ECHOES AND ELECTRIC DIPOLE ECHOES IN GLASSES AT LOW TEMPERATURES

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ECHOES AND ELECTRIC DIPOLE ECHOES IN GLASSES AT LOW TEMPERATURES

L. Piche

To cite this version:

L. Piche. ECHOES AND ELECTRIC DIPOLE ECHOES IN GLASSES AT LOW TEMPERATURES.

Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1545-C6-1552. �10.1051/jphyscol:19786598�.

�jpa-00218091�

JOURNAL DE PHYSIQUE Colloque C6, supple'n~erzl au no 8, Tome 39, aolit 1978, page C6- 1545

ECHOES AND ELECTRIC DIPOLE ECHOES IN GLASSES AT LOW TEMPERATURES

Centre de Recherches sur Zes TrDs Basses Temperatures, C.N.R.S., B.P. 166X, 38042 GrenobZe Cddes, France.

R6sumd.- Les propridtds thermiques, acoustiques et didlectriques des verres B basse tempdrature sont expliquges en supposant l'existence de ddfauts (glastiques ou dlectriques) B deux niveaux d'dnergie. On utilise pour les ddcrire le mEme formalisme que pour la particule de spin 112. REcem- ment, il s'est avdrd que la technique des dchos, appliqu6e au problsme des verres, pouvait Etre une mine de renseignements nouveaux et originaux. Ici, nous passons en revue les diffdrents types d'gchos observds dans les amorphes et insistons davantage sur les dchos dipolaires dlectriques. tknos mesures nous pouvons ddduire la densit6 d'dtats, les composantes longitudinales et transverses des moments, la valeur du potentiel de ddformation, nous mettons aussi en 6vidence la variation en temperature du temps de relaxation longitudinal (T ) et transverse (T2).

1

Abstract.- The thermal, acoustic and dielectric properties of glasses at low temperature are attri- buted to two level defects either elastic or electric. These obey to the same description as that of the spin 112 particle. Amongst the various methods of probing such defects, the technique of echoes has recently proven to be most efficient. A review is made of the different types of echoes observed in glasses, with emphasis given to electric dipolar echoes. From the results of these measurements, values are obtained for the density of states, the longitudinal and transverse com- ponents of the dipole moment, the coupling to the strain field, also the temperature behaviour of the longitudinal (T1) and transverse (T2) relaxation times are measured.

1.INTRODUCTION.- It is now well admitted that the low temperature thermal and acoustic properties of glasses can be explained through the existence of additional low energy states /I/. These are identi- fied as two level systems (T.L.S.) and it has been shown experimentally that they have a broad and flat energy spectrum and that they come into strong resonant interaction with phonons of same energy.

If ionic impurities are imbedded in the glass,there results a material having dielectric properties which themselves are explained in terms of electri- cal T.L.S. with a broad energy spectrum 121.

The microscopic nature of T.L.S., elastic or electric, is still an unknown but it is usually thought of in terms of configuration defects, defects to which would correspond two energy levels.

The quantum state of the T.L.S. is then specified by the knowledge of the four elements of a 2 x 2 matrix ; under the influence of an external field, the probability that transitions will occur between the two eigenstates is given by the off diagonal elements of a perturbation hamiltonian which des- cribes the interaction. The picture is completed by the density matrix which gives the state of po- pulation.

From this, the step is short to see the strong similarity between the problem of the T.L.S.

in a glass and that of the spin 112 particle. In

particular, as for the spin 112, the dynamics of the T.L.S. may be described, in a first approach, using the formalism of Bloch equations which invol- ves two characteristic times : a spin-lattice or longitudinal relaxation time TI and a phase memory or transverse relaxation time T2. The first, TI, gives the time scale of the process by which the population of the T.L.S. is reestablished into ther- mal equilibrium. In glasses, this concept is some- what well understood and at low temperature (T < 1 K), it finds it's origin in the direct one phonon process 1 3 1 . As for Tp, not very much is known.

From a phenomenological point of view, the relati- vely large concentration of defects and the small distances that exist between them has lead to the idea / 4 / that there could exist indirect spin-spin interactions via the phonon field. Such a coupling will allow the transfer of energy between spins and T2 is the time, shorter than T1, whereby equi- librium is established inside the spin systems ; in terms of a 2 x 2 matrix description, the Tp me- chanism tends to destroy the transverse component of the magnetization. The very fact that the T.L.S.

can absorb energy in a resonant way naturally im- plies the existence of T2 but, if we are to under- stand its origin, definite assumptions must be made on the internal structure of the "spin" system and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786598

C6- 1546 JOURNAL DE PHYSIQUE

on the mechanism of broadening and relaxation.

Hope to help progress in this direction initia- ted a number of experiments. For example, the cri- tical intensity which describes the onset of satu- ration in ultrasonic absorption depends on the pro- duct T T ; TI being known, T can be estimated / 4 /

1 2 2

: at a temperature T = 0.5 K and a frequency w/2n =

730 MHz, T is of the order of 3 x p.s. Bur- 2

ning hole experiments 151 are more demonstrative and give similar results. However, the most signi- ficant break-through comes from the performing of echoe experiments 161 with the elastic dipoles (T.L.S.); similarily echoe experiments were made on the electric dipoles /7,8,9/ of glasses containing ionic impurities. The principle involved is essen- tially the same as that well-known in NMR and EPR work, and allows the measurements /lo/ of both T1 and TI.

Our aim is to give an outlook on the develop- ments which echoe experiments have recently provi- dedconcerning the description of the assembly of

defects in glasses and of their dynamic properties.

2. ECHOES IN GENERAL AND ECHOES IN GLASSES.- 2.1. Echoes in general (Fig. 1 )

echoe signal.

Fig. I : Schematic, illustrating in the rotating In order to help understand the results which frame the different phases for the

polarization-(^)

in it's evolution towards the formation of the spon- will be reported, we need a clear picture of the taneous echoe in a (712 ; n) pulse sequence. Ini- technique through which they are obtained. We shall tially, P lies along the z axis, during phase I, it

is rotated 90' onto the x axis : phase I' shows the thus recall, using a simple and pragmatic approach, effect of inhomogeneous broadening : some packets the general principles which are involved. precess faster (PF) than others (PS). A second

pulse (phase I") causes a 180" rotation of the x A first high frequency field pulse is applied to

componentS of the vectors : fast diDoles are then the assembly of spins, it will cause those spins

which are at resonance to rotate, i.e. those which have an energy splitting E = Mu, w being the angular frequeacy of the field. A coherent state is created to which corresponds a macroscopic transverse di- pole, it's amplitude will be maximum when the pulse (712 pulse) brings the spins 90' from their initial state. Now, if there is a spread in the resonant frequencies of the individual spin packets within the resonance line (inhomogeneous broadening), and this is to be expected in a glass because of the random nature of it's structure, 'the individual spin packets will rapidly get out of phase : this is the free decay. However such is an effect of static broadening and is thus reversible so that if at time T after the first pulse, a second pulse

(preferably a n pulse) is applied, it reverses the dephasing process and at time t = 2 ~ , the individual dipoles are again in phase. The new state of the magnetization gives rise to the coherent two pulse

lagging $low dipoles ; all meet at t h e 2~ to form an echoe.

The observation of an echoe signal implies that the dipoles retain memory of their original phase rela- tions. In practice this memory is finite, there always exists a mechanism which causes an irrever- sible decay of the transverse magnetization with a time constant T2. It is this irreversible process

(dynamic homogeneous broadening for example) with which we are concerned and the measurement of the echoe decay rate as a function of T will give the measure of T2.

If in a double pulse sequence, the second pulse is not strong enough to bring about the full time reversal of all the dipoles, there will result in the magnetization having a longitudinal compo- nent which is not subject to dephasing but tends

towards thermal equilibrium in a characteristic time T1. A third pulse applied at time T + t, can complete the time reversal of that part of the

longitudinal magnetization which has not yet rela- xed and causes an echoe (stimulated echoe) to appear at time T + t + T. The measure of the amplitude of

the three pulse echoe with increasing t will allow that of TI.

2.2. Echoes in glasses.-Three types of echoe experi- ments have been performed : a) backward phonon-

echoes, b) elastic dipole echoes and c) electric dipole echoes.

In the first case, coherence is established within the phonon assembly and the T.L.S. acts as a cata- lyzer in the echoe producting, whereas in the other cases, coherence is within the assembly of T.L.S.

We will report briefly on a) and b) while being more concerned with c) (section 3) mainly since it has been, up to now, the most thoroughly studied.

a) The backward wave phonon echoe (phonon echoe) phenomenon was unique for piezoelectric substances /Ill until1 it was also observed in glasses 112, 131.

In the usual case, a piezoelectric material is sub- mitted to an R.F. electric field pulse whereby pho- nons of the same frequence (a) are produced-forthese, there corresponds within the sample a topology of equiphase surfaces: the pulses have created a cohe- rent phonon population. For the rest, it is essen- tial that there are impurities which couple to both the strain field and the electrical field in a non- linear way. A second electrical pulse (a) triggered at time T will cause the movement of these defects which will have a frequency component at 2w. A col- lective mode (2w, k = 0) is thus established and its interaction with a phonon (w, + k) of the cohe- rent population will involve another phonon (w,

-

^k).

This second phonon created in the parametric process will travel back in the crystal and give rise tothe phonon echoe at time 2 ~ . It is in this sense that the second RF pulse serves to reverse the k vectors of the coherent phonons created during the first pulse.

For the experiment that was carried out in a glass, a transducer bonded to the sample served to produce the coherent phonon population. The measu- rements *ere done with w/2n = 9 GHz and T + 1.25 K ; the main resplts can be sunrmerized as follows: i) that an echoe is observed means that there exist in the glass anharmonic oscillators which couple to both the strain and the electrical field. ii) a series of experiments on a variety of samples show that the amplitude of the echoe scales with the

concentration of unassociated O H ions so that the origin of the echoe is attributed to OH- impurities.

iii) The acoustic attenuation which is obtained from the echoe decay rate compares to that estimated for the resonant ultrasonic attenuation by T.L.S. in the glass.

This leads to understand that the hydroxyl groups which are at the source of non-linear electro- elastic effects could very well be at the origin of T.L.S. which couple to both the electric and the strain fields. In fact, the existence of defects having this double nature is also demonstrated in experiments 1141 performed by crossing and electro- magnetic and an acoustic wave (no more can be con- cluded for in this last case, the ionic impurities in the sample were not OH-).

b) The phenomenon of elastic dipole echoes in glasses is quite analogous to that of magnetic spin echoes : The broad distribution of T.L.S. acts as the resonant spin medium in which a window is framed by the frequency spectrum of the ultrasonic field pulses used for the excitation.

The experiments were performed 161 at very low temperatures where TI and especially T2 become long

-

according to usual electronic standards. The pro- cedure is well illustrated by figure 2 : the sample is Suprasil W /IS/, pulses have a frequency w / 2 ~ = 680 MHz and equal widths Atl = At2 = 0.1 us., T

-

20 mK. The first part (Fig. 2a) shows the usual acoustic attenuation pattern in a single pulse (PI) experiment ; figure 2 b shows what happens when a second identical pulse (P2), is added at a time T

after P 1 : there is of course a new reflection pat- tern delayed by T with regard to the first but, one notes the presence of additional pulses such as E12. E I 2 is the spontaneous elastic dipole echoe signal. The amplitude decay of the echoe E12 with increasing values of T .is a measure of T2. In most cases, the decay which appears to be exponential is described by exp(-2.r/T2) ; for example, at 20 mK, T2 = 14 p.s

.

^{and at 45 mK, T2 = 3 p.s}

. ^,

^and

these values are consistent with results of burning hole experiments at the same temperature 161.

The same technique allows to study the stimula- ted or three pulses echoe : at T 2 20 mK, the decay is exponential with a characteristic time TI = 200 U.S., also in good agreement with the value obtained from the burning hole technique.

This elastic dipole echoe experiment where both

c6-1548 JOURNAL DE PHYSIQUE

a spontaneous and a stimulated echoe were observed in an amorphous material is a first and it opens a new field for the investigation of the population of defects in glasses and of their dynamic proper- ties.

Fig. 2 : Redrawn from oscilloscope photographs (ref 161) showing the decay of 0.1 u.S., 680 MHz ultrasonic pulses in Suprasil W at 20 mK. The top trace shows the attenuation pattern for a single pulse (P1 which comes from the attenuated leakage signal) ; the time between two reflections R and R1, is 2.2 U.S. The bottom trace shows the ekkect of two pulses (P1 and P ) with a separation T =

0.6 U.S. : the echoe E12 appears at a time 2 T = 0.6 v.s. after the first reflection (R21) of pulse P2.

at a time 2~ = 1.2 v.s., after Rll.

However the ultrasonic technique is difficult and there are some drawbacks : i) it is only possible to estimate the effective acoustic energy which is seen by the T.L.S. ii) need is to consider the acoustic power regime under which the echoe is ob- served : high acoustic powers leed to saturation and this must be taken into account when measuring the amplitude of the echoe. (This last point is illustrated in fig. 2-b where the amplitudes of R21 and E12 are larger than that of R l l , this comes from that R21 and E12 correspond to pulses which have travelled in a region partially saturated by Rll).

3. ELECTRIC DIPOLE ECHOES.- In a glass having a cer- tain ionic content, at least part of these impurities behave like T.L.S. with an associated dipole moment

(electric T.L.S.) and a broad excitation spectrum.

This constitutes the resonant medium on which the electric d i ~ ~ l e echoe experiment is performed, using electric field pulses for the excitation.

There was a first observation of an electric echoe in a glass sample produced by sputtering 171, however very little data exist and in order to do any physics, a more thorough and detailled study of the phenomenon is needed. We shall thus concentrate on an experiment that was performed /8,9/ on a bulk sample of a known glass, Suprasil I 1151 at a fre- quency w/2n = 370 MHz in the temperature range of 4 mK to 30 mK. We will study the behaviour of 1) the spontaneous echoe, 2) the stimulated echoe.

3.1. Spontaneous echoe.- The spontaneous echoe is a non linear phenomenon, the amplitude E(2T) of the echoe signal depends on the angles

el

and O2 through which the polarization is rotated during the first and second pulse. It can be shown 1101 that :

E(~T)= sinel sin2e2/2 with pxgbt = (1) where At is the width of the pulse,g the electrical field which it creates and Px the value of the trans verse (induced) dipole moment.

The results on the study of the echoes signal (E~(~T)) versus input pulse power (Pin) are shown in figure 3 : pulses have equal heights and At =

2 2Atl = 1.4 p.s. At 4 mK, the separation between pulses was T = 5 p.s., the same at 10 mK for the upper curve whereas for the lower one T = 20 p.s. In all cases, for small values of P. (small angle li-

In

mit) the signal is proportional to P! as predicted.

In

Also, for the smallest values of T, a single maximum occurs for P. 1 -28 dBm, while, when T increases,

In

the overall amplitude is less and another maximum appears at Pin= -38 dBm.

Because of the random nature of the amorphous structure, it is very plausible that there is a distribution in the values of the dipole moments, this would explain the existence of the double maxi- mum : the first maximum at P! =

-

38 dBm corresponds

In

to a n12 rotation of large dipoles and the second at P. In

- ^-

28 dBm to a n12 rotation of small dipoles.

Given the characteristics of the electrical circuit, the corresponding values of the dipole moments are found to be PxL = 2.8 Debye and PxS 2 0.9 Debye.

Also figure 3 shows that the contribution of small dipoles decreases much faster than that of the lar- ger dipoles which would mean that the small dipoles have a tendancy to relax faster and that they are more numerous. In another experiment 191 a DC pulse 'is added to the train of RF pulses and this allows

to probe the distributionof the longitudinal (per- ,rnanent) dipole moment pZ : the distribution is a

Lorentzian having a width hpZ = 0.2 Debye and c e n t e red on zero. That there exists a distribution for pZ supports the idea that the transverse dipole mo- ments px are also distributed.

1 ^{Signal (d B)}

Fig. 3 Plot of the amplitude (power dB's) of spon- taneous echoe signal versus input R.F. pulse power (P. ) ; the second pulse has a width (At2 = 1.4 p.s) tw?ze that of the first (Atl) ; the distance between pulses is T = 5 p.s. for the curve at 4.17 mK while the curves at 10.9 mK show the effect of increasing

T from 5 p.s.to 20 u.s.(for clarity intermediate data is omitted). The dashed line is there for com- parison with the theoretical P? behaviour in the low power limit. in

Finally, the amplitude of the echoe signal is given by the polarization 181 within the sample which is proportional to the density of states of the dipoles (n) : a value n =

lo3'

erg-l cm-3 is found. This is to be compared with the value erg-l cm3 obtained from specific heat data /I 61 which includes the contributions of both the intrin- sic and the OH- defects : the electric T.L'.S.

account for 10 % of the defects in glasses.

Another purpose of an echoe experiment is to measure the homogeneous phase memory time T2, re-

sults of such measurements at T = 4.17 mK are shown in figure 4.' The decay is exponential only in the low power limit (PQ, regime). For larger values of Pin the amplitude shows a rapid initial decay while with time,ittends towards the limiting case of low

The assumption that there is a distribution for the dipole moments can also explain the'"power dependence" of T2 : for large field values, small and large dipoles contribute to the signal ; also the small dipoles are more numerous and they relax faster, so their contribution becomes less as T is increased up to a point where only the large dipoles account for the echoe amplitude (as in the case of P. small).

In

Signal

(d

B ) Floating S c a l e

rn

Pin a- 23 dBm

-5 \ T="7mK

⁰

=-35dBm

Fig. 4 : Decay of the spontaneous echoe signal (pulsing conditions identical to those of figure 3) with Z T , the time between the first pulse and the echoe. The results obtained at T = 4.17 mK are shown for different values of R.F. pulse power P The memory time T2 is that in which the signal h k ' decreased by 8.6 dB. The shortest values of T2 are given by the slope at the origin of the decay

(dashed line) in the high power limit, long T2's are those obtained from the decay at low powers.

The temperature behaviour of T' is illustrated in figure 5 ; the uppe; curve corresponds to short T's (initial slopes of high power curves, in figure

2

4) and the lower curve to long T's (exponential

2

decay time at small Pin values). Figure 5 shows that T2 = T-I ; that this is true for long as well as for short T2's seems to indicate that T2 has the same origin in both cases.

Now the question : what is the mechanism which in glasses governs T2 ? On this, it has been suggested /4/ that the phonon field mediates an indirect interaction between defects resulting in pulse power.

C6-1550 JOURNAL D E PHYSIQUE

an exchange coupling between two defects which could be quite strong since the T.L.S. are them- selves strongly coupled to the strain field. On the other hand thermal phonons can produce fluc- tuations in the orientation of spins with which they are at resonance. Then, a given spin "sees"

via the exchange coupling, the thermal fluctuations of unlike spins, from this would result that the local field of the spin evolves in time. This me- chanism of "spectral diffusion" /17,18/ would have as main consequences :

i) the echoe signal does not decay exponentially, instead it evolves with time first very slowly then, much faster. ii) T2 follows a quadratic temperature

law : T2 T - ~

Fif. 5 : Temperature behaviour of phase memory time TI ; top curve is the behaviour observed for short TzS1s and bottom curve for long TIL1s (see figure 4); in both cases, T?' a T; a line is lncluded to compa- re with an eventual T' behaviour, also included da- ta points (G) obtained from phonon echoe measure- ments (ref. 161).

As can be seen in figure 4 and figure 5, neither i) nor ii) correspond to the behaviour which is observed. It appears then that the theory of spectral diffusion cannot be used at least in its

us with a very elegant method to measure T1, through the time decay of the stimulated echoe signal in a three pulse sequence. For electric dipoles in glas- ses, one observes that in the high power regime (Pin =

-

23 dBm), the decay is not exponential : initially, the signal decreases rapidly and then, it slowly levels off (in a millisecond time scale).

In order to describe such a behaviour, a distribu- tion of relaxation times is needed which extends from a minimum value Tl,,,in) obtained by the initial decay to values more than an order of magnitude larger. Another interesting feature is that when the pulse power is reduced, only large dipoles are accessible and contribute to the signal ; they are the ones with the longest relaxation times. By ope- rating in this way, it is virtually possible to

"select" the dipoles which relax in a given time and thus to correlate the value of their moment to that of relaxation time.

At low temperature, if T1 originates from the direct one phonon process, for a T.L.S. of energy splitting E = Ho, it is given by 131 :

l2 E

~;1 (E) = (- + 2 C;

where p is the mass density and cl and ct the lon- gitudinal and transverse sound velocities (p = 2.2 g . ~ m - ~ , cl 5.87 x

lo5

cm S-I, ct = 3.6 x

lo5

cm.

s-I). The formula also serves as a definition of the coupling constant M for the resonant interaction between a T.L.S. and the elastic strain field.

In figure 6, the results obtained for Tlmin are plotted against x = h/2kgT ; the dotted line represents the function 4.0 coth x which shows to fit the data points reasonably well. This is a good indication that T1 is governed by the direct process. Using the factor 4.0 found from the fit and the approximation M~~ ² 2 M~~ /1,20/, we evaluate the maximum value for the coupling constant : M = 3.2 eV, M ² 2.2 eV. This is in agreement with

1 t

the value furnished by phonon echoe measurements 161 which falls on the curve of figure 6, in agree- ment also with the results of direct measurements of T for the elastic T.L.S. in burning hole experiments

1 16,211.

present form to explain T- for the problem of elec-

L

tric dipole echoes at very low temperatures 1191.

3.2. Stimulated echoe.- The echoe technique furnishes

4.CONCLUSION.- As it appears, the technique of echoes proves to beapowerfull tool for studies in a field of amorphous materials. In the special case of glasses, it allows to examine the dynamic pro- perties of the T.L.S., both elastic and electric and provides for a new look on the nature of defects in glasses. If it has furnished some answers, it has also stimulated a number of questions which have yet to be answered.

A first question comes from that most of the results ,on T and T where obtained through expe-

l 2

riments on electric T.L.S., would they still hold for the care of intrinsic, or purely elasticT.L.S.?

Concerning T1

,

the direct one phonon process go- verns the relaxation of the electric T.L.S. as it does for the intrinsic defect. In fact, the cou- pling to the strain field appears to be as strong in both cases and, this puts forth the double iden- tity of the electric defect. As for T2, does i t originate from a purely elastic dipole-dipole in-

0

teraction or if in the case of ionic defects must an electric field coupling also be considered in parallel ? It may be that more elaborate results on phonon echoe measurements will help in finding an answer.

At any rate, restricting to electric T.L.S., there seems to be no model yet which could describe T2 nor its temperature dependence. So, what is then the mechanism by which phase coherence is established in an environment where a defect is

x = f i u ~ / 2 k g ~

I I I I

surrounded by so few neighbours with which he is on speaking terms ? Actually the same question holds for other disordered materials such as spin-glasses where EPR and NMR lines are still unaccounted for.

Finding a model for T2 means that definite assump- tions must be made on the interngl structure of the assembly of defects, and for this, a better know- ledge of spatial correlation might be helpful.

What is the true nature Of the defects ? Does it involve a single atom or a whole group, a clus- ter ? What is the meaning behind' the fact that the values for the elastic and electric dipoles are so

0 0.5 1 1.5 2 ⁷

Fig. 6 : Temperature behaviour of T obtained from stimulated echoe measurement fmfge plot is that of T ~ versus coth ~ ~ x , x ~= Hw/2kgT (here, - ~ Hw/kg E 17.76 mK) to compare with the theoretical prediction of the one phonon process. The data is seen to fit equation 4.0 coth x (dotted curve).

Also included data point (G) from phonon echoe measurements (ref. 161).

high and very close to those found for the problem of ionic impurities in a single crystal (KC1 : OH for example) ?

In the immediate future, it might be interes- ting to try using the different distribution found for the parameters (p,M,E) which describe T.L.S., in an approach to a more microscopic model, to a clear image of disorder.

ACKNOWLEDGEMENT.- I consider it a pleasure to acknowledge the friendly help of Professor 3 . Joffrin, Professor R. Maynard and Dr. R. Rammal.

JOURNAL DE PHYSIQUE

References

/I/ See for example Hunklinger S., Arnorl W., in Phy- sical Acoustics, ed. R.N. Thurston and W.P. Mason (Academic, New-York), 1976 vol. 12 qnd references therein.

/2/ Schickfus M.v., Hunklinger S., Piche L., Phys.

Rev. Lett.

35

(1975) 876.

/3/ Jgckle J., Piche L., Arnold W., Hunklinger S., J.

Non-Cryst. Solids

20

(1976) 365.

/4J Joffrin J., Levelut A., J. Physique

36

(1975) 811.

/5/ Arnold W., Hunklinger S., Solid State Commun. 17 (1975) 883 ; Bachellerie A., Doussineau P., Leve- lut A., Ta T-T, J. Physique 38 (1977) 69.

/ 6 / Golding B., Graebner J.E., Phys. Rev. Lett.

37

(1976) 852.

171 Golding B., Graebner J.E., Haemmerle W.H., Proc.

7th Int. Conf. on Amorphous and Liquid Semiconduc- tors, Edinburgh, 1977.

181 Bernard L., Piche L., Schumacher G., Joffrin J., Graebner J., J. Physique Lett.

2

(1978) L-126.

191 Bernard L., Piche L., Schumacher G., Joffrin J., J. Physique Colloq., LT 15 Proc. Grenoble (1978).

/lo/ See for example, MIMS W.B., Electron Paramagnetic Resonance, Chap. IV, S. Geschwind Ed. (Plenum Press, New-York) 1 972.

/11/ Billmann A., Frenois Ch., Joffrin J., Levelut A., Ziolkiewicz S., J. Physique

24

(1973) 747.

/12/ Shiren N.S., Arnold W., Kazyaka T.G., Phys. Rev.

Lett.

2

(1977) 239.

1131 Arnold W., Kazyaka T.G., Shiren N.S., Proc. Int.

Conf. Amorphous Semiconductors Edinburgh (1977).

1141 Laermans C., Arnold W., Hunklinger S., J. Phys.

C

10

(1977) L 161 ; Doussineau P., Levelut A., Ta T-T., J. Physique Lett.

38

(1977) L-37.

1151 Suprasil W. and Suprasil I are both vitreous silica glasses, the first contains less than 1.5 ppm OH- ions and the other roughly 1200 ppm OH-, both contain less then 0.5 ppm metallic ions.

/16/ Lasjaunias J.C., Ravex A., Vandorpe M., Hunklinger S., Solid State Comun

2

(1975) 1045.

/17/ Black J.L., Halperin B.I., Phys. Rev. B

16

(1977) 2879.

1181 Hu P., Walker L.R., Solid State Commun

26

⁽¹⁹⁷⁷⁾

813.

1191 Another approach suggests that spectral diffusion would be important only in the 1 K temperature range : Rammal R., Maynard R., J. Physique Lett.

39 (1978) L-195.

-

1201 Hunklinger S., Piche L., Solid State Commun 7!- (1975) 1189.

/ 2 1 / Hunklinger S.,Schickfus,M.v., Arnold W., Piche L.,

Dransfeld K., Proc. L.T. 14, ed. M. Krusius and M.

Vuorio (North-Holland and American Elsevier), (1975) vol. 3, p. 5.