High frequency ultrasonic relaxations in smoky quartz

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High frequency ultrasonic relaxations in smoky quartz

M. Saint-Paul, R. Nava

To cite this version:

M. Saint-Paul, R. Nava. High frequency ultrasonic relaxations in smoky quartz. Journal de Physique,

1978, 39 (7), pp.786-792. �10.1051/jphys:01978003907078600�. �jpa-00208814�

HIGH FREQUENCY ULTRASONIC RELAXATIONS IN SMOKY QUARTZ

M. SAINT-PAUL

(*)

^{and R. NAVA}

Instituto Venezolano de

Investigaciones Científicas, Physics

^Center,

Aptdo, 1827,

Caracas, ^Venezuela

(Reçu

^{le 6 mars 1978,}

accepté

le 5 avril

1978)

Résumé. ²⁰¹⁴ Des mesures de l’atténuation d’ondes

longitudinales

ultrasonores

pulsées

de très haute

fréquence ^ont été effectuées dans des cristaux de quartz naturels brésiliens de coupe X irradiés par rayonnement gamma. Des

pics

de relaxation de type

Debye

^{ont été mis en} évidence à basse tem-

pérature

(Tm

^{= 18} K) ^{et à haute} temperature

(Tm

^{= 155 et 255} K). ^Ces pics peuvent etre attribués

aux transitions effectuées par ^{un trou} (défaut d’électron)

piégé

^{sur un} oxygène

proche

voisin d’une

impureté substitutionnelle d’aluminium. Les valeurs des énergies d’activation observées sont en

accord avec celles obtenues antérieurement par des ^mesures de perte

diélectrique

^et acoustique.

Par contre les temps de relaxation observés sont d’un ordre de grandeur plus faible à haute tem-

pérature ^{et de} cinq ^{ordres de} grandeur plus

faibles

à

basse

temperature. ^Pour des doses d’irradiation

supérieures ^{à un} Mrad, l’augmentation ^{de dose} provoque la

disparition

progressive ^du

pic

^{de basse} température. ^Cette

disparition

peut être attribuée à un blocage des transitions tunnels des niveaux intermédiaires de faible

énergie,

^{dû aux} modifications de l’asymétrie ^{du double}

puits

^de

potentiel

provoquées par des interactions

électrostatiques

^avec des défauts ionisés. L’absence de couplage ^entre

l’onde sonore et les niveaux dedoubles de l’état fondamental du trou peut être attribuée à l’anisotropie

de la constante du

potentiel

^de deformation.

Abstract. ²⁰¹⁴ Pulse-echo ultrahigh

frequency

ultrasonic measurements in gamma irradiated natural

brazilian X-cut quartz crystals reveal the existence of

Debye

relaxation peaks ^{at low} (18 K) ^and

high

(155 ^{and 255} K) temperatures ^{which can be} attributed to transitions of the hole

trapped

^{at the} oxygen ion nearest to the substitutional aluminum impurities. The observed activation energies ^{are in} agreement ^with

previous

anelastic and dielectric loss measurements but the relaxation times are found to be respectively ^{one to} two, and five orders of magnitude smaller for the high ^{and low} temperatures peaks. ^The low-temperature peak ^{shows an} apparent ^radiation annealing for gamma doses in ^excess of one megarad attributed to the closing ^of

tunnelling

transitions of the hole from low-lying ^inter-

mediate energy levels due ^to electrostatic shifts of the asymmetry of the double-well

potential by

ionized defects.

Uncoupling

of the sound waves to the split ground ^{state levels} of the hole may be due to the anisotropy ^{of the} deformation

potential

^constant.

Classification Physics ^Abstracts 62.80 - 61.80E

1. Introduction. ^- Considerable _progress has been made in

understanding

^{the structure} of the colour centre that is formed when quartz

crystals containing

substitutional aluminum

impurities

^are

exposed

^to

ionizing

^radiation

[1].

^{The centre} is created when one

of the _oxygen ions of the

A’04

distorted tetrahedron loses an electron under the action of the radiation.

Recent

experimental

^data

[2, 3] support

the view that at low temperatures

(

¹⁰⁰

K)

the hole thus created

moves in a double-well

asymmetrical potential

ⁱⁿ

which

the

particle

may

jump

^{over or tunnel}

through

the

potential

^barrier

separating

^the

inequivalent

(*) Présent ^{address : Centre de} Recherches sur les très basses températures, ^{38042 Grenoble} Cedex, France.

minima at the two oxygen ions nearest to the alu- minum

impurity.

^At

higher

temperatures,

jumps

^over

to the _oxygen atoms

slightly

^further away become

possible through

thermal activation of the hole.

As aluminum ions would be

charge

deficient when substituted for silicon in the host lattice it is

thought

that

charge compensation,

in the unirradiated state, is achieved

by

the association of monovalent

impurity

ions with the aluminum centre. After irradiation the

charge compensation

^{role is taken over}

by

^{the hole ;}

the interstitial ions

diffusing

away because of their considerable

mobility

in the open quartz ^structure.

The presence of these

charge

compensators ^have been detected

by

dielectric loss and EPR measure-

ments in irradiated quartz

[4, 5].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003907078600

787

In recent

experiments,

^{de Vos and}

Volger [2]

^studied

dielectric losses in natural irradiated

quartz crystals

in the

frequency

range from 20 Hz to 3 MHz at low temperatures. ^Their

results

are described in terms of three

independent

relaxation mechanisms of the hole in the double-well

asymmetrical potential :

a)

^{Below 15 K}

one-photon

^assisted

tunnelling transitions

in the

ground

state, with ^a characteristic inverse temperature

dependent

relaxation time in the millisecond _{range ;}

b)

Between 15 and 80 K,

thermally

^activated

tunnelling transitions,

^{via a}

low-lying

intermediate

state about 8 meV above the

ground

state, with ^{a mean} relaxation time of 30

microseconds,

^and

c)

^{Above 80} K,

thermally

^activated

(presumably tunnelling)

transitions via a

high-lying

intermediate state

roughly

^{85 meV above the}

ground

^{state and with}

a mean relaxation time of 30

picoseconds.

^{The latter}

activation energy agrees well with the values obtained

by

^EPR

linebroadening [6]

^{and anelastic loss}

[7]

measurements.

The above mentioned authors

adopted

^{a model}

where the hole is localized at

liquid

^helium tempera- tures, ^{but in order to account for the} temperature

dependence

^{of the}

experimental

^static differential dielectric

susceptibility

^{a broad}

(Gaussian)

distribution of the

asymmetries

of the double-well

potentials

ⁱⁿ

the lattice was assumed. The

asymmetries e

^were

attributed to strains or Coulomb fields

originating

from

defects, neighbouring dipoles

^or

impurities.

The width of the distribution function a for each

crystal

^{defines a} localization temperature

TL

⁼

a/k

which is taken as a measure of the overall

imperfection

of the

particular sample.

^{For the}

samples

used in the dielectric relaxation

experiments

^a range of localiza- tion temperatures ^{from 2 to 9 K was} deduced. With respect ^{to the hole}

ground

state, this

implies

^the

existence of two-level systems ^{with a} broad distribu- tion of their overall _energy

splitting b

⁼

(E2

^{+ r}

2)l2,

where r is the tunnel _energy

splitting.

^Such distributed systems would constitute resonant scatterers of ultra- sonic and thermal

phonons

and should affect the sound attenuation and the thermal

properties

^of

irradiated quartz with aluminum centres.

The present

experiments

^were undertaken to

study by high frequency

ultrasonic

techniques

the various mechanisms of the

trapped

^hole

[8].

^Of

particular

interest are low temperature transitions from the

ground

^state which offer the

possibility

^to

study tunnelling

entities in non-cubic matrices. The

long

relaxation times involved make their

study by pulse,

methods _very attractive.

2. Ultrasonic losses

by

transitions and relaxations of a

particle

^{in a} double-well

asymmetrical potential.

^-

The quantum mechanical

theory

^of transitions of a

particle

ⁱⁿ double-well

asymmetrical potentials

ⁱⁿ

solids was first studied in detail

by

^{J. A. Sussmann}

[9- 11].

^{The model has been}

successfully applied

^to para-

LE JOURNAL DE PHYSIQUE. - ^T. 39, ^N° 7, JUILLET 1978

electric centres in alkali halides

[12],

^{colour centres in}

smoky

quartz

[2],

and other similar

tunnelling

^entities

in semiconductors

[13].

^Recent

applications

^{to account}

for the anomalous

properties

^of

glasses

^{at low} tempe-

rature have met with great ^success

[14].

FIG. 1. - One dimensional representation of the hole asymme- trical double-well potential. Ground and intermediate levels and

the relevant parameters ^{are indicated.}

Figure

¹ shows how such a

potential

may look for one-dimensional motion and defines its

important

parameters. ^It is assumed that the

separation

^{of the}

energy levels of the

particle

^{in each well is}

large compared

^{to the}

asymmetry

^{e, due to} random fields and

strains,

^{which causes an almost}

complete

^locali-

zation of the

particle

^{in one of the wells. At the lowest}

temperatures,

tunnelling

transitions between the

ground

^{state in the two wells are}

only possible

^{with the}

assistance of a

phonon (thermal

^or

ultrasonic)

^of

energy hm

= b,

^the

particle

relaxation time

being given by [9] :

where B is the deformation

potential,

^{c is the} average sound

velocity

^{and the other}

symbols

have their usual

meaning

^{or have been}

previously

^{defined. For}

à « 2 kT

equation (1) gives

^the characteristics inverse temperature

dependence

^of LI observed in dielectric relaxation measurements in

smoky

quartz ^{and other}

impure dielectrics,

^{and also in the} saturable ultrasonic attenuation observed in

doped

semiconductors and in

glasses,

^{which is}

given by [15] :

Here M is a mean

coupling

energy and

g(hco)

^{is the}

density

^{of states with an} energy

splitting

^{b = hm.}

Besides this direct or resonant transition, at

higher

temperatures, several relaxation mechanisms of the energy levels in the double-well

potential

^are

possible

after their thermal

equilibrium

^{has been}

perturbed by

an external sound wave. These _processes

produce

^an

ultrasonic attenuation whose

frequency dependence

has the form of a

Debye

relaxation

peak.

^The tempera-

ture

dependence

^and

magnitude

of the relaxation time

being

determined

by

the kind of mechanism that ^re- establishes the

equilibrium

^{of the}

perturbed

^levels.

Jackle

[15, 16]

has studied in detail the case for the

split ground

^{state level where the return to}

equi-

librium is achieved

by one-phonon tunnelling

^transi-

tions of the

particle

^{with a} relaxation time

given by equation (1)

^{and the}

corresponding

ultrasonic loss

expressed

^{as :}

Here n is the number of defects _per unit volume and D is a deformation

potential

^constant. This kind of ultrasonic loss has also been observed in

glasses

^and

doped

semiconductors in the

liquid

^{helium tempera-}

ture range

[13-17].

At still

higher

temperatures, thermal activation of the

particle

^{to sets of states} that may lie above

or below the

potential

barrier becomes

possible.

These states act as intermediates for transitions across

the barrier with a relaxation rate

given by

where the activation _energy

En

^is

given by

^the

height

of the intermediate above the

ground

^{state and the}

pre-exponential

^factor

Wo

^is

given by

^the

tunnelling frequency

^{for that set 2}

nb; 1,

^with

bn

⁼

(e2

⁺

r;)1/2 representing

^{the overall} energy

splitting

^{of the set n.}

The

splitting

^{of sets above the}

potential

barrier is

usually

^{of the same order as the}

spacing

between the sets, ^as the

asymmetry

^{is a}

negligible perturbation.

Thus,

^{for these states}

Wo

is like the classical oscillation

frequency

ⁱⁿ

magnitude - 1013 s - 1.

For intermediate states below the

potential

^barrier

(small

^activation

energies)

the effect of the

potential

asymmetry ^{is to}

effectively

increase the overall

splitting Ôn

^{with the}

result that the

pre-exponential

^{factor may be much}

larger

than for the

previous

^{case. Sussmann}

[11]

^has

studied these

tunnelling-controlled

relaxations via

low-lying

intermediate states to obtain a relaxation time

given by :

1 is here the state localization parameter

given

^as

elF,,.

^{We note} that this type of relaxation is _very sensitive to the

magnitude

of the localization _para- meter.

Thus,

external fields ^or strains _{that may}

change

^the asymmetry ^e of the double-wells can cause

large changes

in the relaxation rates.

Thèse thermally

activated transitions will

produce

^an ultrasonic atte- nuation

given by

^{the well}

known Debye

^relaxation

formula :

and while the relaxation

time in

^is

always

^{found to be}

exponentially dependent

^on

IIT

^the

magnitude

^{of the}

pre-exponential

^factor ro may vary

by

several orders of

magnitude depending

^on whether the

perturbed

system relax towards

thermodynamic equilibrium by tunnelling

transitions

through,

^or

by thermally

^acti-

vated

jumps

^{over the}

potential

^barrier.

Thus

far,

^we have considered that all

relaxing

par- ticles or centres in the volume of the

crystal

^{move in}

the same double-well

potential, i.e.,

^their

tunnelling splittings

^rand

asymmetries

^{e are all}

equal.

^This may not be the situation in real

crystals

where local lattice distortions and stray fields from

impurities

^or

imper-

fections will be

randomly

distributed

throughout

^the

volume of the

sample.

^{In such more realistic cases,}

one expects ^a certain distribution function of the

asymmetries

^whose width will be a measure of the lattice

imperfection.

The effect of such a distribution

on the ultrasonic attenuation will be ^to broaden the relaxation

peaks

^{more than is}

prescribed by

^a

single Debye

relaxation

and,

^at the lowest temperatures, ^to allow

phonons

^{over a wider}

frequency

range ^to interact

resonantly

with the manifold of

split ground

states.

Equations (2), (3)

^and

(5)

would in this case

have to be

integrated

^{over the}

appropriate

distribution functions.

Experimental

evidence of such distributed effects have been obtained not

only

ⁱⁿ

glasses [14]

but also for some off-centre ions in alkali halides

[12]

for both

tunnelling

transitions and thermal activations.

3.

Experiment.

^{- The}

samples

^{used in} these expe- riments which are in the

form

of

cylindrical

^X-cut

rods of

nominally

pure natural brazilian quartz ^were

polished

^to microwave ultrasonic

specifications.

^Con-

tents of

major impurities

determined

by spectroscopic analysis [18]

^are

given

^{in table I.} Ultrasonic measure- ments were carried out

using

^standard

pulse-echo techniques by

direct surface excitations of the rods in

TABLE 1

Spectroscopically

determined

impurity

^content

of

^X-cut

samples

789

re-entrant microwave cavities or non-resonant

sample

holders. The

experimental frequency

^and tempera-

ture ranges ^were 57 to 1 200 MHz and 1.5 to 300 K

respectively.

^The attenuation was measured

by

^an

automatic attenuation recorder

[19].

^{For low}

pulse repetition

^{rates or}

peak

powers the

signal

^was pro- cessed in a sensitive box-car

integrator [20].

^The temperature ^{of the}

sample

^was determined

by

^{means of}

calibrated Ge-thermometers.

Experimental

accuracy in the determination of the temperature ^{and the} attenuation were

respectively

^{7 and 5} per ^{cent. Gamma} irradiation was carried out at room temperature ^with

a 60CO

source at a rate of 100

krads/h

^{with the}

samples slowly rotating

about their axes. Doses range from 105 to 2 ^x 10’ rads. The radiation induced coloration

was assessed

by measuring

^the

optical absorption

relative to unirradiated

samples

^{of the same batch}

in a

recording spectrophotometer [21].

^The

samples

were

subjected

^{to various}

cycles

^of

y-irradiation, UV-bleaching

^{in a 1 kW Hanovia source, and thermal}

annealing

^{in air for one hour at 400}

^OC,

^{the attenua-}

tion, optical absorption

^{and EPR} spectra

being

measured at various stages ^{of these} treatments. Two

crystals

^{of a batch of 5 were} studied in detail

by

^the

various

experimental techniques

^{and showed the}

same behaviour after successive irradiation or bleach-

ing

treatments.

4. Results and discussion. ^- The attenuation of both

longitudinal

^{and shear waves was measured in}

the same X-cut

samples.

Since similar results were

observed for both

polarizations

^we concentrated our

efforts on

longitudinal

^{waves whose}

generation by

surface excitation is

simpler

^{in X-cuts.} The attenuation of

longitudinal

^{waves in the as received}

crystals

was found to be

typical

^of

good quality

^{quartz. The}

absolute attenuation versus temperature ^{curves shows}

a well defined temperature ^-

independent

^residual

attenuation at helium temperatures ^{followed for} T > ^{10 K}

by

^the steep ^{rise due to thermal}

phonon viscosity.

^At

temperatures

^{in excess of 60 K this} contribution

again

^becomes

relatively independent

^of

temperature.

^{The relative}

attenuation,

^obtained

by substracting

from the measured value the low

tempe-

rature residual part, ^{shows a T"}

dependence

^with

n 7 characteristic of quartz ^{with a low concentra-} tion of

point impurities

^{or defects}

[22].

Upon

irradiation the absolute attenuation curves

show a

plateau

^{in the} temperature

region

^{between 10}

and 30 K

[8]

^{and also} an excess attenuation for T greater than 100 K. The residual attenuation was not

noticeably changed by

^the irradiation.

Figure

^{2 shows}

the relative attenuation versus temperature ^for 410 MHz of a

sample

irradiated to 106

rads,

^and after

subsequent

^{UV - or thermal}

bleaching.

^The

attenuation for

heavily ( > 106 rads)

^irradiated

samples nearly

coincides with the curve for bleached

ones. The same behaviour was observed at different

experimental frequencies

^{but with the} temperature

FIG. 2. ^- Temperature dependence ^of the relative ultrasonic attenuation of one X-cut quartz sample irradiated with y-rays and

after bleaching.

of the maximum excess attenuation

displaced

^accor-

dingly.

The relative attenuation of the

virgin crystal

falls below that of the bleached one for T 15 K but it is greater ^{for T} > 23 K.

The attenuation in excess of the value for the bleached

crystal

^is

plotted

ⁱⁿ

figure

3 for the whole temperature range for two doses in the irradiation series. It can be noted that while the

high-tempera-

ture double

bump

increases with

dose,

^{the low-} temperature

peak

decreases for doses in excess of

106

rads. This is shown more

clearly

ⁱⁿ

figure

^{4 that}

illustrates the radiation behaviour of both anomalies for the

experimental

^dose range. It has also been observed that for

crystals

^{with 106 rads the low-}

temperature

peak disappears

^{when the}

sample

^is

FIG. 3. ^- Temperature dependence ^{of the excess} attenuation of

a y-irradiated ^X-cut quartz sample ^{at low- and} high-temperatures.

FIG. 4. ^- Radiation behaviour of the low- and high-temperature

ultrasonic relaxation peaks ⁱⁿ y-irradiated ^X-cut quartz. T. ^indi-

cates the experimental peak temperatures. ^{Curve 1} represents ^first irradiation results. Curve II was obtained after thermal annealing

of the heavily irradiated sample.

further

subjected

^to

optical

^{or thermal}

bleaching.

Successive

irradiations of the

crystals

^{bleached after}

a 2 ^x 10’ rads dose makes this attenuation

peak

reappear but the recovery of its

magnitude

^is

only

50 _per cent of the

original.

Both the low- and

high-temperature

^anomalies

were

analysed

^{in terms of}

Debye

relaxations, _equa- tion

(5).

^{While the}

low-temperature

^excess attenuation could be fitted well to a

single Debye

^{curve this was not}

possible

^for

high

temperatures

because

^{of the}

mixing

of the two

adjacent peaks.

^{In both}

tèmperature

ranges

a linear

frequency dependence

^was found for the various maxima.

Figure

⁵

displays

the characteristic linear

logr vs 1/7"

^{curves of}

thermally

^activated

processes. ^In table II the various

parameters

^that define each relaxation ^are listed for a radiation dose of

106

rads. The concentration of Al-centres shown in the table was obtained

by assuming

^{D 1 eV in}

equation (5)

^and agrees well with the value of

1016 cm^3

determined

by

^Smekula’s

equation

^from

the C-band

optical absorption.

A search was made from 1.3 to 4 K for _any

possible

saturation effects in the ultrasonic attenuation. No

change

^was observed in the residual attenuation

FIG. 5. ^- Temperature dependence ^of the relaxation times _in for the ultrasonic attenuation peaks.

throughout

^the

frequency

range and for microwave

input

power reductions of _up to 20 dB from an esti- mated zero acoustic _power level of one

mW/cm2.

Neither was there _any indication of a temperature

dependence

of the residual attenuation at either

high

or low _power

inputs

^{or of the} presence of relaxation

peaks

^{of the} type ^discussed

by

^Jâckle

[16]

^and

given by equation (3).

Changes

^{in the}

optical absorption

^{as a} function of irradiation doses were monitored relative ^to unirra- diated

samples.

Both the broad

AI

^and

A2

^{bands in}

the visible and the better resolved ultraviolet C- band

[23]

increase with dose. The A-bands showed the

beginning

of saturation for a dose of 5 x

106

rads whereas the C-bands

continues

to increase. The

amplitude

^{of the}

high

temperature ultrasonic

peaks

was found to be

directly proportional

^{to the}

optical absorption

in the visible between

106

and 5 ^x

106

rads while the

low-temperature

relaxation shows a more

complex

behaviour. The

ratio, Aa/pj,

^{of this}

TABLE II

Experimental

^values

of

^the parameters

for

the ultrasonic relaxation

peaks

The concentration n was calculated from equation (5) ^{for COT = 1 and an assumed D of 1 eV}

791

excess attenuation over the

optical absorption

pj (j

⁼

^{AI, A2}

^or

^AI

⁺

^A2)

^{is constant for doses}

smaller than 106 rads but it

strongly

decreases with pj ^for greater doses. This

dependence

^{will be comment-}

ed _upon below.

EPR

spectra

^{at 9 GHz and 77 K were taken on the} ultrasonic

samples

^{at various} stages of irradiation or

bleaching.

^{For doses} up ^to 106 rads two groups of six well defined lines were observed whose intensities increase with _gamma dose. The best resolution was

obtained when the external

magnetic

^{field was} pro-

perly

oriented in the

plane perpendicular

^{to the}

sample’s

X-axis. This

hyperfine

^{structure identifies}

the Al-hole centre

previously

observed in irradiated

quartz [3].

^{This EPR} spectrum ^was also found in the bleached

crystals

^{and in one}

virgin crystal

^studied

which indicated the existence of some

uncompensated

Al-centres in these very pure

samples

^{even before}

being exposed

^to y-rays. ^In

fact,

from table 1 it can be

seen that the concentration of the

possible

compensa- tors, Cu+ and

Mg’+,

^{is below one} ppm.

For

samples

irradiated to 5 ^x 106

rads,

^{the EPR} spectrum

changes drastically. Many

^{more lines are}

observed which cannot be resolved

by

rotation of the external

magnetic

field. The calculated

g-value

^for

the

trapped

hole is in agreement ^{with that}

reported by

Schnadt and Schneider

[3]

^for

X-ray

^irradiated

samples

^{with a}

larger (150 ppm)

^Al concentration.

The

complex

^EPR

signal

observed for the

heavily

irradiated

samples

^{indicates an} increase in the aniso- tropy ^{of the}

g-factor

^which may be due to radiation induced

changes

ⁱⁿ the local environment of the hole

spin.

It can be deduced from the present ^{results that the} ultrasonic

peaks

^{are caused}

by thermally

^activated

transitions of

light particles trapped

^{in color centres}

formed

by

the ionization of defects

already present

ⁱⁿ the quartz lattice. The

pre-exponential

^{factors ro}

for the various relaxations are of the order

of,

^or smaller than classical oscillation

frequencies.

^The

small activation

energies found,

the fact that the

peaks

^are

readily

^bleached

by

^{heat or}

UV-light,

^and

the reversible radiation behaviour of the bleached

crystals

^{allow us to} discard thermal activations of interstitial ions or

charge compensators

^{as the cause} of the excess attenuation

[4].

^{Since the}

only

parama-

gnetic species

detected in our

samples

^{is the Al}

trapped

hole centre we will try ^to

interpret

^{our results in}

terms of the known relaxations of this centre.

The activation _energy found for the

high-tempera-

ture relaxation

peak

agrees well with

previously reported

^values from anelastic

loss,

^EPR line broaden-

ing

^and dielectric loss measurements where the ano-

malies were attributed to

thermally

activated transi- tions via a

high-lying

intermediate state. From this state the hole _may tunnel

through

^or

jump

^{over the}

potential

^{barrier to one of the two} oxygen ^atoms further _away from the substitutional aluminum ion.

Our

pre-exponential

^{factor To are,}

^however,

^{one to}

two orders of

magnitude

smaller than those found in the dielectric relaxation measurements. This _sug- gests ^{that in our}

samples

^the intermediate state is close to or above the

potential

^barrier

height

^and

that the

perturbed

hole relaxes to

equilibrium mainly by thermally

activated transitions between _oxygen ions near and far from the Al-ions. Since two

peaks

are observed there seems to be

slightly

^different

potentials

for the hole

trapped

^at aluminum sites

with

different local environment ^or associated com-

pensators.

^{For the}

highest

temperature

peak

^tunell-

ing-controlled

relaxation of the hole cannot be

entirely

ruled out since the measured _r, is

nearly

^{one order of}

magnitude

smaller than for the lower

temperature peak.

The ultrasonic

peak occurring

^at the lowest

tempe-

rature has also a relaxation time of the classical

exponential

form with its

pre-exponential

^factor

somewhat smaller than the classical value.

Thus,

^a

tunelling-controlled

relaxation of the hole can

again

^be

presumed

^{from an} intermediate state to which it has been

thermally

activated from the

ground

^state.

Such a process has been observed in the dielectric relaxation measurements with ^an activation _energy E c.-- 8.4 meV in

good

agreement ^{with the} present results.

However,

^{in our case a}

pre-exponential

^factor

5 orders

of magnitude

smaller is found for

crystals

irradiated to

10’

rads. This

implies

^{that for our}

samples

^the double-well

potential

^that controls the

hopping

^{of the} hole between the two oxygen ions nearest to the aluminum ion has a smaller barrier

height k

^E and is much more

symmetrical.

^From

equation (4), assuming

^{a small}

localization il

⁼

2,

the _energy

split

between the intermediate states in each well

ô,,

would be of the order of 2

meV, roughly

half of which is due to the

relatively large overlap

of the wave functions of each

potential

^well.

We must also discard the influence of a wide dis- tribution of

asymmetries

of the double-wells for our

samples

^{since no evidence was found of a} distribution of relaxation times for the lowest temperature

peak.

Therefore,

^the localization temperature

TL

^should

be much smaller than for the dielectric loss

samples [2].

The unusual radiation

annealing

of the low tempera-

ture

peak

for doses greater ^than

106

rads _may be related to the

quality

^{of our}

crystals

^{and to the nature}

of the

tunnelling-controlled

transitions of the hole.

From

equation (4)

^{we note that the}

pre-exponential

factor for this

peak

’fo

depends strongly

^{on the} asym- metry ^{of the}

potential

^{well T}

through

^the localization parameter 11. Gamma irradiation may

change F by

electrostatic effects associated with

charged

centres, strains

originating

^from

displaced

^{atoms or broken}

bonds,

^or

by piezoelectric

^fields.

Furthermore,

^the

sensitiveness of _{LO to} electrostatic fields

f

^as

given by [2J dïo/dy

²

pIkTL,

^{where p is the electric}

dipole

moment of the

trapped hole,

^{would be}

greatly

^enhanc-

ed if the

slightly

irradiated

samples

^{have a small .}

localization temperature. There is insufficient ^know-

ledge

about the centre structure to attempt ^{here a}

quantitative

^{account of the effect of} radiation induced electrostatic effects on To but

by

way of a

rough

estimate we note that to

produce

^a 100-fold increase

in L 0’

which will shift the temperature of the relaxation maximum to a range where it will be masked

by

^the

thermal

phonon

attenuation

(T it

⁴⁰

K)

^{and thus}

cause an apparent ^radiation

annealing

^{of the relaxa-}

tion

peak,

^would

require

an q of 200. This value is

comparable

^{to what can} be deduced from de Vos and

Volger

^{results for their} r,. For ^a

dipole

^{moment of}

6

Debye

^{and an} assumed localization temperature ^of 0.2 K such a 100-fold increase in _ro would be _pro- duced

by

^{a field of 70}

kV/cm,

^a

strength

^{to be}

expected

from

singly charged impurities

^{with a} concentration

of 1018 cm - 3 .

The strong

dependence

of the ration

A(x/

^{on the}

optical absorption

coefficient J1 j ^{for doses in excess} of 106

rads,

supports ^{the view that as the}

ionizing

radiation continues to create

charged

^{centres their}

Coulomb fields

drastically

affect the

original high

symmetry ^{of some double}

potential

^{wells. It thus}

seems very reasonable to attribute the observed apparent ^radiation

annealing

^{of the}

low-temperature

relaxation

peak

^{to the}

closing

^{of the}

tunnelling paths

from the

low-lying jntermediate

^state

by

electrostatic shifts in the

potential

asymmetry

arising

^{from ioniza-}

tion of

impurities by

^the gamma rays.

Finally,

^{as a}

certain amount of permanent structural

damage,

^such

as oxygen

vacancies,

^{will be}

produced

ⁱⁿ quartz

by Compton electrons,

^one expects ^a fraction of the hole centres to become

permanently

localized with suc-

cessive irradiations which will in term prevent ^{the full} recovery of the

low-temperature

relaxation

peak.

It is somewhat

surprising

that transitions between the

split ground

^{state levels are not detected}

by

^the

present ultrasonic

experiments.

Even in the case where saturation effects

preclude

^the observations of reso-

nant attenuation of the sound waves, the relaxation part of the attenuation as

given by equation (3)

^should

be observed. In fact if we assume n _ 1016 cm- 3 and D - 1 eV the maximum of this contribution would amount to 0.3

dB/ps

^{at 2 K for l5 kT and}

should be

easily

detected above the

typical

^residual

attenuation value of 3 ^x 10-2

dB/gs.

^{It is}

possible

that due to the

anisotropy

^{of the} deformation poten- tial tensor waves

along

^{the X-axis do no}

couple

^{to the}

ground

^{state levels. It} would be convenient to extend the present measurements to include waves of diffe-

rent

propagation

^direction.

5. Conclusion. ^- The

high frequency

^ultrasonic

relaxations observed in

gamma

irradiated

nàtural

quartz

crystals

have been attributed to transitions of the hole

trapped

^at substitutional aluminum sites.

Activation

energies

deduced from the

experimental

results agree well with those found

by

other authors but the relaxation times are

respectively

^{one to two,}

and five orders of

magnitude

smaller for the

high-

and

low-temperature

relaxation

peaks.

^{The short}

relaxation times and the observed radiation

annealing

of the

low-temperature peak

have been related to the influence of the

high purity

^and

perfection

^{of the}

samples

^{on the nature of}

tunnelling-controlled

^transi-

tions of the hole _among the _oxygen ions of the

AI04

tetrahedrons. Sound waves

along

the X-axis do not seem to be

coupled

^{to the}

split ground

^{state levels of}

the hole.

Acknowledgment.

^{- We are}

grateful

^{to Drs. R.}

Calvo and S. Oseroff for

taking

^{the EPR} spectra ^and for

helpful

discussions.

References

[1] ^{For an extensive} bibliographical ^{review see :} WEIL, ^J. A., Radiat. Eff. ²⁶ (1975) ^261.

[2] ^DE Vos, W. J. and VOLGER, J., Physica ⁴⁷ (1970) ^13.

[3] SCHNADT, ^R. and SCHNEIDER, J., Phys. ^{Kondens. Mater. 11} (1970) ^19.

[4] STEVELS, J. M. and VOLGER, J., Philips ^Res. Rep. 17 (1962) ^283.

[5] MACKEY, ^J. H., ^J. Chem. Phys. ³⁹ (1963) ^74.

[6] SCHNADT, ^R. and RÄUBER, A., ^Solid State Commun. 9 (1971)

159.

[7] KING, ^J. C., ^Bell Syst. ^{Tech. J. 38} (1959) ^573.

[8] ^For preliminary ^{results see :}

NAVA, ^R. and RODRIGUEZ, M. in Proc. Internat. Conf. ^Phonon Scattering ⁱⁿ Solids, edited by ^{H. J.} Albany (La ^Docu- mentation Frangaise, Paris) 1972, p. 280.

[9] SUSSMANN, J. A., Phys. ^{Kondens. Mater. 2} (1964) ^146.

[10] ^{SUSSMANN, J. A.,} Phys. ^{Lett. 225A} (1967) ^146.

[11] ^{SUSSMANN, J.} A., Phys. Chem. Solids 28 (1967) ^1643.

[12] ROLLEFSON, R. J., Phys. ^{Rev. B 5} (1972) ^3235.

[13] ORTLIEB, E., SCHAD, Hp. ^and LASSMAN, K., Solid State Commun.

19 (1976) ^599.

[14] HUNKLINGER, ^S. and ARNOLD, W., in Physical ^Acoustics, edited by ^{W. P. Mason and} R. N. Thurston (Academic,

New York) 1971, Vol. XII, Chap. ^3.

[15] JÄCKLE, J., PICHÉ, L., ARNOLD, ^{W. and} HUNKLINGER, S., J. Non-Cryst. ^{Solids 20} (1976) ^365.

[16] JÄCKLE, J., ^Z. Phys. ²⁵⁷ (1972) ^212.

[17] SCHAD, Hp. ^and LASSMAN, K., Phys. ^{Lett. 56A} (1976) ^409.

[18] ^Polished crystals ^and impurity ^contents supplied by Valpey-

Fisher Corp. (Hopkinton, Mass., U.S.A.).

[19] Matec Inc. Providence, ^R. I., U.S.A.

[20] ^Princeton Applied ^Research Corporation (Princeton, ^N.J., U.S.A.).

[21] ^Hitachi Perkin-Elmer, Tokyo, Japan.

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[23] PAIGE, E. G. S., Philos. Mag. ² (1957) ^864.