Mean Excitation Energies Stopping Powers in Various ... · PDF fileexcitation levels and corresponding oscillator strengths is the main hindrance. It was suggested by Dalgarno (ref

NASA Tech n ica I

I Paper ~ 2271

1984

Nallonal Aeronautics and Space Administration

Scientific and Technical Information Branch

Mean Excitation Energies -

for Stopping Powers in Various Materials Using Local Plasma Oscillator Strengths

John W. Wilson Langley Research Center Hampton, Virginia

Y. J. Xu Old Dominion University Norfolk, Virginia

E fs t a t hios Kamara t os and C. K. Chang Christopher Newport College Newport News, Virginia

https://ntrs.nasa.gov/search.jsp?R=19840013417 2018-05-14T15:58:44+00:00Z

SUMMARY

The basic model of Lindhard and S c h a r f f , known as t h e l o c a l plasma model, i s u t i l i z e d t o s t u d y the e f f e c t s on s t o p p i n q power of the chemical and p h y s i c a l s t a t e of t h e medium. Unlike p r e v i o u s work with t h e l o c a l plasma model, i n which i n d i v i d u a l e l e c t r o n s h i f t s i n the plasma frequency were estimated e m p i r i c a l l y , the P i n e s correc- t i o n d e r i v e d f o r a d e g e n e r a t e Fermi gas is shown h e r e i n t o provide a reasonable esti- mate, even on t h e atomic scale. Thus, t h e model is moved t o a complete theoretical base r e q u i r i n g no empirical ad jus tments , as c h a r a c t e r i s t i c of p a s t a p p l i c a t i o n s . The p r i n c i p a l remaining error is i n t h e o v e r e s t i m a t i o n of t h e low-energy a b s o r p t i o n prop- ert ies t h a t are c h a r a c t e r i s t i c of t h e plasma model i n t h e r e q i o n of t h e atomic d i s - crete spectrum, a l though hiqher-energy phenomena are a c c u r a t e l y r e p r e s e n t e d , and even e x c i t a t i o n - t o - i o n i z a t i o n ra t ios a r e given to f a i r accuracy . Mean e x c i t a t i o n e n e r g i e s f o r covalent-bonded gases and s o l i d s , f o r i o n i c g a s e s and c r y s t a l s , and f o r metals are c a l c u l a t e d u s i n g f i r s t - o r d e r models of the bonded states, f o r which r e a s o n a b l e aqreement wi th t h e r e c e n t l y e v a l u a t e d data of S e l t z e r and Berger i s obta ined . Hence, t h e methods d e s c r i b e d h e r e i n a l l o w reasonable estimates of mean e x c i t a t i o n energy f o r any physical-chemical combinat ion of m a t e r i a l media f o r stopping-power a p p l i c a t i o n s .

INTRODUCTION

The impact of e n e r g e t i c i o n s w i t h atoms and molecules is a fundamental p h y s i c a l phenomenon w i t h numerous a p p l i c a t i o n s i n a r e a s such as a s t r o p h y s i c s , medical p h y s i c s , high-power lasers, and plasma f u s i o n d e v i c e s , and is of special i n t e r e s t f o r r a d i a - t i o n p r o t e c t i o n of people and m a t e r i a l s i n space. Most of what i s known i n modern p h y s i c s is u l t i m a t e l y r e l a t e d t o o b s e r v a t i o n s of c h a r g e d - p a r t i c l e impact phenomena, by which par t ic le c h a r a c t e r i s s tud ied , s t a r t i n g w i t h t h e e a r l y works of Ruther ford u s i n g r a d i o a c t i v e s o u r c e s . To advance o u r knowledge of such p r o c e s s e s i s both t i m e l y and t e c h n o l o g i c a l l y impor tan t . The h i s t o r i c a l c o n t e x t i n which p r e s e n t t h e o r e t i c a l developments are to be undefstood i s d i s c u s s e d i n the f o l l o w i n g section.

HISTORICAL PERSPECTIVE

I t w a s recognized e a r l y i n t h e c l a s s i c a l t r e a t m e n t of charged p a r t i c l e s lowing down t h a t t h e f r e e - e l e c t r o n , . long-range Coulomb i n t e r a c t i o n l e a d s t o d i v e r g e n c i e s i n t h e energy- loss rate. These d i v e r g e n c i e s i n d i c a t e t h a t t h e r e i s a need f o r a long- range s a t u r a t i o n e f f e c t . me s a t u r a t i o n i n gases w a s d i s c u s s e d by Bohr ( r e f . 1) i n terms of E h r e n f e s t ' s p r i n c i p l e . Bohr proposed t h a t t h e s a t u r a t i o n i n g a s e s i s caused by t h e bonding of t h e e l e c t r o n s . To e f f e c t energy t r a n s f e r , t h e i n t e r a c t i o n t i m e 'G = b/v (where b i s t h e impact parameter and v t h e i o n v e l o c i t y ) must be s h o r t compared w i t h t h e o s c i l l a t i n g p e r i o d of t h e bonded e l e c t r o n . ( A l is t of symbols and a b b r e v i a t i o n s u s e d i n t h i s paper a p p e a r s a f t e r t h e r e f e r e n c e s . ) long-range c o l l i s i o n s p r o v i d e t h e n e c e s s a r y s a t u r a t i o n , and a n upper l i m i t i s estab- l i s h e d fo r t h e e f fec t ive impact parameters . Most of o u r modern u n d e r s t a n d i n g stems

Hence, t h e a d i a b a t i c

from B e t h e ' s d e t a i l e d quantum t h e o r y ( r e f . 2 ) based on the Born approximation. S topping power for gaseous media wi th t h i s approximation is g iven by

where Z, i s t h e project i le charge , n i s t h e number of t a r g e t s p e r u n i t volume, Z2 is t h e number of e l e c t r o n s per t a r g e t , m i s t h e e l e c t r o n mass, v i s t h e pro- j e c t i l e v e l o c i t y , p = v/c, c i s t h e v e l o c i t y of l i g h t , C i s t h e v e l o c i t y - dependent s h e l l - c o r r e c t i o n t e r m ( r e f . 3), and I2 i s t h e mean e x c i t a t i o n energy g iven by s o l v i n g

z I n 1 = I f I ~ E n n 2 2 n

where f n is t h e e l ec t r i c d i p o l e osc i l la tor s t r e n g t h of t h e t a r g e t and En i s the cor responding e x c i t a t i o n energy. The sum i n e q u a t i o n (2) i n c l u d e s d i s c r e t e and continuum l e v e l s . E m p i r i c a l l y , it w a s observed t h a t molecular s t o p p i n g p o w e r is reasonably approximated by t h e sum of t h e cor responding e m p i r i c a l l y d e r i v e d "atomic" s t o p p i n g powers ( r e f . 4 ) . Equat ions ( 1 ) and ( 2 ) imply

where Z and I p e r t a i n to t h e molecule , Z j and I j are t h e cor responding atomic v a l u e s , and n j r e p r e s e n t s t h e s t o i c h i o m e t r i c c o e f f i c i e n t s . This a d d i t i v i t y r u l e , g i v e n by e q u a t i o n ( 3 1 , is c a l l e d Bragg's rule.

Sources of d e v i a t i o n s from Bragg 's a d d i t i v i t y r u l e f o r molecules and t h e condensed phase a r e d i s c u s s e d by Platzman ( r e f . 5 ) . Aside from s h i f t s i n e x c i t a t i o n e n e r g i e s and ad jus tments i n l i n e s t r e n g t h s as a r e s u l t of molecular bondinq, new terms i n t h e s topping power are caused by t h e c o u p l i n g of v i b r a t i o n a l and r o t a t i o n a l modes. A d d i t i o n a l l y , i n t h e condensed phase, some discrete t r a n s i t i o n s are moved i n t o t h e continuum, and c o l l e c t i v e modes among va lence e l e c t r o n s i n a d j a c e n t atoms produce new terms to be d e a l t w i t h i n t h e a b s o r p t i o n spectrum. Platzman proposed t h a t t h e e x p e r i m e n t a l l y observed a d d i t i v i t y rule may n o t show t h a t molecular S topping power i S t h e sum of a tomic p r o c e s s e s b u t r a t h e r t h a t it demonst ra tes t h a t molecular bond s h i f t s f o r covalent-bonded molecules are r e l a t i v e l y independent of the molecular combinat ion. On t h e b a s i s of such arguments, Platzman s u g g e s t e d ionic-bonded sub- s t a n c e s a s t h e ones which should be s t u d i e d as a r igid test of t h e a d d i t i v i t y r u l e because of t h e r a d i c a l d i f f e r e n c e i n bonding type . bond s h i f t s could change t h e s t o p p i n g p o w e r by as much as 50 p e r c e n t .

H e f u r t h e r estimated t h a t i o n i c -

Among t h e e a r l y i n d i c a t o r s of t h e v i o l a t i o n of t h e Bragg r u l e w a s t h e C a l C U l a -

t i o n Of 15 e V f o r t h e mean e x c i t a t i o n energy of a tomic hydrogen ( u s i n g eq. ( 2 ) w i t h t h e e x a c t l y known o s c i l l a t o r s t r e n g t h s and e x c i t a t i o n l e v e l s ) compared w i t h a r a t h e r f i r m l y e s t a b l i s h e d e x p e r i m e n t a l va lue f o r molecular hydrogen of about 1 8 e V . S i n c e accurate values of atomic mean e x c i t a t i o n e n e r g i e s have been c a l c u l a t e d for numerous

2

e lements by I n o k u t i and co-workers (refs. 6 to 8) f o r t h e purpose of e v a l u a t i n g chem- i c a l bonding e f f e c t s i n molecules, empirical v a l u e s have been s u b s t a n t i a l l y p e r t u r b e d by e f f e c t s of t h e chemica l bonds. Although t h e mean e x c i t a t i o n energy f o r g a s mole- c u l e s cou ld be e v a l u a t e d i n p r i n c i p l e from e q u a t i o n (21, t h e l a c k of knowledge of t h e e x c i t a t i o n l e v e l s and co r re spond ing o s c i l l a t o r s t r e n g t h s i s t h e main h ind rance .

It was sugges t ed by Dalgarno (ref. 9) t h a t t h e o s c i l l a t o r s t r e n g t h d i s t r i b u t i o n s cou ld be de te rmined e m p i r i c a l l y from t h e pho toabso rp t ion s p e c t r a ( a s i d e from e x p e r i - mental u n c e r t a i n t y ) . Much of t h e s e d a t a are o b t a i n e d by energy- loss exper iments by e l e c t r o n impact s c a t t e r i n g a t forward ang le s . Values of mean e x c i t a t i o n energy f o r a number of s imple molecules have i n t h i s way been e s t i m a t e d , and have demonst ra ted t h e s h i f t i n atomic v a l u e s caused by chemical bonding ( r e f s . 10 and 1 1 ) .

T h e o r e t i c a l c a l c u l a t i o n of mean e x c i t a t i o n e n e r g i e s is h inde red by t h e d i f f i - c u l t y of s o l v i n g f o r t h e complete e x c i t a t i o n spectrum of complex quantum systems. Dalgarno w a s a b l e t o s i m p l i f y t h e c a l c u l a t i o n by i n t r o d u c i n g a g e n e r a l i z e d f u n c t i o n , which i s r e l a t e d t o the e x c i t a t i o n spectrum a s fo l lows :

However, t h i s f u n c t i o n can be e v a l u a t e d wi thout e x p l i c i t l y forming t h e i n d i c a t e d sum. Thus, Dalgarno ( r e f . 1 2 ) was a b l e to reduce e q u a t i o n ( 4 ) t o

w i t h

L

( H - E + w)”X + ;i$o = 0 0 i=l

where is the ground-state+wave f u n c t i o n , Eo i s the cor responding energy , w i s an energy e i g e n v a l u e , and x is the cor responding e i g e n v e c t o r . Chan and Dalgarno ( r e f . 1 3 ) c a l c u l a t e d I as 42 e V f o r helium and Kamikawai e t a l . ( r e f . 14) ca lcu - l a t e d 18.2 e V f o r molecular hydrogen by t h e same method. These va lues are i n e x c e l - l e n t agreement wi th exper iments .

S imul taneous w i t h the development of the microscop ic theo ry of s t o p p i n g p o w e r w a s t h e macroscopic e l ec t rodynamic d e s c r i p t i o n of energy loss as r e q u i r e d f o r the d e s c r i p t i o n of t h e long-range p a r t of the in t e rac t ion i n condensed phase. T h i s is because t h e i n t e r a c t i o n i s s imul taneous among many c o n s t i t u e n t s . The s lowing down i s th rough the f o r c e e x e r t e d on t h e p a s s i n g particle by t h e e lec t r ic f i e l d induced i n

3

t h e medium by t h e passage ( r e f . 1 5 ) . I t is customary t o assume tha t the e lec t r ic d isp lacement v e c t o r is l i n e a r l y r e l a t e d to t h e t ime-varying electric f i e l d as

I f o r which t h e d i e l e c t r i c c o n s t a n t i s

The shor t - ranqe c o l l i s i o n s are s t i l l t r e a t e d by Bethe t h e o r y wi th t h e r e s u l t f o r t o t a l s toppinq power (see r e f . 16 f o r d e t a i l s ) of

where 6 is a d e n s i t y - e f f e c t c o r r e c t i o n a p p l i c a b l e a t h i g h e n e r g i e s ( B 2 > 1 /E ( 0 ) ) . A l s o ,

00

2 I n 1 = I w Im[&] ln (hw) dw 2 2n2ne2 o

where I m ( Z ) d e n o t e s t h e imaginary p a r t of 2. A r e s u l t of d i s p e r s i o n t h e o r y i s

( 1 0 )

where f ( w ) is t h e d i p o l e o s c i l l a t o r s t r e n g t h per u n i t c e l l of t h e medium, so t h a t

which reduces t o t h e u s u a l Bethe e x p r e s s i o n (eq. ( 2 ) ) i n a s p a r s e gas f o r which €(a) = 1.

I f t h e long-range s a t u r a t i o n e f f e c t i s i n terms of a d i a b a t i c l i m i t s f o r a g a s and i n terms of t h e medium p o l a r i z a t i o n r e s p o n s e €or condensed d i e l e c t r i c s , t h e s a t u r a t i o n e f f e c t f o r a f r e e - e l e c t r o n g a s i s related t o t h e tendency of a n e u t r a l

4

plasma t o s c r e e n a l o c a l charge imbalance a t l a r g e d i s t a n c e s ( r e f . 1 7 ) . The d i e l e c - t r i c f u n c t i o n of a f r e e - e l e c t r o n gas is der ived by Lindhard ( ref . 1 8 ) and a p p l i e d t o t h e stopping-power problem f o r a classical e l e c t r o n gas and f o r t he i n t e r a c t i o n - f r e e

1 Sommerfeld e l e c t r o n qas model. For a f r e e - e l e c t r o n gas a t rest, Lindhard a r r i v e s a t t h e equa t ion

2 4 4nZ1 e p s = 2 l n ($)

mv \ P I ( 1 3 )

where p is t h e e l e c t r o n d e n s i t y and w i s t h e c lass ica l plasma frequency g iven by P

2 2 4ne P 0 = -

P m ( 1 4 )

S t r i c t l y speaking , equa t ion (13 ) a p p l i e s only when t h e e l e c t r o n gas is a t rest, b u t it a l s o a p p l i e s i n t h e l i m i t of h igh p r o j e c t i l e v e l o c i t y compared wi th the ave rage motion of t h e e l e c t r o n s .

A d i scove ry which p a r a l l e l e d t h e Lindhard i n v e s t i g a t i o n s w a s made by Ebhm and Pines ( r e f s . 19 t o 2 1 ) , i n which c o l l e c t i v e long-range i n t e r a c t i o n s i n a quantum e l e c t r o n gas w e r e s e p a r a t e d from i n d i v i d u a l e l e c t r o n motion through a c a n o n i c a l t r a n s f o r m a t i o n , a f t e r which t h e normal c o o r d i n a t e s of c o l l e c t i v e o s c i l l a t i o n appea r . T h i s s e p a r a t i o n of t h e Hamil tonian i n t o c o l l e c t i v e and i n d i v i d u a l e l e c t r o n motion i s accomplished because of t h e e f f e c t i v e sc reen ing of t h e Coulomb f i e l d s of i n d i v i d u a l e l e c t r o n s f o r d i s t a n c e s g r e a t e r t h a n t h e s c r e e n i n g d i s t a n c e motion t o g i v e a major c o n t r i b u t i o n t o t h e Hamiltonian, t h e i n d i v i d u a l e l e c t r o n wave- l e n g t h must be g r e a t e r t h a n A . Bohm and Pines ( r e f . 2 1 ) found the average c o l l e c - t i v e plasma f requency t o be

hc. For c o l l e c t i v e

C

where h i s t h e ave rage e l e c t r o n s e p a r a t i o n and x i s t h e ra t io of t h e ave rage e l e c t r o n wavelength t o t h e s c r e e n i n g d i s t ance . P ines ( r e f . 22) s u g g e s t s t h a t t h e s c r e e n i n g pa rame te r x shou ld be chosen t o minimize t h e e l e c t r o n long-range c o r r e l a - t i o n energy ( t h a t i s , t h e e l e c t r o n i c Coulomb ene rgy) , which, f o r plane-wave s ta tes a p p r o p r i a t e t o t h e i r d e g e n e r a t e electron gas , is g iven by

S

4 - 0.866% 0 . 4 5 8 ~ + 0.019x 3

S h

S - A

- Elr, corr Al. 5

S

(1 6)

P i n e s ( r e f . 22) d e r i v e d the s topp ing power i n t h i s degene ra t e e l e c t r o n gas and showed t h a t t h e u s u a l c lass ical plasma frequency w is r ep laced by <w>, which i n c l u d e s c o r r e c t i o n s f o r i n d i v i d u a l e l e c t r o n motion. P

5

A r a t h e r bold s u g q e s t i o n w a s made by Lindhard and Schar f f ( r e f . 23) t h a t equa- t i o n ( 1 3 ) could be a p p l i e d on t h e atomic scale i f t h e a p p r o p r i a t e averaqe o v e r t h e atomic e l e c t r o n d e n s i t y w a s made. They f u r t h e r suqqes ted t h a t t h e e f f e c t s of i n d i - v i d u a l bonding of t h e e l e c t r o n s i n t h e i r a tomic o r b i t a l s could be i n c o r p o r a t e d through t h e added f a c t o r y = )12 as

I Following t h i s i n i t i a l s u c c e s s of t r e a t i n g atoms as l o c a l i z e d e l e c t r o n plasmas , Lindhard and Winther ( r e f . 24) ex tended e q u a t i o n ( 1 7 ) by u s i n g t h e more g e n e r a l ve loc i ty-dependent d i e l e c t r i c f u n c t i o n d e r i v e d by Lindhard ( r e f . 181, and demon- s t r a t e d t h e a b i l i t y of t h e Lindhard t h e o r y t o p r e d i c t t i g h t bonding c o r r e c t i o n s o f s imi la r c h a r a c t e r t o t h o s e of Walske ( r e f . 25) i n c o n n e c t i o n w i t h t h e Bethe t h e o r y .

From e q u a t i o n ( 1 7 ) , t h e mean e x c i t a t i o n enerqy i s qiven by

I AS noted by Dehmer e t a l . ( r e f . 6 ) e q u a t i o n (18 ) may be r e w r i t t e n as

Lindhard and Schar f f e s t i m a t e d t h e mean e x c i t a t i o n energy f o r atomic Hg as 768 e V compared with =800 e V from exper iments . For H e , t h e y g o t 37 e V compared w i t h 35 e V from q u o t e d experiments. They f u r t h e r approximated molecular hydrogen by t a k i n g t h e e f f e c t i v e charge t o be 2 = 1 . 2 and o b t a i n e d 1 6 eV.

Chu and Powers ( ref . 26) made e x t e n s i v e u s e of t h e work of Lindhard and S c h a r f f ( r e f . 23) t o demonstrate Z2 o s c i l l a t i o n s i n t h e mean e x c i t a t i o n energy. T h i s work gave r ise t o corresponding Z o s c i l l a t i o n s i n s t o p p i n g power from which p e r i o d i c v a r i a t i o n s are a s s o c i a t e d w i t 2 t h e a tomic s h e l l s t r u c t u r e ( r e f . 26) . d e t a i l e d c a l c u l a t i o n s of Rousseau e t a l . ( r e f . 27) u t i l i z e d t h e v e l o c i t y - d e p e n d e n t Lindhard-Winther t h e o r y ( ref . 24) and Bonderup’s s i m p l i f i e d form of t h e Lindhard t h e o r y ( r e f . 28) and show good agreement w i t h 2-MeV a l p h a - p a r t i c l e stopping-power d a t a ( ref . 29) . Throughout t h e s e e f f o r t s , t h e parameter y i s t a k e n a s t h e s q u a r e r o o t of 2, as sugges ted by Lindhard and S c h a r f f ( r e f . 23) .

The more

Chu e t a l . ( r e f . 301, u s i n q t h e t h e o r y of Lindhard and Winther i n which i n d i v i d - u a l e l e c t r o n c o r r e c t i o n s t o t h e l o c a l c o l l e c t i v e e x c i t a t i o n were t r e a t e d e m p i r i c a l l y by t a k i n q y as an a d j u s t a b l e parameter , e v a l u a t e d t h e a g g r e g a t i o n e f f e c t s f o r condensed noble gases and metals . The condensed-gas c a l c u l a t i o n s determined e l e c t r o n d e n s i t i e s accordinq t o a tomic Hartree-Fock d e n s i t i e s , i n c l u d i n q o v e r l a p from t h e n e a r e s t neighbors i n t h e condensed phase. Metallic wave f u n c t i o n s w e r e t aken from t h e muff in- t in c a l c u l a t i o n s of Moruzzi e t a l . ( r e f . 31 ). I n most cases, y w a s i n t h e ranqe from 1 .2 t o 1.3. (See r e f . 32.)

from which can be obta ined

where 6 ( x ) i s t h e d e l t a f u n c t i o n . It i s seen from e q u a t i o n ( 2 0 ) t h a t , i n t h e local plasma approximation, t h e volume of plasma wi th c u t o f f f requency ywp = w approxi - mates t h e t o t a l o s c i l l a t o r s t r e n g t h of t h e system a t f requency w. No e x a c t equiva- lence i s impl ied between t h e osc i l la tor frequency d i s t r i b u t i o n g i v e n by e q u a t i o n ( 2 0 ) and t h e osci l la tor f requency d i s t r i b u t i o n of a quantum system. (This i s t r u e because eq. ( 2 0 ) e x h i b i t s a cont inuous spectrum, al though quantum systems g e n e r a l l y e x h i b i t a series of poles a s s o c i a t e d w i t h t h e d i s c r e t e quantum levels as w e l l as a continuum a t h i g h e r f r e q u e n c i e s . ) Some i n s i g h t may be ga ined by comparing d i s p e r s i o n r e l a t i o n s f o r atomic sys tems w i t h t h o s e f o r a re la ted plasma. The d i s p e r s i o n r e l a t i o n f o r a c lass ica l plasma i s g i v e n by t h e d i e l e c t r i c c o n s t a n t E ( W ) as

where w i s the u s u a l plasma f requency and e q u a t i o n ( 2 1 ) r e s u l t s from t h e plasma c o n d u c t i v i t y ( r e f . 33) . Indeed, t h e same pole a p p e a r s i n m e t a l s as the r e s u l t of t h e conduct ion e l e c t r o n s from which t h e c h a r a c t e r i s t i c o p t i c a l properties of metals d e r i v e ( r e f s . 33 and 34) . The more g e n e r a l d i s p e r s i o n r e l a t i o n , d e r i v e d from equa- t i o n s (8) and ( 1 1 1 , is

P

2 47rZ e f ( x ) dx

€ ( W ) = 1 - m P J - 2 2 x - w 0

( 2 2 )

where P d e n o t e s the p r i n c i p a l va lue a t t h e s i n g u l a r i t y . I n a tomic systems, the osc i l la tor s t r e n g t h s are b r o a d l y s e p a r a t e d i n f r e q u e n c i e s a c c o r d i n g to s h e l l s ; t h e o u t e r s h e l l s appear a t the lowest f r e q u e n c i e s , and the innermost shel l a p p e a r s a t the h i q h e s t f r e q u e n c i e s . The l a c k of o s c i l l a t o r s t r e n g t h a t f r e q u e n c i e s between s h e l l s r e s u l t s i n l a r g e gaps i n the spectrum. L e t w be a frequency i n the broad gap between t w o s u c c e s s i v e s h e l l s - t h e f i r s t c e n t e r e d a t w and t h e second a t w Then the d i s p e r s i o n r e l a t i o n (eq. ( 2 2 ) ) becomes 2 - 1

7

where

w 2

1 f ( x ) dx 0

2 4~re Z w 2 =

PI 1 m ( 2 4 )

s o t h a t u p , l s h e l l . Although e q u a t i o n s ( 2 3 ) and ( 2 4 ) provide m o t i v a t i o n f o r u s i n g t h e local plasma approximation (eq. ( 2 0 ) 1, t h e r e is p l e n t y of room f o r a more complete under- s t a n d i n g as t o why t h e model works as w e l l as it does i n p r a c t i c a l c a l c u l a t i o n s ( r e f . 35) .

i s t h e plasma f requency a s s o c i a t e d w i t h t h e e l e c t r o n s of t h e outermost

I n prev ious i n v e s t i g a t i o n s , we cons idered t h e use of t h e local plasma model t o e v a l u a t e molecular bonding e f f e c t s on t h e mean e x c i t a t i o n energy of molecules of c o v a l e n t l y bonded hydrogen and carbon ( r e f . 35) as w e l l as i o n i c c r y s t a l s and g a s e s ( r e f . 36) , i n which q u i t e s e n s i b l e c o r r e c t i o n s t o t h e u s u a l Bragg r u l e were o b t a i n e d . The chemical bond s h i f t s were unambiguously d e f i n e d i n terms of a tomic i n t e g r a l s and molecular parameters . I n t h e u s u a l implementat ion of t h e local plasma model (eq. ( 1 8 1 1 , y c o r r e c t s f o r a s h i f t i n t h e l o c a l plasma f requency caused by i n d i v i d - u a l e l e c t r o n e f f e c t s . Lindhard and Schar f f ( r e f . 23) s u q g e s t y = E; however, y a 1.2 y i e l d s atomic mean e x c i t a t i o n e n e r g i e s from t h e l o c a l plasma model i n be t te r agreement with t h e a c c u r a t e atomic va lues c a l c u l a t e d by Dehmer e t a l . ( ref . 6). The f a c t t h a t t h e l a r q e r value ( y = @) g i v e s b e t t e r aqreement wi th e m p i r i c a l . d a t a suq- q e s t s t h a t t h i s l a r q e r value corrects ( i n a d d i t i o n t o i n d i v i d u a l e l e c t r o n s h i f t s ) f o r t h e chemical s h i f t s as w e l l . Such chemical s h i f t s w e r e e s t i m a t e d s e p a r a t e l y fo r c o v a l e n t and i o n i c bonds i n r e f e r e n c e s 35 and 36.

Encouraqed by t h e s m a l l n e s s ( t 3 0 p e r c e n t ) of t h e e m p i r i c a l i n d i v i d u a l e l e c t r o n c o r r e c t i o n s t o t h e c o l l e c t i v e plasma f requency ( r e f s . 32, 35, and 3 6 ) , a c a l c u l a t i o n ( r e f . 37) i n which i n d i v i d u a l e l e c t r o n s h i f t s were e s t i m a t e d a c c o r d i n g t o t h e t h e o r y f o r plane-wave s ta tes i n an extended plasma, as c a l c u l a t e d by Pines ( r e f . 221, y i e l d s r e s u l t s t h a t a r e i n qood agreement wi th Dehmer e t a l . ( r e f . 6 ) . Consequent ly , t h e local plasma model i s p laced on a parameter - f ree basis i n which chemical s h i f t s are determined from atomic molecular parameters a l o n e , and effects of i n d i v i d u a l e l e c t r o n motion are e v a l u a t e d i n terms of t h e P ines c o r r e c t i o n , t h e combined e f f e c t s of which are on t h e order of t h e plasma frequency s h i f t of y - J2 sugqes ted by Lindhard and Scharf f .

The Pines c o r r e c t i o n makes a remarkable improvement i n t h e p r e d i c t i o n Of the local plasma model, and f u r t h e r ad jus tments i n t h e t h e o r y , to account for t h e plasma f requency s h i f t s r e s u l t i n g from t h e atomic s h e l l s t r u c t u r e , should b r i n g t h e model i n t o a c c u r a t e p r e d i c t i v e c a p a b i l i t y . To f u r t h e r e l u c i d a t e t h e r e l a t i o n s h i p between t h e l o c a l plasma model and t h e more e x a c t quantum t r e a t m e n t of bonded sys tems, r e l a t e d q u a n t i t i e s of both t h e o r i e s i n t h e case of one- and two-electron systems are examined i n the fo l lowinq s e c t i o n of t h i s r e p o r t . A t o m i c mean e x c i t a t i o n e n e r g i e s and s t r a g q l i n q parameters , based on t h e local plasma model, a r e compared w i t h accu- ra te c a l c u l a t i o n s of I n o k u t i e t a l . ( refs . 6 t o 8 ) i n the s e c t i o n e n t i t l e d "Stopping and S t r a q q l i n q Parameters of A t o m s . " The use of t h e Gordon-Kim e l e c t r o n qas model of molecular bondinq ( r e f . 3 8 ) to de termine the e f fec ts of c o v a l e n t chemical hond s h i f t s on mean e x c i t a t i o n energy f o r e lements of t h e f i r s t t w o rows is p r e s e n t e d i n t h e s e c t i o n e n t i t l e d "Covalent-Bond E f f e c t s . " C a l c u l a t i o n s of mean e x c i t a t i o n e n e r g i e s of ionic-bonded s u b s t a n c e s are d i s c u s s e d i n t h e s e c t i o n e n t i t l e d "Ionic-Bond

8

E f f e c t s , " and t h e mean e x c i t a t i o n e n e r g i e s of metals are d i s c u s s e d i n t h e s e c t i o n e n t i t l e d "Metallic-Bond E f f e c t s . "

EXCITATION SPECTRA OF ONE- AND TWO-ELECTRON SYSTEMS

The hydrogen atomic e x c i t a t i o n spectrum i n t h e d i p o l e approximation i s w e l l known a s

(ha > R )

f .(w) = H k exp(- ; a r c t a n k

where n is t h e p r i n c i p a l quantum number, R is Rydberg's c o n s t a n t , W is g iven n by

2 ( 2 6 ) f i w = R(l - 1/n 1

n

and

Rk2 = i i w - R

The cor responding spectrum f o r t h e l o c a l plasma model (eq. ( 2 0 ) ) is given as

f ( w ) = P 1 ( w < wo)

Where 3U = 55.12 eV. The cumulat ive o s c i l l a t o r s t r e n g t h 0

( 2 7 ) I

( 2 8 )

i s shown i n f i g u r e 1 for each of t h e two models. S i m i l a r l y , t h e e x c i t a t i o n spectrum of t h e hel ium atom h a s been e v a l u a t e d f o r screened wave f u n c t i o n s and is shown i n f i g u r e 1. The f r a c t i o n a l e x c i t a t i o n s of the t w o models never d i f f e r by more t h a n =I5 p e r c e n t above t h e e x c i t a t i o n t h r e s h o l d . A s noted by Dehmer e t a l . ( r e f . 61, t h e main error i n t h e local plasma model is t h e c o n t r i b u t i o n t o a b s o r p t i o n b e l o w e x c i t a - t i o n t h r e s h o l d a l l t h e way down to zero. This error is also e v i d e n t i n t h e energy

9

moments of the plasma model. The moments of atom are shown i n f i g u r e 2, where

t h e energy spectrum f o r t h e hydrogen

(30)

requency c o n t r i b u t i o n s a s s o c i a t e d wi th and m i s a cont inuous parameter . The low- t h e local plasma model cause a d ivergence i n e q u a t i o n ( 3 0 ) a t m = -2 which is n o t p r e s e n t i n t h e quantum system. Atomic p o l a r i z a b i l i t y and t h e low-frequency r e f r a c - t i v e index a r e a f f e c t e d t h e most. Other atomic p r o p e r t i e s , such as t h e t o t a l i n e l a s - t i c c r o s s sec t ion , the mean e x c i t a t i o n energy, t h e s t r a g g l i n g parameter, and t h e mean e l e c t r o n i c k i n e t i c energy, are reasonably r e p r e s e n t e d by t h e plasma model. A l s o shown i n f i g u r e 2 are r e s u l t s , i n c l u d i n g t h e P ines c o r r e c t i o n ( r e f . 22) t o t h e plasma frequency, which i n d i c a t e s u b s t a n t i a l improvement i n t h e p r e d i c t i o n of atomic proper - t i e s , a l thouqh low-energy atomic properties are s t i l l beyond t h e scope of t h e model.

The plasma model is expec ted t o be more a c c u r a t e as more e l e c t r o n s are added t o t h e system. This occurs i n t w o ways, as seen i n f i g u r e 1 . F i r s t , a g r e a t e r c o n t r i - b u t i o n comes from t h e continuum, which is most l i k e t h e plasma. Second, t h e e x c i t a - t i o n t h r e s h o l d s s h i f t to r e l a t i v e l y lower e n e r g i e s and f i l l i n t h e low-frequency r e g i o n , f o r which t h e plasma model normally t e n d s t o e r r o r . A c o n s i d e r a b l e improvement i n t h e energy moments of hel ium f o r t h e local plasma approximation are c l e a r l y shown i n f i g u r e 3.

The moments of t h e e x c i t a t i o n spectrum of H2 have been e v a l u a t e d e m p i r i c a l l y by u s i n g exper imenta l o s c i l l a t o r s t r e n g t h s ( r e f . 39) and t h e o r e t i c a l l y ( re f . 1 4 ) u s i n g t h e Dalgarno sum r u l e s of r e f e r e n c e 1 2 (eqs. ( 4 ) t o ( 6 ) ) . These are compared i n f i g u r e 4 with an "atomic" approximat ion t o Bragg 's r u l e ( r e f . 1 1 ) . Also shown i n f i g u r e 4 are v a l u e s f o r H c a l c u l a t e d u s i n g t h e l o c a l plasma model w i t h t h e P ines c o r r e c t i o n and w i t h t h e Gorion-Kim model of t h e molecular wave f u n c t i o n s ( r e f . 38) g i v e n a s

H2 t a k e n as a g e n e r a l i z a t i o n of

-+ where pH(:) is t h e atomic hydrogen e l e c t r o n d e n s i t y and R i s t h e d i s p l a c e m e n t vector of l e n g t h 1.4a between t h e t w o c e n t e r s . It i s clear from f i g u r e 4 t h a t , even w i t h t h e simple Gordon-Kim approximat ion , t h e plasma model is a c o n s i d e r a b l e improvement over t h e Bragg r u l e , e x c e p t f o r t h e lowest-energy molecular properties (i.e., m < -0.5). Figure 4 a l so shows t h a t t h e Gordon-Kim approximat ion i n t r o d u c e s minor errors compared w i t h t h e i n h e r e n t l i m i t a t i o n s of t h e local plasma model.

0

The mean e x c i t a t i o n energy f o r s t o p p i n g power may l i k e w i s e be e v a l u a t e d . A t o m i c and molecular hydroqen a r e compared i n table I w i t h a r e c e n t c o m p i l a t i o n of e x p e r i - mental d a t a ( re f . 40) . Quite r e a s o n a b l e estimates of atomic and molecular p r o p e r t i e s of importance t o i o n i z i n g r a d i a t i o n are o b t a i n e d by t h i s local plasma model if the Pines c o r r e c t i o n i s inc luded . O p t i c a l and o t h e r low-frequency p r o p e r t i e s , however, are p o o r l y represented . The plasma model should become more a c c u r a t e f o r more com- p l e x many-electron systems, e s p e c i a l l y t h o s e i n which t h e o p t i c a l p r o p e r t i e s are more i n l i n e w i t h those p r e d i c t e d by t h e plasma model.

With t h e p r e s e n t r e s u l t s , it is now clear what approach should be t a k e n t o improve t h e plasma model a p p l i c a t i o n s . C l e a r l y , a c o r r e c t i o n factor s imilar to tha t

1 0

of P i n e s should be i n t r o d u c e d t o s u p p r e s s a b s o r p t i o n b e l o w e x c i t a t i o n t h r e s h o l d and, cor respondingly , t o enhance f r e q u e n c i e s j u s t above t h r e s h o l d . A number of p o s s i b i l i -

~ t ies are open t o implement such a correction, which would appear as a f i r s t - o r d e r I quantum c o r r e c t i o n f o r t h e d i s c r e t e spectrum. P r e l i m i n a r y work by Walecka ( r e f . 41) ‘ development . on t h e s t u d y of c o l l e c t i v e atomic o s c i l l a t i o n s may be a s t a r t i n g p o i n t f o r f u r t h e r

I STOPPING AND STRAGGLING PARAMETERS OF ATOMS

I I I n t h i s s e c t i o n , parameters are cons idered f o r atoms a s s o c i a t e d w i t h t h e s t o p -

p i n g of charged particles and f l u c t u a t i o n s i n t h e i r energy t r a n s f e r . The energy moment i s

and t h e r e l a t e d q u a n t i t y i s

I n terms of t h e s e q u a n t i t i e s , t h e mean e x c i t a t i o n energy is

and t h e s t r a g g l i n g parameter r e l a t e d to f l u c t u a t i o n s i n energy loss is

The mean e x c i t a t i o n enerqy (eq. (34)) has been e v a l u a t e d i n t h e c o n t e x t of t h e l o c a l plasma model and is compared i n f i g u r e 5 with t h e va lues computed by I n o k u t i and co-workers ( refs . 6 t o 8) f o r atoms through krypton. Hartree-Fock wave f u n c t i o n s ( r e f . 42) have been used through neon and sodium through krypton are r e p r e s e n t e d by s c r e e n e d wave f u n c t i o n s ( r e f . 43).

The values f o r t h e s t r a g g l i n g parameter w e r e s i m i l a r l y e v a l u a t e d and are com- p a r e d i n f i g u r e 6 w i t h t h e v a l u e s o b t a i n e d by I n o k u t i and co-workers. A l s o shown are v a l u e s f o r n o b l e gases compiled by I n o k u t i e t al . ( r e f s . 6 t o 8) and v a l u e s o b t a i n e d by Zeiss e t a l . ( r e f . 11 ) . Tne present v a l u e s t e n d t o be about 25 p e r c e n t l o w a t Z 36, w i t h improvements a t lower v a l u e s of Z , which may be caused by t h e lack of s h e l l s t r u c t u r e c o r r e c t i o n s i n t h e plasma f r e q u e n c i e s of t h e K and L s h e l l s .

I t is clear from t h e s e atomic c a l c u l a t i o n s t h a t t h e plasma model wi th t h e P ines c o r r e c t i o n g e n e r a l l y p r o v i d e s good r e s u l t s f o r mean e x c i t a t i o n energy and r e a s o n a b l e estimates f o r t h e s t r a g g l i n g parameter. Although Hartree-Fock wave f u n c t i o n s are r e q u i r e d for l o w atomic numbers, r e a s o n a b l e r e s u l t s are o b t a i n e d u s i n g screened wave f u n c t i o n s f o r atoms h e a v i e r t h a n argon. It is mainly t h e low-energy atomic

1 1

p r o p e r t i e s which r e q u i r e improvements beyond t h e P i n e s c o r r e c t i o n . These p r o p e r t i e s emphasize t h e need f o r a f i r s t - o r d e r quantum c o r r e c t i o n t o t h e atomic s t r u c t u r e .

COVALENT-BOND E F F E C T S

Ear ly exper imenta l work w i t h i o n i z a t i o n energy loss w a s conducted i n c o v a l e n t - bonded g a s e s ( a l s o noble g a s e s ) , from which Bragg 's r u l e w a s der ived . Although more r e c e n t exper imenta l work, beginning w i t h Thompson ( r e f . 44) , h a s shown s y s t e m a t i c v a r i a t i o n from Bragg ' s r u l e , such r u l e s s t i l l seem a p p r o p r i a t e f o r f i x e d molecular s t r u c t u r e s ( r e f s . 45 and 46) . A s t h e r e s u l t of the t h e o r e t i c a l e f f o r t s o f I n o k u t i and co-workers ( r e f s . 6 t o 81, it i s clear t h a t chemical-bond s h i f t s i n t h e mean e x c i t a t i o n enerqy have o c c u r r e d , and, a s sugges ted by Platzman ( r e f . 51, a l l c o v a l e n t s h i f t s a r e of s imilar magnitude.

I n any molecular dynamic c a l c u l a t i o n , t h e r e i s a t r a d e - o f f between model accu- r a c y and computat ional e f f i c i e n c y . A s p e r t a i n s t o t h e r a d i o l y s i s of l a r g e molecular s t r u c t u r e s , t h e most u s e f u l model i s t h e lowest o r d e r possible. It i s clear t h a t t h e u s e of s e l f - c o n s i s t e n t f i e l d methods t o de termine molecular wave f u n c t i o n s would s e r i o u s l y l i m i t t h e a b i l i t y t o s t u d y systems of p r a c t i c a l i n t e r e s t . C o n s i d e r i n g t h e r e l a t i v e success of t h e Gordon-Kim e l e c t r o n g a s model of molecular bonding ( r e f s . 38, 47, and 481, a s imple method f o r t h e c a l c u l a t i o n of chemical-bond e f f e c t s on t h e mean e x c i t a t i o n e n e r g i e s i s sugges ted . As sugges ted by Gordon and K i m , t h e molecular e l e c t r o n d e n s i t y a s a s u p e r p o s i t i o n of t h e unper turbed atomic s ta tes i s g i v e n by

f o r d i a t o m i c molecules. There is a n obvious g e n e r a l i z a t i o n of e q u a t i o n ( 3 6 ) f o r t h e polyatomic case. Whereas Gordon and K i m u s e d e q u a t i o n ( 3 6 ) t o c a l c u l a t e t h e molecu- l a r p o t e n t i a l (see r e f s . 47 and 48 f o r i o n i c and c o v a l e n t a p p l i c a t i o n s ) from which R i s t h e o r e t i c a l l y o b t a i n e d , d i s t a n c e s . S u b s t i t u t i n g e q u a t i o n 6 ) t n t o e q u a t i o n s ( 1 4) and (1 8) and r e d u c i n g r e s u l t s i n

i s t a k e n h e r e from observed e x p e r i m e n t a l bond R1 3 12

where I, and I2 are t h e c o r r e s p o n d i n g atomic v a l u e s , which are a c c u r a t e l y known ( r e f s . 6 t o 8 ) . The chemical bonding c o r r e c t i o n i s g e n e r a l l y

1 + + + l n ( 1 t 6. . ) = p. (r) l n [ l + p . (r - R d 3 r

1 1 i j i 1 3 ( 3 8 )

G e n e r a l i z i n g equat ion ( 3 7 ) y i e l d s

12

where t h e sum over j i n c l u d e s e v e r y bond i n which Z is a t t a c h e d i n t h e molecule. C o r r e c t i o n f a c t o r s have been c a l c u l a t e d ( r e f . 35) f o r hydrogen and carbon molecules w i t h t h e bond parameters i n table 11. Carbon sp3 h y b r i d o rb i t a l wave f u n c t i o n s were used i n t h e s e c a l c u l a t i o n s , a l though s2p2 values were only s l i g h t l y d i f f e r e n t . The t e t r a h e d r a l o r b i t a l s w e r e s p h e r i c a l l y symmetr ical i n their e l e c t r o n d e n s i t i e s . Therefore , s p h e r i c a l symmetry w a s assumed throughout subsequent c a l c u l a t i o n s .

Recommended va lues of mean e x c i t a t i o n e n e r g i e s ( r e f . 40) are compared i n table I11 w i t h t h e o r e t i c a l v a l u e s c a l c u l a t e d u s i n g atomic mean e x c i t a t i o n e n e r g i e s from Dehmer e t a l . ( r e f . 6 ) w i t h t h e bond c o r r e c t i o n s i n table 11. Bragg 's r u l e i s also used w i t h t h e atomic v a l u e s of Dehmer e t a l . f o r comparison. Although t h e theo- r e t i c a l v a l u e s are w i t h i n 4 p e r c e n t of t h e exper imenta l and empirical v a l u e s , t h e Bragg 's r u l e v a l u e s are from 17 to 21 p e r c e n t l o w , i n d i c a t i n g a s u b s t a n t i a l a d j u s t - ment as t h e r e s u l t of chemical bonding.

Mean e x c i t a t i o n e n e r g i e s have been c a l c u l a t e d f o r c o v a l e n t gases of the f i r s t t w o r o w s u s i n g t h e local plasma model and t h e P i n e s c o r r e c t i o n . R e s u l t s are compared w i t h empirical v a l u e s ( r e f . 40) i n t a b l e I V . Corresponding v a l u e s f o r c o v a l e n t sol- ids are shown i n table V.

Moments f o r t h e N 2 molecule u s i n g t h e plasma model are compared i n f i g u r e 7 w i t h v a l u e s c a l c u l a t e d from t h e o s c i l l a t o r s t r e n g t h s compiled by Dalgarno e t a l . ( r e f . 4 9 ) . As can be seen , good agreement between t h e present simple plasma model c a l c u l a t i o n s and t h e osc i l la tor s t r e n g t h d i s t r i b u t i o n of r e f e r e n c e 49 i s o b t a i n e d e x c e p t f o r t h e lowest-f requency phenomena.

IONIC-BOND EFFECTS

Although c o v a l e n t bond s h i f t s were found t o be r e l a t i v e l y small c o r r e c t i o n s t o atomic v a l u e s , such a s e p a r a t i o n as i n equat ion ( 3 7 ) i n terms of n e u t r a l atomic va l - ues are n o t p o s s i b l e for i o n i c bonds. Using t h e Gordon-Kim model e l e c t r o n d e n s i t y of t h e par t ia l i o n i c ( d i a t o m i c ) system,

( 4 0 )

where and B ( - p ) r e f e r t o p a r t i a l l y i o n i c s ta tes of t h e t w o c o n s t i t u e n t s , itAB i s t h e i r n u c l e a r s e p a r a t i o n , and p i s the p a r t i a l i o n i c f r a c t i o n . The elec- t r o n d e n s i t y of a p a r t i a l i o n i c atom i n e q u a t i o n ( 4 0 ) i s

-b -b (r) is t h e elec- where p A ( r )

t r o n d e n s i t y of t h e atomic ion . w i t h t h e a i d of e q u a t i o n s ( 4 0 ) and (41 1 , s h i f t s i n

I P f

is t h e electron d e n s i t y of the n e u t r a l atom and A

13

t h e mean e x c i t a t i o n energy caused by i o n i c and c o v a l e n t e f f e c t s can be e v a l u a t e d . As shown i n r e f e r e n c e 36,

w i t h

where y i s t h e P ines c o r r e c t i o n g i v e n by e q u a t i o n ( 1 5 ) or e s t i m a t e d e m p i r i c a l l y as i n r e f e r e n c e 46. l a t e d u s i n g t h e P ines c o r r e c t i o n and t h e a t o m i c wave f u n c t i o n s of Clementi and Roetti ( r e f . 42) are shown i n f i g u r e 8. In a d d i t i o n t o t h e t h e ionic-bond s h i f t s , t h e r e are s h i f t s caused by c o v a l e n t - l i k e c h a r a c t e r , as g iven by

Mean e x c i t a t i o n e n e r g i e s f o r v a r i o u s s t a g e s of i o n i z a t i o n c a l c u -

Mean e x c i t a t i o n e n e r g i e s f o r p a r t i a l ionic-bonded s u b s t a n c e s are shown i n t ab le V I w i t h t h e cor responding bond parameters used i n t h e model. Also shown are v a l u e s f o r a p u r e c o v a l e n t bond and Bragg v a l u e s u s i n g t h e n e u t r a l atomic mean e x c i t a t i o n e n e r - g i e s of Dehmer e t a l . ( r e f . 6) as w e l l a s Bragg v a l u e s of t h e c o r r e s p o n d i n g p a r t i a l i o n i c s ta tes . “he ionic-bond f r a c t i o n s a r e t a k e n from P a u l i n g ( r e f . 50) as experi- menta l d a t a f o r HF and L i H . Bond l e n g t h s are f o r i o n i c c r y s t a l s e x c e p t f o r t h e HF gas. A t o m i c mean e x c i t a t i o n e n e r g i e s are shown f o r p a r t i a l i o n i c states i n f i g u r e 8, and d i f f e r from v a l u e s i n r e f e r e n c e 3 6 because of t h e P i n e s c o r r e c t i o n .

It is c l e a r from t a b l e V I t h a t t h e main c o n t r i b u t i o n t o c o r r e c t i o n s t o t h e Bragg r u l e i s t h e ad jus tment from atomic n e u t r a l t o a t o m i c i o n mean e x c i t a t i o n e n e r g i e s as proposed by Platzman (ref. 5) . Indeed, when t h e r e i s l i t t l e d i f f e r e n c e between t h e u s u a l Bragg va lue and t h e p a r t i a l l y i o n i c Bragg v a l u e , t h e c o v a l e n t v a l u e i s i n near agreement wi th t h e p r e d i c t e d v a l u e of I f o r HF and L i H i n t h e table. For LiF, t h e r e l a t i v e l y l a r g e ad jus tment from t h e u s u a l Bragg v a l u e (81.6) t o t h e p a r t i a l l y i o n i c Bragg v a l u e (92.6) l e a v e s a l a r g e d i f f e r e n c e between t h e c o v a l e n t v a l u e ( 8 3 . 4 ) and t h e p r e d i c t e d v a l u e of I (93.6). The a d j u s t m e n t of t h e ionic-bond s h i f t c a u s e d by t h e c o v a l e n t - l i k e c h a r a c t e r f o r LiF i s 1 e V compared w i t h a d j u s t m e n t s of t h e n e u t r a l s ta tes caused by t h e p u r e c o v a l e n t bond of 1.8 eV. T h i s comparison shows t h e g r e a t e r r o l e of t h e Coulomb a t t r a c t i o n i n forming t h e bond o f t h e i o n i c molecules r e l a t i v e t o t h e two-electron i n t e r a c t i o n i n forming t h e c o v a l e n t bond.

Calcu la ted mean e x c i t a t i o n e n e r g i e s f o r i o n i c c r y s t a l s u s i n g t h e P i n e s correc- t i o n a re shown i n t a b l e V I I , a l o n g w i t h recommended v a l u e s ( r e f . 4 0 ) . The c r y s t a l parameter and f r a c t i o n a l i o n i c c h a r g e have been t a k e n from P a u l i n g ( r e f . 5 0 ) . The LiF v a l u e is t h e only one w i t h a n e x p e r i m e n t a l basis ( ref . 3 6 ) .

1 4

METALLIC-BOND EFFECTS

The f i r s t approach t o metals i n t h i s paper i s s imilar t o t h a t t a k e n by Chu e t a l . ( r e f . 30) , i n which t h e y employed t h e muff in- t in wave f u n c t i o n s ( ref . 31) and stopping-power t h e o r y a c c o r d i n g t o Lindhard and Winther ( r e f . 24) . I n d i v i d u a l elec- t r o n c o r r e c t i o n s t o t h e l o c a l plasma frequency are t r e a t e d e m p i r i c a l l y through a n a d j u s t a b l e parameter y. (See t a b l e I of ( r e f . 32) and r e l a t e d d i s c u s s i o n . ) Unlike t h i s p r e v i o u s work, t h e p r e s e n t work i n c l u d e s e s t i m a t e s of s h i f t s i n t h e plasma f r e - quency a c c o r d i n g t o t h e P ines c o r r e c t i o n i n e q u a t i o n (15) and i s i n t h a t s e n s e com- p l e t e l y de termi n i s t i c .

The metall ic wave f u n c t i o n s f o r l i t h i u m metal approximated by t h e Wigner-Seitz model ( r e f . 51) are c o n s i d e r e d f i r s t . I n d e r i v i n g t h e s e wave f u n c t i o n s , t h e l i t h i u m i o n core p o t e n t i a l w a s t a k e n from t h e screened wave f u n c t i o n s of Clementi and Raimondi ( r e f . 431, and t h e c a l c u l a t e d c r y s t a l - v a l e n c e wave f u n c t i o n s (aside from n o r m a l i z a t i o n ) w e r e found t o be a s l i g h t p e r t u r b a t i o n (mainly due t o boundary condi- t i o n s ) of t h e f r e e hydrogenic ( 2 s ) o r b i t a l i n s i d e t h e Wigner-Seitz sphere . %e f i n a l c r y s t a l wave f u n c t i o n s used were c o n s t r u c t e d from unper turbed Hartree-Fock o r b i t a l s (ref. 42) i n t h e c o r e r e g i o n w i t h a small p e r t u r b a t i o n o u t s i d e t h e c o r e . This per- t u r b a t i o n matched t h e boundary c o n d i t i o n s on t h e s u r f a c e of t h e Wigner-Seitz s p h e r e ( re f . 51 1. This w a s fo l lowed by n o r m a l i z a t i o n of t h e v a l e n c e - s h e l l wave f u n c t i o n s ( t o make t h e v a l e n c e e l e c t r o n d e n s i t y add u p t o g i v e t h e c o r r e c t number of v a l e n c e e l e c t r o n s ) . These wave f u n c t i o n s are q u i t e s imi la r t o t h e m u f f i n - t i n model ( r e f . 31) and y i e l d mean e x c i t a t i o n e n e r g i e s i n s u b s t a n t i a l agreement w i t h Z i e g l e r ( r e f . 32) when y is t a k e n a s h i s e m p i r i c a l v a l u e . The mean e x c i t a t i o n e n e r g i e s fo r metals o f t h e second and t h i r d row u s i n g Wigner-Seitz wave f u n c t i o n s ( t r e a t i n g a l l v a l e n c e e l e c t r o n s as s p a t i a l l y e q u i v a l e n t ) and t h e Pines c o r r e c t i o n are compared i n t a b l e V I 1 1 w i t h empirical v a l u e s ( r e f . 40) .

The p r e s e n t r e s u l t s c l e a r l y demonst ra te t h a t t h e e f f e c t s of t h e metall ic bond i n l i t h i u m and b e r y l l i u m are l a r g e and a r e mainly t h e r e s u l t o f c o l l e c t i v e o s c i l l a t i o n s i n t h e f r e e - e l e c t r o n g a s formed by t h e va lence e l e c t r o n s . Although s imi l a r good agreement s h o u l d be expec ted f o r sodium and magnesium, i t i s emphasized h e r e o n l y t h a t t h e s e e m p i r i c a l v a l u e s are i n t e r p o l a t i o n s w i t h o u t a n e x p e r i m e n t a l b a s i s , and smaller empirical v a l u e s more i n l i n e w i t h t h e p r e s e n t r e s u l t s s h o u l d n o t be e l i m i - nated. The s m a l l v a l u e p r e d i c t e d f o r aluminum (149 e V ) i s i n doubt , as t h e empirical v a l u e (166 f 4 ev) i s based on one of t h e most e x p e r i m e n t a l l y s t u d i e d q u a n t i t i e s S i n c e aluminum s e r v e d as a s t a n d a r d i n stopping-power experiments f o r many years . The f a u l t c o u l d w e l l l i e i n t h e use of t h e Wigner-Seitz model f o r group I11 meta ls . I t i s well-known t h a t t h e s u c c e s s of t h e Wigner-Seitz theory rests mainly on a p p l i c a - t i o n t o a l k a l i metals. Although some hope f o r a p p l i c a t i o n t o group I1 metals e x i s t s , t r e a t i n g t h e t h r e e va lence e l e c t r o n s of group I11 as s p a t i a l l y e q u i v a l e n t i s c l e a r l y i n error. C o r r e c t i o n t o metals from a n a l t e r n a t e model, proposed by P a u l i n g (ref. 50) f o r metal l ic o r b i t a l s and implemented h e r e i n s i m p l i f i e d f a s h i o n , are c o n s i d e r e d n e x t .

I n X-ray d i f f r a c t i o n exper iments , even bery l l ium metal shows a c o n s i d e r a b l e d e g r e e of c o v a l e n t q u a l i t y , as s u s p e c t e d from bulk m a t e r i a l p r o p e r t i e s ( r e f . 5 2 ) . I n t h i s view, a model i s c o n s i d e r e d i n which t h e valence-bond e f f e c t s can be i n c l u d e d e x p l i c i t l y . I n t h e s p i r i t of t h e Paul ing valence-bond t h e o r y ( r e f . 50) and t h e Gordon-Kim model (ref. 38) of va lence bonding, t h e e l e c t r o n d e n s i t y a b o u t t h e i o n cores i s assumed t o be a s u p e r p o s i t i o n of p a r t i a l i o n i c core s ta tes among n e a r e s t core ne ighbors . A d d i t i o n a l c o n t r i b u t i o n s from n e x t - n e a r e s t ne ighbors are assumed t o add t o t h e e l e c t r o n continuum s ta tes i n a manner analogous t o t h e Paul ing

15

unsynchronized resonances i n l i t h i u m c r y s t a l s (ref. 50) . The e l e c t r o n d e n s i t y of t h e p a r t i a l l y i o n i c c o r e charge p i s

where v i s t h e number of v a l e n c e e l e c t r o n s , p A ( ? ) i s t h e e l e c t r o n d e n s i t y of t h e a t o m i c n e u t r a l s t a t e , and i s t h e e l e c t r o n d e n s i t y of t h e v a l e n c e - s t r i p p e d

i o n cores. We have used t h e o b s e r v a t i o n by S l a t e r t h a t r a d i a l wave f u n c t i o n s of t h e L s h e l l are n e a r l y t h e same f o r b o t h v a l u e s of !I as a r e s u l t of exchange i n t e r a c - t i o n be tween t h e ( 2 s ) and ( I S ) o r b i t a l s . The same i s t r u e f o r t h e M s h e l l . In t h e p r e s e n t c a l c u l a t i o n , each metal i o n c o r e h a s been p l a c e d i n t o a Wigner-Seitz c e l l and t h e e l e c t r o n d e n s i t y from n e a r e s t ne ighbors h a s been approximated by r e f l e c t i n g t h e e x t e r i o r core d e n s i t y f u n c t i o n a c r o s s t h e c e l l boundary. The continuum e l e c t r o n d e n s i t y is then t a k e n as

+ p ( + v ) ( r ) A

where 6 i s t h e next -neares t -ne ighbor c o n t r i b u t i o n t o t h e continuum. The v a l u e of 6 i s determined by r e q u i r i n g a f u l l complement of v va lence e l e c t r o n s p e r ce l l . The r e s u l t a n t e lec t ron d e n s i t y p ( r ) w a s used t o c a l c u l a t e t h e l o c a l plasma f r e - quency and mean e x c i t a t i o n energy p e r cel l . The P i n e s c o r r e c t i o n w a s u s e d f o r i n d i - v i d u a l p a r t i c l e s h i f t s . The r a d i i rs f o r t h e Wigner-Seitz c e l l are g i v e n i n t ab le V I I I . !he ion-core wave f u n c t i o n s were c a l c u l a t e d from t h e Hartree-Fock wave f u n c t i o n s of Clementi and Roetti ( r e f . 4 2 ) . A s l i g h t dependence on t h e i o n - c o r e charge a p p e a r s ( r e f . 53) i n which t h e r e i s some i n c r e a s e i n mean e x c i t a t i o n t o I = 155 e V f o r aluminum. However, t h e r e a re some u n r e s o l v e d q u e s t i o n s , c o n c e r n i n g p e r i o d i c i t y a t t h e ce l l boundar ies , which l e a v e t h e v a l u e of t h i s model i n doubt .

The mean e x c i t a t i o n energy f o r aluminum r e q u i r e s t h e r e c o n s i d e r a t i o n of t h e d a t a on which i t i s based and t h e cor responding a n a l y s i s . I n a n a n a l y s i s by Andersen and Z i e g l e r ( r e f . 5 4 ) , 162 e V was assumed as t h e mean e x c i t a t i o n energy f o r aluminum. A r e d u c t i o n i n I t o 150 e V r e s u l t s i n a 3-percent i n c r e a s e i n s t o p p i n g power a t 1 MeV, which leaves i t w i t h i n t h e s t a t e d u n c e r t a i n t y l i m i t s o f t h e Andersen and Z i e g l e r parametric c u r v e s ( r e f . 54) . These c u r v e s cor respond t o t h e u n c e r t a i n t y i n t h e exper imenta l d a t a used i n t h e a n a l y s i s of r e f e r e n c e 54. (See f i g . 9.) Indeed, a number of a u t h o r s have r e p o r t e d mean e x c i t a t i o n e n e r g i e s f o r aluminum i n l i n e w i t h t h e p r e s e n t r e s u l t s ( r e f s . 55 t o 6 0 ) , a l t h o u g h more r e c e n t a n a l y s e s a r e h i g h e r . A r e c e n t s t u d y of aluminum o p t i c a l p r o p e r t i e s i n d i c a t e s t h a t a v a l u e of I s e v e r a l e V l o w e r than 166 e V i s n o t i n c o n s i s t e n t wi th t h e empirical d i e l e c t r i c f u n c t i o n ( r e f . 61) . The s h i f t of s e v e r a l e V i s a s s o c i a t e d w i t h p o l a r i z a t i o n of t h e AX3+ c o r e by t h e va lence e l e c t r o n s i n t h e i r metall ic orb i ta l s . Such core p o l a r i z a t i o n e f f ec t s a r e n o t c a l c u l a t e d i n t h e p r e s e n t model. Furthermore, quantum c o r r e c t i o n s t o K- and L - s h e l l d i s c r e t e spectrum may c a u s e f u r t h e r small a d j u s t m e n t s . In any case, t h e a p p a r e n t d i screpancy i s due t o t h e e l e c t r o n i c wave f u n c t i o n s used i n t h e p r e s e n t c a l c u l a t i o n , t o t h e i n a d e q u a t e t r e a t m e n t of c o r r e c t i o n s t o t h e Bethe formula , from which I i s e x t r a c t e d from t h e e x p e r i m e n t a l d a t a (see f o r example r e f s . 62 and 631, t o quantum c o r r e c t i o n s , or t o a combinat ion of t h e s e .

To f u r t h e r c l a r i f y t h e r e l a t i o n s h i p between t h e mean e x c i t a t i o n e n e r g y for a l u - minum and exper imenta l d a t a , a band i s shown i n f i g u r e 9 which b r a c k e t s t h e experi-

1 6

m e n t a l d a t a i n r e f e r e n c e s 62 and 64 t o 68 f o r p r o t o n e n e r g i e s between 0.5 and 10 MeV. t h e - Andersen and Z i e g l e r empirical s h e l l c o r r e c t i o n s ( r e f . 54) . The o l d e r d a t a of Kahn ( r e f . 6 4 ) , which would have lowered t h e band c o n s i d e r a b l y , w e r e exc luded from t h e f i g u r e . The mean e x c i t a t i o n e n e r g i e s e x h i b i t e d i n t h e f i g u r e are t h e 167 e V used as i n p u t t o S h i l e s e t a l . ( r e f . 611, t h e 1 6 2 e V de te rmined by Andersen and Z i e g l e r ( r e f . 54) , t h e 155 e V e s t i m a t e d u s i n g one form o f valence-bond t h e o r y , and t h e 149 e V c a l c u l a t e d a c c o r d i n g t o t h e p r e s e n t ( s i m p l i f i e d ) Wigner-Seitz model. While i t i s n o t clear t h a t t h e 167-eV curve i s s u p e r i o r t o t h e 149-eV curve , a modest s h i f t i n t h e empirical s h e l l c o r r e c t i o n s can b r i n g any of t h e f o u r c u r v e s i n t o a n e q u a l l y good f i t t o t h e d a t a . It i s f u r t h e r emphasized t h a t s h e l l c o r r e c t i o n s are n o t e x a c t l y known n o r i n empirical a n a l y s i s a re s h e l l c o r r e c t i o n s u s u a l l y d i f f e r e n t i a t e d from o t h e r c o r r e c t i o n s t o t h e Bethe formula (eq. (1 ) 1.

These e n e r g i e s are compared w i t h t h e reduced s t o p p i n g power c a l c u l a t e d from

DISCUSSION

The p r e s e n t r e s u l t s are combined i n f i g u r e 1 0 w i t h t h e e v a l u a t e d d a t a of S e l t z e r and Berger ( r e f . 40). Care i s t a k e n where p o s s i b l e t o model t h e same p h y s i c a l - chemical s ta te . (See s p e c i f i c tables f o r d e t a i l s . ) R e s u l t s f o r f r e e atoms (Hartree- Fock wave € u n c t i o n s f o r Z < 10, and screened wave f u n c t i o n s e l sewhere) are a lso com- p a r e d i n f i g u r e 1 0 w i t h t h e a c c u r a t e atomic v a l u e s of Dehmer e t a l . ( r e f . 6 ) . It i s clear t h a t t h e t r e n d s i n t h e f i r s t - and second-period e lements a re w e l l approximated by t h e p r e s e n t a p p l i c a t i o n of t h e local plasma model, e s p e c i a l l y when t h e P i n e s cor- r e c t i o n i s a p p l i e d . The p r e s e n t r e s u l t s are g e n e r a l l y i n f a i r agreement w i t h t h e c o m p i l a t i o n a n d recommendations of S e l t z e r and Berger ( r e f . 401, a l t h o u g h small d i s - c r e p a n c i e s i n t h e t h i r d p e r i o d remain t o be reso lved .

Perhaps t h e g r e a t e s t cri t icism of t h e p r e s e n t a p p l i c a t i o n of t h e l o c a l plasma model c a l c u l a t i o n s i s t h e u s e of t h e Gordon-Kim approximat ion t o t h e covalent-bonded wave f u n c t i o n s . When t h e moments of t h e energy spectrum are cons idered , it i s c lear t h a t t h e Gordon-Kim model approximate ly a d j u s t s t h e e x c i t a t i o n spectrum i n t h e r e g i o n of g r e a t e s t impor tance t o i o n i z i n g r a d i a t i o n , and a p p e a r s no more i n error t h a n t h e basic plasma model i n which i t i s used. (See f i g . 4.) Of c o u r s e , a c c u r a t e u s e of t h e plasma model implies t h e necessary use of t h e P i n e s c o r r e c t i o n , as demonstrated f o r t h e hydrogen atom i n f i g u r e 2 and u s e d throughout t h e p r e s e n t c a l c u l a t i o n s . Although t h e P i n e s c o r r e c t i o n produces marked improvements i n t h e p r e d i c t i v e capabil- i t y of t h e model, f u r t h e r quantum c o r r e c t i o n s fo r t h e d i s c r e t e spectrum would produce a d d i t i o n a l c o r r e c t i o n s and would h o p e f u l l y remove most of t h e remaining error i n t h e plasma model. F u r t h e r improvement i n e l e c t r o n i c wave f u n c t i o n s would be h e l p f u l i n i d e n t i f y i n g t h e remaining c o r r e c t i o n s r e q u i r e d f o r t h e plasma model.

ACKN OWLED GMENTS

The a u t h o r s wish t o e x p r e s s t h e i r g r a t i t u d e t o Govind S. Khandelwal of Old Dominion U n i v e r s i t y , W i l l a r d E. Meador of t h e Langley Research Center , Mito I n o k u t i

17

of t h e Argonne N a t i o n a l Labora tory , and Stephen S e l t z e r of t h e N a t i o n a l Bureau of S tandards f o r h e l p f u l d i s c u s s i o n s and encouragement. S p e c i a l thanks are o f f e r e d t o Stephen S e l t z e r , Mart in Berger, and Mito I n o k u t i f o r s h a r i n g unpubl i shed d a t a .

Langley Research Center N a t i o n a l Aeronaut ics and Space A d m i n i s t r a t i o n Hampton, VA 23665 February 3, 1984

REFERENCES

1. Bohr, N.: On t h e Decrease of V e l o c i t y of S w i f t l y Moving E l e c t r i f i e d P a r t i c l e s i n P a s s i n g Through Matter. P h i l o s . Mag. & J. S c i . , ser. 6, vol. 30, no. 175, J u l y 191 5 , pp. 581-61 2.

2. Bethe, Von H.: Zur Theor ie d e s Durchgangs s c h n e l l e r K o r p u s k u l a r s t r a h l e n durch Materie. Ann. Phys., 5 Folge, Bd. 5 , 1930, pp. 325-400.

3. Walske, M. C.; and Bethe, H. A.: Asymptotic Formula f o r S topping Power of K-Electrons. Phys. Rev. , vo l . 83, 1951, pp. 457-458.

4. Bragg, W. H.; and Kleeman, R.: On t h e a Par t ic les of Rad ium, and T h e i r Loss of Range i n P a s s i n g Through Various Atoms and Molecules. Phi los . Mag. & J. S c i . , ser. 6, vo l . 10, no. 57, Sept . 1905, pp. 318-340.

5. Platzman, R o b e r t L.: I n f l u e n c e s of Details of E l e c t r o n i c Binding on P e n e t r a t i o n Phenomena, and t h e P e n e t r a t i o n of Energe t ic Charged P a r t i c l e s Through Liquid Water. Symposium on Radiobiology - The B a s i c Aspects of R a d i a t i o n E f f e c t s on Liv ing Systems, James J. Nickson, ed. , John Wiley & Sons, Inc. , c.1952, pp. 139-176.

6. Dehmer, J. L.; I n o k u t i , Mitio; and Saxon, R . P.: Sys temat ics of Moments of Dipole O s c i l l a t o r - S t r e n g t h D i s t r i b u t i o n s f o r A t o m s o f t h e First and Second Row. Phys. Rev. A, t h i r d ser., vol. 12, no. 1 , J u l y 1975, pp. 102-121.

7. I n o k u t i , Mitio; Baer, T.; and Dehmer, J. L.: Addendum: Sys temat ics of Moments of Dipole O s c i l l a t o r - S t r e n g t h D i s t r i b u t i o n s f o r A t o m s i n t h e F i r s t and Second Row. Phys. Rev. A , v o l . 17, no. 3, Mar. 1978, pp. 1229-1231.

8. I n o k u t i , M.; Dehmer, J. L.; Baer, T.; and Hanson, J. D.: O s c i l l a t o r - S t r e n g t h Moments, Stopping Powers, and T o t a l I n e l a s t i c - S c a t t e r i n g Cross S e c t i o n s o f A l l A t o m s Through Stront ium. Phys. Rev. A, vol. 23, no. 1 , Jan. 1981, pp. 95-109.

9. Dalgarno, A.: The Stopping Powers of Atoms. Proc. Phys. SOC. (London), v o l . 76, pt . 3, no. 489, Sept . 1 , 1960, pp. 422-424.

10. Zeiss, G. D.; and w a t h , W . J.: The H 2 0 - H 2 0 Dispers ion Energy Cons tan t and t h e D i s p e r s i o n of t h e S p e c i f i c R e f r a c t i v i t y of D i l u t e Water Vapour. M o l . Phys., v o l . 30, no. 1 , J u l y 1975, pp. 161-169.

1 1 . Zeiss, G. D.; Meath, W i l l i a m J.; MacDonald, J. C . F.; and Dawson, D. J.: Accu- rate E v a l u a t i o n of Stopping and S t r a g g l i n g Mean E x c i t a t i o n m e r g i e s f o r N , 0, H 2 , N 2 , 02, NO, N H ~ , H 2 0 , and N 2 0 Using Dipole Osc i l la tor S t r e n g t h D i s t r i b u - t i o n s - A T e s t of t h e V a l i d i t y of Bragg's Rule. Radiat . Res., vol. 70, no. 2, May 1977, pp. 284-303.

12. Dalgarno, A.: Sum Rules and Atomic S t r u c t u r e . Rev. Mod. Phys., V O ~ . 35, no. 3, J u l y 1963, Pp. 522-524.

13. Chan, Y. M.; and Dalgarno, A.: The Stopping Powers of A t o m s and Molecules. Proc. Phys. SOC. (London), vol. 85, 1965, pp. 457-459.

19

14. Kamikawai, Ryotaro; Watanabe, Tsutomu; and Amemiya, Ayao: Mean E x c i t a t i o n Energy f o r Stopping Power of Molecular Hydrogen. Phys. Rev., second ser., v o l . 184, no. 2, Aug. 10, 1969, pp. 303-311.

15. Landau, L. D.; and L i f s h i t z , E. M.: Electrodynamics of Continuous Media. Addison-Wesley Pub. Co. , Inc. , 1960.

16. Ahlen, Steven P.: T h e o r e t i c a l and Experimental Aspects of t h e Energy L o s s of R e l a t i v i s t i c Heavi ly I o n i z i n g P a r t i c l e s . Rev. Mod. Phys., v o l . 52, no. 1, Jan. 1980, pp. 121-173.

17. Kramers, H. A.: The S topping Power of a Metal f o r a-Particles. Phys ica , vole 13, 1947, pp. 401-412.

18. Lindhard, J.: On t h e P r o p e r t i e s of a Gas of Charged P a r t i c l e s . Mat.-Fys. Medd. - K. Dan Vidensk. Se lsk . , v o l . 28, no. 8, 1954.

19. Bohm, David; and Pines , David: A C o l l e c t i v e D e s c r i p t i o n of E l e c t r o n I n t e r a c - t i o n s . I. Magnetic I n t e r a c t i o n s . Phys. Rev., second ser., v o l . 82, no. 5, June 1 , 1951, pp. 625-634.

20. Pines , David; and Bohm, David: A C o l l e c t i v e D e s c r i p t i o n of E l e c t r o n Interac- t i o n s . 11. C o l l e c t i v e v s I n d i v i d u a l P a r t i c l e Aspects of t h e I n t e r a c t i o n s . Phys. Rev. , second ser. , vol. 85, no. 2, Jan. 15, 1952, pp. 338-353.

21. Bohm, David; and Pines , David: A C o l l e c t i v e D e s c r i p t i o n of E l e c t r o n Interac- t i o n s . 111. Coulomb I n t e r a c t i o n s i n a Degenerate E l e c t r o n Gas. Phys. Rev., second ser., v01. 92, no. 3, Nov. 1, 1953, pp. 609-625.

22. Pines , David: A C o l l e c t i v e D e s c r i p t i o n of E l e c t r o n I n t e r a c t i o n s : I V . E l e c t r o n I n t e r a c t i o n i n Metals. Phys. Rev., second ser., v o l . 92, no. 3, Nov. 1, 1953, pp. 626-636.

23. Lindhard, J e n s ; and S c h a r f f , Morten: Recent Developments i n t h e Theory of Stop- p i n g Power. 1. P r i n c i p l e s of t h e S t a t i s t i c a l Method. P e n e t r a t i o n of Charged P a r t i c l e s i n Matter, Edwin A l b r e c h t Uehling, ed. , Nucl. S c i . Ser . Rep. NO. 29 (Publ . 7521, N a t l . Acad. Sei.-Natl. R e s . Counc., 1960, pp. 49-55.

24. Lindhard, J.; and Winther, A.: S topping Power of E l e c t r o n Gas and E q u i p a r t i t i o n Rule. Mat.-Fys. Medd. - K. Dan. Vidensk. S e l s k . , v o l . 34, no. 4, 1964.

25. Walske, M . C . : The S topping Power of K-Electrons. Phys. Rev. , second ser. I

v o l . 88, no. 6, Lkc. 15, 1952, pp. 1283-1289.

26. Chu, W . K.; and Powers, D.: C a l c u l a t i o n s of Mean E x c i t a t i o n Energy f o r A l l E l e - ments. Phys. L e t t . , v o l . 4 0 A , no. 1, June 19, 1972, pp. 23-24.

27. Rousseau, C. C.; Chu, W. K. ; and Powers, D.: C a l c u l a t i o n s of S t o p p i n g Cross S e c t i o n s f o r 0.8- t o 2.0-MeV Alpha P a r t i c l e s . Phys. Rev. A, t h i r d ser., vol . 4, no. 3, Sept . 1971, pp. 1066-1070.

28. Bonderup, E.: Stopping of S w i f t Pro tons Evalua ted From S t a t i s t i c a l A t o m i c Model. Mat.-Fys. Medd. - K. Dan. Vidensk. S e l s k . , vol . 35, no. 17, 1967.

20

29.

30.

31.

3 2.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

Chu, W. K.; and Powers, D.: Alpha-Part ic le S topping Cross S e c t i o n i n S o l i d s From 400 KeV t o 2 MeV. Phys. Rev. , second s e r . , v o l . 187, no. 2, Nov. 10, 1969, pp. 478-490.

Chu, W. K. ; Moruzzi, V. L.; and Z ieg le r , J. F.: C a l c u l a t i o n s of t h e Energy Loss of 4He Ions i n S o l i d Elements. J. Appl. Phys., vo l . 46, no. 7, J u l y 1975, pp. 281 7-2820.

Moruzzi, V. L.; Janak, J. F.; and W i l l i a m s , A. R.: Ca lcu la t ed E l e c t r o n i c Proper- t i es of Metals. Pergamon Pres s , Inc., c.1978.

Z ieg le r , J. F.: The Stopping of Ene rge t i c Ions i n S o l i d s . Nucl. Instrurn. &

Methods, v o l . 168, nos. 1-3, Jan. 1-15, 1980, pp. 17-24.

Hubbard, J.: The Dielectric Theory of E l e c t r o n i c I n t e r a c t i o n s i n S o l i d s . Proc. Phys. SOC. (London), vo l . 68, p t . 1 1 , no. 431A, Nov. 1 , 1955, pp. 976-986.

.. Froh l i ch , H.; and Pe lze r , H.: Plasma O s c i l l a t i o n s and Energy Loss of Charged

P a r t i c l e s i n So l ids . Proc. Phys. SOC. (London), vol . 68, pt . 6, no. 426A, June 1 , 1955, pp. 525-529.

Wilson, J. W.; and Kamaratos, E.: Mean E x c i t a t i o n Energy f o r Molecules of Hydro- gen and Carbon. Phys. L e t t . , vo l . 85A, no. 1 , Sept . 7, 1981, pp. 27-29.

Wilson, J. W.; Chang, C. K.; Xu, Y. J.; and Kamaratos, E.: I o n i c Bond E f f e c t s on t h e Mean E x c i t a t i o n Energy f o r Stopping Power. J. Appl. Phys., vo l . 53, no. 2, Feb. 1982, pp. 828-830.

Wilson, J. W.; and Xu, Y. J.: Metallic Bond E f f e c t s on Mean E x c i t a t i o n Energ ies f o r Stopping Power. Phys. Le t t . , vo l . 9OA, no. 5, J u l y 1 2 , 1982, pp. 253-255.

Gordon, Roy G.; and Kim, Yung Sik: Theory f o r t h e Forces Between Closed-Shel l Atoms and Molecules. J. C h e m . Phys., vol . 56, no. 6, Mar. 15, 1972, pp. 31 22-31 33.

Dalgarno, A.; and W i l l i a m s , D. A.: P r o p e r t i e s of t h e Hydrogen Molecule. Proc. Phys. SOC. (London), v01. 85, p t . 4, no. 546, Apr. 1965, pp. 685-689.

S e l t z e r , S. M.; and Berger, M. J.: Evaluat ion of t h e C o l l i s i o n Stopping Power of Elements and Compounds f o r E lec t rons and Pos i t rons . I n t . J. Appl. Radia t . & I s o t . , vol . 33, no. 1 1 , Nov. 1982, pp. 1189-1218.

Walecka, J. D.: C o l l e c t i v e E x c i t a t i o n s i n Atoms. Phys. L e t t . , vo l . 58A, no. 2 , AUg. 9, 1976, pp. 83-86.

Clementi , m r i c o ; and R o e t t i , Carla: Roothaan-Hartree-Fock Atomic Wavefunctions. A t . D a t a & Nuc1. Data Tables , vo l . 1 4 , nos. 3-4, Sept.-Oct. 1974, pp. 177-478.

Clementi , E.; and mimondi , D. L.: Atomic Screening Cons tan ts From SCF Func- t i o n s . J. Chem. Phys., vo l . 38, no. 1 1 , June 1, 1963, pp. 2686-2689.

Thompson, Theos J a r d i n : E f f e c t of Chemical S t r u c t u r e on Stopping Powers f o r High-Energy Protons. UCRL-1910, Univ. of C a l i f o r n i a , Aug. 1 1 , 1952.

21

45. Lodhi, A. S.; and Powers, D.: Energy Loss of a P a r t i c l e s i n Gaseous C-H and C-H-F Compounds. Phys. Rev. A, vol. 10, no. 6, D e c . 1974, pp. 2131-2140.

46. Neuwirth, Wolfgang; P i e t s c h , Werner; and Kreutz , Ronald: Chemical I n f l u e n c e s o n t h e Stopping Power. Nucl. Instrum. & Methods, vol. 149, nos. 1-3, Feb. 15- Mar. 1 , 1978, pp. 105-113.

47. Tossell, J. A.: C a l c u l a t i o n of Bond Dis tances and Cohesive Energ ies f o r Gaseous Hal ides Using t h e Modified E l e c t r o n Gas I o n i c Model. C h e m . Phys. L e t t . , vo l . 67, no. 2-3, Nov. 15, 1979, pp. 359-364.

48. Waldman, Marvin; and Gordon, Roy G.: Sca led E lec t ron Gas Approximation f o r In t e rmolecu la r Forces . J. C h e m . Phys., vol . 71, no. 3, Aug. 1 , 1979, pp. 1325-1 339.

49. Dalgarno, A.; Degges, T.; and W i l l i a m s , D. A.: Dipole P r o p e r t i e s of Molecular Nitrogen. Proc. Phys. SOC. (London), vo l . 92, p t . 2, no. 576, O c t . 1967, pp. 291-295.

50. Paul ing , Linus: The Chemical Bond. C o r n e l l Univ. P r e s s , c.1967.

51. Wigner, E. P.; and S e i t z , F.: On t h e C o n s t i t u t i o n of M e t a l l i c Sodium. 11. Phys. Rev., second ser., vol . 46, no. 6, Sept . 15, 1934, pp. 509-524.

52. Brown, P. J.: A Study of Charge Dens i ty i n Beryll ium. Ph i los . Mag., vol . 26, no. 6, Dec. 1972, pp. 1377-1394.

53. Kamaratos, E.; Chang, C . K.; Wilson, J. W. ; and Xu, Y. J.: Valence Bond E f f e c t s on Mean E x c i t a t i o n Energ ies f o r Stopping Power i n Metals. Phys. L e t t . , vo l . 92A, no. 7, Nov. 29, 1982, pp. 363-365.

54. Andersen, H. H.; and Z i e g l e r , J. F.: Hydrogen Stopping Powers and Ranges i n A l l Elements. Pergamon Pres s , Inc . , c. 1977.

55. Bakker, C. J.; and Segr;, E.: Stopping Power and Energy Loss f o r Ion P a i r Pro- d u c t i o n for 340-Mev Protons. Phys. R e v . , second ser., vol . 81, no. 4, Feb. 15, 1951, pp. 489-492.

56. Simmons, D. H.: The Range-Energy R e l a t i o n f o r P ro tons i n Aluminum. €'roc. Phys- SOC. (London), vo l . 65, p t . A, 1952, pp. 454-456.

57. Mather, R.; and Segr;, E.: Range Energy R e l a t i o n f o r 340-Mev Protons. Phys- Rev., second ser., vol. 84, no. 2, Oct . 15, 1951, pp. 191-193.

58. Sachs, Donald C.; and Richardson, J. Reginald: Mean E x c i t a t i o n P o t e n t i a l s . Phys. Rev., second ser., vol . 89, no. 6, Mar. 15, 1953, pp. 1163-1164.

59. Bogaardt, M. ; and Koudijs , B.: On t h e Average E x c i t a t i o n P o t e n t i a l s and t h e Range-Energy R e l a t i o n s i n L i g h t Elements. Phys ica , vol. 18 , Apr. 1952, pp. 249-264.

60. V a s i l e v s k i i , I. M.; and Prokoshkin, Yu. D. : I o n i z a t s i o n n y e P o t e n t s i a l y Atomov ( I o n i z a t i o n P o t e n t i a l s of A t o m s ) . Dubna:Ob'edinen. I n s t . Yadernykh Is s ledovanni , 1 960.

22

61. S h i l e s , E.; S a s a k i , Taizo; I n o k u t i , Mitio; and Smith, D. Y.: Se l f -Cons i s t ency and Sum-Rule T e s t s i n t h e Kramers-Kronig Ana lys i s of O p t i c a l D a t a : Appl ica- t i o n s t o Aluminum. Phys. Rev. B, vo l . 22, no. 4, Aug. 15, 1980, pp. 1612-1628.

62. Andersen, H. H.; Bak, J. F.; Knudsen, H.; and Nie l sen , B. R.: S to p i n g Power of AI, Cu, Ag, and Au f o r MeV Hydrogen, Helium, and Li thium Ions z13 and Z14 Pro- p o r t i o n a l Dev ia t ions From t h e Bethe Formula. Phys. Rev. A , vol. 16, no. 5, NOV. 1977, pp. 1929-1940.

63. Khandelwal, G. S.: S topping Power of K and L E l e c t r o n s . Phys. Rev. A, vo l . 26, no. 5, Nov. 1982, pp. 2983-2986.

64. Kahn, David: The Energy Loss of Pro tons i n Metallic F o i l s and Mica. Phys. Rev., second ser., vol . 90, no. 4, May 15, 1953, pp. 503-509.

65. Ne i l sen , L. P.: Energy Loss and S t r a g g l i n g of Pro tons and Deuterons. Mat.-Fys. Medd. - K. Dan. Vidensk. Se l sk . , vol. 3 3 , no. 6 , 1961.

66. Leminens, E.; F o n t e l l , A.; and B i s t e r , M.: S topping Power of A l , Zn and I n f o r 0.6-2.5 MeV Pro tons . Ann. Acad. S c i . Fenn. , no. 287, 1968.

67. Nakata, H i rosh i : Analys is of Energy-Loss Data f o r 0.2-0.5 MeV/amu p, a, and N i n Se. Phys. Rev. B, vo l . 3, no. 9 , May 1 , 1971, pp. 2847-2851.

68. Sorensen , H.; and Andersen, H. H.: Stopping Power of A l , Cu, Ag, Au, Pb, and U f o r 5-18-MeV Pro tons and Deuterons. Phys. Rev. B , vo l . 8, no. 5, Sept . 1, 1973, pp. 1854-1863.

23

SYMBOLS

Bohr ' s r a d i u s

s h e l l c o r r e c t i o n

v e l o c i t y of l i g h t i n vacuum

e l e c t r i c d i sp l acemen t v e c t o r

system e x c i t a t i o n energy

e l e c t r i c f i e l d v e c t o r

e i g e n v a l u e of n t h s t a t e

long-range c o r r e l a t i o n energy

magnitude of e l e c t r o n charge

accumulated o s c i l l a t o r s t r e n g t h wi th angu la r f requency below W

Dalgarno sum

o p t i c a l o s c i l l a t o r s t r e n g t h

o p t i c a l o s c i l l a t o r s t r e n q t h of hydrogen atom

plasma o s c i l l a t o r s t r e n g t h of hydrogen atom

p o l a r i za t i o n response

Hamil tonian o p e r a t o r

P lanck ' s c o n s t a n t

h/2n

mean e x c i t a t i o n energy

atomic mean e x c i t a t i o n energy

molecular mean e x c i t a t i o n energy f o r Bragg ' s r u l e w i th n e u t r a l atoms

mean e x c i t a t i o n energy of c o v a l e n t bonded molecule

molecular mean e x c i t a t i o n energy f o r Bragg ' s r u l e wi th p a r t i a l l y i o n i c atoms

mean e x c i t a t i o n energy of system j

wave number of e j e c t e d e l e c t r o n

l o c a l plasma model

mth moment of l o g mean e x c i t a t i o n energy

i n d i v i d u a l e l e c t r o n orbi ta l a n g u l a r momentum quantum number

e l e c t r o n mass

d e n s i t y of atoms

s t o c h i o m e t r i c c o e f f i c i e n t

Rydberg's c o n s t a n t

i n t e r n u c l e a r d i s t a n c e i n molecule composed of atoms A and B

e l e c t r o n p o s i t i o n v e c t o r

r a d i u s of Wigner-Seitz sphere

s t o p p i n g power

mth moment of o s c i l l a t o r s t r e n g t h s

t i m e

i o n v e l o c i t y

i o n charge number

a tomic number

r a t i o of i o n v e l o c i t y t o v e l o c i t y of l i g h t

plasma frequency s h i f t

s t r a g g l i n g parameter

p o l a r i z a t i o n c o r r e c t i o n

chemica l bond s h i f t i n mean e x c i t a t i o n energy

d ie lec t r ic f u n c t i o n

average e l e c t r o n

e lectron d e n s i t y

e l e c t r o n d e n s i t y

e l e c t r o n d e n s i t y

s e p a r a t i o n

i n hydrogen atom

i n hydrogen molecule

s c r e e n i n g parameter

ground s t a t e e l e c t r o n i c wave f u n c t i o n

a n g u l a r f requency of o s c i l l a t o r

25

w P

c l a s s i c a l plasma frequency

< > averaqed va lue

An arrow over a symbol denotes a vec to r . A prime over a symbol denotes a v a r i a b l e of i n t e g r a t i o n .

26

PG

6 PI

W z H PI PI

cn Ix R

2

i3

z

8

Sr W

z w z B a: B H

W z a: W E

El W 0

E X

I

H

W

a: B

d

- I

k 0 W

k

E a m c

.rl ch

4J 0

k 0 w

3

5 a, . k Q) c a,

c 0

.rl c, Id 4J .rl U X a,

c (d Q) E c a, m 0 k

X a -

4J c 01 E .?I k

B W

a, In 0

+I

Ln a, 7

(ZI

w c o $ Q 1 c -

aw co c

cv x x

~

cn

B i3

2 .%

3 d 3 u W d 0 r: z

u i2 a 2 El W 0

E X

I

H H

W

a: B

. d

u II u

Y u

X I u

Y X

X I

5:

U E ' r l k

w c o p w o cv

. (ZI

w I -

(v $ 9

m c v r r

cv 9 9

m ( u r n 9 4 cv

m c o o c v r cv

cv r r w

cv ? 9

e c o w

cv 9 9

cv c o w

cv 9 4

0 (d

. m 2 Lo

27

TABLE 111.- MOLECULAR MEAN EXCITATION ENERGY

Chemica 1 s p e c i e s Pres e n t

t h e o r y

44.7

(CH2lx 55.0

'gH6 60.6

I

H4

18.9 H2

Graphi te 76.1

Expe r ime n t a 1 e mp i r i ca 1

42.8

53.4

61.4 2 1.9

18.5 f 0.5

78.5 f 1.5

Bragg's r u l e

35.1

43.5

50.6

15.0

62.0

TABLE 1V. - MOLECULAR MEAN EXCITATION ENERGIES FOR COVALENT GASES

r Chemical s p e c i e s

I H2

N2

R a AB 0

1.40

2.08

2.34

2.67

3.76

I, eV

(a 1

18.9

85.0

99.6

114.2

170.8

I, e V

(b 1

19.2 f 0.4c

82.0 2 1.6'

95 f 1.gC

115 +_ 10

171 f 14

I :: I2

aLocal plasma model. bReference 40.

These va lues a r e s t r o n g l y i n f l u e n c e d by r e f e r e n c e 11. C

28

TABLE V.- MEAN EXCITATION ENERGIES FOR COVALENT-BONDED CRYSTALS

Chemical species

B ( t e t r a g o n a l )

C ( d i a m o n d )

C ( g r a p h i t e )

S i ( d i a m o n d )

P ( b l a c k )

S ( r h o m b i c )

R a AB 0

3.06

2.94

2.68

4.42

4.16

3.85

I, e V

67.3

75.3

76.1

151.0

155.7

162.7

aLocal plasma model. bRe f erence 40. C U n s p e c i f i e d a l l o t rop ic f o r m .

TABLE VI . - IONIC-BOND PARAMETERS

I 1.72 I HF

I, e V

( b )

76 f 7.6

78 f 2.3

173 f 4

181 f 14c

190 f 15'

0.50 97.6 91.7 91 .o 96.4

.25 27.8 25.2 25.9 26.7

.90 83.4 92.6 81.6 93.6

29

TABLE V I 1 . - MEAN EXCITATION ENERGIES O F I O N I C CRYSTALS

Local plasma model. a

b R e f e r e n c e 40. C P a u l i n g p a r t i a l i o n i c character f u n c t i o n .

TABLE V I I 1 . - METALLIC PARAMETERS FOR SELECTED METALS O F F I R S T TWO ROWS

Chemical species

L i th ium

Be r y 1 l i u m

Sodium

Ma gne s i u m

A l u m i n u m

Iat, e V

( a 1

34.0

38.6

123.6

121.2

124.3

a R e f e r e n c e 6. bWigne r -Sei t z mode 1. C R e f e r e n c e 40.

30

rs'

3.260

2.375

3.99

3.34

2.991

I, e V

(b 1

4 5

6 0

140

144

149

I, e V

(C 1

41.5 f 3.7

63.7 f 3.2

162 f 8

164 f 8

166 f 3

- Quantum ---Local Plasma

Figure 1 .- Cumulative o s c i l l a t o r s t r e n g t h d i s t r i b u t i o n f o r a tomic hydrogen and helium.

I I I I

\ d x a c t

_ _ _ _ _ _ Local Plasma, YpinqS Local Plasma, Y=

' 1

1 I I - 1 0 1

m

F i g u r e 2.- Moments of o s c i l l a t o r s t r e n g t h s of hydrogen atom.

31

I I I

-2 -1 0 1 01

m

F i q u r e 3 . - Moments of o s c i l l a t o r s t r e n g t h s of he l ium atom a c c o r d i n g t o quantum osc i l l a to r s t r e n g t h s u s i n g s c r e e n e d wave f u n c t i o n s . Z e f f = 1.7.

- Exact ---- Local Plasma

I I 1

-1 0 1

m

F i g u r e 4.- Moments of osc i l l a to r s t r e n g t h s of hydrogen molecule f o r several models.

32

olnokuti et al. (refs. 6 to 8) OLocal Plasma UEmpirical (ref. 40)

1 8 @ t

1

Z

Figure 5.- Comparison of atomic mean e x c i t a t i o n e n e r g i e s accord ing t o quantum c a l c u l a t i o n s of I n o k u t i e t a l . and local plasma model. v a l u e s are from r e f e r e n c e 40.

Empirical

lnokutl et el. (refs. 6 to 81 8 Local lasma nCompl6tion (refs. 6 to 81 A Ziess et ai. (ref. 111

Figure 6.- Comparison of atomic s t r a g g l i n g parameters a c c o r d i n g t o quantum c a l c u l a t i o n s of I n o k u t i e t a l . ( r e f s . 6 t o 8 ) and l o c a l plasma model.

33

- Empirical -- -- Local Plasma

!

m

F i g u r e 7.- Moments of osc i l la tor s t r e n g t h s from e m p i r i c a l v a l u e s of r e f e r e n c e 49 compared w i t h local plasma model u s i n g Gordon-Kim molecu la r model d e n s i t i e s .

0 Reference 6

- Local plasma model

2001

5 0 -

0 - 1 0 1

I o n i c c h a r g e

F i g u r e 8.- Mean e x c i t a t i o n e n e r g i e s for p a r t i a l l y i o n i c atoms.

34

L Q)

0 P

0 C

P P 0

v)

U Q) 0 3 U Q)

3

.-

4-

a

25-

20

5-41

AAtomic values by LPM ODehmer et a/. [refs. 6 to 8) OMolecular values by LPM ASelfzer-Berger (ref. 40)

4 A Q

-

5.01 1 I I I I I I I I I I I I

0 . 5 1 2 5 10

E, M e V

Figure 9.- Comparisons of reduced s t o p p i n g power f o r aluminum for s e v e r a l mean e x c i t a t i o n e n e r g i e s w i t h range of e x p e r i m e n t a l d a t a . (See r e f s . 60 and 6 2 t o 66.) Shaded area i s band of e x p e r i m e n t a l d a t a .

51 I I I I I I I I 1

Z

0 2 4 6 8 10 72 74 76 18 i 3

F i q u r e 10.- Mean e x c i t a t i o n e n e r g i e s f o r atoms, m o ~ e c u ~ e s , s o l i d s , and metals. S p e c i f i c d a t a taken from tables I11 t o V and t a b l e VIII.

35

1. Report No. 2. Government Accession No.

NASA TP-2271 4. Title and Subtitle

MEAN EXCITATION ENERGIES FOR STOPPING POWERS I N VARIOUS MATERIALS USING LOCAL PLASMA OSCILLATOR STRENGTHS

7. Authorls) John W. Wilson, Y. J. Xu, E f s t a t h i o s Kamaratos, and C. K. Chang

9. Performing Organization Name and Address

NASA Langley Research Center Hampton, VA 23665

3. Recipient's Catalog No.

5. Report Date

6. Performing Organization Code

March 1984

506-55-23-03

8. Performing Organization Report No.

L-15731

10. Work Unit No.

11. Contract or Grant No.

12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered

' Jkchnical Paper 1

14. Sponsoring Agency Code

I 15. Supplementary Notes

John W . Wilson: Langley Research C e n t e r , Hampton, V i r g i n i a . Y. J. XU: Old Dominion U n i v e r s i t y , Norfo lk , V i r g i n i a (NASA Grant NSG-1614). E. Kamaratos and C. K. Chang: C h r i s t o p h e r Newport Col lege , N e w p o r t N e w s , V i r g i n i a

(NASA Cooperat ive Agreement NCC1-42).

16. Abstract The basic model of Lindhard and S c h a r f f , known as t h e local plasma model, i s u t i l i z e d t o s t u d y t h e e f f e c t s on s t o p p i n g power of t h e chemica l and p h y s i c a l s t a t e of t h e medium. Unlike p r e v i o u s work w i t h t h e local plasma model, i n which i n d i v i d u a l elec- t r o n s h i f t s i n t h e plasma f requency w e r e e s t i m a t e d e m p i r i c a l l y , t h e P i n e s c o r r e c t i o n d e r i v e d f o r a d e g e n e r a t e Fermi g a s i s shown h e r e i n t o p r o v i d e a r e a s o n a b l e e s t i m a t e , even on t h e atomic scale. Thus, t h e model i s moved t o a complete t h e o r e t i c a l base r e q u i r i n g no e m p i r i c a l a d j u s t m e n t s , as c h a r a c t e r i s t i c of p a s t a p p l i c a t i o n s . The p r i n c i p a l remaining error i s i n t h e o v e r e s t i m a t i o n of t h e low-energy a b s o r p t i o n prop- e r t ies t h a t a r e c h a r a c t e r i s t i c of t h e plasma model i n t h e r e g i o n of t h e atomic d i s - crete spectrum, a l t h o u g h h i g h e r - e n e r g y phenomena a re a c c u r a t e l y r e p r e s e n t e d , and even e x c i t a t i o n - t o - i o n i z a t i o n ra t ios a re g iven t o f a i r accuracy . Mean e x c i t a t i o n e n e r g i e s for covalent-bonded g a s e s and s o l i d s , f o r i o n i c g a s e s and c r y s t a l s , and f o r metals are c a l c u l a t e d u s i n g f i r s t - o r d e r models of t h e bonded s ta tes , f o r which r e a s o n a b l e agreement with t h e r e c e n t l y e v a l u a t e d d a t a of S e l t z e r and Berger i s o b t a i n e d . Hence, t h e methods d e s c r i b e d h e r e i n allow r e a s o n a b l e estimates of mean e x c i t a t i o n energy f o r any phys ica l -chemica l combinat ion o f material media f o r stopping-power a p p l i c a t i o n s .

19 Security Classif. (of this report1

U n c l a s s i f i e d Unc las s i f i e d 36

20. Security Classif (of this page) 21. No. of Pages

17. Key Words (Suggested by Author(r) I

Stopping power Molecules Pro tons

22. Rice

A 0 3

18. Distribution Statement

U n c l a s s i f i e d - Unl imi ted

Documents

Mean Excitation Energies Stopping Powers in Various ... · PDF fileexcitation levels and corresponding oscillator strengths is the main hindrance. It was suggested by Dalgarno (ref