QUANTUM TUNNELING OF TRAPPED HYDROGEN IN Nb

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QUANTUM TUNNELING OF TRAPPED

HYDROGEN IN Nb

E. Drescher-Krasicka, A. Granato

To cite this version:

JOURNAL DE PHYSIQUE

C o l l o q u e

C10,

s u p p l 6 m e n t

au

11-12,

Tome

4 6 , dbcembre 1 9 8 5 p a g e C10-73

QUANTUM TUNNELING

OF

TRAPPED HYDROGEN IN Nb

University of Illinois at Urbana-Champaign, Urbana,

1110

W.

Green Street, Urbana, IL 61801, U.S.A.

A b s t r a c t

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Quantum t u n n e l i n g of hydrogen t r a p p e d by oxygen i n a Nb l a t t i c e i s found a t low temperatures. Information o b t a i n e d by v a r i o u s measurement t e c h n i q u e s w i l l be reviewed and compared. U l t r a s o n i c measurements of a t t e n u a t i o n and v e l o c i t y a s a f u n c t i o n of p o l a r i z a t i o n , t e m p e r a t u r e , f r e q u e n c y , d e f e c t c o n c e n t r a t i o n , i s o t o p e and c o o l i n g r a t e p r o v i d e d e t a i l e d q u a n t i t a t i v e i n f o r m a t i o n p a r t i c u l a r l y concerning t h e symmetry and dynamics of i s o l a t e d systems i n Nb where t h e normal t o superconducting t r a n s i t i o n is used t o e s t a b l i s h t h e e f f e c t of conduction e l e c t r o n s on t h e t u n n e l i n g r a t e . In t h e OH system a peak a t 2.25 K a t 10 MHz i n t h e s u p e r c o n d u c t i n g s t a t e d i s a p p e a r s i n the normal s t a t e . In a second system produced by r a p i d c o o l i n g , a peak a t 6 K a t 10 MHz a l s o moves d r a m a t i c a l l y , but i n t h i s c a s e t h e response can be f u l l y measured i n t h e normal s t a t e . It is found t h a t a two l e v e l system (TLS) formalism which t a k e s i n t o account t h e r e l a x a t i o n s t i m u l a t e d by i n e l a s t i c s c a t t e r i n g of e l e c t r o n s g i v e s a good q u a n t i t a t i v e d e s c r i p t i o n of t h e quantum behavior of t h e OH system. The theory used f o r a n a l y z i n g the d a t a is s i m i l a r t o t h a t f o r m e t a l l i c g l a s s e s but s i m p l e r . U l t r a s o n i c experiments, p a r t i c u l a r l y high accuracy v e l o c i t y measurements, a r e s u f f i c i e n t f o r t h e complete e v a l u a t i o n of a l l t h r e e parameters of t h e TLS; A.

-

t h e minimum gap, a

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t h e c o u p l i n g t o t h e s t r a i n f i e l d , and E

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t h e average a b s o l u t e s t r a i n magnitude. For t h e OH system, t h e r e l a x a t t o n r a t e i n t h e normal s t a t e cannot be measured. For t h e quenching peak, t h e r e l a x a t i o n r a t e a s w e l l a s t h e quantum d e p l e t i o n of t h e r e l a x a t i o n s t r e n g t h , a r e d i r e c t l y a c c e s s i b l e .

I. Brief Review

Measurements by S e l l e r s , Anderson, and Birnbaum [ I ] of a l a r g e i s o t o p e e f f e c t i n t h e e x c e s s s p e c i f i c h e a t of niobium c o n t a i n i n g hydrogen o r deuterium showed convincingly t h a t hydrogen i n niobium formed a quantum t u n n e l i n g system. More d e t a i l e d measurements of s p e c i f i c h e a t by Morkel, Wipf, and Neumaier [ 2 ] showed t h a t i f t h e r e is no oxygen o r n i t r o g e n i m p u r i t y , t h e n t h e r e is no e x c e s s s p e c i f i c h e a t . This i m p l i e s t h a t t h e t u n n e l i n g system c o n s i s t s of a complex made up of hydrogen t r a p p e d a t an oxygen o r n i t r o g e n impurity.

More s p e c i f i c i n f o r m a t i o n has s i n c e been o b t a i n e d from neutron s c a t t e r i n g [3-61 and u l t r a s o n i c measurements [7-181. I n e l a s t i c n e u t r o n s c a t t e r i n g experiments by Magerl, e t a 1 [ 3 ] and a l s o by R i c h t e r and Shapiro [41 showed t h a t t h e r e a r e two v i b r a t i o n a l l e v e l s c e n t e r e d a t 0.11 eV and 0.16 w i t h an i n t e n s i t y r a t i o of 112, i n d i c a t i n g t h a t t h e h i g h e r v i b r a t i o n a l l e v e l

i s

doubly d e g e n e r a t e , a s would be expected f o r a t e t r a h e d r a l p o s i t i o n . L a t e r measurements by Rush, e t a 1 [ 5 ] showed t h a t t h e r e is n o t much change i n t h e v i b r a t i o n a l l e v e l s f o r a-phase hydrogen and hydrogen t r a p p e d a t n i t r o g e n o r oxygen. Using i n e l a s t i c n e u t r o n s c a t t e r i n g measurements a t 0.15 K i n niobium c o n t a i n i n g 1900 ppm 0, Wipf, e t a1 [ 6 ] found r e s u l t s which could be i n t e r p r e t e d w i t h a quantum two l e v e l system (TLS) formalism w i t h a t u n n e l i n g m a t r i x element o r gap of J = 0.21

2

.03 meV (2.4 K). This was i n f a i r agreement w i t h t h e v a l u e of 0.19 f -02 m e V they o b t a i n e d from f i t t i n g s p e c i f i c h e a t data.

*permanent a d d r e s s : IFTR, P o l i s h Acad. of S c i . , Warsaw, P O U N D

C10-74

JOURNAL

DE

PHYSIQUE

Poker, e t a 1 [ 7 , 8 ] measured t h e u l t r a s o n i c a t t e n u a t i o n and v e l o c i t y i n niobium c o n t a i n i n g oxygen and hydrogen as a f u n c t i o n of temperature, frequency,

p o l a r i z a t i o n , c o n c e n t r a t i o n and i s o t o p e . They found a r e l a x a t i o n peak a t 2.3 K f o r 10 MHz i n t h e C' mode, but not i n t h e Cg4 mode. The r e l a x a t i o n time was found t o be A r e n i u s w i t h an a c t i v a t i o n energy of 1.8 meV ,and a frequency f a c t o r of 3.9 x

lofB. The r e l a x a t i o n s t r e n g t h did not follow a c l a s s i c a l 1/T temperature dependence, but began t o decrease a t t h e lowest t e m p e r a t u r e s , i n d i c a t i n g a low temperature quantum d e p l e t i o n of t h e s t a t e involved i n t h e r e l a x a t i o n t r a n s i t i o n . Also a l a r g e i s o t o p e e f f e c t was found, and a d e c r e a s e i n t h e v e l o c i t y below t h e r e l a x a t i o n peak was observed. The d e f e c t r e s p o n s i b l e f o r t h e peak was i d e n t i f i e d a s hydrogen t r a p p e d a t an oxygen i n t e r s t i t i a l . They d i s c u s s e d t h e c o n s t r a i n t s t h e s e r e s u l t s p l a c e on a t u n n e l i n g model, and found t h a t a l l t h e r e s u l t s could not be explained u s i n g only phonon a s s i s t e d t r a n s i t i o n s i n a TLS model, b u t could be explained u s i n g t u n n e l i n g i n a m u l t i - l e v e l model (MIS) of a t l e a s t f o u r e q u i v a l e n t s i t e s . Evidence a g a i n s t a phonon a s s i s t e d TLS model came p r i n c i p a l l y from ( 1 ) t h e temperature dependence of t h e r e l a x a t i o n time, and (2) t h e e x i s t e n c e of a r e l a x a t i o n a t a low (- 100 ppm) oxygen content. The e x p o n e n t i a l temperature dependence of t h e r e l a x a t i o n t i n r e could be explained by an Orbach process, but such a process i s not contained i n a TLS model. Also, f o r low i n t e r n a l s t r a i n corresponding t o low oxygen c o n t e n t , no r e l a x a t i o n e f f e c t is expected f o r a symmetric TLS. For high hydrogen c o n c e n t r a t i o n s , a second peak near 5.5K a t 10 MHz was o b t a i n e d with t h e same symmetry.

Huang, e t a l . [ 9 ] found t h a t when t h e specimen i s cooled r a p i d l y , a l a r g e peak a t 6K a t 10 MHz i n t h e C' mode is o b t a i n e d which i s u n s t a b l e a g a i n s t a n n e a l i n g near 90K. A s t h i s peak a n n e a l s , t h e OH peak a t 2.3K grows. The peak a n n e a l s a t t h e same temperature as t h a t found by Hanada [ l o ] i n h i s r e s i s t i v i t y quenching experiments, which he i n t e r p r e t e d a s a r i s i n g from OHn complexes.

C a n n e l l i and C a n t e l l i [ l l ] a l s o found a peak a t low temperature, but a t much lower f r e q u e n c i e s . T h i s s p e a k appears t o be not a s s o c i a t e d w i t h t h e OH peak, but may be r e l a t e d t o the second peak observed bj Poker, e t a l . , a t high c o n c e n t r a t i o n s . B e l l e s s a [12] measured t h e v e l o c i t y a t 200 MHz down t o 50 mK i n an u n o r i e n t e d specimen c o n t a i n i n g 700 ppm 0 and 1,700 ppm H, and found t h a t he could d e s c r i b e h i s r e s u l t s u s i n g t h e TLS theory given e a r l i e r by J a c k l e , e t a l . [13]. He o b t a i n e d 0.13 mev f o r t h e t u n n e l i n g gap.

Wang, e t a l . [14] measured t h e a t t e n u a t i o n i n niobium c o n t a i n i n g 1500 ppm of n i t r o g e n and 2500 ppm of hydrogen i n both t h e normal and superconducting s t a t e s between 15 and 195 MHz. In t h e superconducting s t a t e a peak was found i n agreement with t h e Poker, e t a l . r e s u l t s f o r Nb/O, but t h e peak disappeared i n -the normal s t a t e . The disappearance of t h e peak i s e x p l a i n e d by supposing t h a t t h e r e l a x a t i o n r a t e i n c r e a s e s g r e a t l y i n the normal s t a t e because of t r a n s i t i o n s s t i m u l a t e d by e l e c t r o n s c a t t e r i n g . T h i s Korringa t y p e p r o c e s s had been pfoposed e a r l i e r by Golding, e t a l . [15] t o e x p l a i n a r e l a x a t i o n r a t e of TLS i n m e t a l l i c g l a s s e s f o u r o r d e r s of magnitude l a r g e r t h a n t h a t expected f o r phonon processes. Evidence f o r an enhanced r e l a x a t i o n r a t e which d e c r e a s e s below t h e t r a n s i t i o n temperature a s t h e conduction e l e c t r o n s f r e e z e i n t o t h e BCS ground s t a t e i n amorphous superconductors has been given bj Weiss, e t a l . [16]. Supposing t h a t t h e r e l a x a t i o n r a t e is s t i l l dominated by e l e c t r o n s c a t t e r i n g i n t h e superconducting s t a t e , t h e r e l a x a t i o n r a t e is then given by [17]

where B is a constant g i v i n g t h e normal s t a t e r a t e and

A(T)

i s the BCS gap f u n c t i o n . The BCS gap A(0) a t low temperature of 1.5 meV f o r niobium i s i n reasonable agreement with t h e 1.8 mev reported by Poker, e t a l . [7]. The

e x p e r i m e n t a l peaks found by Wang, e t a l . , a r e j u s t a l i t t l e broader than t h o s e they c a l c u l a t e f o r a s i n g l e energy s p l i t t i n g . They suppose t h a t t h e agreement would be improved by i n t r o d u c i n g t u n n e l i n g systems w i t h asymmetric p o t e n t i a l s .

a l s o found t h a t t h e r e l a x a t i o n peak d i s a p p e a r s i n t h e normal s t a t e . I n a d d i t i o n , v e l o c i t y measurements showed t h a t t h e f u l l r e l a x a t i o n is achieved below 2.3 K p r o v i d i n g p o s i t i v e e v i d e n c e t h a t t h e r e l a x a t i o n r a t e was g r e a t l y i n c r e a s e d i n t h e normal s t a t e . Also, they found t h a t w h i l e t h e peak was u n a f f e c t e d by magnetic f i e l d s below Tcl, t h e peak s t r e n g t h decreased w i t h o u t changing p o s i t i o n with i n c r e a s i n g magnetic f i e l d i n t h e mixed s t a t e . In t h e second system produced by r a p i d c o o l i n g , t h e peak a l s o moves t o lower t e m p e r a t u r e s , but i n t h i s c a s e t h e response can be f u l l y measured i n t h e normal s t a t e .

The p o s s i b i l i t y t h a t e l e c t r o n s c a t t e r i n g d e t e r m i n e s t h e r e l a x a t i o n r a t e i n t h e superconducting s t a t e removes one of t h e o b j e c t i o n s l i s t e d by Poker, et a l . [81 t o t h e u s e of a TLS d e s c r i p t i o n f o r OH. A second o b j e c t i o n t h a t no r e l a x a t i o n peak s h o u l d be expected f o r t h e symmetric p o t e n t i a l expected f o r low s t r a i n (low oxygen c o n t e n t ) samples can be t e s t e d q u a n t i t a t i v e l y , s i n c e t h e v e l o c i t y measurements by themselves a r e s u f f i c i e n t t o determine a l l the parameters of t h e system, i n c l u d i n g t h e i n t e r n a l s t r a i n . These e f f e c t s w i l l be d e s c r i b e d i n some d e t a i l i n t h e r e s t of t h i s a r t i c l e . It is found t h a t t h e i n t e r n a l s t r a i n i s s u r p r i s i n g l y l a r g e , s o t h a t some r e l a x a t i o n o c c u r s even f o r 100 ppm 0 specimens, and t h a t a TLS formalism can indeed d e s c r i b e t h e r e s u l t s . T h i s does n o t prove t h a t t h e system is a two s i t e system, s i n c e only two l e v e l s of a MLS may be e f f e c t i v e .

Recently, Yu and Granato [ 191 have found t h a t t h e gap i n a TLS i n niobium s h o u l d be about 25% s m a l l e r i n t h e normal s t a t e t h a n i n t h e s u p e r c o n d u c t i n g s t a t e . T h i s e f f e c t should be most r e a d i l y observed i n t h e resonance p a r t of t h e e l a s t i c c o n s t a n t change a t t h e lowest temperatures. Evidence f o r a gap change i s found f o r t h e second (quenching) peak and shown a t t h e end of t h i s a r t i c l e .

11. Experiments

In what f o l l o w s , measurements a r e given of a t t e n u a t i o n and v e l o c i t y i n a specimen of niobium c o n t a i n i n g 100 ppm 0 and 700 ppm H. The 100 ppm 0 c o n c e n t r a t i o n d e s i g n a t i o n i s nominal. An impurity a n a l y s i s i n d i c a t e d 200 ppm Ta, 70 ppm C, 49 ppm N, and 64 ppm 0. T h i s specimen i s t h a t r e f e r r e d t o a s Specimen 2 by Poker, e t a l .

[ 8 ] , and a s Nbl by Huang, e t a l . [ 9 ] . The r e s u l t s found i n t h e s u p e r c o n d u c t i n g s t a t e a r e i n agreement w i t h those found by Huang, e t a l . The v e l o c i t y measurements a r e made w i t h an i n t e r f e r o m e t r i c method [20] i n which t h e e l a s t i c c o n s t a n t change i s d e t e c t e d a s a frequency change w i t h

The frequency s e n s i t i v i t y f o r changes w i t h t e m p e r a t u r e and s t a t e i s about 10 Hz a t 10 MHz and 20 Hz a t 30 MHz. Absolute e l a s t i c c o n s t a n t s a r e not o b t a i n e d , and a r e f e r e n c e v a l u e must be e s t a b l i s h e d from t h e measurements. A t t e n u a t i o n and v e l o c i t y a r e measured s i m u l t a n e o u s l y .

R e s u l t s f o r a t t e n u a t i o n and v e l o c i t y v e r s u s t e m p e r a t u r e a r e shown i n Fig. 1 f o r a C44 mode, which is u n a f f e c t e d by hydrogen. The l o g decrement A is o b t a i n e d from t h e a t t e n u a t i o n a measured i n db/usec through A = .I15 a/f(MHz). At low

t e m p e r a t u r e , t h e decrement c o n s i s t s of a n e l e c t r o n i c component p r o p o r t i o n a l t o t h e e l e c t r i c a l c o n d u c t i v i t y and t h e f r e q u e n c y , and a second, t e m p e r a t u r e independent component given by bond l o s s e s and d i f f r a c t i o n . The' e l e c t r o n i c component d i s a p p e a r s i n t h e s u p e r c o n d u c t i n g s t a t e below t h e t r a n s i t i o n t e m p e r a t u r e of 9.2 K and a s i s g i v e n by as/a, = 2 / ( 1

+

exp A(T)/kT), t h e BCS gap f u n c t i o n [21]. From t h e s e d a t a , t h e t e m p e r a t u r e dependence of t h e gap and t h e p u r i t y of t h e specimen can be determined. There is a l s o a change of t h e v e l o c i t y between t h e normal and

s u p e r c o n d u c t i n g s t a t e s , shown i n t h e lower p a r t of Fig. 1. I n t h e normal s t a t e , t h e e l a s t i c c o n s t a n t is of t h e form (22,231

(210-76 JOURNAL DE

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l a t t i c e f r e e energy. I n t h e s u p e r c o n d u z t i n g s t a t e , t h e r e is a d e c r e a s e below t h i s v a l u e of

-

2 KHz at 30 MHz o r

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7 x 10- f o r 6 f / f . For C ' , t h i s change i s much s m a l l e r , competing i n magnitude w i t h t h e v e l o c i t y changes a r i s i n g from t h e t u n n e l i n g hydrogen system. For OH r e l a x a t i o n 6 f / f

-

400 Hz i s o b t a i n e d . The f r e q u e n c i e s can be measured very a c c u r a t e l y , b u t t h e e f f e c t s to. be s t u d i e d a r e s m a l l and s o p r o p e r account of a l l t h e background e f f e c t s is c r i t i c a l t o t h e a n a l y s i s .

I n Fig. 2 a r e shown a t t e n u a t i o n (decrement) and v e l o c i t y ( f r e q u e n c y ) changes i n t h e C' mode a t 10 MHz. % A s m a l l peak is observed a t 2.3 K i n t h e s u p e r c o n d u c t i n g s t a t e , b u t no peak is s e e n i n t h e normal s t a t e . The r e l a x a t i o n s t r e n g t h of t h i s peak is about t h r e e o r d e r s of magnitude s m a l l e r t h a n t y p i c a l v a l u e s f o r d i p o l a r p o i n t d e f e c t s . Also t h e d e f e c t c o n c e n t r a t i o n s a r e m c h s m a l l e r t h a n u s u a l ( t o minimize i n t e r n a l s t r a i n e f f e c t s ) . N e v e r t h e l e s s , t h e measurement accuracy i s s u f f i c i e n t t o d e f i n e t h e peak c l e a r l y . By comparison w i t h Fig. 1, one s e e s t h a t i n t h e normal s t a t e , t h e r e is an e l a s t i c c o n s t a n t r e d u c t i o n which i n c r e a s e s w i t h d e c r e a s i n g temperature. The e f f e c t i n t h e s u p e r c o n d u c t i n g s t a t e i s more

complicated, but one r e c o g n i z e s a d i s p e r s i o n which accompanies t h e r e l a x a t i o n peak i n t h e decrement, a s w e l l a s a r e d u c t i o n of t h e e l a s t i c c o n s t a n t below t h a t f o r t h e normal s t a t e due t o t h e e l e c t r o n i c e f f e c t . The r e p r o d u c i b i l i t y of t h e measurements i s e v i d e n t i n t h e o v e r l a p p i n g of t h e c u r v e s above TC.

Fig. 1 Decrement A (upper c u r v e s ) and Fig. 2 Same a s Fig. 1, but f o r frequency change 6 f ( l o w e r c u r v e s , a C' mode a t 10 MHz. e l a s t i c c o n s t a n t change 6C/C = 26f / f )

v e r s u s t e m p e r a t u r e i n t h e normal N and s u p e r c o n d u c t i n g s t a t e S f o r a C44 mode a t 30 MHz.

Fig. 3 Same a s Fig. 2, b u t a t 30 MHz a f t e r r a p i d cooling.

I

d,

_--..

Fig. 4 Decrement and modulus change

I

(6C/C = 26f I f ) v e r s u s temperature. AS1 and A R 1 a r e t h e resonance and h e l a x a t i o n s t r e n g t h s , normalized by a £ / d o . Below t h e r e l a x a t i o n peak, AS1 is measured. Above t h e peak t h e

f u l l e q u i l i b r i u m AS'

+

AR1 i s measured.

111. T I S Formalism

The theory of a t t e n u a t i o n and e l a s t i c c o n s t a n t changes f o r TLS i n c r y s t a l s is b a s i c a l l y t h e same a s t h a t f o r amorphous systems, but simpler. For amorphous systems t h e r e i s not only a d i s t r i b u t i o n of s t r a i n s , b u t a l s o a d i s t r i b u t i o n of t u n n e l i n g gaps, extending down t o zero. For T I S i n c r y s t a l s , t h e r e i s a minimum g a p , Ao, and f o r u l t r a s o n i c measurements w i t h $A

<

A t h e r e i s no d i r e c t r e s o n a n t a b s o r p t i o n . The theory was given a l r e a d y bj ~ a c k l e , O e t a l . 1131 i n 1972, and a p p l i e d immediately t o amorphous systems. The same r e s u l t s f o r c r y s t a l l i n e systems a r e d e r i v e d by Granato, e t a l . 1241 i n a n o t h e r way which f a c i l i t a t e s a d i s c u s s i o n of t h e meaning of t h e r e s u l t s , p a r t i c u l a r l y of t h e s t r a i n dependence.

For TLS, made asymmetric by a s t r a i n e , t h e gap 28 i s g i v e n by t h e t h r e e parameters A O , a , E i n

where 2A0 is t h e t u n n e l s p l i t t i n g and 2 a ~ is t h e s t r a i n induced asymmetry e n e r g y , where a is a s t r a i n c o u p l i n g c o e f f i c i e n t and e i s t h e s t r a i n . The thermal e q u i l i b r i u m e l a s t i c c o n s t a n t change 6C is given bj d e f i n i t i o n a s

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where Z =

1

exp(Ei/kT) ( 7 )

and f ' i s t h e number of d e f e c t s . For a TLS,

z

= 2 cosh A / ~ T ( 8 ) Using Eqs. 4,6,7, and 8 i n 5, one o b t a i n s

where

and

The f a c t o r i n round b r a c k e t s i n Eq. 10 i s t h e f a m i l i a r r e s u l t f o r t h e c l a s s i c a l Snoek e f f e c t f o r El = T a € . The sech2 A/kT f a c t o r cuts-of f t h e r e l a x a t i o n a t temperatures low en6ugh f o r t h e upper s t a t e t o become depopulated. What i s new f o r a quantum system $s t h e square b r a c k e t f a c t o r i n Eq. 10 f o r AR and Eq. 11 f o r A

.

The f a c t o r (aE/A) i n Eq. 10 shows e x p l i c i t l y t h a t f o r a symmetric p o t e n t i a l , t g e r e w i l l be no r e l a x a t i o n because t h e r e is no energy s h i f t f o r a small p e r t u r b i n g u l t r a s o n i c s t r a i n . For l a r g e s t r a i n s a&/A + 1. The new term A, i s t h e resonant response. Even though t h e r e i s no resonant a b s o r p t i o n f o r FIW

<?

Ao, t h e r e is s t i l l a response in-phase w i t h an a p p l i e d s t r a i n . A t z e r o temperature, where only t h e ground s t a t e is populated AS i s g i v e n by t h e s q u a r e bracket i n Eq. 11, a s is r e a d i l y c a l c u l a t e d u s i n g F =

+

fE1. From Eq. 5, A S is t h e c u r v a t u r e of t h i s s t a t e a s a f u n c t i o n of s t r a i n . This c u r v a t u r e goes t o z e r o f o r l a r g e s t r a i n s where El becomes l i n e a r i n s t r a i n . A t f i n i t e temperature, t h e r e i s some p o p u l a t i o n of E2 w i t h o p p o s i t e c u r v a t u r e , reducing t h e s i z e of t h e e f f e c t by t h e f a c t o r t a n h A/kT.

At f i n i t e frequency, o r low t e m p e r a t u r e s , t h e r e l a x a t i o n is incomplete and t h e r e l a x a t i o n response i s given by t h e Debye form

For a random d i s t r i b u t i o n of s t r a i n , t h e s e r e s u l t s must be averaged over a s t r a i n d i s t r i b u t i o n f u n c t i o n , w i t h t h e r e s u l t s expressed now i n terms of s o , t h e average a b s o l u t e value of t h e s t r a i n . This is i l l u t r a t e d i n Fig. 4, which shows t h e

1

e l a s t i c c o n s t a n t change (6C/C1) = (GC/C)/(a f/CAo) a s a f u n c t i o n of temperature (normalized by A / k ) f o r a norm l i z e d random s t r a i n 6

₁

= a€,/AO = 1. A t h i g h t e m p e r a t u r e s 6 ~ / 8 = AS

+

A R = a f/CkT, t h e c l a s s i c a l value. Below t h e temperature f o r which

wr

= 1, t h e e l a s t i c c o n s t a n t change i s given by t h e resonant response AS. In what f o l l o w s , we focus mainly on t h e v e l o c i t y measurements.

Fig. 5 i l l u s t r a t e s t h e n a t u r e of t h e response t o be expected f o r t h e d i f f e r e n t l e v e l s of i n t e r n a l s t r a i n of 0 = 0.3 and 5. For s m a l l s t r a i n (B = 0.3) t h e

resonance i s l a r g e and r e l a x a t i o n is a s m a l l f r a c t i o n of t h e t o t a l e l a s t i c c o n s t a n t change a t high temperature. For l a r g e s t r a i n (6 = 5.0), t h e resonance i s g r e a t l y reduced, and the r e l a x a t i o n is a l a r g e r f r a c t i o n of t h e t o t a l change a t h i g h temperatures. The t o t a l change i s t h e same f o r both s t r a i n s a t high temperature. For kT

<

A,, 6C/C d e v i a t e s from t h e 1/T c l a s s i c a l high temperature response and becomes c o n s t a n t f o r kT

<<

A,.

These f e a t u r e s provide a means f o r a n a l y z i n g d a t a i n terms of a TLS formalism. In t h e superconducting s t a t e 6C/C changes from A R

+

AS a t h i g h

IV. OH Systems

The d a t a of Fig. 2 a r e p l o t t e d vs. 1/T i n Fig. 6. The d a t a have been c o r r e c t e d f o r t h e l a t t i c e c o n t r i b u t i o n and t h e e l e c t r o n i c c o n t r i b u t i o n i n t h e s u p e r c o n d u c t i n g s t a t e . The d a t a cannot be f i t by a s i n g l e ( d e l t a d i s t r i b u t i o n o f ) s t r a i n . However, they a r e w e l l f i t by a random d i s t r i b u t i o n ( s o l i d l i n e s ) w i t h t h e f i t t i n g p a r a m e t e r s - -

of A, = -85 K and 13 = as,/A0. T h i s i s a lower v a l u e of A, t h a n t h a t g i v e n e a r l i e r by Poker e t a l . [ 8 ] , u s i n g a d e l t a f u n c t i o n s t r a i n d i s t r i b u t i o n f i t . These v a l u e s a r e i n r e a s o n a b l e agreement w i t h t h o s e found from i n e l a s t i c n e u t r o n s c a t t e r i n g and show t h a t u l t r a s o n i c measurements a r e complementary t o n e u t r o n measurements i n some a s p e c t s . l& f i t t i n g a l s o t h e magnitude of t h e v e l o c i t y changes, one o b t a i n s a l l 3 parameters

v d e s c r i b i n g t h e TLS s e p a r a t e l y . These a r e

i .

i/'

a of = s t r a i n 51 m e V i s and somewhat E = 7.2 h i g h e r x t h a n T h i s t h a t v a l u e which

A,+&

might have been expected f o r a 100 ppm 0

Fig. 5 Resonant and r e l a x a t i o n i m p u r i t y l e v e l , b u t e x p l a i n s how one can s t r e n g t h s , A

'

and A

',

v e r s u s s t i l l o b t a i n a r e l a x a t i o n f o r such a

R

t e m p e r a t u r e Bor two v a l u e s of r e l a t i v e l y p u r e specimen. normalized s t r a i n 6=0.3 and 5.0,

where 8 = aeo/Ao.

V. Second System (Rapidly Cooled Specimens)

F i n a l l y , t h e v e l o c i t y d a t a f o r t h e second peak ( r a p i d l y cooled specimens) are p l o t t e d i n t h e same way i n Fig. 7. One s e e s t h a t t h e resonance i s s m a l l e r t h a n t h e r e l a x a t i o n i n t h i s case. These d a t a have n o t y e t been f u l l y a n a l y z e d , and w i l l be t h e s u b j e c t of a s e p a r a t e paper. However, independent of t h e n a t u r e of t h e t u n n e l i n g system and t h e s t r a i n d i s t r i b u t i o n , t h e t e m p e r a t u r e independent v e l o c i t y change a t low t e m p e r a t u r e must be a resonance e f f e c t . The f a c t t h a t t h i s is d i f f e r e n t i n t h e N and S s t a t e s i m p l i e s t h a t t h e t u n n e l i n g gap must a l s o have changed.

Fig. 6 F i t of d a t a i n Fig. 2 t o TLS Fig. 7 Frequency change vs 10/T f o r formalism f o r A, = 0.85 K and d a t a i n Fig. 3. The r e l a x a t i o n s t r e n g t h normalized random s t r a i n

8

= 0.5. i s l a r g e compared t o t h e resonance

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V I . Summary

1. The u l t r a s o n i c e f f e c t s observed i n slowly cooled niobium 100 ppm0, 700 ppmH can be d e s c r i b e d w i t h a TIS formalism i n which e l e c t r o n s c a t t e r i n g g r e a t l y i n c r e a s e s t h e t r a n s i t i o n r a t e i n t h e normal s t a t e .

2. From an a n a l y s i s of t h e u l t r a s o n i c v e l o c i t y d a t a a l o n e of t h e t e m p e r a t u r e dependence and r a t i o of r e l a x a t i o n t o resonance s t r e n g t h s , one may o b t a i n a l l of t h e t h r e e p a r a m e t e r s s p e c i f y i n g t h e system. These a r e A. = - 8 5 K, E = 7.2 x and a = 51 meV. In t h i s r e s p e c t , t h e OH system i n niobium is a mode? TW.

3. The low t e m p e r a t u r e resonance of t h e second r e l a x a t i o n system o b t a i n e d by r a p i d c o o l i n g changes w i t h t h e N-S t r a n s i t i o n . This is e x p e r i m e n t a l e v i d e n c e t h a t t h e t u n n e l i n g gap changes w i t h t h e d e n s i t y of normal e l e c t r o n s .

Acknowledgement

We thank H. K. Birnbaum f o r p r o v i d i n g t h e specimen, W. Johnson and

E.

Johnson f o r a s s i s t a n c e w i t h t h e measurements and G. de Lorenzi f o r a s s i s t a n c e w i t h computer c a l c u l a t i o n s . T h i s work was s u p p o r t e d by t h e N a t i o n a l S c i e n c e Foundation under g r a n t number DMR84-09396. E. D.-K. was an IBM Fellow i n 1984-85.

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