Fast electrons in a polar crystal: Interpretation of the mobility measurements of Chollet and Rossel

Volume 22, number 3

PHYSICS LETTERS

l e v e l s , This factor is the s a m e as for the J = t e r m of Sm 3+ in a cubic field and has b e e n d e r i v e d for that t e r m by F r a n k [4]. Thus in the m o l e c u l a r field a p p r o x i m a t i o n , the s u s c e p t i b i l i t y (using Hex =-3kM) is found f r o m eq. (1) to be (C/x) - ~ = T / f ( T )

(3)

where ot = hC is an a d j u s t a b l e p a r a m e t e r . In an a n a l y s i s of e x p e r i m e n t a l s u s c e p t i b i l i t y data of the 4-f i n t e r m e t a l l i c compounds, allowa n c e s m u s t be made for a finite c o n t r i b u t i o n to the m a g n e t i z a t i o n (and hence s u s c e p t i b i l i t y ) f r o m the conduction e l e c t r o n spin p o l a r i z a t i o n a r i s i n g f r o m an s - f exchange i n t e r a c t i o n [e. g. 5]. The effect of the conduction e l e c t r o n m a g n e t i z a t i o n is to change the value of the C u r i e - W e i s s c o n s t a n t f r o m the expected f r e e ion v a l u e s . Thus, in g e n e r a l the effective n u m b e r of Bohr m a g n e t o n s of the s a m ple will not equal the f r e e ion value. F o r this r e a s o n , in c o m p a r i n g theory and e x p e r i m e n t it is p r e f e r a b l e to c o m p a r e C/× and not l/X, i . e . , c o m p a r e (Cex/Xex) with (C/x) f r e e ionFig. 1 shows the r e s u l t s of a c a l c u l a t i o n for C / X given by eq. (3) for v a r i o u s values of the c r y s t a l field splitting h v c. T h e p a r a m e t e r a ~ 8OK was chosen to be c o n s i s t e n t with the C u r i e - W e i s s t e m p e r a t u r e of CeP obtained by e x t r a p o l a t i o n of C/× to zero f r o m the high t e m p e r a t u r e region [6]. The open c i r c l e s shown in

FAST

ELECTRONS MOBILITY

15 August 1966

fig. 1 a r e taken f r o m ref. 1 (C =0.885). It is e v i dent f r o m fig. l that s a t i s f a c t o r y a g r e e m e n t is obtained for hv c ~ 200°K. In conclusion, the a g r e e m e n t between theory and e x p e r i m e n t shown in fig. 1, indicates that the o b s e r v e d a n o m a l o u s d e c r e a s e in the value of the C e P magnetic s u s c e p t i b i l i t y , at low t e m p e r a t u r e s , can be s a t i s f a c t o r i l y accounted for by including the effects of a c r y s t a l field i n t e r a c t i o n upon the Ce3+(2F~) ground state. A c r y s t a l field s p l i t t i n g of 200OK for CeP is also in a c c o r d a n c e with the 250OK r e p o r t e d by White et al. [2] for the m e t a l l i c Cubic Laves s y s t e m Ce:LaA12. Thus there is no need to invoke a valancy change for the Ce3+ ion at low t e m p e r a t u r e s . F u r t h e r m o r e , it is to be noted that extrapolating the s u s c e p t i b i l i t y data shown in fig. 1 to zero will give e r r o n e o u s values for the C u r i e - W e i s s t e m p e r a t u r e 0.

Refewences

1. G. Busch and O. Vogt. Physics Letters 20 (1966) 152. 2. J.A. White. H.J. Williams, J.H. Wernick and R.C. Sherwood, Phys. Rev. 131 (1963) 1039. 3. E.D. Jones, to be published. 4. A. Frank, Phys. Rev. 48 (1935) 767. 5. C. Kittel, Quantum theory of solids (John W~l~y and Sons, Inc., New York 1963). 6. E.D.Jones, to be published.

IN A P O L A R C R Y S T A L : ; I N T E R P R E T A T I O N OF MEASUREMENTS OF CHOLLET AND ROSSEL

THE

J . YAHIA Oklahoma State University, Stillwater. Oklahoma, U . S . A .

Received 7 July 1966

An interpretation of mobility data for AgBr obtained by Chollet and Rossel is given in terms of the interaction of a f a s t electron with the polarization waves in the crystal, rather than an explanation in terms of slow electron theories.

The b e h a v i o u r of the e l e c t r o n i c mobility in p o l a r m a t e r i a l s has r e c e i v e d a g r e a t deal of a t tention in the p a s t few y e a r s , both t h e o r e t i c a l l y and e x p e r i m e n t a l l y [1]. P r a c t i c a l l y all the theor e t i c a l t r e a t m e n t s c o n s i d e r the e l e c t r o n to be a s l o w p a r t i c l e i n t e r a c t i n g with the p o l a r i z a t i o n field, giving r i s e to a w e l l - k n o w n t e m p e r a t u r e

v a r i a t i o n of the mobility: exponentially d e c r e a s i n g with i n c r e a s e in t e m p e r a t u r e . T h e r e a r e s e v e r a l e x p e r i m e n t a l methods for d e t e r m i n i n g m o b i lity; Hall data can be combined with e l e c t r i c a l conductivity data [e. g. 2], e l e c t r i c a l conductivity and t h e r m o g r a v i m e t r i c data may be combined [3] or a time of t r a n s i t m e a s u r e m e n t may be used [4].

267

Volume 22, number 3

PHYSICS

LETTERS

15 August 1966

Table 1 Energy levels of X - r a y spectra for 47Ag and 35Br, v/R ; (R is the Rydberg constant in same units as frequency v) MI

MII

MIII

MIV

Br

19.48

13.60

13.21

5.27

MV 5.13

Ag

52.98

44.49

42.21

27.56

27.13

NI

N II

N III

NIV

NV

YI

YII

0.36

2.32

0.16

0.5 7.15

4.28

0.45 0.39

Values from Handbook of Physics, Ed. Condon and Odishaw, McGrawHill Book Company, New York, 1957 T h i s note is c o n c e r n e d w i t h the i n t e r p r e t a t i o n of r e s u l t s of the l a s t t y p e of m e a s u r e m e n t , p a r t i c u l a r l y with the e x p e r i m e n t of C h o l l e t and R o s s e l [e. g. 4]. In t h i s e x p e r i m e n t , a p u l s e of 8 k i l o v o l t X - r a y s g e n e r a t e s c h a r g e c a r r i e r s at one f a c e of the c r y s t a l and the m o b i l i t y of t h e s e c a r r i e r s is m e a s u r e d a s t h e y t r a v e r s e the c r y s t a l . T h e s e c h a r g e s a r e t a k e n to be s l o w e l e c t r o n s by C h o l l e t and R o s s e l , but it w o u l d s e e m m o r e r e a s o n a b l e to e x p e c t t h e m to b e v e r y e n e r g e t i c p a r t i c l e s , f o r the p r e p o n d e r a n t X - r a y s c a t t e r i n g p r o c e s s e s o c c u r r i n g f o r i n c i d e n t r a d i a t i o n ~ 1.5 • a r e p h o t o e l e c t r i c p r o c e s s e s [5]; m a n y s u c h p r o c e s s e s m a y o c c u r in the i n t e r a c t i o n of an 8 k i l o v o l t X - r a y w i t h a c r y s t a l , s u c h a s e j e c t i o n of an e l e c t r o n f r o m the K, L, M . . . s h e l l and f l u o r e s c e n c e w i t h f u r t h e r e l e c t r o n e j e c t i o n and a l l of t h e s e e l e c t r o n s w i l l h a v e v e r y high e n e r g i e s a s c o m p a r e d w i t h flu) (¢~ a c h a r a c t e r i s t i c l a t t i c e f r e q u e n c y ) . The o r i g i n a l t h e o r y of the i n t e r a c t i o n of a f a s t e l e c t r o n w i t h the p o l a r i z a t i o n f i e l d of an i o n i c l a t t i c e is due to Frt~hlich [6]. In this p a p e r , the c o n c e p t of a p o l a r i z a t i o n w a v e was f i r s t i n t r o d u c e d and the p r o c e d u r e f o l l o w e d w a s to e x p r e s s the t o t a l H a m i l t o n i a n as the s u m of t h r e e t e r m s : the l a t t i c e H a m i l t o n i a n , the f r e e - e l e c t r o n H a m i l t o n i a n and a p e r t u r b a t i o n t e r m d e r i v a b l e f r o m a P o i s s o n - t y p e e q u a t i o n f o r the p o l a r i z a t i o n p o t e n t i a l . P e r t u r b a t i o n t h e o r y w a s u s e d to o b t a i n t r a n s i t i o n p r o b a b i l i t i e s and e v e n t u a l l y an e x p r e s s i o n f o r the r e l a x a t i o n t i m e . T h i s p r o c e d u r e is v a l i d so long a s the e n e r g y of the p a r t i c l e is v e r y m u c h g r e a t e r t h a n / ~ w ( w h e r e co is the f r e q u e n c y of the p o l a r i z a t i o n w a v e s ) . The e x p r e s s i o n f o r the r e l a x a t i o n t.ime i s : 7 = To t a n h ( h u / 2 k T ) , w h e r e To = = 2 rn~ M a 5 uE~2/2~ n2e4]~. H e r e 7 is the r e l a x a t i o n t i m e , e is the e l e c t r o n ' s c h a r g e , m is the f r e e e l e c t r o n m a s s , M is the r e d u c e d m a s s f o r the two i o n s of d i f f e r e n t s i g n , E is the e n e r g y of the e l e c t r o n and u s t a n d s f o r the r e s t s t r a h l e n f r e q u e n c y . F r o m t h i s e x p r e s s i o n f o r 7, the m o b i l i t y ~ is r e a d i l y o b t a i n e d a s 1~ = ( e / m ) r . We now c o m p a r e the r e s u l t s of t h i s t h e o r y w i t h m o b i l i t y d a t a in A g B r [4], s t a r t i n g w i t h a d i s c u s s i o n on the a b s o l u t e v a l u e of the m o b i l i t y . At 100°K, 1

268

Z

.

the m e a s u r e d v a l u e of ~ is ~ 300 c m 2 / V s e c . In t e r m s of the F r 0 h l i c h t h e o r y [6], u s i n g v a l u e s f o r M, a, v f o r A g B r * one o b t a i n s the r e s u l t that the e n e r g y of the p a r t i c l e g i v i n g r i s e to this m o b i l i t y is ~ 100 eV. The q u e s t i o n c o m e s up of the o r i g i n of t h e s e e l e c t r o n s . We r e c a l l that the A g B r crystal was bombarded with 8 kilovolt X-rays; t h i s e n e r g y ( v / R = 593, w h e r e R is the R y d b e r g c o n s t a n t in f r e q u e n c y units) is l e s s than the K s h e l l e n e r g y f o r e i t h e r Ag (1 879) o r B r (993) s o that no p r i m a r y K p h o t o e l e c t r o n is p r o d u c e d ; h o w ever, primary L or M etc., photoelectrons may be f o r m e d , the e n e r g y of t h e s e b e i n g v e r y high. T h e l o w e s t e n e r g y E of the p r i m a r y L p h o t o e l e c t r o n (the one e j e c t e d f r o m the L 1 s u b s h e l l in Ag) is E = 593 - 280 = 313 in u n i t s of v / R . T h i s c o r r e s p o n d s to an e n e r g y of 3.9 k e V and in t e r m s of the t h e o r y r e p r e s e n t s a r e l a x a t i o n t i m e of 5.4 × 10 -12 s e c . The ( r e l a t i v i s t i c ) v e l o c i t y of a 3.9 k e V e l e c t r o n is 3.7 × 109 c m / s e c which m e a n s that the m e a n f r e e p a t h of this p a r t i c l e is of the o r d e r of the d i m e n s i o n s of the c r y s t a l , 2-5 m m , and the t r a n s i t t i m e a c r o s s the c r y s t a l ~ 10 -10 s e c , m u c h s h o r t e r than any t i m e s m e a s u r e d in the e x p e r i m e n t . T r a n s i t t i m e s m e a s u r e d a r e f r o m 0.5 to 2 ~ s e c [4] and t h i s is in l i n e with the s t a t e m e n t s m a d e h e r e c o n c e r n i n g the e n e r g y of the p a r t i c l e s r e s p o n s i b l e f o r the c o n d u c t i o n . A s s o c i a t e d with the p r i m a r y p h o t o e l e c t r i c p r o c e s s e s d i s c u s s e d a b o v e , t h e r e is a n o t h e r p r o c e s s , that of f l u o r e s c e n c e o r r a d i a t i v e d e - e x c i t a t i o n of the atom; conceivably this radiation could once again t a k e p a r t in a n o t h e r p h o t o e l e c t r i c p r o c e s s , but it s e e m s m o r e n a t u r a l to invoke at t h i s p o i n t the A u g e r m e c h a n i s m [8] f o r the p r o d u c t i o n of s e c o n d a r y ~ - r a y s , a m e c h a n i s m c o m p e t i n g with and p e r h a p s b e i n g m o r e d o m i n a n t than that of r a d i a t i v e d e - e x c i t a t i o n of the a t o m [9]. If one l o o k s at the l o n g - w a v e - l e n g t h a b s o r p t i o n e d g e s in Ag and * The reduced mass M for AgBr is calculated from given values of Ag and Br atomic weights, the d i s tance a is obtained from X-ray data on AgBr (Leighton [5] pp. 441, 442) and v is from Jones et al. [7]. The values a r e M = 77 x 10-24 gram, a = 2 . 8 8 and v = 2.75 × 1012 sec ~1.

Volume 22, n u m b e r 3

PHYSICS

B r , o n e f i n d s v/R v a l u e s a s in t a b l e 1. T h e e n e r g y of r e l e v a n t A u g e r e l e c t r o n s i s g i v e n b y E 1 = = hv L - 2h~M, (2); E 2 = hv M - 2h~N, (4) e t c . ( T h e b r a c k e t e d n u m b e r s r e f e r t o t h e n u m b e r of A u g e r electrons). One can see from table 1 that E 2 for b o t h Ag a n d B r i s a v a l u e of e n e r g y c l o s e to t h a t deduced from mobility analysis. The measured t e m p e r a t u r e v a r i a t i o n of m o b i l i t y s e e m s to s h o w roughly the temperature variation obtained theor e t i c a l l y : t a n h (hv/2kT). A t 1 0 0 ° K it i s ~ 300 c m 2 / V s e c a n d a t 2 0 0 ° K it i s ~ 100 c m 2 / Vsec; furthermore, a t low t e m p e r a t u r e s i t a p p e a r s to b e i n d e p e n d e n t of t e m p e r a t u r e , a s e x p e c t e d . T h i s r e s u l t i s in m a r k e d c o n t r a s t to t h e c a s e f o r a slow e l e c t r o n in a p o l a r l a t t i c e w h e r e ~cc [ e x p ( O / T ) - l ] , @ b e i n g t h e D e b y e t e m p e r a t u r e . 1. H. F r 5 h l i c h , H. P e l z e r and S. Zieneau, Phil. Mag. 41 (1950) 221; H. F r 5 h l i c h , Adv. in P h y s i c s 3 (1954) 325. Also, H. F r 5 h l i c h and F. Brown in P o l a r o n s and e x -

ON

THE

FIELD-DEPENDENT SPIN

LETTERS

2.

3. 4.

5.

6, 7, 8. 9.

15 August 1966

citons, eds. C.G. Cooper and G. D. Whitfield (Oliver and Boyd, Edinburg, 1963). R.G. B r e c k e n r i d g e and W . A . H o s l e r , Phys. Rev. 91 (1953) 793; J . Yahia and H. P. R. F r e d r i k s e , Phys. Rev. 123 (1961) 1257 for data on oxides of titanium. J . Y a h i a , J . P h y s . Chem. Solids 25 (1964) 881. J . R . Haynes and W. Shockley, Phys. Rev. 81 (1951) 835; M . B . P r i n c e , Phys. Rev. 91 (1953) 271; R.Hofstadter, Nucleonics 4, No. 4 (1949) 2; 1, No. 5 (1949) 29; L. Chollet and J. Rossel, Helv. P h y s . A c t a 33 (1960) 627. W . H e i t l e r , The quantum theory of radiation (Oxford Univ. P r e s s , 3rd ed., 1954) p. 204; Also, R i c h t m y e r and Kennard, Introduction to m o d e r n P h y s i c s (McGraw-Hill Book Company, New York, 3rd ed., 1947) Ch.X; R.B. Leighton, P r i n c i p l e s of m o d e r n physics (McGraw-Hill Book Company, New York, 1959) Ch. XII. H.Frtihlich, P r o c . Roy. Soc, A160 (1937) 280. J o n e s et a l . , P r o c . Roy. Soc., Lond. A261 (1961) 10. P . A u g e r , J, Phys. 6 (1925) 205. Leighton[5], p.427.

SUSCEPTIBILITY 1 ~ FERROMAGNET

OF

THE

HEISENBERG

G. A. B A K E R , J r . Brookhaven National Laboratory, Long Island, New York H. E . G I L B E R T

University of California, Los Alarnos Scientific Laboratory, New Mexico J . E V E a n d G. S. R U S H B R O O K E

University of Newcastle upon Tyne, England Received 11 July 1966

The f i r s t eight coefficients in the high t e m p e r a t u r e ' e x p a n s i o n s of the t e r m s proportional to H 4, H 6 and H 8 in the f r e e - e n e r g y of a H e i s e n b e r g f e r r o m a g n e t have been d e t e r m i n e d for the f a c e - c e n t r e d , bodycentred and simple cubic lattices. F o r all t h r e e l a t t i c e s the s e r i e s diverge at the Curie point with expo_ nents close to 5.05, 8.70 and 12.3 respectively.

We r e c e n t l y r e p o r t e d [1] h i g h t e m p e r a t u r e e x pansion coefficients for the zero-field specific h e a t a n d z e r o - f i e l d s u s c e p t i b i l i t y of H e i s e n b e r g spin ½ cubic ferromagnetic models, the work bei n g b a s e d on t h e f i n i t e c l u s t e r m e t h o d ( s e e r e f e r e n c e s i n [1] ). F r o m t h e s t a r t of t h e c a l c u l a t i o n s o u r a i m h a s b e e n to c a l c u l a t e t h e s u s c e p t i b i l i t y c o e f f i c i e n t s a s f u n c t i o n s of t h e a p p l i e d f i e l d , a n d f o r t h i s r e a s o n we f o u n d t h e r e q u i s i t e i r r e d u c i b l e r e p r e s e n t a t i o n s , [2], a n d l a t t i c e o c c u r r e n c e f a c -

tors for many more basic graphs than were strictly needed for the results already presented. We have now used this material, in conjunction w i t h eq. (1.6) of [2], to d e r i v e t h e f i e l d d e p e n d e n c e of t h e s e c o e f f i c i e n t s . Starting with the customary Heisenberg Hamiltonian

9 t = -½J ~ a i . a j - ttH ~ azi (i,j) i

(1)

269