Hungarian ^Academy of Sciences

OLVASD PÉLDÁNY

TU~ A bb 18 L\

KFKI-1981-33

K, v l a d á r

A . Z A W A D O W S K I

THEORY OF RESONANT ELECTRON SCATTERING IN AMORPHOUS METALS

Hungarian ^Academy of Sciences C E N T R A L

R E S E A R C H

I N S T I T U T E F O R P H Y S I C S

B U D A P E S T

KFKI-1981-33

THEORY OF RESONANT ELECTRON SCATTERING IN AMORPHOUS METALS

K. Vladár and A. Zawadowski

Central Research Institute for Physics II—1525 Budapest 114, P.O.B. 49, Hungary

Submitted to Phys. Rev. Lett.

HU ISSN 0368 5330 ISBN 963 371 814 7

The crossover temperature TK is calculated at which the motion of the tunneling atom and the conduction electron charge screening cloud is gradual­

ly coupled together. The first theoretical estimation is given to show that the formation of the resonance provides a realistic explanation for the

electrical resistivity minimum and for the inelastic electron scattering rate relevant in localization theory if T^ г 1 - 5 К.

АННОТАЦИЯ

Вычислена температура кроссовера Т|^, в окрестности которой туннелирующий атом и экранирующее облако электронов проводимости постепенно связываются.

Дается теоретическая оценка, согласно которой возникновение резонансного рас­

сеяния может объяснить минимум электрического сопротивления, если Тк ~ 1 - 5 К, а также число соударений при неупругом рассеянии электронов, имеющее важное значение в теории локализации.

KIVONAT

Kiszámitjuk azt a crossover hőmérsékletet, amely környékén az alagutazó atom és a vezetési elektronok töltésleárnyékoló felhője fokozatosan egymáshoz csatolódik. Első Ízben végzünk olyan elméleti becslést, amely igazolja, hogy a rezonanciaszórás kialakulása T^ : 1 - 5 К körül reális magyarázat lehet az elektromos ellenállás minimumára, és a rugalmatlan elektronszórásból adódó ütközési számra, mely utóbbi a lokalizáció-elméletben játszik fontos szerepet

The c o n c e p t of two l e vel s y s t e m s - ^ (TLS) has p r o v e d to be a m i l e s t o n e in u n d e r s t a n d i n g the l o w t e m p e r a t u r e b e h a v i o r of i n s u l a t i n g and m e t a l l i c g l a s s e s as w e l l — . C o n s i d e r i n g the2 / r e l a x a t i o n rate of TLS o b s e r v e d by u l t r a s o u n d m e a s u r e m e n t s on m e t a l l i c g l a s s e s the i m p o r t a n c e of i n t e r a c t i o n b e t w e e n c o n d u c t i o n e l e c t r o n s and TLS has f i r s t be e n p o i n t e d out by G o l d i n g et al— 3 / s u g g e s t i n g a K o r r i n g a - l i k e r e l a x a t i o n m e c h a -

4 /

nism. E a r l i e r , C o c h r a n e et a l — c a l l e d the a t t e n t i o n to the p o s s i b i l i t y th a t the e l e c t r o n - T L S i n t e r a c t i o n m a y c o n t r i b u t e to the f o r m a t i o n of e l e c t r i c a l r e s i s t i v i t y m i n i m u m o b s e r v e d

S / 6 /

in m a n y a l l o y s — . The r e s i s t i v i t y has b e e n s t u d i e d by K o n d o — and by the a u t h o r s — 7/ up to f o u r t h o r d e r in the c o u p l i n g a n d l o g a r i t h m i c c o n t r i b u t i o n has been o b t a i n e d in s o m ew h a t s i m i l a r m a n n e r as in the case of d i l u t e m a g n e t i c alloys. In o r d e r to get l o g a r i t h m i c c o n t r i b u t i o n it was n e c e s s a r y to take i n t o

a c c o u n t the a n g u l a r d e p e n d e n c e of the c o u p l i n g on the d i r e c t i o n of the in- and o u t g o i n g e l e c t r o n s . R e c e n t l y , the p o s s i b l e role of TLS in l o c a l i z a t i o n t h e o r y has b e e n e m p h a s i z e d in the

c o n t e x t of the i n e l a s t i c e l e c t r o n s c a t t e r i n g rate w h i c h by m a k i n g h o p p i n g b e t w e e n l o c a l i z e d s t a t e s l i m i t s the a p p l i c a - b i l i t y of the s c a l i n g a r g u m e n t s — .8 /

9 /

One of the p r e s e n t a u t h o r s — h a s s u g g e s t e d a s i m p l i f i e d m o d e l to d e m o n s t r a t e that in the l e a d i n g l o g a r i t h m i c a p p r o x i -

в

m a t i o n the e l e c t r o n - T L S i n t e r a c t i o n s c a l e s to a s trong c o u p l i n g p r o b l e m , namely, to the i s o t r o p i c ant i f e r r o m a g n e t i c K o n d o

p r o b l e m . This r e s u l t has b e e n i n t e r p r e t e d as the i n d i c a t i o n

that at low t e m p e r a t u r e a s t r o n g l y c o r r e l a t e d state is formed in w h i c h the t u n n e l i n g of the TLS and the c h a r g e p o l a r i z a t i o n c lo u d e x h i b i t i n g the F r i e d e l o s c i l l a t i o n are m o v i n g r i g i d l y t i g h t t o g e t h e r . The a i m of the p r e s e n t l e t t e r is to give the first t h e o r e t i c a l e v i d e n c e that at low t e m p e r a t u r e s the r e s o ­ nan t e l e c t r o n s c a t t e r i n g on TLS m a y c o n t r i b u t e to the e l e c t r i ­ cal r e s i s t i v i t y and to the i n e l a s t i c s c a t t e r i n g rate just in the range of v a l u e s o b s e r v e d in e x p e r i m e n t s . T h e s e e s t i m a t i o n s are b a s e d on s e c o n d o r d e r s c a l i n g a r g u m e n t s a n d it is shown tha t the c r o s s o v e r t e m p e r a t u r e d i v i d i n g the w e a k and s trong c o u p l i n g r e g i o n s is v e r y s e n s i t i v e on the v a l u e s of the c o u p ­ l ings but it m a y be in the r a n g e of 1K°.

The f o l l o w i n g H a m i l t o n i a n is c o n s i d e r e d

H *

I

1

ks 5TLS + Ло °TLS *

+ J , V s 1 (k'> Vaß £B (kl) aks < 4 s KK s

aß, i

(1 )

w h e r e i = x , y , z , a is the a n n i h i l a t i o n o p e r a t o r for c o n d u c t i o n К s

e l e c t r o n w i t h spin s and e n e r g y e , a* (i=x,y,z) is the p s e u d o -

К TLS

Í г i

spin P a u l i m a t r i c e s for TLS and V, = ) f (к1) V „ f„ (k) is

k'k n a aß p

a8 i

the c o u p l i n g b e t w e e n e l e c t r o n s and p s e u d o s p i n о f (k)'s form a c o m p l e t e set of f u n c t i o n s (e.g. s p h e r i c a l h a r m o n i c s ) , w h ich d e p e n d on the d i r e c t i o n of к o nly. It can be shown, that in the s t a r t i n g H a m i l t o n i a n one can tak e V ^=0 6 '7^ and V X d e s c r i b e s

X Z X

the e l e c t r o n a s s i s t e d t u n n e l i n g t h u s V /V << 1 as V is p r o ­ p o r t i o n a l to the t u n n e l i n g rate e ^<< 1 . F u r t h e r m o r e , it can be

s e e n — ^ by t a k i n g a r e p r e s e n t a t i o n w h e r e V Z is d i a g o n a l th a t o n l y those two i n d i c e s a , 8 are of i m p o r t a n c e for w h i c h the

z z

d i f f e r e n c e V -V„„ is the l a r ge s t . In a simple m o d e l w h ere ota 88

one atom m o v e s in a s y m m e t r i c d o u b l e w e l l p o t e n t i a l f^ and f 2 have been f o u n d as l i n e a r c o m b i n a t i o n s of s- and p - and d - l i k e f u n c t i o n s . The s e c o n d o rder s c a l i n g e q u a t i o n s are d e r i v e d by c h a n g i n g the e l e c t r o n ban d w i d t h c u t - o f f fro m v a l u e D to D' and they can be o b t a i n e d in the f r a m e w o r k of m u l t i p l i c a t i v e r e n o r m a l i z a t i o n g r o u p — ^ . In the f o l l o w i n g onl y t w o i n d i c e s a = l f2 are k e p t and p V1„ = v1 a1,, h o l d s in the r e p r e s e n t a t i o n

о aß aß

u s e d w h ere p Q is the c o n d u c t i o n e l e c t r o n d e n s i t y of s t a t e s at the F e r m i level for one sp i n d i r e c t i o n and v 1 is the

d i m e n s i o n l e s s c o u p l i n g . The s c a l i n g e q u a t i o n s for the s y m m e t r i c cas e A=o are

3 i. . . j, . k, . _ i, , , j. .2 k. .2 s x — v (x) = - 4 v (x) v (x) + 8 v (x) ( v (x) + v (x) )

oX

w i t h i / j / k

(2)

x *r*—■ £n Д (x) = 8 ( vZ (x)2 + vY (x)2 )

d X О (3)

w h e r e x= D ' / D . In the g e n e r a l c a s e Д/O s c a l i n g e q u a t i o n can

2 2 1/2

be d e r i v e d for the e n e r g y s p l i t t i n g E(D') = ( Д(х) + Äq (x) ) >

but the r a t i o Д/Д is c h a n g i n g as w e l l - ^ ^ . The s e c o n d o r d e r о

s ca l i n g e q u a t i o n s are c o r r e c t as far as the s c a l e d c o u p l i n g s are m u c h s m a l l e r th a n unity, vX (x)á о,2 . For eq. (2) there is

* * *

X V z

an i s o t r o p i c f ixed p o i n t v = v = v = 1/4 , but a c c o r d i n g to A n d e r s o n ' s a r g u m e n t the e x act s c a l i n g e q u a t i o n m u s t have stable

* * *

x v z

f i x e d p o i n t o n l y in the i n f i n i t y (or at zero) , th u s v =v =v =°°.

By i n t e g r a t i n g the s c a l i n g eq. (2) ^v1 < < 1 one get for the t y p i ­ cal v a l u e of the c u t o f f b e t w e e n the w e a k and s t r o n g c o u p l i n g r e g i o n s w h i c h is k n o w n as the c r o s s o v e r (Kondo) t e m p e r a t u r e --12/

1 x 4v

T = D ( ——— ) (vX vZ)x z,l/2

К (4)

4v x v z

at w h i c h v =v =v D i_T ~l/8 . The c h a n g e s of the c o u p l i n g s are К

d e p i c t e d in Fig. 1. The r e n o r m a l i z a t i o n of Aq is shown also in the s y m m e t r i c c a s e w h ere Л = 0 in the r e p r e s e n t a t i o n used.

Aq is s c r e e n e d by s e v e r a l o r d e r of m a g n i t u d e .

D e p e n d i n g on the v a lue of the p a r a m e t e r s of an i n d i v i d u a l TLS two c a s e s m u s t be d i s t i n g u i s h e d : (i) T >> E (T ) thus a

К к

c o r r e l a t e d state is formed, (ii) T £ E ( T ), t h e r e f o r e the corre-

К к.

l a t e d state can n o t be f o r m e d c o m p l e t e l y , b e c a u s e the s c a l i n g s t o p s at D ' ~ E ( T = D ' ) , thus the s i t u a t i o n e x i s t i n g at T'= D' is p r e s e r v e d even to v e r y low t e m p e r a t u r e s .

In case (i) o n e can tur n to the a n a l o g y of the K o n d o p r ob- x у

l e m for m a g n e t i c i m p u r i t i e s . At T=T v =v ~l/8 thus t h e y are К

l a r g e and the e l e c t r o n s c a t t e r i n g a m p l i t u d e is large as well.

For T << T one m u s t c o n s i d e r the s c a t t e r i n g a m p l i t u d e r a t h e r К

tha n the coupl i n g . Its d i a g o n a l p a r t p r o p o r t i o n a l to u n i t

o p e r a t o r I i n c r e a s e s and t e nds to the u n i t a r i t y limit, while TLS

the t e r m s p r o p o r t i o n a l to have a m a x i m u m at T=T and

T L S К

the y tend to zer o as the t e m p e r a t u r e is lowered. T h ese l i m i t s m e n t i o n e d can be a c h i e v e d o n l y if E=0.

5

In case (ii) the e l a s t i c (e.g. v for the case Aq= 0) and

x у

the i n e l a s t i c (v and v ) s c a t t e r i n g a m p l i t u d e s (or c o u p l i n g s ) r e m a i n r o u g h l y the same ev e n at v e r y low t e m p e r a t u r e s .

2

C o n s i d e r i n g d i f f e r e n t p h y s i c a l q u a n t i t i e s in c a s e (ii) a f a i r l y go o d a p p r o x i m a t i o n can be o b t a i n e d by c a r r y i n g out the c a l c u l a t i o n s in the l o w e s t o r d e r s of p e r t u r b a t i o n t h e o r y but w i t h the e n h a n c e d c o u p l i n g s v 1 <l/8 (i=x,y) d e p i c t e d in F i g . 1.

In o r d e r to e s t i m a t e T one can use the m o d e l w h e r e only К

one a t o m is m o v i n g . For this m o d e l B l a c k et a l -- o b t a i n e s13/

v 2 ~ Uvp (k d) w h e r e U is the p s e u d o p o t e n t i a l of the atom, v o F

is the a t o m i c volume, d is the s e p a r a t i o n d i s t a n c e b e t w e e n the two e q u i l i b r i u m p o s i t i o n s and is the F e r m i m o m e n t u m . C o n s i ­ d e r i n g the e l e c t r o n a s s i s t e d t u n n e l i n g t h r o u g h the b a r r i e r

14/ x 2 1/2

one o b t a i n s -- v ~ (k d) AvUp Л /V w h e r e A = w ( 2 M V ) / wit h

F о о

the w i d t h and h e i g h t of the s q u a r e p o t e n t i a l b a r r i e r , w and V r e s p e c t i v e l y , a n d M is the mas s o f the atom, f u r t h e r m o r e , the f a c t o r (k d) 2 is due to the fa c t t h a t the c h a n g e in the b a r r i e r

F

h e i g h t must be m e a s u r e d fr o m the a v e r a g e of the two p o t e n t i a l minima. U s i n g t y p i c a l v a l u e s p q v= 0.2 eV ^ , U=leV, d = 0 . 3-0.5,

0 — 1 IS/ Z V 7 — 3

M= 5ОМ , k =lA , A = 6 , t h e n v =0.2( k _ d ) and v / v ~10

proton F F

— 4 о z

-10 . In o r d e r to get T ~ 1 K one m u s t have v ~0.3 f o r D=lOeV.

К

It m u s t be e m p h a s i z e d th a t T m a y c h a n g e by s e v e r a l o r d e r s of K.

m a g n i t u d e due to small c h a n g e s of the p a r a m e t e r s , e s p e c i a l l y

. z

i n V

Let us c o n s i d e r n o w d i f f e r e n t p h y s i c a l q u a n t i t i e s . M i n i m u m in the e l e c t r i c a l r e s i s t i v i t y ; A s s u m i n g a u n i f o r m d i s t r i b u t i o n

2 2 1/2

P ( E ) = P for TLS e n e r g i e s Е = ( Д + Д ) , the n u m b e r of TLS for

о о

w h i c h cas e (i) h o l d s is T P . For t h e s e TLS the total s c a t t e - К. о

ring a m p l i t u d e is in the r a n g e of u n i t a r i t y limit. For T > T К the t e m p e r a t u r e d e p e n d e n c e of the r e s i s t i v i t y can be e s t i m a t e d by a s s u m i n g tha t t h ere is o n l y one e l e c t r o n in the f i n i t e state thus the s c a t t e r i n g rate is p r o p o r t i o n a l to the e n h a n c e d coup-

x 2 у 2 z 2 I

lings, a c t u a l l y to (v ) + ( v ) +(v ) I X = T/D ' In th;’-s r a n 9 e of t e m p e r a t u r e the r e s i s t i v i t y c h a n g e s l o g a r i t h m i c a l l y at l e a s t in one d ecade of T. As T->-0, the t o t a l i n c r e a s e ÄR in the r e s i s t i v i t y can be e s t i m a t e d as the s c a t t e r i n g a m p l i t u d e in the u n i t a r i t y l i m i t 2 р ^ / т \ m u l t i p l i e d by the n u m b e r of TLS in case (i ), thus

Д R~ m ne2

, -*1

(po 2P T

о К

p T

О К N

1

~2 8 ír

к (5)

w h e r e m and e a r e the e l e c t r o n m a s s and ch a r g e , N is the total n u m b e r of e l e c t r o n s and the f a c t o r 2 is due to the two c h a n n e l s

m k F . -7

a = l,2, f u r t h e r m o r e , p =-- у . In o r d e r to e x p l a i n Ä R ~ l O ftcm

° 2 ír

w i t h T K = 5 K ° and k F ~l8 ^ a n d N = 10 2^/cm2 one n e e d Pq=2 .1 0 ^ K ° ^cm 2,——/

w h i c h is an a c c e p t a b l e a m o u n t of TLS. It s h o u l d be n o t e d that Д R i n c r e a s e s w i t h T .

К

I n e l a s t i c e l e c t r o n l i f e t i m e : Black et a l -- s u g g e s t e d t h a t the13/

i n e l a s t i c e l e c t r o n s c a t t e r i n g rate d u e to TLS must be p r o p o r ­ t ional to those n u m b e r of TLS w h i c h c a n be e x c i t e d by t h e r m a l e l e c t r o n s ( E < T ) . U s i n g the g o l d e n r u l e e.g. for the c a s e Aq= 0

A All , . X . A , V . A . “ A „ m

T. = T— { (v ) + (v*) } p P T

m ri о о ⁽⁶⁾

1 1 »y ^ ^ »y /

In o r d e r to e s t i m a t e t. one can use P ~4.10 К cm —

in о and

3 4 — 3 — 1 x y

p =0.6 10 cm erg and a s s u m i n g T~T , t h u s v ~v ~ 0 . 1 2

о к

the one gets т 7 1 = 1 .2.1 0 l 0 s 1K° 1 .T. The e x p l i c i t e f a c t o r T

3

xn

x у

d o m i n a t e s the w e a k d e p e n d e n c e of the c o u p l i n g s v a n d v on the t e m p e r a t u r e . The o r d e r of m a g n i t u d e is the one s u g g e s t e d on the basis of r e s i s t i v i t y m e a s u r e m e n t s on t h i n w i r e s .18/

If one uses bar e c o u p l i n g s , the e s t i m a t e d x./ is le s s by a f a c t o r 50 and th a t is the s o u r c e of d i s c r a p a n c y p r e v i o u s l y

. .18/

q u o t e d -- .

. “1 r, *-1 r , Уч 2 , x. 2 . 2 / 2 z.2 .2.2-, , „— ,19/

TLS r e l a x a t i o n rate T^ = 8ti R { (чг) +(v ) A E +(v ) Aq/E ^x=T/DkT—

can be o b s e r v e d in u l t r a s o n i c m e a s u r e m e n t s w h i c h p r o v i d e the 20/

m o s t d i r e c t i n f o r m a t i o n on the c o u p l i n g s . U s i n g the n o t a t i o n N ( E )К = v N ( E ) f o r the e x p r e s s i o n in the c u r l y b r a c k e t above

e

(where N(E ) = 2 p and the o t h e r f a c t o r of two is due to the Pauli

2 0/ o p e r a t o r s i n s t e a d of spin 1/2 o p e r a t o r s in eq. (1)), NK^ has p r e v i o u s l y b e e n found in the o r d e r of 0.2 for s e v e r a l a l l o y s

2 1/

It has r e c e n t l y bee n p o i n t e d o u t -- that the e x p e r i m e n t on s u p e r c o n d u c t i n g m a t e r i a l s are e s p e c i a l l y c o n c l u s i v e if the a n o m a l y at the s u p e r c o n d u c t i n g t r a n s i t i o n is c o n s i d e r e d .

In Pd Zr the o b s e r v e d c o u p l i n g is N ( E )К =0.9, th u s if Е~Д

О ■ / О • j с

then v X ~ v y = 0 . 1 6 w h i c h c o r r e s p o n d to e n h a n c e d c o u p l i n g s near T .K. in Fig. 1. or if E~A and v X ~ v y < < v Z then v Z= 0 .32 for w h i c h T m u s t be large. These e x p e r i m e n t s give the f i rst d i r e c t e v i d e n c e

that the e n h a n c e d c o u p l i n g s can r e a l l y be f o u n d in the i n t e r ­ m e d i a t e s t r o n g c o u p l i n g r a nge thu s T ;, 1-5K° m a y o c c u r .

The c o n c l u s i o n of o u r l e t t e r is that if v 2 is l a r g e e n o u g h then a c harge p o l a r i z a t i o n c l o u d s t r o n g l y c o u p l e d to the TLS a t o m can be b u i l t up at T=T . If T ~1K ° t h e n the c o u p l i n g s are

К к

e n h a n c e d a n o m a l o u s l y and that may r e s u l t in we l l o b s e r v a b l e c o n t r i b u t i o n to the bul k r e s i s t i v i t y m i n im u m , to r e l e v a n t in l o c a l i z a t i o n t h e o r y a n d to T 1 d e t e r m i n i n g the u l t r a s o u n d a b s o r b t i o n . As the f i r s t two of t h e s e e x p e r i m e n t a l r e s u l t s m i g h t have d i f f e r e n t e x p l a n a t i o n , t h e r e f o r e , the p e r f o r m a n c e of t h e s e e x p e r i m e n t s on s a m p l e s p r e p a r e d or t r e a t e d ( a n nealing, by r a d i a t i o n , by h y d r o g e n ) in the same w a y w o u l d be of c r u c i a l i m p o r t a n c e to t e s t the p r e s e n c e of a b s e n c e of the c o r r e l a t e d state. It m u s t f i n a l l y be e m p h a s i z e d th a t t h e s e e f f e c t s s h o u l d not o c c u r in all of the m a t e r i a l s b e c a u s e a small c h a n g e in the c o u p l i n g s can p u s h T out of the r a n g e of i n t e r e s t . One m a y

К 2

r a i s e the idea that the l a r g e s t v 2 can be e x p e c t e d w h e n in the e l e c t r o n - t u n n e l i n g a t o m s c a t t e r i n g the d - l e v e l r e s o n a n c e

s c a t t e r i n g d o m i n a t e s at the Fermi e n e r g y and the in n u m e r i c a l 2 2 / c a l c u l a t i o n s vp ~0.5 can be r e a c h e d e.g. for Zr b a s e d a l l o y s -- .

о

We t h a n k all of t h o s e with w h o m we d i s c u s s e d the t h e o r y d u r i n g this work, e s p e c i a l l y J.L. B l a c k , B.L. G y o r f f y a n d J. Sólyom, f u r t h e r m o r e , W. A rnold, E. BabiS, K. D r a ns f e l d , N. G i o r d a n o , B. G o l d i n g , H.U. H a b e r m e i e r , S. H u c k l i n g e r and G. W ei s s for d i s c u s s i n g the e x p e r i m e n t a l results.

9

R E F E R E N C E S

1. P.W. A n d e r s o n , B.I. H a l p e r i n and C.M. Varma, Philos. Mag.

2 5 , 1 (1972)

W.A. P h i l l i p s , J. Low Temp. Phys. 1_, 351 (1972)

2. See for an e x c e l l e n t r e v i e w J.L. B l a c k in " M e t a l l i c G l a s s e s "

e d i t e d by H.J. G ü n t h e r o d t , ( S p r i n g e r - V e r l a g N.Y. 1980).

3. В. Golding, J.E. G r e a b n e r , A.B. K a n e and J.L. Black, Phys.

Rev. Lett. 4 1 , 1487 (1978)

4. R.W. C o c h r a n e , R. Har r i s , J.O. S t r o m - O l s e n a n d M .J .Z u c k e r m a n , Phys. Rev. Lett. 3j^, 676 (1975) .

5. See for r e f e r e n c e s R.W. C o c h r a n e , J . de P h y s i q u e 3_9, C 6 - 1 5 4 0 (1978) and G. M i n n i g e r o d e in " L i q u i d and A m o r p h o u s M e t a l s "

ed. by e. L ö s c h e r and H. C o u f a l ( S i j t h o f f a n d N o o r d h o f f , G e r m a n t o w n M a . USA 198 0 p. 399).

6. J. Kondo, P h y s i c a 8 4 B , 207 (1976) .

7. K. V l a d á r a n d A. Z a w a d o w s k i , S o l i d State C o m m u n . 3J^, 217 (1980).

8. See for the rol e of i n e l a s t i c s c a t t e r i n g in l o c a l i z a t i o n yPhyis . Re v . Le t t .

t h e o r y D.J. T h o u l e s s ^ 3 9 , 1167 (1977) and P.W. A n d e r s o n , E. A b r a h a m s a n d T.W. R a m a k r i s h n a n , Phys. Rev. Lett. 4_3, 718

( 1 9 7 9 ) . The p o s s i b l e r o l e of TLS h a s been s u g g e s t e d by P .Lee q u o t e d in D.J. Touless, S o l i d S t a t e Commun. 34, 683 (1980).

9. A. Z a w a d ow s k i , Phys. Rev. Lett. АЪ_, 211 (1980) . 10. K. V l a d á r a n d A. Z a w a d o w s k i to be p u b l i s h e d .

11. See for the m e t h o d a p p l i e d J. s ó l y o m , J. Phys. F. 4_, 2269 (1974) and s p e c i a l a p p l i c a t i o n for the c o m m u t a t i v e m o d e l of TLS J.L. B lack, B.L. G y o r f f y , K. V l a d á r and A . Z a w a d o w s k i to be p u b l i s h e d and Ref. 7.

12. The f a c t o r (vX v Z ) 1//2 is the c o r r e c t i o n to T d e r i v e d Iv

in l e a d i n g l o g a r i t h m i c a p p r o x i m a t i o n in Ref. 9, f u r t h e r m o r e , a n u m e r i c a l e r r o r in eq. lO of Ref. 9 has b e e n c o r re c t e d . 13. J.L. Black, B.L. G y o r f f y a n d J. J ä c k l e , P hilos. Mag. B 4 0 ,

331 (1979).

14. See Ref. 9. and the r e s u l t s are s i m i l a r as t h o s e in Ref. 6.

15. A=6 c o r r e s p o n d s to До =1К ° on the b a s i s Aq= exp (-A) w i t h fl ш =400K°.

d

16. Thi s v a l u e is five t i mes l a r g e r th a t the v a l u e q u o t e d in Ref. 17. for Z r Q

7P d 0 . 3 '

17 . J.E. G r e a b n e r , B. G o l d i n g , R.J. S c h u t z , F .S .L . Hsu and H .S . C h e n , P h y s . Re v . Lett. 39, 1 4 8 0 (1977).

18. P. C h a u d h a r i and H . U. H a b e r m e i e r , S o l i d State C o m m u n . 34, 687 (1980) and N. G i o r d a n o , Phys . Rev. B2_2, 563 5 (1980) . 19. In o r d e r to d i a g o n a l i z e the TLS p a r t of the H a m i l t o n i a n

g i v e n by eq. (1) one m u s t p e r f o r m a r o t a t i o n a r o u n d the у ax i s by angle a for in i = x , y , z , space a n d t g a = A / A

T L S О

20. See Ref. 3. and f u r t h e r r e f e r e n c e s c a n be f o u n d in the p a p e r s q u o t e d in R e f.s 2. and 21.

21. The N ( E )К = 0 . 9 v a l u e is o b t a i n e d for Zr Pd_ by W . A r n o l d ,

e О . / О . 3

P. D o u s s i n e a u , Ch. F r e n o i s and A. L e v e l ű t (preprint) and G. W e i s s and S. H u n k l i n g e r (private c o m m u n i c a t i o n ) on the b as i s of t h e o r y g i v e n in J.L. B l a c k and P. F u l d e , Phys.

Rev. Lett. 43^, 453 (1979).

22. B.L. Gyorffy, p r i v a t e c o m m u n i c a t i o n .

11

F I G U R E C A P T I O N

Fig. 1. The s c a l i n g t r a j e c t o r i e s c a l c u l a t e d u s i n g e q . s (2)

z x z “* 3

and (3) are shown for v =0.2 a n d v /v = 1 0

S o l i d (dotted) c u r v e s show the p a r t s w h e r e the s econd o r d e r s c a l i n g is (is not) valid. T is

К

c a l c u l a t e d fro m e q . (4). The l o g a r i t h m i c b e h a v i o r is a p p e a r e n t a r o u n d T . The c h a n g e in Д (x) is

К о

d e p i c t e d as well.

Pig. 1.

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Kroó Norbert

Szakmai lektor: Sólyom Jenő Nyelvi lektor: Sólyom Jenő

Példányszám: 670 Törzsszám: 81-300 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly

Budapest, 1981. május hó