THEORY OF TEMPERATURE DEPENDENCE OF KNIGHT SHIFTS AND SPIN-LATTICE RELAXATION TIMES IN LIQUID LITHIUM AND SODIUM

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THEORY OF TEMPERATURE DEPENDENCE OF

KNIGHT SHIFTS AND SPIN-LATTICE

RELAXATION TIMES IN LIQUID LITHIUM AND

SODIUM

P. Heitjans, A. Coker, T. Lee, T. Das

To cite this version:

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JOURNAL DE

PHYSIQUE

CoZZoque C8, suppldment au n08, Tome 41, aoct 1980, page

C8-403

THEORY OF TEMPERATURE DEPENDENCE OF KNIGHT S H I F T S AND S P I N - L A T T I C E RELAXATION T I P E S I N L I Q U I D L I T H I U M AND SODIUM

P.

Heitjans, A . ~oker', T. ~ee' and

T.P.

as+

Fachbereich Physik, Universitat

Marburg,

R.F.A.

and

Institut Laue-Langevin, Grenoble, France '~e~artment

of

Physics, State University of New-York, Albany,

N. Y.

12222, U.

S.

A .

Abstract.- Using a first-order pseudopotential perturbation procedure we have investigated the Knight shifts K and spin-lattice relaxation times T I in liquid lithium and sodium both for their values at the melting points and the temperature dependences. The analysis uses temperature-depen- dent structure factors obtained from n e u t r o n - diffraction data. Our results for K and T I at the

melting point agree well with available data for both metals, the T I data for lithium referring to 8 ~ i nucleus being taken from polarized-neutron capture beta-decay technique and the other data from nuclear magnetic resonance measurements. For the temperature dependences, our calculations give positive slopes for K and (T,T)-I, a trend in agreement with experiment. However, while the slopes in sodium are in reasonable agreement with experiment with respect to magnitudes, they are substantially over-estimated for lithium. Arguments are presented for second-order effects of the pseudopotential in lithium as being the likely source for this difference between theory and experiment.

I. INTRODUCTION

The theoretical study

[I]

of hyperfine inter- experimental errors are relatively sizeable and do action

-

related properties such as Knight shifts

(K) and relaxation times (TI) in solid metals provides a valuable opportunity to test our under-

s c o i ; d i ~ i g oL tile

rlecrron-distributions

in them. In the case of liquid metals 12-41 one needs information about structure factors which are the counterparts of the lattice structures in the solid metals. While theoretical procedures 153 for ca,lculating the structure factors are available, it is helpful if the latter are deter- mined experimentally because it then allows theory to concentrate only on the nature of the electron distribution. In addition to attempting to explain the absolute magnitudes of K and T I , there is an added dimension and challenge to theory when experimental data on the temperature dependence of K and T , are available. In the metal lithium, of major interest in the present work, relaxation data are available 161 over an extensive temperature range, from the melting point 454 K up to 600K higher, from the polarised-neutron capture beta-decay technique 171. Experimental data are also available [8] for the structure factors over a large temperature range from neutron diffraction data which makes this metal a good vehicle for the examination of the theory of relaxation rates in liquid alkali metals. Knight-shift data are also available [4,9lfor liquid lithium over a smaller range from the melting point to 673

K

from conventional magnetic resonance measurements but in view of the smallness of

K

in this metal, the

not allow a definitive determination of the temperature dependence of

K.

We shall also study the temperature-dependence of the relaxation rate and Knight shift in sodium, for which T I and K data arc available [lo,]

11

over a more limited range above the melting point (370 K). Accurate neutron diffraction data are also available 181 as a function of temperature in sodium.

The study of Knight shifts and relaxation rates requires [12,131 a knowledge of the average spin density B ='lykF(o)12>at the nucleus due to electrons at the Fermi-surface and respectively the uniform field and non-uniform field spin susceptibilities xs, or in Fourier space, xs(o) and x~(Y). For the spin density s from the Fermi-surface electrons, we shall follow the conventional procedure 1141 of perturbation of plane waves in the liquid state by the pseudo-potential to get the pseudo-functions for the Fermi-surface electrons and then orthogonalising these to the core- functions of the ions to get the actual wave- functions in terms of orthogonalised plane-wave functions. Following the initial formulations of this type 1 \ 5 1 for studying changes in the Knight- shift on melting, there have been a number of theoretical investigations. Among these is the work of Watabe et a1 [I01 and Srivastava and Sharma [I61 who utilized a first-order local pseudopotential and went up to q vecLors (repre-

+ -+

senting admixture of states of momentum kF+q to a Fermi momentum state

zF)

of length 2kF.

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JOURNAI. DE PHYSIQUE

Subsequently, Ritter and Gardner [17], in their study of temperature dependence of Knight shifts of sodium also employed a first-order perturbation treatment but using both local and non-local components of the pseudopotential and more extensive integration in q using theoretical structure- factors [5]. Jena et a1 [14], in their treatment of the absolute value and temperature dependence of the Knight shift in cadmium, have developed a perturbation treatment up to second-order in the pseudopotential including local and non-local components, the incorporation of the second-order effect with non-local pseudopotential involving substantial effort. In subsequent work on K in liquid metal alloys. Perdew and Wilkins 1181 used a first-order local pseudopotential treatment, removing an approximation of a constant enhancement factor in going from a plane-wave to the corres- ponding OPW function used in the work of Jena et a1

make use of available experimental data at the melting points [23], limited temperature-dependence data available for liquid lithium [24] and pressure dependence data in solid statss 17.61 in both metals. In evaluating the relaxation time, one needs in addition a difference in the enhancement effects for the Fourier components

x

(K) of the non-uniform spin susceptibility and that for the uniform field, namely Xs(o). Earlier theoretical results [20] for this effect in the literature shall also be utilized for this purpose.

Section I1 presents briefly the procedure of calculation and our results using the direct spin- density contribution

T I ]

using first-order pseudopotential perturbation procedure. Section 111 discusses the nature of Lhe agreement between theory and experiment and possible effects that could improve this agreement for the temperature dependences.

[141. Recently in studying K at melting points in

11. PROCEDURE AND RESULTS a number of metals, Styles and Tranfield [I91

The direct contributions to the Knight shift extended Perdew and Wilkins' [I81 treatment co

and relaxation times are given [1,20] respective- include pseudopotentials with local and non-local

l y by tne equations: components. In our present work, we shall make

use of the treatment of Jena et a1 [I41 up to first-order but with the q-dependence of the enhancement factor which is similar to the procedure of Styles and Tranfield [19]. However, we shall make u6e of some of the conclusions with respect

to second-order pseudopotential perturbation effects found by Jena et a1 1141 with respect to temperature dependence of K in cadmium in discus- sing the nature of agreement between the present work and experimental results [6,9-111 in liquid lithium and sodium.

In addition to this direct contribution to the spin density and hence the Knight shifts and relaxation rates, there can also be small but significant contributions from other effects such as the exchange polarization of core electrons 11,201 and manybody or correlation type effects r21.221, both of which have been studied extensi- vely in atomic systems and the former in solid metals. We shall use the conclusions from these investigations in the Discussion section (Section 111) to consider their possible effects o n K and T and their temperature dependence.

I

For the spin susceptibilities and their temperature dependences needed in the calculations of the Knight shifts and relaxation times, we shall

and

where

x

is the uniform field spin susceptibility per unit volume,

R

is the atomic volume, y N and ye the nuclear and electronic gyromagnetic ratios, k the Boltzmann constant and ~-'(a) a factor

B

11,201 arising from the different exchange

enhancementsof

x

(K) and

x

(0) referred to earlier.

From equations

(I)

and (2) one gets the modified Korringa relation [l3,20

1

The expression for the spin density < ( Y _(o)12>

kF in the liquid considering first-order effects of the pseudo-potential, following the steps in [I41

*

(but not making the assumption of Ikl-independent overlaps between plane-wave and core-functions), is given by

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S ( q ) ~ , ( k , . k ' ) v ( k , , k ' ) 3, = P

(

d3Z1 ( 6 ) t e m p e r a t u r e r a n g e of 470 K t o 725 K i n o u r work, k; - k t 2 t r a n s l a t e s t o l e s s t h a n 0 . 2 % . F o r l i q u i d sodium, no e x p e r i m e n t a l d a t a on X: above m e l t i n g p o i n t a r e s(q)d,,(k,,k1 )Y ( k F , k ' ) J,, = P

(

d32*

--

2 ( 7 ) a v a i l a b l e . Using p r e s s u r e dependence a t a [ 2 6 1 a s kF - k 1 a l n x t -$-I i n t h e c a s e of l i t h i u m l e a d s t o (-) =1.4x10

,

a T P w i t h 3 3 + b i ( k ) = .li(;)eik-r d3: The S ( q ) , w i t h t h e < > d e n o t i n g ensemble a v e r a g e , a r e t h e i n t e r f e r e n c e f u n c t i o n s which i n o u r work a r e t a k e n from n e u t r o n d i f f r a c t i o n measurements r8

1.

The Y . ) (; r e f e r t o a l l t h e c o r e s t a t e f u n c t i o n s of t h e m e t a l i o n s , n s r e f e r r i n g t o o n l y 3 c o r e s s t a t e s . The b i ( k ) a r e i n g e n e r a l d e p e n d e n t 3 on t h e d i r e c t i o n of k when t h e s t a t e s i r e f e r t o non-s s t a t e s , b u t s i n c e one stlrrls o v e r compl , A t e s h e l l s , t h e r e i s no dependence on t h e d i r e c t i o n of

k-

i n t h e summation o v e r i i n e q u a t i o n ( 5 ) . The w ( k F , k t ) r e f e r t o t h e p s e u d o p o t e n t i a l m a t r i x - a l e o i e n t s w i t h t h e l o c a l t e r m s d e p e n d i n g o n l y on + + q = [ k t - k F I w h i l e t h e n o n - l o c a l t e r m s depend on

c'

a s w e l l . We have used t h e p s e u d o p o t e n t i a l of Animalu and Heine [ 2 5 ] j n o u r work. The y ( k F , k l ) term i n e q u a t i o n ( 1 0 ) i s a consequence of removing t h e r e s t r i c t i o n b ( k ' ) = bns(kF) used i n e a r l i e r n s work [ 1 4 ] . A For t h e a t o m i c s p i n s u s c e p t i b i l i t i e s

xS

= xSR a t t h e m e l t i n g p o i n t , e x p e r i m e n t a l l y a v a i l a b l e v a l u e s from s p i n - r e s o n a n c e measurements were used

126,231. For h i g h e r t e m p e r a t u r e s , i n t h e c a s e o f l i t h i u m , e x p e r i m e n t a l measurements [ 2 4 ] from 300 K i n t h e s o l i d t o 493 K , a b o u t 40 K above t h e m e l t i n g p o i n t , i n t h e l i q u i d i n d i c a t e d t h a t t h e r e was no change i n XA w i t h i n a r a n g e of 1 % and s o X A was assumed t o be e s s e n t i a l l y t e m p e r a t u r e - i n d e p e n d e n t i n o u r work. T h i s a s s u m p t i o n a l s o r e c e i v e s s u p p o r t from t h e e x p e r i m e n t a l o b s e r v a t i o n t h a t

xA

i n t h e s o l i d s t a t e i n l i t h i u m 1261 a r e v i r t u a l l y p r e s s u r e - i n d e p e n d e n t . On t r a n s l a t i n g t h e volume changes i n t h e s e p r e s s u r e measurements t o t e m p e r a t u r e c h a n g e s u s i n g t h e c o e f f i c i e n t of volume e x p a n s i o n , one f i n d s a d i f f e r e n t i a l t e m p e r a t u r e c o e f f i c i e n t , a l n x A (--+) o f a b o u t 0 . 0 7 x I O - ~ K - ' w h i c h , o v e r t h e P

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JOURNAL DE PHYSIQUE F o r sodium we have c a r r i e d o u t o u r i n v e s t i g a - t i o n s a t 3 9 0 K, c l o s e t o t h e m e l t i n g p o i n t o f 370 K , and a t 590 K. Our c a l c u l a t e d K a t 390 K , u s i n g t h e e x p e r i m e n t a l

Xs

o f 1 . 1 3 x 1 0 - ~ c g s volume u n i t s [261, i s 0.101%, i n r e a s o n a b l e agreement w i t h t h e e x p e r i m e n t a l v a l u e [ 4 1 o f 0.116% a t t h e m e l t i n g p o i n t . The e f f e c t o f t h e p s e u d o p o t e n t i a l i s now r e l a t i v e l y weaker compared t o l i t h i u m , p r o d u c i n g a b o u t 30

X

r e d u c t i o n o f t h e one-OPW r e s u l t . A l s o t h e n o n - l o c a l p s e u d o p o t e n t i a l i s now l e s s dominant, p r o d u c i n g a J N L a b o u t t w i c e t h e s i z e of JL and a g a i n w i t h o p p o s i t e s i g n . For 6

,

u s i n g K = ( T ) ~ o u r c a l c u l a t e d s p i n d e n s i t i e s o f t h e two tempera- A t u r e s s t u d i e d and t h e v a l u e o f ( a l n X s ) e x t r a c t e d

aT

P from p r e s s u r e - d e p e n d e n c e d a t a f o r

,

x!

we f i n d t h e -4 -1 t h e o r e t i c a l bK = 2 . 4 x 10 K

.

o n l y a b o u t a f a c t o r 1.5 h i g h e r t h a n t h e e x p e r i m e n t a l -4 -1 6 = 1 . 8 ~ 10 K [ l o ] , t h e agreement b e i n g m u c h K b e t t e r t h a n i n t h e c a s e of l i t h i u m . For T I T a t 390 K u s i n g Eqs . ( 2 ) and ( 4 ) , f o r t h e 23Na n u c l e u s , we f i n d t h e t h e o r e t i c a l v a l u e o f 5.33 sK, i n r e a s o n a b l y good a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e [ I l l o f 4.41 sK. F o r t h e l o g a r i t h m i c d e r i v a t i v e 1 -4 - 1 6 (

-

) we have o b t a i n e d a v a l u e 4.8 x 10 K

,

TIT a l s o i n f a i r l y good agreement w i t h t h e e x p e r i m e n t a l -4 - 1 v a l u e of 3 . 9 x 10 K a t 390 K and 590 K o b t a i n e d from t h e e m p i r i c a l e x p r e s s i o n found from e x p e r i m e n t [ I l l . 111. DISCUSSION C o n s i d e r i n g f i r s t t h e K and T I T n e a r t h e ~ e l t i n g p o i n t , t h e r e i s r e a s o n a b l y good agreement between e x p e r i m e n t and t h e o r y f o r b o t h m e t a l s . T h i s s u p p o r t s o u r s p i n - d e n s i t y r e s u l t s and j u s t i f i e s t h e u s e o f t h e Animalu-Heine [ 2 5 ] p s e u d o p o t e n t i a l i n t h e p r e s e n t work. There a r e of c o u r s e a number of a d d i t i o n a l c o n t r i b u t i o n s t o t h e s p i n - d e n s i t y b e s i d e s t h e d i r e c t one c a l c u l a t e d h e r e , among them t h e exchange c o r e p o l a r i z a t i o n (ECP) [20],manybody e f f e c t s 121,223 and a l s o o r b i t a l e f f e c t s [ 2 8 - 301

,

t h e l a t e r more l i k e l y f o r l i t h i u m which h a s s u b s t a n t i a l p - c h a r a c t e r a t t h e Fermi s u r f a c e . The ECP e f f e c t on K i n s o l i d l i t h i u m i s s m a l l 1201 b e c a u s e t h e r e i s s i g n i f i c a n t c a n c e l l a t i o n between

t h e p o s i t i v e and n e g a t i v e ECP c o n t r i b u t i o n s from t h e s and p c o n p o n e n t s o f t h e w a v e - f u n c t i o n s n e a r t h e F e r m i - s u r f a c e . A s i m i l a r c a n c e l l a t i o n would be e x p e c t e d f o r t h e l i q u i d b e c a u s e t h e p s e u d o p o t e n t i a l and hence t h e r e a l p o t e n t i a l , i s q u i t e s t r o n g f o r l i q u i d l i t h i u m and t h e r e i s s i g n i f i c a n t s h o r t r a n g e o r d e r i n t h e l i q u i d , a t l e a s t w i t h r e s p e c t t o t h e n e a r n e i g h b o u r s two c a u s e s t h a t l e a d t o s i g n i f i c a n t p - c h a r a c t e r i n t h e s o l i d , t h e l a t t e r t h r o u g h t h e r e l a t e d i n f l u e n c e o f t h e B r i l l o u i n zone i n t h e s o l i d s t a t e . C o r r e l a t i o n e f f e c t s a r e known t o be l e s s t h a n 5 p e r c e n t i n l i t h i u m atom [211 and t h e r e i s no r e a s o n f o r them t o be more pronounced i n t h e m e t a l . Thus ECP and c o r r e l a t i o n e f f e c t s c a n n o t a d v e r s e l y i n f l u e n c e t h e good agreement between t h e o r e t i c a l and e x p e r i m e n t a l K. The r e l a x a t i o n r a t e i s i n f l u e n c e d o n l y s l i g h t l y [ 2 0 ] by t h e p-type ECP e f f e c t and s i n c e t h e r e i s no l o n g e r any c a n c e l - l a t i o n between p and s components, t h e l a t t e r by i t s e l f c o u l d s i g n i f i c a n t l y i n f l u e n c e T T, r e d u c i n g I i t t o a b o u t 60% o f t h e c a l c u l a t e d d i r e c t v a l u e from t h e d i r e c t s p i n d e n s i t y , and a b o u t 52 % of t h e e x p e r i m e n t a l v a l u e [ 6 ] , b u t t h i s i s n o t t o o s e r i o u s a d i f f e r e n c e i n view of t h e a p p r o x i m a t i o n s used i n - I o b t a i n i n g [ 2 0 ] t h e r e d u c t i o n f a c t o r R (a) i n E q . ( 2 ) . For sodium, t h e ECP e f f e c t i s a g a i n e x p e c t e d t o b e s i m i l a r t o t h a t of t h e s o l i d s t a t e , t h e r e b e i n g m a i n l y s - t y p e ECP e f f e c t which would i n c r e a s e b o t h K and t h e r e l a x a t i o n r a t e i n t h e d i r e c t i o n o f

b r i d g i n g t h e s m a l l gap between t h e o r y and e x p e r i m e n t . C o r r e l a t i o n e f f e c t s i n sodium [ 2 2 1 a t o E a r e somewhat s t r o n g e r t h a n i n l i t h i u m [21

1

b u t s t i l l l e s s t h a n

10

X

o f t h e e x p e r i m e n t a l h y p e r f i n e c o n s t a n t . The o r b i t a l c o n t r i b u t i o n t o ( T ~ T ) - ' i n s o l i d L i was e s t i m a t e d by Haea and Kaeda [ 3 1 ] t o be a b o u t 5 %.

We c o n s i d e r n e x t t h e n a t u r e of agreement I

between o u r c a l c u l a t e d 6K and 6(- ) and e x p e r i - T ~ T

ment. For b o t h m e t a l s , t h e o r y and e x p e r i m e n t [ 6 , 1 0 I I ] a g r e e w i t h r e s p e c t t o t h e t r e n d of i n c r e a s i n g K and r e l a x a t i o n r a t e w i t h i n c r e a s e o f t e m p e r a t u r e . However, i n t e r m s of m a g n i t u d e s , w h i l e t h e o r y and e x p e r i m e n t a g r e e r e a s o n a b l y w e l l f o r sodiurc, i n t h e I c a s e o f l i t h i u m t h e c a l c u l a t e d 6 (

-

) i s more T ~ T t h a n a f a c t o r of f i v e l a r g e r t h a n e x p e r i m e n t . I n c o r p o r a t i o n o f ECP and c o r r e l a t i o n e f f e c t s i s n o t I e x p e c t e d t o s i g n i f i c a n t l y a f f e c t 6(- ) b e c a u s e they i n f l u e n c e b o t h t h e d i f f e r e n c e i:lTI/T T a t two

1 t e m p e r a t u r e s a s w e l l a s t h e a b s o l u t e v a l u e s . The f a c t t h a t t h e p s e u d o p o t e n t i a l h a s much s t r o n g e r i n f l u e n c e on t h e s p i n d e n s i t y i n l i t h i u m a s corcpared t o sodium s u g g e s t s t h a t s e c o n d - o r d e r e f f e c t s of t h e p s e u d o p o t e n t i a l may have t o b e c o n s i d e r e d i n l i t h i u m t o b r i d g e t h e gap between e x p e r i m e n t a l and t h e o r e -

(6)

d e n s i t y i s c o n c e r n e d , t h e y made c o m p a r a b l e c o n t r i - [ I 4 1 P . J e n a , T.P. Das, G.D. G a s p a r i , N.C. H a l d e r , b u t i o n s of o p p o s i t e s i g n t o d i f f e r e n c e s a t two P h y s . Rev.

g,

2158 ( 1 9 7 1 ) .

t e m p e r a t u r e s . T h e s u b s t a n t i a l c h a n g e i n s p i n d e n s i t y [ I 5 ' F. Laclanann-C~rOt* P h ~ s . KOndens. Materie

3,

75 ( 1964)

.

-

due t o t h e s e c o n d - o r d e r e f f e c t w i t h t e m p e r a t u r e

I16

1

S . K . S r i v a s t a v a , P.K. Sharma, Nuovo Cimento

i s u n d e r s t a n d a b l e b e c a u s e t h e s t r u c t u r e f a c t o r s ,

-

708, 9 0 ( 1 9 7 0 ) .

w h i c h a r e a l s o t e m p e r a t u r e - d e p e n d e n t , o c c u r t w i c e [ I 7 1 A.L. R i t t e r , J.A. G a r d n e r , P h y s . Rev.

E, 46

i n t h i s c a s e [ 1 4 1 . S i n c e l i t h i u m a p p e a r s t o h a v e ( 1 9 7 1 ) . a s t r o n g e f f e c t f o r t h e s p i n d e n s i t y f r o m t h e [ I 8 1 J . P . Perdew, J . N . V i l k i n s , P h y s . Rev.

z,

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