

and

1-c

1 + (2 w c )^ (l+ 1/w)

d (s in ^ J /2 )-

I d (s in ^ e /2 )

I + 2w

-,

2

(36)

= 1 [ i + j k a f l- c 2

s in ( 8 / 2 ) .

The la s t step Is e a s ily v e r if ie d by s u b s titu tin g fo r sin (8 /2 ) (Eq, 36), Thus the d if f e r e n t ia l s c a tte rin g cross section (Eq. 33) becomes

2 i+. l- c '

1 2w

CSC ( 8 / 2 ) , ₍₃₇₎

r (b /4 ) CSC (8 /2 ) = |a _ j^ e s c l 0 / 2 )

(fe)

which allows us to rew rite Eq. 37 In the form

dii-1-c^ 1 +2w

(fi).

w^ l à j \

d W

.1/2

where k = 2 w ( l- c ^ ) /( l+ 2 w )

With the a id of Eq. 36 , the k value becomes

k - l + b / ( 2 a )

1 + b/a + b V (4 a ^ s in 8 /2 )

and the d if f e r e n t ia l cross section is

d ^ =

dcL

ab (a + b /2)

4a sIn^e/2) (a + b) + b^ z

(38)

(9 )

(

8

)

39

APPENDIX 111

TRANSFORMATION FORMULAE BETWEEN LAB AND CM FRAMES

Because the e x p e rim e n ta lis t takes measurements in the laboratory (la b ) system, and the th e o re tic ia n finds i t sim pler to make his c a lc u la tio n s in the Center-of-Mass (CM) system, conversion formulae from one frame of reference to the other are necessary.

Consider a n o n r e la tiv is tic binary e la s t ic c o llis io n which asym pto tically appears in the lab system as in F ig . 11.

’'08

cm

CM

'e m

— >

(a ) before c o llis io n (b) a f t e r c o llis io n F ig . 11. Asymptotic views of an e la s t ic c o llis io n in the lab system.

I n i t i a l l y a p a r t i c l e o f mass is m oving w i t h v e l o c i t y v^^^ and

impact parameter p towards a p a r tic le of mass m ^ a t rest in the lab

frame. A fte r s c a tte rin g through angles dj^ and (|l^, m^ and ra^ have v e lo c itie s v,g and re s p e c tiv e ly . In the lab frame the center of mass of the

41

two p a rtic le s moves w ith constant v e lo c ity « The magnitude and d ire c tio n of are determined by the conservation theorem fo r to ta l lin e a r

momentum

(m^+ = m, Vo4

The CM system moves w ith respect to the lab system so th a t the o rig in of the CM system is always coincident w ith the center of mass of the c o llid in g p a r tic le s . In the CM frame the to ta l energy is

E = Ejj - i ( m i f mg^)vjn = ""z Eg #1+ m%

Because the to ta l lin e a r momentum is zero a t a l l times in th is frame, the p a rtic le s are tr a v e llin g in opposite d ire c tio n s to one another before and a f t e r the c o llis io n . Hence they share a common angle of s c a tte rin g , e (F ig . 12).

m 'e m

'em

m

'em

/(CM

(a ) before col 1 i s i on (b) a f t e r col 1is i on F ig , 12. Asymptotic views of an e la s t ic c o llis io n in the CM system.

42

From a vector ad d itio n diagram re la tin g the f in a l v e lo c itie s in the CM and lab frames, the re la tio n s h ip between the CM and lab s c a tte rin g angles can be obtained.

Vcm

'cm

'12

F ig . 13. A vector a d d itio n diagram of the v e lo c itie s of the p a rtic le s in the CM and lab frames.

tan = (%i *Vg.m) s in 6 _ s in 6

0

m^^/m^+cos e

The second step follow s by the d e fin itio n of the center of mass v e lo c ity : "’zycm" N ote t h a t i f 6 — 6^ and th e la b and CM

systems are almost id e n tic a l. For the case 6jj = 0|_= 1 3 6.4 °, the CM

43

Also from F ig , 13 we see th a t

2 sin (T T /2 -(|)^ )

and 2 t t- 0 .

Hence 2 s in (6 /2 )

and the k in e tic energy tra n s fe rre d to the ta rg e t I f the incident p r o je c tile is e la s t ic a lly scattered through 6 is

T«= = 4m, E s in ^ (e /2 ). (27) (m, + m^)

111:1 R elationship of the Cross Sections

In an actual experiment the counting rate of a fix e d d etecto r is independent of the reference frame of the observer,

i . e . I i ^ \ dxL - / ^ \ dr^e \d W Vdnjji

For s c a tte rin g by s p h e ric a lly symmetric p o te n tia ls , the s c a tte rin g d is tr ib u tio n is independent of the azimuthal ang le, so th a t

drig_ sin 6g dQa . dj^ “ s i n e de

S u b s t i t u t i n g In 0^ = 0|_= 1 3 6 .4 and th e CM s c a t t e r i n g a n g le s d e te r m in e d

above, the ra tio s (d-ajj/dn _) fo r ^H, ^He, ^Li and on Au are found to be 1.0074, 1.0302, 1.0538 and 1.0956 re s p e c tiv e ly . The cross sections c a lcu lated in the CM system are converted to the lab system by d iv id in g by these fa c to rs .

APPENDIX IV

EXPERIMENTAL DIFFERENTIAL SCATTERING CROSS SECTION

Consider a th in ta rg e t f i l m of thickness t and density N atoms per u n it volume. In a th in f i l m no ta rg e t atom is blocked by any o th e r. Suppose a co llim ated beam of monoenergetic p ro je c tile s and cross-sectional area A s trik e s the f i l m a t normal incidence. The nimiber of s c a tte rin g centers in the path of the beam Is given by the product AtN. Denoting the d if f e r e n t ia l cross section per ta rg e t atom fo r s c a tte rin g in to a u n it s o lid angle in the d ire c tio n 8^ in the laboratory system by d q ~( 6 f . Ef ) . the e ffe c tiv e ta rg e t area presented to the beam fo r such

d ^

s c a tte rin g is

AtN û £ i § j L J k ) d s i .

where E_g is the lab energy. The fra c tio n of the in cident p a rtic le s backscattered in to the u n it s o lid angle is

Nt d q - ( 6 i.E j) d XL

I f a detecto r subtends a lab s o lid angle AXi_, and the fra c tio n of the incident ions detected is denoted by B (the backscatter y ie l d ) , then

B ^ N t dcr(6fl . E a ) A x l . d XL

The lab d if f e r e n t ia l s c a tte rin g cross section is then

. (32)

d N (A ix jt

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45

V IT A AUCTORIS

Born November 6 , 1946 in Leamington, Ontario.

Received elementary and secondary school education at Harrow, Ontario.

Graduated from University of Windsor with B. Sc. in Physics in 1 9 6 9 .