Boundary scattering and the thermal conductivity of paramagnetic salts

Volume 20, number 3

PHYSICS LETTERS

BOUNDARY

15 February 1966

SCATTERING AND THE THERMAL OF PARAMAGNETIC SALTS D. L. HUBER

CONDUCTIVITY

*

Department of Physics, University of Wisconsin, Madison, Wisconsin Received 14 January 1966

The thermal conductivity of a paramagnetie s a l t is calculated. It is assumed that the effect of the spinphonon i n t e r a c t i o n i s to remove from the thermal current those p h o n o n s w h o s e mean free paths a r e l e s s than the dimensions ofthe sample.

In p r e v i o u s t r e a t m e n t s [1] of the t h e r m a l conductivity of p a r a m a g n e t i c s a l t s it was a s s u m e d that the i n c r e a s e in the r e s i s t i v i t y a r i s i n g f r o m the spin-phonon i n t e r a c t i o n is given by the r e c i p r o c a l of a conductivity AK which s a t i s f i e s the equation AK :

ks

cksvksxgs

"

(1)

H e r e C k s is the s p e c i f i c heat of the l a t t i c e mode d e s i g n a t e d by k and s, V k s is the c o r r e s p o n d i n g v e l o c i t y , and Xks is the phonon m e a n f r e e path which is taken to be d e t e r m i n e d s o l e l y by the o r b i t - l a t t i c e coupling. As has a l r e a d y been noted [2] this a p p r o a c h has the defect that the change in r e s i s t i v i t y is e x t r e m e l y s e n s i t i v e to the c h o i c e of the line shape function which e n t e r s into the e x p r e s s i o n f o r the m e a n f r e e path. Indeed, if one a s s u m e s a G a u s s i a n d i s t r i b u t i o n f o r the density of final s t a t e s one obtains an infinite value f o r AK. An e x a m i n a t i o n of the r e l e v a n t e q u a t io n s i n d ic a t e s that the difficulty is a s s o c i a t e d with the long m e a n f r e e path of the phonons whose f r e q u e n c i e s a r e on the wings of the a c o u s t i c r e s o nance line. In all s i t u a t i o n s of e x p e r i m e n t a l i n t e r est, h o w e v e r , the m e a n f r e e path is a l w a y s l i m i t e d by the b o u n d a r i e s of the s a m p l e . A s i m p l e way of taking t h i s into account [3] is to w r i t e the r e c i p r o c a l of the e f f e c t i v e m e a n f r e e path as the sum of the r e c i p r o c a l of a length c h a r a c t e r i s t i c of the d i m e n s i o n s of the s a m p l e and the r e c i p r o cal of the m e a n f r e e path d e t e r m i n e d by the o r b i t l a t t i c e coupling. Only a s m a l l band of m o d e s have m e a n f r e e paths which a r e m u c h l e s s than the s a m p l e d i m e n s i o n s . B e c a u s e of this the e x p r e s * Alfred P. Sloan Fellow 230

sion f o r the t o t al conductivity t a k e s the f o r m

K:

cksvks

,

(2)

w h e r e L is on the o r d e r of the s a m p l e t h i c k n e s s . The p r i m e i n d i c a t e s that only those m o d e s whose ( o r b i t - l a t t i c e ) m e a n f r e e paths a r e g r e a t e r than L a r e to be eounted. S p e c i a l i z i n g to a t w o - l e v e l p a r a m a g n e t i c spin s y s t e m we find that (2) can be w r i t t e n K : A T 3 - I z 2 A w L w 4 exp(l~wo/kT) × 2~2kT2 ~2

(3)

× (exp(l~Wo/kT) - 1) -2 . H e r e A T 3 is the conductivity in the a b s e n c e of a spin-phonon i n t e r a c t i o n (boundary s c a t t e r i n g is a s s u m e d to be dominant), ~w o is the doublet splitting, and A¢o is the bandwidth of phonons with m e a n f r e e p at h s l e s s than L. We have a l s o a s s u m e d that/~Aw i s much l e s s than kT. P l o t t e d as a function of ~w o the r e s i s t i v i t y obtained f r o m (3) h as a m a x i m u m in the v i c i n i t y of ~w o = = 4 k T , in a g r e e m e n t with the t h e o r y d ev el o p ed in ref. 1. It is a p p a r e n t that a p a r t f r o m the constant A, which m ay be r e g a r d e d as being fixed by e x p e r i ment, the significant p a r a m e t e r in (3) is the bandwidth Aw. The c a l c u l a t i o n of ~w depends on a knowledge of the a b s o r p t i o n in the wings of the line. We can obtain what a r e roughly l o w e r and u p p er bounds on A0~ by a s s u m i n g G a u s s i a n and L o r e n t z i a n shapes, r e s p e c t i v e l y . In the c a s e of a G a u s s i a n line ~ we find AcoG = 2~/~y(ln(L/Xo))½ We assume that the doublet splitting is sufficiently l a r g e s o that the anti-resonance terms in the line shape function can be neglected.

Volume 20, number 3

PHYSICS LETTERS

15 February 1966

effective in t r a n s p o r t i n g excitation to the bound a r i e s of the s a m p l e .

where Xo is the m e a n f r e e path of the phonons at the c e n t e r of the line and V is the linewidth. With the Lorentzian__. shape we obtain the r e s u l t AwL = = 2V(L/Xo)~. A c r u d e c o m p a r i s o n with e x p e r i m e n t s u g g e s t s that the m e a s u r e d v a l u e s of Aw lie c l o s e r to the G a u s s i a n value than the L o r e n t z i a n . F i n a l l y we would like to r e m a r k that the a s s u m p t i o n that the phonons in the band Aw do not c o n t r i b u t e to the t h e r m a l c u r r e n t is supported by the work of Holstein [4]. In studying the i m p r i s o n m e n t of r e s o n a n c e r a d i a t i o n he found that only the r a d i a t i o n e m i t t e d in the wings of the line is

The author would like to thank Prof. R. Orbach for s t i m u l a t i n g his i n t e r e s t in the p r o b l e m .

References 1. R.Orbach, Phys. Rev. Letters 8 (1962) 393; Physics Letters 3 (1963) 269. 2. D.L.Huber, Physics Letters 12 (1964) 309. 3. J. Callaway, Phys. Rev. 113 (1959} 1046. 4. T.Holstein, Phys.Rev. 72 (1947) 1212.

* * * * *

VACANCY

FORMATION ENERGY AND DEBYE IN C L O S E P A C K E D M E T A L S

TEMPERATURE

N. H. M A R C H *

Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York Received 8 January 1966

Screening theory is shown to lead directly to the relation between vacancy formation energy and Debye temperature proposed empirically by Mukherjee.

Mukherjee [1] has r e c e n t l y p r o p o s e d the e m pirical relation 1 1 1 0 = C ETMZ ~ (1) between the Debye t e m p e r a t u r e 0 and the vacaney f o r m a t i o n e n e r g y Ev, for elose packed m e t a l s . Here M is the atomic m a s s , f~ is the atomie volume and C i s a eonstant. The p u r p o s e of this c o m m u n i c a t i o n is to show that the f o r m (1) follows d i r e c t l y f r o m s c r e e n i n g theory in m e t a l s . We r e q u i r e two r e s u l t s ; the f i r s t for the long wavelength b e h a v i o u r of the phonons and the second for the e n e r g y change on r e m o v i n g an atom. Taking up the second point, we can a r g u e that in the s i m p l e s t approximation, for a m e t a l of v a l e n c y Z, the vacancy can be r e p r e s e n t e d a s a s c r e e n e d c e n t r e of charge - Z . The s c r e e n e d p o t e n t i a l V(r) round the vacaney can be c a l c u l a t e d f r o m the equation

V2V(r) = [ V ( r ' ) J l ( 2 k f ! r - r ' ] ) d r '

J

(2)

It_r,] 2

* On leave from the University of Sheffield, England.

given by M a r c h and M u r r a y [2], which is equiv a l e n t ot d i e l e c t r i c s c r e e n i n g t h e o r y [3]. Here kf is the F e r m i wave n u m b e r . Then the e n e r g y change due to the s c r e e n e d potential is obtained f r o m f i r s t - o r d e r p e r t u r b a t i o n theory as

AE : 0 / v a t

(3)

where p, the n u m b e r of density of e l e c t r o n s , is s i m p l y kf3/3n 2. But by i n t e g r a t i n g (2) over r we find [4] = g2Z/kf- , (4)

fvdr

and hence f r o m (3) and (4) it follows that AE is just 2 ZEf, where Ef is the F e r m i energy. If, following F u m i [5], we now c o r r e c t for the fact that when we f o r m a vacancy, we i n c r e a s e the volume of the m e t a l by one atomic volume, neglecting r e l a x a t i o n effects, then the f o r m a t i o n e n e r g y i s reduced by ~-ZEf to ~ ZEf. In v~ew of some u n c e r t a i n t y in the n u m e r i c a l coefficient we choose to write E v = ~ ZEf , (5) where a is a c o n s t a n t s u b s t a n t i a l l y l e s s than unity. We s t r e s s that (5) is valid only for s m a l l Z and m u s t be c o r r e c t e d b y higher o r d e r t e r m s in Z for polyvalent m e t a l s . 231