Acoustical study of tunneling states in magnetic rare-earth glasses : wave propagation and phonon echoes

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Acoustical study of tunneling states in magnetic

rare-earth glasses : wave propagation and phonon echoes

F. Lerbet, G. Bellessa

To cite this version:

F. Lerbet, G. Bellessa. Acoustical study of tunneling states in magnetic rare-earth glasses : wave propagation and phonon echoes. Journal de Physique, 1988, 49 (7), pp.1179-1193.

�10.1051/jphys:019880049070117900�. �jpa-00210800�

Acoustical study ^of tunneling ^{states in} magnetic rare-earth glasses : ^wave propagation ^and phonon ^echoes

F. Lerbet and G. Bellessa

(*) Laboratoire de Physique ^des Solides, ^Bâtiment 510, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France (Requ ^{le 4} janvier 1988, accepté ^{le 1er mars} 1988)

Résumé.

Nous présentons ^{une étude} acoustique des états tunnel dans des verres d’aluminosilicate d’holmium. Les mesures sont faites jusqu’à ^{10 mK,} dans le domaine de fréquences 10-500 MHz et en champ magnétique ^pouvant atteindre 60 kOe. Les paramètres ^de couplage des états tunnel sont obtenus à partir ^des

mesures de vitesse du son et de la transparence auto-induite. La largeur ^{de raie est obtenue à} partir ^{de la} précession libre des états tunnel et d’expériences ^{de hole} burning. Des échos de phonons ^sont observés dans des échantillons de différentes teneurs en holmium, ^{avec et sans} champ magnétique. L’amplitude ^{de l’écho} augmente par ^un facteur 3 quand ^le champ magnétique croit de 0 à 60 kOe. Le temps de relaxation transverse

T2 ^{est obtenu à} partir ^{des échos} spontanés ^et les échos stimulés permettent ^{d’accéder au} temps de relaxation

longitudinal T1. Ce dernier est aussi obtenu à partir d’expériences de désaturation. Enfin, ^une expérience ^de

diffusion spectrale ^est présentée. Nos résultats expérimentaux ^montrent que les paramètres statiques ^{des états}

tunnel (potentiel ^de déformation et densité d’état) changent ^{avec la teneur en ions} magnétiques (entre ^{1.5 % et}

10.1 %), ^{mais ne} changent pas ^en présence ^d’un champ magnétique (jusqu’à ⁶⁰ kOe). ^Au contraire, ^les temps de relaxation T1, ^et T2 ^sont approximativement ^{les mêmes} quelle que soit la ^{teneur en} ions magnétiques ^et T1 augmente par ^un facteur 6 quand ^le champ magnétique croît de 0 à 60 kOe. La relaxation des états tunnel par les ions magnétiques ^est envisagée.

Abstract.

An acoustical study ^of tunneling ^states in aluminosilicate glasses with various holmium contents is

reported. ^The measurements are performed ^{down to 10} mK, ^{in the} frequency range 10-500 MHz and in

magnetic fields up ^to 60 kOe. The coupling parameters ^{of the} tunneling ^{states are} obtained from the sound

velocity measurements and from the self induced transparency. The linewidth is obtained from the free

precession ^{of the} tunneling ^states and from hole burning experiments. Phonon echoes are observed in samples

with various holmium contents, with and without magnetic field. The echo amplitude ^increases by ^a factor of 3

as the magnetic field increases from 0 to 60 kOe. The transverse relaxation time T2 ^is obtained from the spontaneous phonon echoes and the stimulated phonon echoes allow us to reach the longitudinal ^relaxation

time T1. The latter is also obtained from saturation recovery experiments. Lastly, ^an experiment ^of spectral

diffusion is reported. ^Our experimental results show that the static parameters ^{of the} tunneling ^states (deformation potential ^and density ^of states) change ^{with the} magnetic-ion ^content (between ^{1.5 % and}

10.1 %) ^{but are not} changed ^{when a} magnetic ^field (up ^{to 60} kOe) ^{is set} up. On the contrary, the relaxation times T1 ^and T2 ^are roughly ^{the same} whatever the magnetic-ion ^{content and} T1 ^increases by ^a factor of 6 as the

magnetic field increases from 0 to 60 kOe. The relaxation of the tunneling ^states by the rare-earth ions is considered.

Classification

Physics ^Abstracts

43.35

63.20P

75.50K

1. Introduction.

The phenomenological theory of the elastic tunneling

states (TS) ⁱⁿ glasses [1, ^2, 3] ^{has been now well}

achieved in its great ^lines though ^a microscopic description ^{is still} lacking. ^It explains quite ^{well the}

main acoustical and thermal properties ^of glasses.

(*) Laboratoire associe au C.N.R.S.

Till now, the experimental studies of these elastic TS have been extended to many ^areas in various ma-

terials but very few of them ^are concerned with the direct measurement of their relaxation times. This has been achieved in insulating glasses by carrying

out saturation recovery experiments [4] ^and phonon

echoes [5, 6]. In metallic glasses, ^{due to the} strong interaction between the TS and the electrons, ^the

relaxation times are very short and phonon ^echoes

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070117900

cannot be observed [7, 8]. In any case, they ^{are the}

most powerful ^{tool to} study ^{the coherent} properties

of the TS.

In this paper, acoustical measurements and pho-

non echoes in insulating magnetic glasses ^{are re-} ported. ^The experiments ^are performed ^{in holmium-} doped aluminosilicate glasses. ^{The holmium} glasses

are attractive for acoustical studies on account of the

large spin-orbit coupling of the holmium ions [9, 10].

An effect of the magnetic ^{field on the} phonon ^echoes

is reported. ^Our study ^reveals that the static par- ameters of the TS do not change ^{with the} magnetic field, ^{unlike the} dynamic parameters ^which change strongly. ^These properties ^{must be} distinguished

from the effect of magnetism ^on the static parameters of the TS which is known from a long ^time [11] ^and

which has been confirmed recently ⁱⁿ spin glasses [9, 12]. ^{In other} respects, ^{the effects} of the TS on

m4gnetic ^{ions have} already been considered in

photon ^echo experiments ^{in a} rare-earth-doped glass [13, 14]. ^These experiments have shown that the

optical dephasing ^{rates are enhanced in} comparison

with those in crystalline ^materials [14]. This enhance- ment has been assigned ^{to an} effect of the TS on the rare-earth ions [15]. ^{The effect} reported ^{here is}

different from the above mentioned _one, eventhough

it implies ^{the same} coupling. Here, ^{the TS are} directly ^observed and the rare-earth ions are per- turbed by ^the magnetic field, whereas the photon

echoes arise from optical transitions between the

ground ^{state and} excited levels of the rare-earth ions. Preliminary ^{results about our} phonon ^echo experiments ^{have been} published [16].

Our acoustical study ^is performed ^{down to 10} mK,

in the frequency range 10-500 MHz, ⁱⁿ magnetic

fields up ^to 60 kOe and for different magnetic-ion

concentrations between 1.5 % and 10 % at. The

coupling parameters and the relaxation times are

measured and the possible relaxation phenomena

are considered.

2. Theoretical play ^back.

The model of the TS in glasses ^{is now well known} [1, 2]. ^{We recall here a} presentation already partially

done elsewhere [17, 18] and which will be very useful to account for the results of our experimental

coherent acoustic study. That will be followed by ^the

theoretical description ^{of the} magnetic ^rare-earth

ions in glasses.

2.1 THE TUNNELING STATE MODEL. - The TS model assumes that each tunneling system ^{is de-} scribed by ^a double-well potential ^whose parameters

are the asymmetry E ^{and the} overlap energy Eo

h03A9 e- ’ where hd2 is the zero point energy in each well and a is the overlap parameter. ^{The two}

states of lowest energy of the tunneling system

ø a ^and Ob ^are separated by ^an energy splitting

E ⁼ (E2 + E2). ^{Since we are dealing with glasses,}

the parameters ^{of the TS are} randomly distributed and it is usually assumed that the distribution

Ii (E, u) ^{is constant,} which leads to a roughly

constant density ^{of states} n(E) ^{as a function of the} splitting energy E [1, 2]. Each TS of energy E is

coupled ^{to an external strain field e} through ^the

deformation potential B

aelae ^{and the Hamilto-}

nian of the coupled system in the basis ø a’

ø b ^is [3, 17, 19] :

where Sx and Sz ^are the usual Pauli matrices. So we can define the elastic dipoles [17]

and

The distributions of the parameters ^{e and a for a} given packet ^{of TS} of energy E lead ^{to a} distribution of elastic dipoles.

2.2 THE EQUATIONS OF MOTION.

As for the

propagation ^of light pulses in two-level absorber systems [20], the resolution of the acoustic-pulse propagation ⁱⁿ glasses requires ^self consistency ^for e (z, t ), since the elastic polarization p (z, t ) ^induced by the elastic dipoles ^{acts on} the strain e (z, t ) through ^the propagation equation. ^{This treatment}

has been done by ^Shiren [21] ^and by Graebner et al.

[18] for the acoustic case. The propagation equation

of the acoustic wave is :

for a medium where p is the density ^{and c is the}

sound velocity. ^This equation has been solved for the different ranges which appear ^as considering ^the pulse ^{duration with} respect ^{to the} characteristic times of the problem (Tl ^and T2 ^{as defined} below) [18, 20-22]. ^The determination of p ^is formally equivalent ^{to that of the} transverse magnetization ⁱⁿ

NMR and leads to the same effects : linewidth, ^free precession, phonon echoes. The analogy ^{with NMR}

is satisfactory ^as long ^{as the} propagation ^effects (such ^{as the} self-induced transparency) ^{are not}

considered [22].

2.3 RELAXATION PROCESSES OF THE TUNNELING STATES.

On the analogy ^with NMR, ^{two charac-}

teristic relaxation times are considered [4] :

* the longitudinal relaxation time Tl which results from the relaxation of the diagonal ^{terms. In other} words, it is the relaxation time of the energy from the TS towards the lattice ;

* the transverse relaxation time T2 which results from the relaxation of the off-diagonal ^{terms and}

which characterizes the interactions of the TS be- tween themselves.

In insulating glasses ^{at low} temperatures

(T « ¹ K), ^{it is} generally ^assumed [3, ^4, 23] ^{that the} longitudinal relaxation time T, arises from a direct

one-phonon process. ^{For a} TS (E, -0, E) ^the

relaxation rate is [23] :

Since we are dealing ^with glasses, ^{there is a distri-}

bution of relaxation times due to the distribution of Eo for ^a given ^E. Nevertheless, the shortest relaxation time exhibits a E- 2 dependence ^{and a} T-1 depen-

dence when E kT.

The transverse relaxation time T2 ^{of a homo-}

geneous system ^{was first} defined in NMR theory ^as

the intrinsic linewidth of the system and charac- terized the spin-spin interactions. It is better to name

it the phase memory time, ^{in relation to the echo} experiments [5, ^19, 20]. ^{When the} spin-lattice ^time Tl ^is long enough, ^{there can} exist other ways ^to loose the phase ^{of a} precessing ^TS by disturbing its energy E with local strain fields for instance. These interac- tions between TS are characterized by ^this phase

memory time [19].

In glasses, ^{it is} generally believed that interactions between like-spins (i. e. , ^{TS of} roughly ^{the same}

energy E) ^are uneffective in consideration of their relative number. The spectral ^diffusion theory ⁱⁿ glasses ^{assumes that the} thermally ^{active TS} (those

with an energy E , 2 kT) ^act as a source of strain fields [19]. ^The inhomogeneous ^elastic dipole-dipole

interaction induces a shift of the energy E of the resonant TS and so a loss of their phase. ^It predicts

in the short time limit of our interest a non exponen- tial decay ^{of the} phonon ^{echo and a} T- 2 dependence

of T2 [19].

2.4 MAGNETIC RARE-EARTH IONS IN GLASSES.

We have studied the effects of magnetic ^rare-earth

ions on the properties of the TS in glasses. ^The

rare-earth ions (like Ho3 + ) ^{in amorphous materials}

are well described in the framework of the random

anisotropy ^model [24]. ^{In this model, each rare-}

earth ion is subjected ^{to a local} anisotropy ^{field of}

random orientation. The anisotropy energy is gener-

ally ^much larger than the other energies (exchange, dipolar) [25]. Considering ^{the electric} quadrupole plus magnetic dipole ion, it has been showed that,

for large ^{J as for} Ho3 + (J = 8 ), ^{there is an} energy

minimum for one orientation of the quadrupole ^and

that the ground ^{state is} practically ^{a doublet} [26].

The effect of an external magnetic ^{field at low} temperature ^{on such} systems ^{can be} considered by neglecting ^the dipolar ^and exchange energies [25].

For each magnetic ^ion, in the classical limit, ^the

energy has ^to be minimized with respect ^{to the} orientation of the magnetic ^field relatively ^{to the} anisotropy axis. As the ratio of the magnetic energy to the anisotropy energy grows up, all the spins ^tend

to be oriented along ^the magnetic ^field (the ^umbrella

closes up). This model is valid only ^{for low} magnetic

ion contents because it does not consider the interac- tions between the magnetic ^ions.

In addition to the static effects, magnetic ^exci-

tations coupled with the elastic fields exist in such materials. They give ^{rise to} low-energy ^activation

processes that we have indeed observed in our

samples [9,10]. However, these processes ^are quenc- hed in the temperature range of the present study.

Several authors [25-27] have also predicted ^the

existence of magnetic low-energy excitations in these materials. These magnetic excitations could also interact with the acoustic wave or with the TS.

Finally, if the interaction of a spin ^{wave with the TS}

in amorphous ferromagnet ^{can be efficient} [28], ^we

may believe that in ^our compounds, ^at least for the lowest concentrations, ^the spin ^{waves are not} pre- sent.

3. Experimental set up.

3.1 CRYOGENICS. - The low temperatures ^are achieved in a He3-He4 dilution refrigerator ^which

works down to 10 mK. The magnetic ^{field is obtained}

from a superconducting magnet immerged ^{in the} liquid ^{He4 bath and can reach 60 kOe. The} homogeneity ^{of the field} is better than 10- 5 in the whole sample. The thermometer is a 47 fl Matsushita carbon resistance used because of its low mag- netoresistance at low temperatures ^and high ^field [29]. Above 50 mK and without magnetic ^{field, a}

Germanium resistor is also used.

3.2 ELECTRONICS.

The acoustic waves are gener- ated by ^X-cut quartz transducers of 100 MHz funda- mental frequency. ^The experimental ^{device is} phase

sensitive and allows measurements on both the acoustic attenuation and the relative variation of the sound velocity (with ^an accuracy better than 10- 6 for

a reasonnably good signal ^{to noise} ratio). ^{The rise}

and fall times of the generated pulses ^{are not limited} by the electric signals (tR

tF

⁶ ns) but rather by

the mechanical damping ^{of the} quartz transducers

(tR

tF

²⁰ ns). ^{We are able to} generate any

sequence of various pulses (frequency, amplitude,

duration) ^{with or without} phase ^{coherence between}

them in the frequency range 10 MHz-500 MHz. The definition of the terms used below are explicited ⁱⁿ figure ^{1. The} detection device includes a digital

transient analyser which allows recording of very fast events.

Fig. ^1.

Notations for the parameters of the acoustic

pulses ^{and the} phonon ^echoes. E12 ^{is the} spontaneous phonon ^{echo and} E,23 is the stimulated phonon ^echo.

The non-linear attenuation is measured by varying

the input ^acoustic power with an attenuator and

measuring the deviation from the linearity ^{on the} travelling ^acoustic pulses. Hence, ^a good linearity ^of

the whole chain is required ^{in this} type of expe- riment. It is achieved correctly since the deviation from the linearity ^{never exceeds 0.08 dB on the}

whole range 0-60 dB. It must be pointed ^{out that a}

calibration is always possible ^{at 4} K where all the non-linear effects have disappeared.

3.3 THE SAMPLES.

The samples ^{studied are}

aluminosilicate glasses doped ^with Ho3 + magnetic

ions. The four atomic concentrations investigated

are 10.1 %, 6.7 %, 3.4 % and 1.5 % . It is easy ^to obtain good glasses ^{for a holmium content of 10 %} [30]. ^{However, if one tries to} decrease this _one, it is

no longer possible ^{to obtain a} glass ^{from the} liquid phase. ^{In order to overcome this} difficulty, ^{we have} kept ^constant the rare-earth ion content (equal ^to

10 %) ^and partly replaced Ho3 + ^ions by ^non mag- netic La3 + ions. The chemical compositions ^{of the} samples ^are respectively :

1.5 % at. Ho :

3.4 % at. Ho :

6.7 % at. Ho :

10.1 % at. Ho :

The 10.1 % Ho sample ^{exhibits a} spin glass

behaviour with a freezing temperature ^{around 0.5 K} [30].

4. Experimental ^results.

We have performed attenuation and velocity

measurements as a function of various parameters

(acoustical power, temperature, magnetic field, ^hol- mium content). ^{We also} report phonon echo exper- iments. Some preliminary results about these ones

have been already published and will be extended here [16]. Our results are concerned with the static

properties ^{of the TS} (coupling constants) ^{as well as}

the dynamic ^ones (relaxation times).

4.1 COUPLING PARAMETERS.

4.1.1 Sound velocity measurements.

From the measurement of the sound velocity ^{as a} function of temperature, ^the parameter PB2 can be deduced [4].

Due to the resonant interaction of the TS with the acoustic wave the sound velocity ^{c varies} according

to [31] : ^;

where To ^{is an} arbitrary ^reference temperature.

Since the TS involved in this mechanism are not those in resonance with the acoustic pulse ^{in the} high temperature range h03A9 -, 2 kT, they ^{are not con-}

cerned with the problem of the coherent or incohe- rent regime. ^{It is} only sufficient that the pulse length

Tii is greater ^{than the} period ^{of the wave. We have}

used acoustic frequencies such that the regime

1íl2 2 kT is valid down to 10 mK. The results for four holmium contents in zero magnetic ^{field are}

shown in figure ^{2a. For} comparison, there is also the sound velocity variation in a a-Si02 sample ^{that we}

have measured. At the highest concentrations

(10.1 % ^and 6.7%) ^the slope ^is slightly frequency dependent ^{and is no} longer ^{constant above 150 mK} [9, 10]. ^{It will be} proved ^{below, with} studying ^the

non-linear effects, ^{that the} determination of PB2

from the slope ^at these concentrations is valid. Our results are reported ⁱⁿ table I. There is a strong variation of PB2 with the concentration of the Holmium ions. This coupling ^{factor is reduced} by ^a

factor of 3 from the 1.5 % to the 10.1 % Ho content.

This variation of PB2 with magnetism ^has already

been pointed ^out [11, ^9, 12]. ^{It seems} here that the effects arise at a rather low concentration (c :- ⁵ % ),

are not linear with the Ho content and tend to saturate at both ends 1.5 % and 10.1 %. It must be noticed that, ^{in a cubic} lattice, magnetic ^nearest neighbours appear from ^a concentration of 12 %.

With the magnetic ^{field, no sensible variation of}

PB2 is detected at least in the low concentration

Fig. ^2.

^Relative velocity variation of longitudinal

acoustic waves as a function of temperature in different holmium aluminosilicate glasses ^{and in a silica} glass, ⁱⁿ

zero magnetic ^field (a) ^{and in a} magnetic field of 60 kOe

(b) ; (+) ^{Ho 1.5 % at. at 50} MHz ; (x) ^{Ho 3.34 % at. at}

50 MHz ; (V) ^{Ho 6.7 % at. at 10} MHz ; (0) ^{Ho 10.1 % at.}

at 10 MHz ; (*) Si02 ^{at 50 MHz.}

regime, ^as is shown in figure ^{2b. For the 1.5 % Ho} compound, the relative variation of FB 2 is less than 5 % and for the 10.1 % Ho compound it is less than

10 %. Hence, ^{it seems that} the presence of ^a spin glass phase in the 10.1 % compound ^{does not affect} drastically ^the PB2 value.

4.1.2 Self induced transparency.

There is another way ^to find out the value of PB2. It takes into account the amplitude ^{of the} non-linear attenuation due to the resonant interaction of the TS with the acoustic wave. It is well known that in the equilib-

rium regime where the pulse duration Tii > Tl, T2’, ^{the resonant} interaction is non-linear because of the incoherent saturation of the TS [3, 4]. ^This

attenuation is given by :

where J is the acoustical _{power and} Jj ^{is a critical}

power which is proportional ^to 1IM2 T, T2 [4].

These results have already ^{been used} particularly ⁱⁿ amorphous metals where the approximations ^of equilibrium ^{are valid} [8]. ^{However this} regime

cannot be reached in our glasses, ^{at least at} very low temperature where the non-linear effects are suffi-

ciently important (see below the values of T, ^and T2’). ^{On the} contrary, ^{we can reach at T 100 mK} the coherent regime ^where Tii Tl, T2. ^This regime

has also been well studied [20, 18]. ^{It leads to an}

attenuation which is also non-linear due to the self induced transparency (SIT) well known in optical

resonance [20, 22]. ^The amplitude of the non-linear attenuation in this coherent process is still given by

an equation ^{similar to} equation (8), except ^{for the} power dependence which is somewhat sharper [20, 21]. ^Then, the critical power from which the ^non- linear attenuation arises is calculated with the rele- vant parameter 6 (the pulse area) ^because 03B8cc =

Table I.

Coupling parameters ⁱⁿ aluminosilicate glasses ^with different ^{holmium contents and in} fused ^silica.

7T’/2 [20, 22]. ^{For a square} pulse ^{of duration}

Tii, the critical _power lee is ^{found to be} proportional

to 1/MZ 1 [20]. The difference between the two

saturation processes ^{is that} Jcc ^{is not} temperature dependent ^whereas Ja depends ^{on it} through Tl ^and T2. ^The experimental ^results clearly give

evidence of the coherent processes ^as does the observation of phonon echoes in the same exper- imental conditions.

Figure ³ displays ^the variations of the acoustic attenuation as a function of the input power in the 1.5 % Ho sample, ^{at 450} MHz, ^{at 11.5} mK, ^{with and}

without magnetic ^field respectively. ^{The two results}

are quite ^{similar. The} attenuation is indeed non-

linear and decreases as the acoustical input power increases between - 50 dB and - 20 dB. The linear

regime ^{is reached at} the lowest acoustical _powers.

Hence, the values of PB2 can be deduced. They ^are reported ^{in table I. As} compared with those obtained

Fig. ^3.

Attenuation variation as a function of the acoustic power in the 1.5 % Ho sample ^at 450 MHz, 11.5 mK, ^{for a} pulse duration ’Tn

60 ns and for two

magnetic ^fields. (*) ^{is for 0 k0e ;} (+) is for 60 kOe.

from the variations of the sound velocity ^{as a}

function of temperature, ^the agreement is very good

and confirms our interpretation. ^{It must be} pointed

out that the curves in figure ^{3 exhibit a non-linear}

variation between - 50 dB and - 20 dB which is less

sharp than in the optical ^resonance experiments [20], surely ^{due to} the distribution of the elastic dipoles ^M

and so of the critical _powers lee ^{inside the} packet ^of

resonant TS of energy E

hn.

In addition to the determination of the coupling

factor PB2, ^{it would be} possible, ⁱⁿ principle, ^to

determine B (and then P) ^{from the SIT. This could}

be done from the determination of jcc (MTii

Jl:c/h = ^{w /2 [20]). The first} difficulty arises from the distribution of the dipoles ^{M so that the exact}

value of lee ^{is not} well defined. The second and more

important ^{reason is the} impossibility ^of evaluating experimentally ^{the absolute} acoustic power Jcc ^{with a}

reasonable accuracy. Indeed, ^the electro-mechanical

efficiency ^{of the} piezoelectric transducers and the intrinsic attenuation of the sample make any ^com-

parison ^of Jcc ^{between two} experiments very doubt- ful. Nevertheless, ^{it is} possible ^to determine the variations of the deformation potential ^{M without} changing any of the experimental parameters ^de- tailed above. This is the case when we set up the

magnetic ^{field at the same} frequency ^and tempera-

ture for the same sample. Then the variations of

Jcc ^are those of M. Figure ³ displays ^{in the same} experimental conditions the non-linear attenuation with a magnetic ^{field of 60 kOe and without} magnetic

field. It shows that Jcc ^decreases slightly with mag- netic field (about 2 dB). This would mean that M would increase with magnetic ^field by ^{a factor} 10°.1 = 1.25. If this variation of M was due to a

corresponding ^{variation of B} (according ^to Eq. (3)),

there would be a variation of the slope of the sound

velocity by ^{a factor} (1.25 )2 ^{= 1.6.} However, ^{there is}

no variation of this slope ^with magnetic ^field (para- graph 4.1.1). ^Another possibility ^{would be a} change

of the dipole distribution (through 80 ⁱⁿ Eq. (3)), ^but

this is very speculative. Presently, ^{we have no} explanation ^for this effect which is small.

In the other samples ^with larger ^{Ho contents the}

results are similar but less clear because of the smaller values of PB2. _So, we can conclude that there is no appreciable ^{variation of B and P with} magnetic ^field.

4.1.3 Pulse propagation delay ^time.

^{The SIT} theory ^{and the} corresponding experiments ^have

indeed shown that there is a propagation delay ^{of a} light pulse, ^{due to the coherent} processes [20]. ^The

crucial problem ^{to define the sound} velocity ^{in the}

critical regime ^is beyond the scope of the present paper. The final ^result is that the maximum _propa-

gation delay ^{time of a} pulse ^is given by [20] :

where L is the length ^of the acoustic path. Ta is obtained for 0

2 03C0. At low and high powers the

delay ^{time vanishes.}

Experimentally, ^we have observed such a propa-

gation delay ^{with an} acoustical pulse ^{in our} samples.

The results are reported ⁱⁿ figure ^{4. It can be seen}

that _{7-d --} 65 ns. This value is to be compared ^with

that expected ^using equation (9). ^{From the known} parameter PB2, ^we expect ^{for a} pulse length

Tii

60 ns a value of _{7-d --} 150 ns which is about three times larger ^{than the} experimental delay. ^We

can explain ^this discrepancy putting forward the

broad distribution of the elastic dipoles ^{inside the}

Fig. ^4.

Third acoustic echo in the 1.5 % sample ^at 450 MHz, 11.5 mK and 60 kOe for different acoustic powers. The time T

0 is arbitrary, but is the same for the three curves. From left to right : - 10 dB, - ^{30 dB and}

-

35 dB.

packet ^{of resonant TS. This one acts as a} broadening

of the non-linear attenuation in the SIT due to the

dependence of lee ^{with M} (see paragraph 4.1.2). ^The

summation over the dipoles (over Eo/E) ^{has to be}

done. The maximum delay ^{time is not} easy ^to calculate but goes through the maximum Td for ^a definite value od

ir/2 [20] ; the summation over

the dipoles ^{would be} only achievable if the variation of Td with 0 ^was known. Nevertheless, ^it surely ^acts

as a decrease of Td and ^a spreading ^{with the}

acoustical power because ^{all the} dipoles ^{do not reach}

the maximum delay time for the same power.

4.2 LINEWIDTH MEASUREMENTS. - Here, ^{we are} only dealing ^{with the} measurements of the linewidth

T2’- 1 which does not involve propagation, ^{so that a} complete analogy ^{with NMR is available.}

When a strong ^acoustic pulse ^{is sent into the} sample, ^{it saturates} only ^a small fraction of the total linewidth [4, 19]. ^The ability ^{of such a} burned hole to

persist for times as long ^as T, ^{comes from the} independence ^{of the} spins between each other.

There are four sources for line broadening : ^the

relaxation rates TIl ^and T2 1, ^the pulse ^{width and}

the acoustic power. Since T, >> T2, ^{the line broaden-} ing ^{due to} Ti 1 ^is always neglected. ^Two basic ways to determine the linewidth will be described : the free precession and the direct measurement of the acoustical attenuation and the sound velocity ^{as a}

function of frequency. ^{In these two kinds of} exper- iments a direct interpretation ^{and a} good ^measure-

ment of T2 ^{can be} given.

4.2.1 Free precession signal.

Typical free preces- sion signals ^are reported ⁱⁿ figure 5 for three differ-

ent acoustic powers in the 1.5 % Ho glass, ^{at 12} mK, 450 MHz and 60 kOe. The input pulse length

T 11 is 1 03BCs. We have already pointed ^{out that the}

time constant of our experimental ^set up is 20 ^ns

(paragraph 3.2). ^Hence, it appears clearly ⁱⁿ figure ⁵

that an extra exponential decay subsequently ^{to the}

end of the pulse ^is generated ^{in the} sample. ^{A similar} decay ^has already ^{been observed in} glasses [32]. ^{It is} analogous ^{to the} decay ^{observed in} non-linear optics [33]. The caracteristic times of these curves are :

TZ ^{= 155 ns for the low} input power (- 55 dB), T2

^{50 ns for the middle} input power ( - ⁴⁰ dB)

and T2

^{20 ns for the} high input power ( ^{- 10} dB).

The last characteristic time is the damping ^{time of} the apparatus (paragraph 3.2). Performing ^measure-

ments as a function of the input, power ^J and the

Fig. ^5.

^Acoustic free-precession signal in the 1.5 % Ho

sample ^{at 450 MHz,} 12 mK and 60 kOe for a pulse

duration of 1 03BCs and for three acoustic powers : - 10 dB,

-

40 dB and - 50 dB (from ^{the fast} decay ^{to the slow} one).

pulse duration Tll, ^we have found that when

T2’ ^{T11, the} characteristic time follows the law :

At the highest powers ^{we are} limited by ^the damping ^{time of the} apparatus ^{and at the lowest}

powers by ^the pulse length ^{because we must avoid}

the overlap ^{of the free} precession signal ^{with the} following acoustic echoes. Hence, ^{the extend of our}

measurements is : 40 ns T2 ^{480 ns.}

The free precession signal ^{and the} J- 1/2 depen-

dence of T2 ^{have been} already observed in glasses

but not so clearly because the time constant of the apparatus ^{was not} much shorter than the decay ^time [32]. It has been interpreted ^{in terms of the} optical

resonance theory [33]. However, ^{the use of the}

equilibrium ^Bloch equations ^assumes T11 > Til

T2 and this is not valid here. Although ^this signal ^has

the same origin ^{as for NMR in} homogeneous field,

the results are here different and more similar to those for NMR in very inhomogeneous ^field [34].

The free precession signal gives ^no longer ^{access to}

the dephasing ^time T2. ^{It is} determined by the elastic

polarization (Eq. (5)). ^{The time} dependence ^{of e is}

that of p ^and since p is written as a transverse

magnetization M., , iy, ^{we need only to} know it. This has been achieved for an arbitrary pulse length [34]

and reflects our T11 and J dependences. Then, ^the decay ^{is due to the} dephasing ^{in the} precession ^of

each individual spin ^{of the} spectrum rather than to the interaction dephasing ^{in the usual} NMR. It is

generally faster than 2 T2 ^{which would} arise from a

very small pulse ^of very long ^duration.

It is important ^to regard ^{the variation} of the free

precession ^{with the} magnetic field. No change ^of T2 ^{with the} magnetic field is detected and this is understandable with the preceding results of para-

graph ^4.1.2 concerning the relative insensitivity ^of

the parameter B ^with regard ^{to the} magnetic ^field.

Nevertheless, ^the amplitude of the free precession

increases with the magnetic field like the phonon

echo amplitude (see ^below paragraph 4.3).

4.2.2 Hole burning experiments.

^{There is another}

way ^to obtain the linewidth : it consists in measuring

the width of the hole burned by ^a pulse Pl(f2l, Jl, Til) ^around fit ^{in the} distribution of the TS. This is done with a second weak pulse P2(n2, J2, T22) ^which reads this hole after a time T12 Tl.

Here, reading ^means measuring ^the variations of the sound velocity and of the resonant attenuation caused by the first pulse. ^{In order to} avoid any

perturbation ^{of the hole} by the second pulse, ^{it is}

necessary ^to satisfy r22 > T2 and J2 Jl. ^{So we have}

chosen T12

3 _Rs.

The absorption ^and dispersion ^of P2 ^versus f22 ^are plotted ⁱⁿ figure ^{6 for the 1.5 % Ho} glass ^at

11.5 mK, ^{60 k0e} and around f2l = ^{450 MHz. The}

line is approximately Lorentzian and the width deduced from the velocity ^is roughly equal ^{to the}

one from the attenuation. The same law as function of the acoustic _power (Eq. (10)) has been verified for : 20 ns T2 ^{200 ns.}

We must notice that the experimental points overlapped by ^{the free} precession ^{method and the}

hole burning ^{one are in} perfect agreement.

Moreover, contrary ^{to the free} precession ^{but ac-} cordingly ^to paragraph 4.1.1, ^the amplitude ^{of the} signal ^is independent ^{on the} magnetic ^field.

4.3 PHONON ECHO EXPERIMENTS. - We have ob- served spontaneous ^and stimulated phonon ^echoes

in the four samples, ^{in the} temperature range 10- 50 mK, ⁱⁿ magnetic fields up ^to 60 kOe and for acoustic frequencies around 500 MHz [16].

Fig. ^6.

Variations of the attenuation and the velocity ^of

a pulse P2 ^{as a} function of the frequency ^{after a} pulse P, has saturated the sample. ^The measurements are

performed in the 1.5 % Ho sample, ^at 11.5 mK and 60 kOe. The frequency ^{of the} saturating pulse is 450 MHz.

The duration of P1 ^and P2 ^{is 400 ns.} The relative powers of

P1 and PZ ^{are -} 40 dB and - 55 dB respectively. (+) ^{is for}

the attenuation ; (*) is for the velocity.

4.3.1 Echo amplitude.

^We retain here the no-

tations of Black and Halperin [19] ^and Golding ^and

Graebner [5, 6]. ^{In a} spontaneous phonon ^echo experiment, ^two successive acoustic pulses ^{of suit-}

able intensity ^{and duration at} times 0 and T12 ^create in the sample ^a spontaneous ^{echo at the time} 2 T 12. Here, ^the key parameter is also 0 (the pulse area) ^as in all the coherent processes. It ^must be noticed that generally 0 ^varies during ^the pulse propagation through ^the deformation and the attenu- ation of the acoustic pulse.

If the areas of the first two acoustic pulses ^are 03B81 1 and 82, by analogy ^{with the} optical ^{case, the}

initial echo amplitude ^{can be written} [19, 20] :

where N is the number of radiating TS, Eo ^{is the}

radiation strength ^{of an effective} TS, F (k ) ^{is maxi-}

mum in the forward direction. El2 is maximum for the values 01 = ir/2 and 03B82

^7r. All the results of this paragraph ^are roughly ^{identical for the four} samples, ^{but here we} restrict the analysis ^{to the} sample with 1.5 % holmium content because of its greatest ^{value of} P B 2.

In the present experiments, ^{the form of the} phonon ^{echo is not} determined by T2 ^{as in} optical experiments [20] but rather by ^the pulse ^{duration as}

for NMR in very inhomogeneous ^fields [34]. ^Since

the value of ovaries during ^the propagation, ^{we have} adjusted ^{the relative} amplitude of the first two

pulses ^{in order to} obtain the greatest ^echo amplitude.

This adjustment ^was performed ^{at each} working

temperature. We have observed that the dephasing

between the acoustic echoes and the phonon ^echo E12 ^was independent of T12. For the maximum

phonon ^echo amplitude, ^this dephasing ^{was 60}

70° in the 1.5 % holmium sample.

Figure 7a shows the initial phonon ^echo amplitude

in the 1.5 % Ho sample ^versus the acoustic power with and without magnetic ^field [16]. ^{The time} separation between the two acoustic pulses ^is

T12

200 ns and the pulse ^duration T11 = T22 = 60 ns. There is an important enhancement of the

phonon ^echo amplitude (about ¹⁰ dB) ^{when the} magnetic ^{field is set} up. Moreover, the location of the maximum which corresponds ^{to a determined}

value of 0; ⁱ (and ^{so of} M) ^{seems to be} displaced

towards the low powers by about 2 dB when the

magnetic ^{field is set} up. Regarding its direction and its amplitude, this shift is perfectly consistent with the shift of the attenuation saturation curve displayed

in figure ^{3 and} explained ^{as a} change ^{of M. Here} again, the effect is small and cannot explain ^the

increase of the phonon ^echo amplitude [16].

The variation of the maximum-echo amplitude ^as

a function of the magnetic field is shown in figure ^7b.

The echo amplitude ^increases nearly linearly ^{from 0}

1

Fig. ^7.

(a) : variation of the spontaneous-echo ampli-

tude as a function of the relative acoustic power in the 1.5 % Ho sample, ^at 11.5 mK and 450 MHz without

magnetic ^field (+) ^{and with a} magnetic field of 60 kOe (*).

The duration of the two pulses ^{is 60 ns} and their time

separation ^{is 200 ns ;} (b) : variation of the maximum-echo

amplitude ^{as a} function of the magnetic field. The lines are

guides for eyes.

to 20 kOe and then tends to saturate between 30 and 60 kOe [16]. This behaviour is quite similar in the four samples whatever the holmium content and is not due to any variation of the acoustic attenuation with the magnetic field which is constant between 0 and 60 kOe. The saturation above 30 kOe is also present in the variation of the free precession signal

and will be also present ^{in the} variation of the relaxation time Tl.

4.3.2 The dephasing ^time T2.

^The phase memory time T2 has been measured from the decay ^{of the} amplitude ^{of the} spontaneous phonon ^{echo as a}

function of the time separation T12 between the ^two

pulses. ^{Our results are shown in} figure 8 for the 1.5 % holmium glass, in different magnetic ^{fields at}

11.5 mK and 450 MHz. For each magnetic ^field

value there are two different exponential decays

with a cross-over time which varies continuously

with the magnetic ^field. As for the phonon ^echo amplitude, ^this time variation saturates at high ^fields (H > 30 k0e). ^The characteristic times of the two different decays ^{do not} change ^with magnetic field, except ^{at 0} kOe where it seems that the initial decay

is two times faster than the others. However, ^the

time range of this decay ^{is too small to be} significant.

This study has been made at sufficiently low power to avoid power dependence ^of T2 [6]. Considering

the 1 /e fall time of the initial phonon ^echo ampli- tude, ^{we can} conclude that there is no significant

variation of T2 ^{with the} magnetic field between 0 and 60 kOe (Tab. II). In any ^case and as in fused silica

[6], ^the decay ^never follows the T 2 ^law predicted by

the spectral diffusion in the short time limit [19].

Fig. ^8.

Variation of the spontaneous-echo amplitude ^as

a function of twice the time separation between the two

pulses in the 1.5 % Ho sample, ^{at 11.5 mK, 450} MHz, ^for different magnetic ^{fields :} (*) ^{is for 0} kOe ; (.) ^{is for}

2 kOe ; (0) is for 5 kOe, (x) ^{is for 10} kOe ; (V) ^{is for}

20 k0e ; (+) is for 60 kOe.

We report ⁱⁿ figure ^{9 the} decay ^{of the} amplitude

of the spontaneous phonon ^{echo as a} function of twice the time separation ^{712 in} samples with differ- ent holmium contents at 450 MHz, 11.5 mK and 60 kOe. It can also be _seen, for comparison, ^the

measurement we have performed ^{in fused silica}

under the same experimental conditions. Even

though ^the decays ^{are not} perfectly exponential, ^we

can deduce a T2 value from each curve. They ^are

listed in table II. It appears that the T2 ^{values in our}

holmium doped glasses ^are much smaller (= ³ )JLs)

than in a-Si02 (20 )JLs) ^{and are} roughly independent

Table II.

Relaxation times in aluminosilicate glasses ^with different ^{holmium contents and in} fused ^silica.

The temperature and the acoustic frequency ^are respectively ^{11.5 mK and 450} MHz, except where otherwise stated. Tl, ^{miD is obtained from} the initial slope of the saturation recovery and T, ^{is obtained} from ^the fall ^to lie of ^{the initial} saturation.

Fig. ^9.

Variation of the spontaneous-echo amplitude ^as

a function of twice the time separation between the two

pulses ^{at 11.5 mK, 450 MHz,} 60 kOe and in samples ^with

different holmium contents : (V) is for 1.5 % at. ; (+) ^is

for 3.34 % at. ; (x) is for 6.7 % at. ; (0) is for 10.1 % at.

The experimental points marked with (*) ^{are for a} Si02 glass ^{at the same} temperature ^and frequency.

of the holmium content between 1.5 % and 10.1 %.

Here again, ^{we must} recall that in the same time the value of PB2 is divided by ^a factor of 3. This fact is not very surprising ^{if one} considers the relative number of resonant TS involved in the phonon ^echo

sequences in comparison ^{with the Ho3 + ion content :}

So, ^{there are} for all the doped samples many

Ho3 + ions between two active TS. If one remembers that T2 characterizes the interaction between the TS,

it is not surprising that it does not change ^{with the}

Ho content. We have also measured the temperature dependence ^of T2 ^{in the} sample containing ^{1.5 %} holmium, with and without magnetic ^{field. As} displayed ⁱⁿ figure ^10, T2 ^follows roughly ^a T-1 _law,

at least for the highest temperatures. ^{This is not in} agreement ^{with the} T- 2 law predicted by ^the spectral

diffusion [19] ^and reported for fused silica at higher

temperature [6]. Nevertheless, ^{such a} law has also

been observed for dielectric echoes in a Si02 glass

between 4 mK and 22 mK [35]. ^{An attempt to} predict ^{such a} law has been carried out using ^the spectral ^diffusion theory [36]. However, it has been made for intermediate times and it is not suitable to the present ^case because in our experiments, ^the

value of _{T 12} is always much shorter than the shortest

T1 ^{of the} thermally ^{active TS.}

4.3.3 The TS lattice relaxation time Tí.

^{In order to}

measure the TS lattice relaxation time Tl, ^we firstly performed ^a stimulated phonon ^echo experiment [6, 19]. ^{As for the} spontaneous phonon echoes, ^an

increase of the amplitude ^{of the} stimulated phonon

echo E123 ^{as a} function of the magnetic ^{field has been}

Fig. ^10.

Variation of the transverse relaxation time

T2 ^{as a} function of temperature ^at 450 MHz in the 1.5 % Ho sample (T2 is obtained from the 1/e fall time of the initial echo amplitude). (*) is for 60 kOe and (0) ^{is for}

0 kOe. The straight ^lines correspond ^{to T-1} variations.

detected [16]. Figure ^{11 shows the echo} amplitude ^as

a function of T13 ^at 11.5 mK and 450 MHz for 0 and 60 kOe. The decay ^is non-exponential. ^{This be-}

haviour is also observed in fused silica [6] ^and predicted by ^the theory [19]. ^{If we measure} Tl ^{as the}

time at which the echo falls to 1/e, we can see that

Tl is shorter at 0 kOe (about ¹⁵ 03BCs) ^{than at 60 kOe} (about ⁴⁰ f.Ls). ^{However, it must be} pointed ^{out that}

these values depend crucially ^{on the initial value.}

Here again, they ^are by ^{one order of} magnitude

smaller than those reported ^{in fused silica} [6].

4.4 SATURATION RECOVERY EXPERIMENTS.

Since it is difficult to obtain T, from the stimulated

phonon echoes, ^{we have} performed saturation recov-

ery experiments [16]. They consist in saturating ^the

TS system ^{with a first} pulse ^and afterwards measur-

ing the variation of the attenuation of ^a probing pulse ^{as a function of the} delay time rl2 between the two pulses [4, ^19, 18]. ^If A2, ^and A2, eq ^{are the}

amplitudes of the second pulse P2 ^{with and without}

the first saturating pulse respectively, ^{we can write :}

where f - 1 is given by equation (8) using ^J J, ^and T1 is the relaxation time (supposed single-valued) ^of

the excited TS. Hence, T1 ^{can be} deduced from the measurement of A2, s/ A2, eq as a function _{of T12.} The

advantages ^of measuring T1 in this way ^are manifold :

* firstly, ^the signal-to-noise ^{ratio is better because}

we detect here acoustic echoes which are usually larger ^{than the} phonon echoes ;

* secondly, ^since FB 2 is the same with and without magnetic ^field (see paragraph 4.1.2), ^the signal ^{has the same initial} amplitude ^{in the two}

cases, ^{as can} be seen in figure 12. This makes the

comparison ^{easier ;}

* thirdly, since the measured characteristic time is

only slightly dependent ^{on the chosen} amplitudes ^of

the saturating ^{and the} probing pulses, ^no exceptional

care has to be taken with them ;

* lastly, ^{we measure} indeed the TS-lattice relax- ation time in contrast with the stimulated echo

experiments ^{where some} additional terms due for instance to spectral diffusion processes have ^to be included [19].

Fig. ^11.

Variation of the stimulated-echo amplitude ^{as a}

function of the delay between the first and the third pulse

in the 1.5 % Ho sample, ^{at 11.5 mK, 450} MHz, ^without magnetic ^field (*) ^{and with a} magnetic field of 60 kOe (+).

The time duration of the pulses ^{is 60 ns and the} delay

between the first pulse and the second one is 300 ns.

We have measured the variation of T, ^with temperature ^and magnetic ^{field for} samples ^with

different holmium contents. The decay ^{of the re-}

sonant attenuation is generally non-exponential,

what is characteristic of ^a distribution of relaxation