THE Tγ TEMPERATURE DEPENDENCE OF THE MEAN-SQUARE DISPLACEMENT OF ATOMS IN SOLIDS

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THE Tγ TEMPERATURE DEPENDENCE OF THE MEAN-SQUARE DISPLACEMENT OF ATOMS IN

SOLIDS

B. Kolk

To cite this version:

B. Kolk. THE Tγ TEMPERATURE DEPENDENCE OF THE MEAN-SQUARE DISPLACE- MENT OF ATOMS IN SOLIDS. Journal de Physique Colloques, 1979, 40 (C2), pp.C2-680-C2-682.

�10.1051/jphyscol:19792236�. �jpa-00218618�

JOURNAL DE PHYSIQUE

Colloque C2, supplkment au n o 3, Tome 40, mars 1979, page - C2-680

B. Kolk

Department of Physics, Boston University, 111 C d n g t o n S t r e e t , Boston, Massachusetts, 02215,U.S.A.

R6sumb.- Le d6placement quadratique moyen <x2$ des atomes dans les solides est Qtudib en fonction de la tempbrature, et compare

1

la loi empirique en

[r3h

rbcemment mise en bvidence. Un ajustement aux moindres carrbs des donn6es expbrimentales pour diffgrents mstaux 1 une loi du type <x2$

= a +

bTY conduit

1

des valeurs de y allant de 1,2

1

1,9. On donne les formules exprimant le moment p(-2) et la constante d1anharmonicit6 en fonction des paramstres

a,

b et y .

Abstract.- The recently empirically observed T3A temperature dependence for the mean-square displa- cement, e2$, of atoms in solids is investigated. Least-squares fits of <x23data ofvariousmetals to

<x2$

= a +

bTY, yield values of y ranging from 1.2 to 1.9. Formulas are given to derive the fre- quency moment

u ( - 2 )

and the anharmonic constant

E(-2)

from the parameters

a ,

b and y.

1.

Introduction.- The electric field gradient (EFG) at a nuclear site in various non-cubic metals appears

to decrease with temperature as

[r3'2.

This interesting empirical fact has led to a number of theoretical studies /I-3/ which show that the temperature de- pendence of the EFG can be understood by assuming that the mean-square displacement of the host atoms,

< x 2 $ ,

follows the same T 3 ' 2 law. Empirical studies /3,4/ of <x21) data seem to support such an assump- tion. It is of interest to investigate whether the exponent y of the temperature in

<x2$

= a +

b~~

₍₁₎

is a fundamental constant equal to exactly 3/2, or whether the value 1.5 is only an approximation. Also it is of interest to find relations between the pa- rameters a,b and y and the lattice dynamical proper- ties of the solid (see sec.

3).

It is well known that in the harmonic appro- ximation <x2$ is a linear function of T at high temperatures. Hence, the observed T34 behavior is very likely a result of anharmonic effects. In the following section the anharmonic effects on <x2 are discussed.

d

2. Anhannonic effects on &s.- In the harmonic approximation the motions of N individual atoms in a crystal are described by the superposition of 3N independent normal modes. In this approximation pho- nons have an infinetely long mean life. When the anharmonic terms in the vibrational Hamiltonian are taken into consideration, the energy of each

a

harmonic mode, described by the wave vector k and the polarization index

j ,

is shifted by an amount

+Supported by NSF Grant No. DMR 77-19017

f i A ~ . Moreover, the harmonic modes are no longer independent of each other, so that the phonons do not live infinitely long, but have a mean life

I/r,&..(Hence, for a harmonic crystal kj -

^0.).

^To

keep the following formulas simple a monoatomic iso- tropic crystal is considered. In that case the mean- square displacement can be expressed as /5/

It is important to notice that this result is exact;

no approximation has been made; i.e. anhannonic effects up to all orders are included. However, to evaluate equation

(2)

in closed form seems impossible

In crystals which are not extremely anharmonic the phonons have still a long mean life, so that rzi/qei

<< 1.

In these cases we may take the limit

OF the-right side of equation (2) for

r 2

./a2 k j * o , which yields /5/

coth(hl-. ./2kgT)

I f

2

e2$

=

*

_{N k j}

⁽³⁾

ul%

This expression is similar to that obtained for

<x2$ in the harmonic approximation, however with the difference that the harmonic mode frequencies

wG

in the harmonic expression of a21, are repla- ced by

The frequency shift Azi can be written as /5/

A A

kj/% ^{= EG}

T +

6&

_lu' T2

+

....

⁽⁵⁾

Henceforth, Axi is considered only in first order of T.The mean-square displacement can be expressed

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19792236

i n t e r m s o f f r e q u e n c y moments, v e r y h a r d t o e v a l u a t e . The s t a n d a r d d e v i a t i o n s i n t h e v a l u e s o f a , b and y f o r i r o n i m p u r i t i e s i n m e t a l s ( 6 )

a r e i n c l u d e d , s i n c e t h e e r r o r s i n t h i s G s s b a u e r e f - f e c t d a t a a r e s m a l l and w e l l e s t a b l i s h e d .

and anharmonic c o n s t a n t s o f o r d e r n ,

From t h e v a l u e s o f a and y, d i s p l a y e d i n t a b l e I , ( n =

c Z j w i j n

/

4

^WA k j

" .

( 7 ) t h e f o l l w i n g c o n c l u s i o n s c a n b e d r w n :

k j k j

( i ) The e x p o n e n t y d o e s n o t h a v e a u n i v e r s a l v a l u e of I n t h e low t e m p e r a t u r e l i m i t , T ⁺ 0 , e q u a t i o n ( 3 ) 312, b u t v a r i e s from 1.2 t o 1.9.

becomes ( i i ) The v a l u e s o f a a p p e a r t o b e s m a l l e r t h a n t h e

<xZT> = $ ~ ( - 1 ) / 2 M (8) measured v a l u e s o f t h e n e a r s q u a r e d i s p l a c e m e n t s a t w h e r e a s a t moderate and h i g h t e m p e r a t u r e s *

TL

_-0, <XzO>, because e q u a t i o n (1) d o e s n o t f i t t h e

(OD i s t h e Debye t e m p e r a t u r e ) e q u a t i o n ( 3 ) can be <x2T> d a t a v e r y w e l l a t t e m p e r a t u r e s T + 0.

expanded i n t h e f o l l o w i n g s e r i e s 151; T h e r e i s l i t t l e d i f f e r e n c e b e m e e n t h e c u r v e s

= { k B T / ~ } F ( - 2 ) { I + ~ E ( - 2 ) ~ } + I / 12

( X

/kBT)'+ ab t a i n e d b y f i t t i n g t h e <xzT> d a t a t o e q s . ( I) and ( 9 ) a t h i g h e r t e m p e r a t u r e s . T h i s f a c t e n c o u r a g e d u s t o

-

11720 ($/kg4I4 ~ ( 2 ) { ~ 2 € ( 2 ) T } +

..z

^{( 9 )}

u s e t h e f o l l o ~ i n g p r o c e d u r e i n d e t e r m i n i n g t h e r e l a - r h e r e

M

i s t h e mass o f t h e atom u n d e r c o n s i d e r a t i o n .

The terms i n e q u a t i o n ( 9 ) ~ h i c h c o n t a i n f r e q u e n c y mo- ments o f o r d e r n = 2 o r l a r g e r may i n g e n e r a l b e ne- g l e c t e d a t t h e t e m p e r a t u r e s where e q u a t i o n (9) i s v a l i d . I n most c a s e s <zZT> d a t a c a n b e f i t t e d v e r y w e l l w i t h e q u a t i o n (9) when t h e low t e m p e r a t u r e

(T < OD/2n) p o i n t s a r e l e f t o u t . R e s u l t s of s u c h l e a s t - s q u a r e s f i t s a r e p r e s e n t e d i n t a b l e I , w h e r e i n s t e a d o f p(-2) t h e r e l a t e d q u a n t i t y / 5 /

t i o n s b e t w e e n a , b and y , and t h e l a t t i c e d y n a m i c a l q u a n t i t i e s P(-2) and E ( - 2 ) . Eqs. ( I) and (9) a r e ex- panded i n a p m e r s e r i e s of T

-

^T1

^-

T, w h e r e T1 i s

t h e t e m p e r a t u r e a t w h i c h t h e r e s u l t s of e q s . ( 1) and ( 9 ) c o i n c i d e . B e c a u s e e q s . ( I) and (9) g i v e r i s e t o n e a r l y i d e n t i c a l r e s u l t s , t h e zero- o r d e r , f i r s k o r d e r and s e c o n d - o r d e r t e r m s i n T o f b o t h e x p r e s s i o n s s h o u l d b e e q u a l t o e a c h o t h e r . From t h e s e "matching"

r e l a t i o n s , and u s i n g a

2 e2

>, t h e f o l l w i n g e x p r e r 8(-2) ^a

(HAB)

J 3/p(-2)

i s g i v e n .

(10) s i o n f o r t h e e x p o n e n t y i s d e r i v e d ;

y

Ik-

312 f 112 /I-8<xZ0> / {<x2 < x Z 0 > ) (11) 3. The T~ dependence o f <x2,>.- From l e a s t - s q u a r e s where / 6 /

f i t s o f t h e e Z T > d a t a t o e q u a t i o n ( I ) t h e v a l u e s a,

2

9 < z 2 >

b and y, p r e s e n t e d i n t a b l e I , a r e o b t a i n e d . i s t h e mean-square d i s p l a c e m e n t a t t e m p e r a t u r e T l a s The s t a n d a r d d e v i a t i o n s i n t h e v a l u e s of a , b and g i v e n by e q . ( l ) . Hence, y = 312 f o r < x 2 , > = 9 < ~ ~ ~ > - f o r t h e P u r e m e t a l s a r e o m i t t e d , because the errors The mean-square d i s p l a c e m e n t a t t h e m e l t i n g ~ o i n t , i n t h e < x Z T > d a t a ( m o s t l y from experiments) are which i s r o u g h l y 15 t i m e s t h a t a t T - 0 , g i v e s a n ap-

p r o x i m a t e u p p e r l i m i t o f < x Z 1 > . T a b l e I

R e f e r e n c e s f o r t h e p u r e m e t a l < x Z T > d a t a can b e found i n S i n g h and Sharma, Phys. Rev. (1971) 1141. The

<x2 > d a t a of i r o n i m p u r i t i e s i n v a r i o u s h o s t s g a s c b t a i n e d from Nussbaum e t a l . , Phys. Rev.

173

( 1968) 653.

I n The l e a s t - s q u a r e s f i t t i n g of t h e i r o n i m p u r i t y d a t a t h e <x2 > v a l u e s a t t h e l w e s t t e m p e r a t u r e s w e r e o m i t t e d , s i n c e t h a t r e s u l t e d i n a b e t t e r f i t a t h i g h e r t e m p e r a f u r e s .

1 System

A 2 Cr Cu Ag A u Pb

F e i n Cu F e i n Pd F e i n P t

$

a ( 10- 'A') 0.84 0 . 3 1 9.23

, 1.64

0.47 2. 18

0. 12 t 0 . 0 1 0 . 14

+

0.006 0 . 15 2 0 . 0 0 7

b ( I O - ~ A ~ K ~ / ~ ) 4.4

1.6 2.1 11.0 10.0 0 . 8 8

5. 17 2 0.97 2.79

+

0.26 6.78 ^f 0 . 7 8

( K ) 0(-2) ( K ) o m e q . ( I From eq. ( 9 )

443 460 f 2 8

376 372 f 3

330 336

+

5

222 201 f 5

187 187

+

⁸

165 134

+

7

36 1 374 f 1.3

323 326 2 1.5

367 372

*

3.0

y 1.50 1.47 1. 19 1.28 1. 30 1.94

1 . 2 0 f 0 . 0 3 1.32 ?: 0 . 0 1 1 . 2 0 f 0 . 0 2

E(-2) ( I O - ~ K ~ ) E(-2) ( lo-* ')K From e q . ( I) From e q . ( 9 ) 4 . 0 8 5 . 1

+

1.6 0 . 8 2 1.07 c 0 . 15

1.3 1 1.7 +_ 0 . 6

0 . 5 3 0.4

+

^{0 . 3}

1.68 1.6

+

^0.7

33.8 29.0

+

^3.2

2 . 0 0 2 . 3 8

+

0.10 0 . 8 7 0.92

+

0 . 10

0.82 1.07

+

0. 15

C2-682

^JOURNAL DE PHYSIQUE For 9

5

^<x2 > 15 t h e v a l u e s o f y range from 1.2 t o

1.8 i n f a i r agreement w i t h t h e range of y shown i n t a b l e I.

From t h e "matching" r e l a t i o n s i t can r e a d i l y be d e r i v e d t h a t

E (-2) =

Y-l I ,

( 13)

2-Y 4T1 and t h a t 1.1(-2) =

---

M

k ~ T l

(14) where

The v a l u e s of E ( - 2 ) and 0(-2) ( s e e e q . 1 0 ) . d e r i v e d from t h e p a r a m e t e r s a, b and y w i t h t h e a i d o f t h e f o r m u l a s above, a r e c o l l e c t e d i n t a b l e I. Comparison w i t h r e s u l t s o b t a i n e d from a l e a s t - s q u a r e s f i t o f the same d a t a t o e q u a t i o n ( 9 ) , shows a f a i r agreement w i t h i n t h e e r r o r l i m i t s .

The a u t h o r would l i k e t o thank M r . E . Gawlinski

R e f e r e n c e s

/ I / J e n a , P . , Phys. Rev. L e t t .

2

(1976) 418.

/ 2 / Kolk, B . , J . Physique C o l l o q .

2

(1976) C6-355.

/3/ Nishiyama, K. and R i e g e l

,

^D., Hyperf

.

^{I n t}

. ⁵

(1978) 490.

/ 4 / Bastow, T . J . , Mair, S.L. and W i l k i n s , S.W., J . Appl. Phys.

48

(1977) 494.

/ 5 / A d e t a i l e d d i s c u s s i o n a b o u t anharmonic e f f e c t s on < x ~ ~ > , and t h e r e l e v a n t r e f e r e n c e s , a r e g i v e n by B. Kolk i n "Dynamical P r o p e r t i e s o f S o l i d s "

Vol. I V , Eds. G.K. Horton and A.A. Maradudin, (North Holland/ American E l s e v i e r P u b l i s h i n g Company)(to b e p u b l i s h e d ) .

1 6 1 To f u l f i l l E q u a t i o n 12 t h e p o s i t i v e s i g n i n Equa- t i o n 1 1 h a s t o be chosen when 6a/bT1) > 1 and t h e n e g a t i v e siHn, when 6a/(bT1 Y2) < 1 . F o r v e r y anharmonic s o l i d s m a 2 > i s l a r g e and Tl r e l a t i - v e s m a l l , s o t h a t 6a/(bTl 3/2) > 1 and t h u s y > 312 whereas f o r l e s s anharmonic s o l i d s y < 312 i s e x p e c t e d .

and M r . K. Auerbach f o r t h e i r h e l p i n t h e c o m p i l a t i o n of t a b l e I.