ANOMALOUS PROPERTIES OF GLASSES AT LOW TEMPERATURES

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ANOMALOUS PROPERTIES OF GLASSES AT LOW TEMPERATURES

S. Hunklinger

To cite this version:

S. Hunklinger. ANOMALOUS PROPERTIES OF GLASSES AT LOW TEMPERATURES. Jour-

nal de Physique Colloques, 1978, 39 (C6), pp.C6-1444-C6-1449. �10.1051/jphyscol:19786585�. �jpa-

00218077�

JOURNAL DE PHYSIQUE

CoIloque C6, suppliment au no 8, Tome 39, aozit 1978, page

^C6-1444

ANOMALOUS PROPERTIES OF GLASSES A T LOW TEMPERATURES

S. Hunklinger

kh-PZanck-Institut fUr Festk8rperforschung. 0-7 Stuttgart 80, Federal Republic of Germany

Rbsum6.- Dans ce prbsent papier nous rappelons les propri6tQs thermiques, accoustiques et diblec- triques des solides amorphes 1 basses tempbratures. Pratiquement tous les matdriaux amorphes Studids jusqu'ici contiennent un grand nombre d'excitations de basse Qnergie. En Qtudiant l'absorption rb- sonante des phonons et des micro-ondes dans les verres 1 basse tempdrature, il a Gtb possible d'ob- tenir de nouvelles informations sur la nature de ces excitations, qui se rQvi2lent 8tre des systsmes P deux niveaux et de connaetre leur dynamique et leur interaction mutuelle.

Abstract.- In the present paper we review the thermal, accoustic and dielectric properties of amor- phous solids at low temperatures. Practically all amorphous materials investigated so far, are known to contain a large number of low-energy excitations. By studying the resonant absorption of phonons and microwaves in glasses at low temperatures it,has been possible to gain new information about the two-level nature, the dynamics and mutual interaction of these excitations.

1. INTRODUCTION.- In the past several years there has been an intensive effort to understand the ther- mal properties of glasses below 1

K.

At these tem- peratures thermal hehaviour should be determined by the elastic waves of long wavelength. Below 1 K the wavelength of the corresponding dominant phonons is

0

larger than 1000 A and therefore much larger than the scale of microscopic disorder. As a consequence, at these temperatures and below there should be lit- tle difference between the thermal properties of amorphous and crystalline dielectrics. Although the- re is great regularity in thermal phenomena of all amorphous materials, their behaviour

-

in contrast to expectation

-

is completely different from that of crystalline substances. This contradictory aspect has stimulated not only extensive work on thermal properties but also a new sequence of quite diffe- rent investigations on the mechanical and dielectric properties of glasses below 1 K.

2. THENL4L PROPERTIES.- The starting point of the investigation of the low temperature behaviour of amorphous materials was the discovery by Zeller and Pohl /I/, demonstrating that the specific heat of amorphous solids does not show the usual ~ ~ - d e ~ e n - dence expected on account of the Debye theory, but

varies .linearly with temperature below I K.

This behaviour is shown in figure 1 for two very dif- ferent amorphous solids, namely for the dielectric vitreous silica and for the metallic glass Zr Pd

0.7 0.3' At a temperature of 25 mK the measured specific heat of fused silica /2/ exceeds the Debye contribution of the phonons by a factor of 1000 ! An analogous

behaviour fo the specific heat /3/ is found for the superconducting ZrPd (Tc = 2.53 K) in the temperatu- re range where the contribution of the electrons is negligible (T < 0.3 K)

SPECIFIC HEAT

/ Vitreous

Silica /

/ Crystatline -

/ Quartz

l(r'21 I I I I I I I I

002

0.01

0.1 0.2 0.1 1 2 3

TEMPERATURE [K]

Fig. 1 : Specific heat of vitreous silica /l,2/, glassy Z r o 7 P d o 3 / 3 / and crystalline quartz /I/ as a function' of tkmperature.

Indeed, these differences between the amor- phous and the crystalline state were found to exist in all other amorphous materials investigated so far

141

including amorphous polymers 1 5 1 .

This large extra specific heat demonstrates clearly the existence of a new type of low-energy states which is specific to the disordered state and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786585

does not exist in pure crystals. Since the specific heat is approximately linear in temperature below

1 K, the corresponding density of states must be ne- arly constant in the energy interval below a few Kelvin. Furthermore, from the magnitude of this extra specific heat, one can deduce that all amorphous ma- terials investigated so far apparently contain about

loi7

low-energy states per cm3 in the energy inter- val between sero and 1 K.

The physical nature of this new type of ex- citation is not yet clear. The question arises whe- ther electronical defect states may be responsible for the extra specific heat. Since the specific heat of fused silica was shown to be independent of the magnetic field up to 9 telsa

161,

it seems that elec-

trons do not play an important role, at least in in- sulating glasses. At present, it seems most likely that atoms or atomic clusters are moving in double- well potentials as proposed independently by Anderson et al. / 7 / and Phillips

/a/.

At low temperatures the

"particle" is not able to surmont the barrier in or- aer to move from one well to the other, but quantum mechanical tunneling through the potential barrier can take place. Formally such a system is equivalent to a two-level system (TLS) having an energy split- ting E which is determined by the difference in the depth of the two wells and the overlap of the wave function of the particle at both sites.

Surprising results have also been observedfor their thermal conductivity in amorphous materials. lhe conductivity is small compared to that in crystals and varies approximately as T~ (1,2,4) (see figure 2).

Once more, the magnitude is nearly the same for all amorphous materials. Measurements of the thermalcon- ductivity in samples with restricted geometries /9/

have demonstrated that the heat is carried by pho- nons. Their mean free path, however, is limited by these additional states.

3. ULTRASONIC AND DIELECTRIC ABSORPTION.- Valuable direct information about the physical nature of the excitations can be gained from the study of their interaction with ultrasonic and electromagnetic wa- ves below 1 K.

3.1. Resonant Absorption.- The anomalous ultrasonic absorption of glasses was found independently by Bunklinger et al. /lo/ and Golding et al. /11/. As a typical example, the ultrasonic attenuation of vi- treous silica is shown in figure 3 as a function of temperature. A characteristic dependence on the ac-

coustic measuring power is observed below 2 K : At higher intensities the ultrasonic loss disappears with decreasing temperature ; at very low power le- vels the acoustic absorption increases again.

THERMAL CONDUCTIVITY

TEMPERATURE [K]

Fig. 2 : Thermal conductivity of vitreous silica

/

1,2/ and of the superconducting disordered metal ZrO., Pd0.3

131.

VITREOUS SILICA

-

^{A JSOI} w c m 2

-

T

m CRYSTAL

7

u

TEMPERATURE [K]

Fig. 3 : Temperature dependence of the ultrasonic attenuation in vitreous silica for longitudinal wa- ves of 1 GHz 1121. Curve 1 represents the saturated attenuation, caused by the relaxation process and Curve 2 shows the measured attenuation in the unsa- turated regime.

The saturation effect /13/ is consistent with the assumption that TLS are

-

by resonant absorp- tion of phonons

-

responsible for the attenuation below 1 K. The population of the two states can be

significantly altered, in fact, equalized by pumping

JOURNAL DE PHYSIQUE

enough energy i n t o t h e s e systems. The s a t u r a t i o n of t h e a b s o r p t i o n immediately r u l e s o u t harmonic os- c i l l a t o r s being r e s p o n s i b l e f o r t h e low-energy ex- c i t a t i o n s because t h e a b s o r p t i o n of such an o s c i l - l a t o r does n o t b l e a c h o u t a t h i g h e r power l e v e l s .

A t v e r y s m a l l i n t e n s i t i e s t h e t e m p e r a t u r e dependence of t h e r e s o n a n t a b s o r p t i o n , i . e . t h e dif- f e r e n c e between t h e upper and t h e lower curve i n f i g u r e 3, i s simply g i v e n by t h e f a c t t h a t t h e upper l e v e l of t h e TLS becomes more and more t h e r m a l l y p o p u l a t e d w i t h i n c r e a s i n g t e m p e r a t u r e . The popula- t i o n d i f f e r e n c e between ground s t a t e and e x c i t e d s t a t e i s p r o p o r t i o n a l t o t a n h (El2 k T ) , where E i s t h e energy s p l i t t i n g o f t h e TLS. S i n c e t h i s f u n c t i o n i s unique f o r TLS, measurements of t h e t e m p e r a t u r e dependence of t h e a b s o r p t i o n a t v e r y s m a l l power l e v e l s p r o v i d e a r i g o r o u s t e s t of t h e TLS hypothe- s i s . Experiments down t o 23 mK 1141 have demonstra- t e d c o n v i n c i n g l y t h e agreement between t h e o r y and experiment.

Very r e c e n t l y t h e t a n h (E/2 kT)-temperature dependence of t h e a b s o r p t i o n h a s a l s o been found i n g l a s s y m e t a l s 115,161, b u t no s a t u r a t i o n could b e d e t e c t e d . Apparently i n t h i s c a s e t h e f r e e e l e c t r o n s i n t e r a c t w i t h t h e TLS r e s u l t i n g i n f a s t e r r e l a x a - t i o n r a t e s and t h e r e f o r e i n a s t r o n g i n c r e a s e of the c r i t i c a l a c o u s t i c power.

At h i g h e r t e m p e r a t u r e s t h e TLS c a u s e a s t e e p r i s e of t h e a t t e n u a t i o n w i t h i n c r e a s i n g t e m p e r a t u r e ( s e e f i g u r e 3 ) . T r a v e l i n g t r o u g h a n ensemble of TLS, t h e sound wave d i s t u r b s t h e i r thermal e q u i l i b r i u m , and t h e systems t r y t o r e l a x i n t o t h e new e q u i l i - brium which is r e - e s t a b l i s h e d v i a t h e e m i s s i o n and a b s o r p t i o n of t h e r m a l phonons. The r e s u l t i s a n a b s o r p t i o n t h a t i s i n t e n s i t y i n d e p e n d e n t , s i n c e now a l l TLS which c a n b e t h e r m a l l y e x c i t e d t a k e p a r t i n t h i s r e l a x a t i o n p r o c e s s 113,171.

I f TLS c a r r y e l e c t r i c a l c h a r g e s o r e x h i b i t an e l e c t r i c a l d i p o l e moment, t h e y a r e e x p e c t e d t o i n t e r a c t w i t h e l e c t r o m a g n e t i c r a d i a t i o n i n a com- p l e t e l y analogous way a s t h e y do w i t h e l a s t i c waves.

E s p e c i a l l y a s a t u r a t i o n i n t h e a b s o r p t i o n of e l e c - t r o m a g n e t i c r a d i a t i o n should o c c u r i n t h e microwave r a n g e . Such a n experiment 1181 h a s been c a r r i e d o u t r e c e n t l y . I n f i g u r e 4 t h e d i e l e c t r i c a b s o r p t i o n of microwaves a t 10 GHz i n v i t r e o u s s i l i c a S u p r a s i l I 1191 i s p l o t t e d f o r d i f f e r e n t i n t e n s i t i e s . Comple- t e l y analogous t o t h e a c o u s t i c a b s o r p t i o n a s t r o n g dependence on microwave i n t e n s i t y i s observed. The concept of TLS i s a l s o s u c c e s s f u l i n e x p l a i n i n g ve- r y r e c e n t s t u d i e s of t h e f a r - i n f r a r e d a b s o r p t i o n a t

low t e m p e r a t u r e s 1201 and of t h e d i e l e c t r i c l o s s i n t h e kHz r a n g e /21/ down t o t e m p e r a t u r e s o f a few mK.

TEMPERATURE CK I

Fig. 4 : Temperature- and i n t e n s i t y dependence of t h e d i e l e c t r i c a b s o r p t i o n o f v i t r e o u s s i l i c a Suprasil I a t 10 GHz 1181. The dashed l i n e i n d i c a t e s t h e con- t r i b u t i o n of t h e r e l a x a t i o n p r o c e s s .

However, t h e r e i s an i m p o r t a n t d i f f e r e n c e between t h e a c o u s t i c and d i e l e c t r i c a b s o r p t i o n . I n t h e e l a s t i c c a s e t h e c o u p l i n g p a r a m e t e r s n e c e s s a r y t o d e s c r i b e t h e magnitude of t h e a b s o r p t i o n a r e n o t s e n s i t i v e t o t h e chemical composition of t h e amor- phous s u b s t a n c e , a t l e a s t i n i n s u l a t i n g g l a s s e s . The d i f f e r e n t magnitudes which a r e observed a r e mainly due t o t h e d i f f e r e n t v a l u e s f o r t h e v e l o c i t y o f sound.

I n c o n t r a s t t h e d i e l e c t r i c c o u p l i n g p a r a m e t e r s a r e determined by t h e i m p u r i t y c o n t e n t s ; a r e s u l t t h a t we s h a l l d i s c u s s i n more d e t a i l i n t h e f o l l o w i n g

s e c t i o n .

3 . 2 . V e l o c i t y of Sound and D i e l e c t r i c Constant.- The s t r o n g and t e m p e r a t u r e dependent a c o u s t i c and d i e - l e c t r i c a b s o r p t i o n a l s o l e a d s v i a Kramers-Kronig r e l a t i o n - t o a t e m p e r a t u r e dependence of t h e v e l o - c i t y of sound and d i e l e c t r i c c o n s t a n t , r e s p e c t i v e l y . The v a r i a t i o n of t h e sound v e l o c i t y was f i r s t obser- ved by P i c h 6 e t a l . 1221 i n v i t r e o u s s i l i c a and i s shown i n f i g u r e 5.

On c o o l i n g , t h e v e l o c i t y i n c r e a s e s , p a s s e s t h r o u g h a maximum and d e c r e a s e s l o g a r i t h m i c a l l y w i t h t e m p e r a t u r e . The v a r i a t i o n a t h i g h e r t e m p e r a t u r e s i s caused by t h e r e l a x a t i o n p r o c e s s mentioned above.

The s l o p e of t h e c u r v e below 1 K i s d i r e c t l y d e t e r - mined by t h e s t r e n g t h of i n t e r a c t i o n between phonons and t h e TLS. The magnitude of t h e e f f e c t h a s t u r n e d o u t t o b e s i m i l a r i n a l l d i e l e c t r i c g l a s s e s i n v e s t i - g a t e d s o f a r . The c o u p l i n g p a r a m e t e r s a r e s u f f i c i e n - t l y l a r g e t o a c c o u n t q u a l i t a t i v e l y f o r t h e s c a t t e r i ~ g of t h e r m a l phonons i n h e a t t r a n s p o r t measurements/23/:

The h e a t r e s i s t a n c e below 1 K a r i s e s from t h e reso- n a n t s c a t t e r i n g of a c o u s t i c phonons by t h e l o c a l i z e d low-energy e x c i t a t i o n s . Recently s i m i l a r v a r i a t i o n s

.

of sound v e l o c i t y have been found i n m e t a l l i c glasses 115,241. The e f f e c t i s , however, s m a l l e r t h a n t h a t measured i n d i e l e c t r i c g l a s s e s .

90 MHz

_.

0. me

. .

a8

.

^-

SUPRASIL

I

QUARTZ CRYSTAL LT

9

^-L

02 03 05 1 2 3 5

TEMPERATURE

[K]

Fig. 5 : R e l a t i v e v a r i a t i o n of t h e l o n g i t u d i n a l sound v e l o c i t y Avlv = v(T)/v(T = 0.3 K ) - l i n v i t r e o u s s i l i c a and i n a q u a r t z c r y s t a l 1221.

An analogue v a r i a t i o n of t h e sound v e l o c i t y , a v a r i a t i o n of t h e speed of l i g h t has been observed by Schickfus e t a l . / 2 5 / . The q u a l i t a t i v e behaviour i s t h e same i n a l l amorphous m a t e r i a l s i n v e s t i g a t e d s o f a r / 2 6 / and an example i s shown i n f i g u r e 6, where t h e v a r i a t i o n of t h e d i e l e c t r i c c o n s t a n t ( i n s - t e a d of t h e v e l o c i t y of l i g h t ) i s p l o t t e d a s a func- t i o n of temperature. S i m i l a r o b s e r v a t i o n s have been made i n measurements of t h e d i e l e c t r i c c o n s t a n t i n

t h e kHz-range 121,271 down t o a few mK. I n a d d i t i o n , a s t r o n g n o n - l i n e a r i t y w i t h r e s p e c t t o t h e a p p l i e d e l e c t r i c a l f i e l d s t r e n g t h was found 1271.

The magnitude of t h e v a r i a t i o n of t h e d i e l e c - t r i c c o n s t a n t , however, and hence t h e d i e l e c t r i c coupling depends s t r o n g l y on chemical composition and t h e c o n t e n t of charged i m p u r i t i e s a s a l r e a d y mentioned i n t h e c a s e of d i e l e c t r i c a b s o r p t i o n . T h i s remarkable r e s u l t i s e s p e c i a l l y c l e a r l y seen i n v i - t r e o u s s i l i c a where t h e coupling s t r e n g t h depends l i n e a r l y on t h e c o n c e n t r a t i o n of 0 ~ - - i o n s 1281.

C l e a r l y , of a l l t h e low energy e x c i t a t i o n s p r e s e n t i n t h e g l a s s sample only t h o s e which c a r r y e l e c t r i - c a l d i p o l e s o r charges a r "seen" by e l e c t r o m a g n e t i c waves

.

Here ?the q u e s t i o n a r i s e s ; Are t h e TLS inves- t i g a t e d i n a c o u s t i c and i n e l e c t r i c experiments t h e same ? The answer has been given by t h e "cross"- experiment : TLS a r e e x c i t e d by a r e l a t i v e l y i n t e n s e e l e c t r o m a g n e t i c wave. Simultaneously, t h e a b s o r p t i o n

of a weak a c o u s t i c p u l s e of n e a r l y t h e same frequen- cy i s measured. I f t h e TLS i n t e r a c t w i t h b o t h types of f i e l d s , t h e e l e c t r o m a g n e t i c r a d i a t i o n should cau- s e a d e c r e a s e of t h e a c o u s t i c a b s o r p t i o n and v i c e v e r s a . Indeed, such e f f e c t s have been observed 129, 30/ i n d i c a t i n g t h a t t h e same TLS a r e involved i n both a b s o r p t i o n p r o c e s s e s .

TEMPERATURE [ K I

0"

0

9 o

W W A

a

W

5

I--1-

z

$! F

+ cn

4

0 0 - 2 - Z

0

I-

I!

a: g

^-3

Fig. 6 : Temperature dependence of t h e d i e l e c t r i c c o n s t a n t A E = E(T)

-

E(T = 0.4 K) of c r y s t a l l i n e q u a r t z , v i t r e o u s s i l i c a S u p r a s i l I and amorphous p o l y e t h y l e n e t e r e p h t a l a t e (PET) /25,26/.

1 I I I I

.

Dm m

... *.*. .\

^QUARTZ

. . ^{*. -}

Dm

_=.

me..

..

^a...

_\ ^.*' _-

n PET

D m

l

n

WPRASILI m=

-

¤ l

=m

-

I I I 'h-m== I

_

4. ECHO-PHENOMENA.- On c o o l i n g , t h e r e l a x a t i o n times of t h e TLS i n c r e a s e . Below about 5 0 mK t h e y become longer t h a n t h e d u r a t i o n of a p p l i e d a c o u s t i c o r elec- tromagnetic p u l s e s . Under t h i s c o n d i t i o n coherent

05 1 2 5 10

e f f e c t s can b e observed, s i m i l a r t o well-known co- h e r e n t phenomena i n s p i n resonance experiments. For example, i f two a c o u s t i c o r e l e c t r o m a g n e t i c p u l s e s P and P2 a r e a p p l i e d , s e p a r a t e d by a d e l a y time T

1 12

t h e n a t h i r d p u l s e

-

t h e echo Elg

-

^{i s} d e t e c t e d af- t e r t h e time 2 ~ This echo i s an analogue of t h e ~ ~ . magnetic s p i n echo and.was f i r s t observed a c o u s t i - c a l l y by Golding and Graebner 1311. Very r e c e n t l y t h e e l e c t r i c a l c o u n t e r p a r t h a s a l s o been found 132, 3 3 1 g i v i n g new i n s i g h t i n t o t h e d i e l e c t r i c proper- t i e s of g l a s s e s .

A r e c o r d e r t r a c e o b t a i n e d i n an e l e c t r i c f i e l d echo experiment a t 1.2 GHz / 3 3 / i s shown i n f i g u r e 7.

The echo i s c l e a r l y observed a f t e r twicethe delaytime T ~ = . I n b o t h , t h e a c o u s t i c and e l e c t r i c a l , echo ex- periments t h e decay of t h e echo h e i g h t with i n c r e a - s i n g d e l a y time i s a measure f o r t h e phase me- mory time T i , which i s determined by t h e i n t e r a c t i o n

C6-

1448 JOURNAL DE PHYSIQUE

between t h e TLS. Without going i n t o d e t a i l s we j u s t quencies of b o t h p u l s e s d i f f e r by a s much a s + ~ O M H Z . mention t h a t T i = 15

u s

i n v i t r e o u s s i l i c a a t 20 mK.

Although t h e temperature dependence of T i i s s t i l l under d i s c u s s i o n / 3 2 , 3 3 / , i t seems l i k e l y t h a t T;

can be i n t e r p r e t e d i n terms of s p e c t r a l d i f f u s i o n / 3 4 / ( s e e a l s o Chapter 5 ) . From t h e decay of t h e

" s t i m u l a t e d echoes'' / 3 1 , 3 2 , 3 3 / t h e l i f e t i m e T of the 1

TLS can b e d e r i v e d and t h e r e f o r e t h e i r coupling t o t h e amorphous network.

Fig. 7 : ~ e c o r d e r t r a c e obtained i n an e l e c t r i c f i e l d echo experiment a t 1.2 GHz and 20 mK. The echo E appears a f t e r twice t h e d e l a y time .r / 3 3 / . ~ h e ' l a r - ge width of p u l s e PI and P i s causeh2by t h e overload of t h e a m p l i f i e r . ²

I n t h i s c o n t e x t i t i s of i n t e r e s t t h a t t h e analogue of t h e magnetic s p i n echo i s not t h e o n l y one observed i n g l a s s e s . A t much higher f r e q u e n c i e s

(around 10 GHz) and temperatures above 1 K so-called

"backward-wave phonon echoes" have been observed i n v a r i o u s g l a s s e s by S h i r e n e t a l . /35/. T h i s type of echo i s due t o non-linear coupling of phonons and photons of t h e same frequency v i a t h e TLS and has h i t h e r t o o n l y been found i n p i e z o e l e c t r i c c r y s t a l s . It g i v e s i n f o r m a t i o n on t h e u n s a t u r a t e d u l t r a s o n i c a b s o r p t i o n i n g l a s s e s a t temperatures and f r e q u e n c i e s n o t a c c e s s i b l e t o c o n v e n t i o n a l pulse-echo techniques.

f = 1.2 GHz T= 20mK

-

5 . SPECTRAL DIFFUSION.- The mutual i n t e r a c t i o n b e t - ween neighbouring TLS r e s u l t i n g i n s p e c t r a l d i f f u s i o n

can b e s t u d i e d d i r e c t l y by u l t r a s o n i c t e c h n i q u e s : When a r e l a t i v e l y s t r o n g u l t r a s o n i c p u l s e propagates through a g l a s s sample a t low t e m p e r a t u r e s , one would expect a s a t u r a t i o n o n l y of t h o s e TLS whose energy s p l i t t i n g i s s u f f i c i e n t l y c l o s e t o t h e energy of t h e i n c i d e n t phonons. I f , however, t h e degree of s a t u r a - t i o n i s a c t u a l l y measured (Arnold and Hunklinger / 3 6 / ) u s i n g a second weak probing p u l s e of d i f f e r e n t f r e -

quency, s a t u r a t i o n i s s t i l l observed even i f t h e f r e -

, d

I--%

^{4 4 1 2}

-I

Pl p2 El2

7

S i n c e t h e l i f e t i m e broadening of t h e TLS a t t h e mea- s u r i n g temperature of 0 . 5 5 K and t h e frequency un- c e r t a i n t y of t h e sound p u l s e accounts only f o r rou- ghly 1 MHz, such a l a r g e l i n e w i d t h i s r a t h e r s u r p r i - s i n g / 3 6 , 3 7 / . It has been suggested by J o f f r i n and Levelut / 3 8 / t h a t t h e o r i g i n of t h i s l a r g e s p e c t r a l width may a r i s e from t h e coupling of t h e TLS w i t h each o t h e r . P r e s s u r e f l u c t u a t i o n s a t t h e l o c a t i o n of one TLS may be generated by neighbouring TLS due t o a b s o r p t i o n and r e e m i s s i o n of thermal phonons.

These f l u c t u a t i o n s l e a d t o a modulation of t h e ener- gy s p l i t t i n g of t h e TLS under o b s e r v a t i o n and hence t o t h e broadening of t h e l i n e .

Very r e c e n t l y t h e temporal a s p e c t of t h e in- t e r a c t i o n h a s been d i s c u s s e d / 3 4 , 3 9 / and s t u d i e d ex- p e r i m e n t a l l y / 4 0 / . The width of t h e observed l i n e becomes time dependent when t h e c h a r a c t e r i s t i c time f o r t h e f l u c t u a t i o n o f t h e l e v e l s p l i t t i n g of t h e TLS becomes comparable w i t h t h e time of o b s e r v a t i o n . This e f f e c t i s shown i n f i g u r e 8 where t h e l i n e - width observed i n a c o u s t i c "hole burning" experiments

i s p l o t t e d f o r two d i f f e r e n t o b s e r v a t i o n times. T h i s experimental r e s u l t seems t o prove t h e e x i s t e n c e of s p e c t r a l d i f f u s i o n /34/ and s u p p o r t s t h e i n t e r p r e - t a t i o n of t h e echo phenomena mentioned i n Chapter 4.

VITREOUS SILICA

FREQUENCY OF TljE SATURATING PULSE [MHZ]

F i g . 8 : V a r i a t i o n of t h e r e s o n a n t a b s o r p t i o n of a weak probing p u l s e a s f u n c t i o n of t h e s a t u r a t i n g p u l s e f o r two d i f f e r e n t o b s e r v a t i o n times / 4 0 / .

6. CONCLUSION.- Summarizing we can say t h a t g l a s s e s a t low t e m p e r a t u r e s e x h i b i t many p r o p e r t i e s which a r e fundamentally d i f f e r e n t from t h o s e of t h e i r c r y s t a l l i n e c o u n t e r p a r t s and which'are caused by a new type of two-level s t a t e s p r e s e n t i n a l l d i s o r - d e r e d s o l i d s . To a l a r g e e x t e n t , thermal and acous- t i c p r o p e r t i e s a r e i n s e n s i t i v e t o t h e chemical com- p o s i t i o n of t h e amorphous s u b s t a n c e s and can t h e r e - f o r e be considered a s g e n e r a l c h a r a c t e r i s t i c s o f t h e

g l a s s y s t a t e . The d i e l e c t r i c c o u p l i n g t o t h e two- l e v e l systems, however, i s s t r o n g l y enhanced by po- l a r i m p u r i t i e s . Although t h e formal phenomenological d e s c r i p t i o n of t h e o b s e r v a t i o n s i n terms of "two- l e v e l systems" h a s been v e r y s u c c e s s f u l , we have t o admit t h a t we a r e s t i l l f a r from a n u n d e r s t a n d i n g of t h e low t e m p e r a t u r e p r o p e r t i e s of amorphous ma- t e r i a l s on an atomic s c a l e .

ACKNOWLEDGMENTS.- I am g r a t e f u l t o P r o f . K.Dransfelrl f o r h i s permanent i n t e r e s t and s t i m u l a t i n g d i s c u s - s i o n s and t o D r s . W. Arnold, L. P i c h b and

M.V. S c h i c k f u s f o r many y e a r s of f r u i t f u l c o l l a b o - r a t i o n .

R e f e r e n c e s

/ I / Z e l l e r , R . C . , and P o h l , R.O., Phys. Rev.

(1971) 2029

/ 2 / L a s j a u n i a s , J.C., Ravex, A . , Vandorp, M., and Hunklinger, S . , S o l i d S t a t e Cormnun.

17

(1975)

1045

I 3 1 Graebner, J.E., Golding, B., S c h u t z , R.J., Hsu, F.S.L., and Chen, H.S., Phys. Rev. L e t t . 39 (1977) 1480

-

141 E.g. : S t e p h e n s , R.B., Phys. Rev. (1973)2896 S t e p h e n s , R.B., B E (1976) 852 ; Matey, J.R., and Anderson, A . C . , J . Non-Cryst. S o l i d s . 23 (1977) 129 ; V. Liihneysen, H.V., and s t e g l = h , F., Phys. Rev. L e t t .

2

(1977) 1205

151 S t e p h e n s , R.B., C i e l o s z y k , G.S., and S a l i n g e r , G.L., Phys. L e t t .

A38

(1972) 215

161 Hornung, E.W., F i s h e r , R.A.., B r o d a l e , G.E., and Giauque, W.F., J . Chem. Phys.

50

(1969) 4878 / 7 / Anderson, P.W., H a l p e r i n , B . I . , and Varma, C . ,

P h i l o s . Mag.

25

(1972) 1

/ 8 / P h i l l i p s , W.A., J . Low Temp. Phys. z ( 1 9 7 2 ) 351 191 Z a i t l i n , M.P., and Anderson, A.C., Phys. Rev.

L e t t .

22

(1974) 1158

/ l o / H u n k l i n g e r , S., Arnold, W . , S t e i n , S . , Nava, R., and D r a n s f e l d , K . , Phys. L e t t .

3

(1972) 253 / 1 1 / Golding, B . , Graebner, J . E . , H a l p e r i n , B . I . ,

and S c h u t z , R . J . , Phys. Rev. L e t t .

30

(1973) 223

1121 Hunklinger, S . , A r n o l d , W . , and S t e i n , S . , Phys. L e t t .

45A

(1973) 311

1131 J z c k l e , J . , Z . Phys.

257

(1972) 212

I 1 4 1 Golding B:, Graebner, J.E., and S c h u t z , R.J.,

Phys. Rev. (1976) 1660

1151 Golding, B . , Graebner, J.E., and Haemmerle,W.tt, Proc. I n t . Conf. L a t t i c e Dynamics (Flammarion)

1978, p. 348

1161 Doussineau, P., L e v e l u t , A., B e l l e s s a , G . , and Bethous, O., J . Physique L e t t .

38

(1977) L-483

I181 S c h i c k f u s , M . v . , and Hunklinger, S . , Phys. L e t t . 64A (1977) 144

-

/

191 S u p r a s i l I c o n t a i n s 1200 ppm 0 ~ - - i o n s 1201 Bgsch, M.A., Phys. Rev. L e t t .

40

(1978) 879 1211 F r o s s a t i , G., G i l c h r i s t , J. l e G . ,

L a s j a u n i a s , J . C . , and Meyer, W . , J. Phys. C , I0 (1977) L-515

-

1221 P i c h b , L . , Maynard, R . , H u n k l i n g e r , S., and J z c k l e , J . , Phys. Rev. L e t t .

32

(1974) 1426 1231 Hunklinger, S . , and P i c h b , L . , S o l i d S t a t e

Commun.

17

(1975) 1189

1241 B e l l e s s a , G . , Doussineau, P . , and L e v e l u t , A., J . Physique L e t t .

38

(1977) L-65 ; B e l l e s s a , G., J . Phys. C

10

(1977) L-285, B e l l e s s a , G . , and Bethoux, O . , Phys. L e t t .

62A

(1977) 125 1251 S c h i c k f u s , M . v . , Hunklinger, S . , and P i c h b , L.,

Phys. Rev. L e t t .

35

(1975) 876

1261 S c h i c k f u s , M . V . , T h e s i s , U n i v e r s i t z t Konstanz, Germany ( 1977)

1271 F r o s s a t i , G., Maynard, R., Rammal, R . , and Thoulouze, D . , J . Physique L e t t .

38

⁽¹⁹⁷⁷⁾

L-153

1281 S c h i c k f u s , M.v., and Hunklinger, S . , J . Phys.

C

2

(1976) L-439

1291 Doussineau, P . , L e v e l u t , A . , and Ta, T.T., J.

Physique L e t t .

38

(1977) L-37

1301 Laermans, C . , Arnold, W . , and Hunklinger, S . , J. Phys. C

10

(1977) L-161

1311 Golding, B . , and Graebner, J.E., Phys. Rev.

L e t t .

37

(1976) 852

1321 Bernard, L., P i c h b , L.

,

Schumacher, G.

,

J o f f r i n , J . , and Graebner, J . E . , J. Physique L e t t .

2

(1978) L-126

1331 S c h i c k f u s , M.v., Golding, B . , Arnold, W . , and Hunklinger, S . , i n t h i s i s s u e and unpublished r e s u l t s

1341 B l a c k , J . L . , and H a l p e r i n , B . I . , Phys. Rev. B 16 (1977) 2879

-

1351 S h i r e n , N.S., Arnold, W . , and Kazyaka, T.G., Phys. Rev. L e t t .

2

(1977) 239

1361 Arnold, W . , and H u n k l i n g e r , S., S o l i d S t a t e Commun.

17

(1975) 883

1371 B a c h e l l e r i e , A., Doussineau, P . , L e v e l u t , A., and Ta, T.T., J . Physique

38

(1977) 69 1381 J o f f r i n , J . , and L e v e l u t , A., J . Physique

2

(1975) 811

1391 Rammal, R., and Maynard, R., J . Physique L e t t , 39 (1978) L-195

-

I401 Arnold, W . , Martinon, C . , and Hunklinger, S., i n t h i s i s s u e

1171 J z c k l e , J . , Pichb, L., Arnold, W., and

Hunklinger, S., J . Non-Cryst. S o l i d s

20

(1976) 365