Uniaxial stress effect and spin-lattice coupling in the 3 T1u relaxed excited states of F centres in CaO

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Uniaxial stress effect and spin-lattice coupling in the 3 T1u relaxed excited states of F centres in CaO

Le Si Dang, Y. Merle d’Aubigné, Y. Rasoloarison

To cite this version:

Le Si Dang, Y. Merle d’Aubigné, Y. Rasoloarison. Uniaxial stress effect and spin-lattice coupling in the 3 T1u relaxed excited states of F centres in CaO. Journal de Physique, 1978, 39 (7), pp.760-770.

�10.1051/jphys:01978003907076000�. �jpa-00208811�

UNIAXIAL STRESS EFFECT AND SPIN-LATTICE COUPLING

IN THE 3T1u ^{RELAXED EXCITED STATES OF F} CENTRES IN CaO

LE SI

DANG,

^{Y. MERLE}

D’AUBIGNÉ

and Y. RASOLOARISON

Laboratoire de

Spectrométrie Physique (*),

Université

Scientifique

^et Medicale de Grenoble, BP 53, 38041 Grenoble Cedex, France

(Reçu

^{le 16}

janvier 1978, accepté

^{le 5 avril}

1978)

Résumé. ²⁰¹⁴ Nous avons étudié l’effet des contraintes uniaxiales sur le spectre ^{de résonance}

magné- tique

dans les états excités relaxés

3T1u

^{du centre F} dans CaO. Nous avons bâti un Hamiltonien effectif pour décrire le couplage

spin-réseau

^{dans un niveau}

3T1u couple

^{fortement aux modes} de vibration

Eg

(effet Jahn-Teller

statique).

^{Nous avons} trouvé que le couplage spin-réseau ^{est environ quatre ordres}

de grandeur plus ^faible que le

couplage

orbite-réseau. Trois différents mécanismes du

couplage

spin-réseau ^{ont été} considérés : effet des contraintes uniaxiales sur

(i)

l’interaction

spin-orbite,

(ii) l’interaction

dipôle-dipôle, (iii)

l’énergie de stabilisation Jahn-Teller. Nous avons trouvé que

ce dernier mécanisme est prépondérant ^{dans le} couplage

spin-réseau.

^{Nous avons} aussi estimé l’ordre de grandeur ^des paramètres ^de couplage ^aux modes de vibration

Eg,

de l’interaction

spin-orbite

^{et de}

l’interaction

dipôle-dipôle.

Abstract. ²⁰¹⁴ The effect of uniaxial stress on the EPR spectrum ^{of the}

3T1u

relaxed excited states of F centres in CaO is reported. ^{From symmetry} considerations only, ^an effective Hamiltonian is derived in order to describe the

spin-lattice coupling

^{in a}

3T1u

^level

undergoing

^a strong

coupling

^to

Eg

modes of vibration (static Jahn-Teller effect). It is shown that the

spin-lattice coupling

is about four orders of magnitude smaller than the orbit-lattice

coupling.

^The

physical origin

^{of the}

spin-lattice

coupling ^is

analyzed ;

three mechanisms are considered : variation with the

applied

^{strain of} (i) ^the

spin-orbit

interaction,

(ii)

^the

dipole-dipole

interaction,

(iii)

the Jahn-Teller stabilization energy, this last mechanism

being

^{the most} effective. Estimates for the parameters of the vibrational

coupling

to

Eg

^{modes, of the} spin-orbit interaction, and of the

dipole-dipole

interaction are given.

Classification

Physics ^Abstracts

63.20M - 71.70E - 76.30M - 76.70H

1. Introduction. ^- The F centre in CaO is an

oxygen vacancy

having trapped

^two

électrons.

Iden- tification of its

optical

^{spectra was made}

by

^Henderson

et al.

[1, 2]. They

attributed the strong

absorption

band at 3.1 eV to the electronic

transition 1 A 1g - 1 T 1u,

and the

long-lived

^{emission at 2.1 eV to the}

spin-

forbidden transition

3Tlu

^-+

1 Alg.

^More

^precise

^infor-

mation about the relaxed excited states

3Tlu

^was

obtained

by

Edel et al.

[3] using

^the

technique

^of

optical

detection of

magnetic

resonance. Three

equivalent

electron

paramagnetic

^resonance

(EPR) spectra

^of

tetragonal symmetry

^were

observed, indicating

^a

strong

interaction with the

Eg

^{modes of vibration and}

a static Jahn-Teller effect

[4].

_,

. Recent studies

by

Cibert et al.

[5]

^and

Bontemps-

Moreau et al.

[6]

have shown that the

spin-lattice

relaxation inside the relaxed excited states

3T,u

^is very

complex.

^{It takes}

place through tunnelling

processes

(*) Laboratory associated with the Centre National de la Recherche Scientifique.

between the three

tetragonal

Jahn-Teller

wells,

involv-

ing

^{or not a}

spin

^{reversal. In order to} better understand these relaxation

mechanisms,

^{it is}

important

^to

directly

^{measure the} parameters

defining

^the

spin-

lattice

coupling.

In section

2,

^{we shall}

briefly

recall the Ham

theory [4]

of the static Jahn-Teller effect of an orbital

triplet Tlu coupled

^to

Eg

modes of vibration. Then

using

sym-

metry arguments only,

^we shall write down the effective Hamiltonian

describing

the effect of strains inside the relaxed excited states

3Tlu. Expressions

for the effect of stress on the EPR

spectrum

will be derived for some

particular

orientations of _pressure and

magnetic

^field.

The

experiment

^{and the}

interpretation

of the data

using

the effective Hamiltonian will be discussed in section 3. In section

4,

^we shall show how the

spin-

lattice

coupling

coefficients are related to the _para- meters

defining

^the orbit-lattice

coupling

^{which is}

known from the measurement

by

^{Edel et al.}

[7]

^{of the}

effect of external stress on the zero

phonon

^{line. It will}

be shown also

that, assuming

reasonable values for the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003907076000

vibronic

coupling

^{and the}

spin-orbit interaction,

^a

simple

^{model can account for the}

largest

^{term of the}

spin-lattice coupling.

2. Effective Hamiltonian in the vibronic manifold

3Tl,,. -

^{Z .1} JAHN-TELLER EFFECT FOR AN ORBITAL TRIPLET

T1u

COUPLED TO

Eg

MODES OF VIBRATION. ^-

We shall

briefly

recall the

theory developed by

^Ham

[4]

and at the same time introduce our notations. It will be assumed that the orbital

triplet T 1 u

^is

linearly coupled

^{to a}

single pair

of vibrational modes

(Q8, Qe) belonging

^to the irreducible

representation Eg

^{of the}

point

group

Oh.

Consideration of a more extended set of

Eg

modes would not

change

^the main results of the

theory [4].

The vibronic

problem

^without

spin

^{can be}

solved

exactly.

The vibronic

eigenstates

^are

simple Born-Oppenheimer products

^{of one} of the orbital

states 1 fJ ), fJ

⁼

X,

^{Y or}

Z,

^{and a} vibrational state

1 X., ne, n. > :

1

X,, ^ne,

ne )

^{is an}

eigenstate

^{of a}

displaced

^{two dimen-}

sional

(Q8, Qe)

^harmonic

oscillator,

^the

equilibrium position

of which is different for each

state j8 ).

^The

energies

of the vibronic states

(1)

^{are :}

Eo

is the energy of the orbital

triplet T1u

^{before the}

Jahn-Teller effect has been considered. The Jahn- Teller stabilization _energy,

EJT,

^{is the amount}

by

^which

the

potential

energy is lowered at the new

equilibrium position

^{of the} harmonic oscillator

(see Fig. 1).

COE is the

angular frequency

^{of the}

Eg

modes. The Jahn- Teller

coupling strength

^{is defined}

by

^the dimensionless

Huang

^and

Rhys

^factor

SE

such that :

FIG. 1. ^- Configuration ^curves along the coordinate Q9’ ^Dotted

lines show the effect of the component ee ^of Eg ^strain.

It should be noted that the

ground

vibronic state, i.e. _{n8 = nE} ⁼

0,

^{is still a}

triplet T 1 u. According

^to

(1)

any vibronic matrix element of ^a

given

^electronic

operator 0

^{is the}

product

^{of an} electronic matrix element and an oscillator

overlap integral :

Within the

ground

^vibronic

triplet Tl,,

^the

overlap integral

^is

unity

^for

diagonal

^matrix

elements,

^{and is}

for off

diagonal

matrix elements. As

pointed

^out

by

Ham

[4],

^this

overlap integral

may

change drastically

the first order effect of some

perturbations

^{such as the}

spin-orbit

interaction or the

coupling

^to

T2g ^trigonal

strains, which involve non

diagonal

^orbital operators.

As ^a result of this strong ^Ha/n

reduction,

^{the second} order effect of the

perturbations,

which involves

higher

vibronic levels

(no,

ne =F

0),

may become

larger

than their first-order effect. Then it is convenient to

take into account these second-order effects

by

^an

effective

Hamiltonian

[8, 9] operating

inside the

ground

vibronic manifold

3T lu.

^For instance the

spin-orbit

interaction is defined

by

^the

following

Hamiltonian

operating

ⁱⁿ the electronic manifold

3T lu (1

⁼

1,

S =

1) :

When considered in first and second order of the

perturbation,

^it

gives

^rise to two terms of the effective Hamiltonian

[41 operating

in the vibronic

triplet

where Â’ =

ÂR, and fa and fb

^{are functions of the}

Huang

^and

Rhys

^factor

SE

^{as shown in}

figure

^2.

Explicit

forms for these functions are

given

ⁱⁿ appen- dix II.

The observation of a static Jahn-Teller effect is due

to the

stabilization, by

^internal

strains,

^{of the F centre} in one of the three vibronic

states 1 fi, 0, 0 » [6, 7]

(or

^{in other}

^words,

^{in one} of the three Jahn-Teller wells associated with the

X,

^{Y and Z orbital}

states).

Such strains are

always present

^{in a}

crystal.

^The

effect of

T2g,

^or

^trigonal,

strains is almost

completely quenched

whereas the effect of

Eg,

^or

tetragonal,

strains which is described

by diagonal

operators ^{is not} affected. The threefold vibronic

degeneracy

^{of the}

relaxed excited states

3Tlu

^{is lifted as} indicated in

figure

3. The effect of the

non-diagonal operators

representing

^the

coupling

^to

T2g

strains and the

spin-

FIG. 2. ^- Variations with SE ⁼ EJT/hwE ^of functions of oscillator

overlap integrals involved in the first order (R), second order

(fa, fb), ^and third order (fc, fa) ^of perturbation caiculations (see

text and appendix II).

FIG. 3. ^- Effect of successive perturbations ^{in the} ground ^vibronic manifold 3Tlu. ^{X, Y, Z} stand for vibronic states ] X, 0, 0 )), 1 Y, 0, 0 », Z, 0, ^{0 ».} Figures in brackets indicate level degene-

racies. Wavy ^{arrows show EPR} transitions (AM ^{= ±} 1).

orbit interaction

being strongly reduced,

^{the states}

1 Pl m »> = 1 fil 0, 0 » M X

^{m = +}

^{1, 0, -}

^l,

remain

good approximations

^to the actual

eigenstates.

The reduced

spin-orbit

interaction

(7) only

^{causes a}

small

broadening

^{of the EPR} lines of the order of

À,2/b [10],

where ô is the _average

splitting

^{of the}

vibronic

triplet

^{due to}

Eg

internal strains. The second order effect of the

spin-orbit

interaction

(8)

^which

involve

diagonal

operators

partially

lifts the

spin

degeneracy

^{of each of} the three vibronic states

1 fi, M »>,

^and

gives

^{rise to} the axial character of the EPR spectra. ^The observation of three

equivalent tetragonal

^EPR spectra

[3]

shows that this

theory applies

^to the relaxed excited states

’Tlu

^{of the F}

centre in CaO.

2.2 EFFECTIVE HAMILTONIAN. ^-

Using

symmetry considerations

only

^{we shall now formulate the}

effective Hamiltonian which describes the fine struc- ture, ^the orbit-lattice and the

spin-lattice couplings,

and the Zeeman effect inside the

ground

^vibronic

manifold

3T lu.

^{We retain}

only

^{the terms} linear in the strain components. ^{This is}

justified by

the linear shifts under

applied

^{stress of both the zero}

phonon

line

[7]

^{and EPR} lines observed

experimentally.

^We

do not include the cross terms

involving

^{both the}

strain and the

magnetic

field since their effect on the EPR lines is too small to be observed. The orbital Zeeman effect which is

severely

^reduced

by

^{the Ham}

effect,

^{will be}

neglected

also. Then the effective Hamil- tonian can be written as :

In

(9),

^the first and second terms inside

parenthesis

describe

respectively

^{the fine structure} and the effect

of strains,

^{while the last term is the}

spin

^{Zeeman effect.}

l;y(ST’y’)

^operates inside the

ground

^{vibronic l’ = 1}

(S

⁼

1 )

space, and transforms like the

y(y’)

component of the irreducible

representation r(r’)

^{in the} opera- tions of the _group

Oh. Components

of the strain tensor

e,ll’,,. ^{have the same obvious} symmetry

properties.

Precise definitions of

1 ;y’ Sr,y,

^and er,-y" ^are

given

ⁱⁿ

appendix

^{I. For}

example 1§, 1,’(1’, 1,1’, ¹⁽⁾

^transform

like components ^{of the}

E(T2) representation,

^while

lg

transforms like the

AI representation.

^The

[r, T’]

and

[r, T’, F "]

^{are numerical} coefficients which do not

depend

^on y,

y’, y".

gs is the free

spin gyromagnetic factor,

^and ,uB the Bohr

magneton.

The number of terms in

(9)

is restricted

by

symmetry considerations :

(i)

^{As orbital states and}

spin

^states

belong

^to

Tl,

T and T’ should be contained in the

product

(ii)

^{Terms in}

(9)

should be invariant in the _opera- tions of the cubic _group

Oh.

A necessary condition is that the

product

^{r x r’ for the} first term, ^or r x F’ ^x r" for the second term, contains the

unity representation AI.

(iii)

^The

product 1;’1 Sr,Y,,

^{should be invariant under}

time reversal. l’ and S

being odd,

^{terms with r = E}

and F’ =

T,

^for

example

^{must be}

rejected.

TABLE 1

Effective

Hamiltonian in

3T,,, (l’

⁼

1,

^{S =}

1)

Terms of the effective Hamiltonian

(9) satisfying

these symmetry conditions are shown in table I.

The

first

^{three terms are strain}

independent,

^and

describe the fine structure of the

3Tlu

^{level. Terms 4}

to 6 describe the direct effect of strain on the

ground

vibronic

triplet 1 fui, 0,

⁰

».

^The coefficients

Vi, V2

and

V3

^were determined

by measuring

^{the effect of}

uniaxial stress on the zero

phonon ^line [7]. Terms

⁷

to 11 describe the effect of strain on the

spin

^sublevels.

It will be shown that

they. are

four orders of

magnitude smaller

than

V2. A , complete description

^{of the}

spin-

lattice

coupling

would include seven other terms of the same order of

magnitude

^{as terms 7 to 11. Since}

they

^{involve non}

diagonal

^orbital operators,

they only give

^{rise to} second order shifts of the

spin

^suble-

vels. These shifts are too small to be

observed,

^and

we

only

^{write one of these terms as an}

example

^of

their form

(term 12).

2. 3 STRESS EFFECT ON THE EPR LINES. - In the

following

^we shall show how this effective Hamiltonian

can be used to

analyse

^{the EPR} spectrum ^{and its} variations with uniaxial stress. As illustrated in

figure

3, the

largest

^{terms in}

Jeeff

^are

Vl

^and

V2,

^{which describe}

the

coupling

^{to internal}

(or applied)

strains of

A 1 g

and

Eg

^symmetry

respectively.

^{The latter lifts the}

threefnld degeneracv nf thp vihrnnir tr1nlpt R 0 0 »

The _average

splitting

^{b =}

(Ep - Ep,)

^{due to internal}

strains

Eg

^is

^typically

^{a few} wavenumbers

[10].

^This

is much

larger

^{than the other terms of}

Jeeff

^{which can}

be treated as a

perturbation.

^{Then one} may define

an effective

spin

Hamiltonian

XO f

^{for each of the}

vibronic

states 1 f3, 0, 0 » :

which contains

only spin

operators. ^For

example

the effective

spin

Hamiltonian of the vibronic state

1 Z, 0, 0 ))

^is

given by :

The first line

of (11)

^{is the usual}

spin

Hamiltonian of

a S ⁼ 1 level in axial symmetry. ^{The zero field}

split-

;

ting D (- 6

^x

^10-2

^{cm-’ 1}

[3]),

^{and stress effects}

(AeA - %e, - U)e -

^10-4

cm-l)

^{are smaller than}

the Zeeman effect

(-

^0.3

cm-1).

Therefore the

spin states 1 M),

^{with M = +}

1, o, - 1,

^are

good approxi-

mations to the

eigenstates

^of

XO f.

^The

energies

^{of the}

corresponding

Zeeman sublevels are

given by

For each centre

EPR lines are observed

[3].

^{Table II}

gives

^{the shifts}

of the low field lines calculated for the

magnetic

^{field H}

parallel

^{to the}

[00l],

^or

Oz,

^{direction and the} pressure F

applied along

^the

[100],

^or

^Ox,

direction. _{CA, CE} and _cT

are linear combinations of the

compliance

coefficients which are

given

ⁱⁿ

appendix

^{I. The shift of the}

high

field line is

equal

ⁱⁿ

magnitude

^but

opposite

ⁱⁿ

sign.

3.

Experimental.

^- 3 .1 EXPERIMENTAL TECHNIQUE.

- The

technique

^of

optical

detection of

magnetic

resonance in the excited states has been described elsewhere

[3].

^The

experimental

setup ^{is shown in}

figure

^{4. The emission at 600 nm}

(2.1 eV)

^{was excited}

FIG. 4. ^- Experimental ^set up for simultaneous measurements of stress effect on the EPR spectrum and the emission spectrum.

A ⁼ linear analyzer, ^{F and F’ =} optical filters, ^{M =} semi-trans- parent mirror, ^{PMT =} photomultiplier ^tube.

by illuminating

^the strong

absorption

^{band at 400 nm}

(3.1 eV)

^{with a}

high

pressure mercury

lamp

^{of 200 W}

(HBO-OSA),

^{and a} broadband filter MTO 395 b.

The emission was observed at

right angles

^{to the}

magnetic

field direction

through

another filter MTO 600 b. Both the excitation and emission filters have bandwidths of about 30 nm. The n or t1

polari-

zation of the

emission,

^{i.e. the}

polarization parallel

or

perpendicular

^{to the}

magnetic

^field

direction,

^was

selected with

a linear Polaroid HN 38. Part of the emission

light

^{could be}

separated

^out of the main beam

by

^a

semi-transparent mirror,

and fed into a

Spex

monochromator. This feature allowed a simulta-

neous measurement of the stress effects on the EPR lines and the zero

phonon

^line.

The microwave

frequency

^was about 9.15 GHz.

The

klystron

power ^was modulated at 500 Hz

by

^a

PIN diode. The

sample

^was held between two quartz rods at the centre of a

cylindrical TEon

^mode

cavity.

Light

beams could _go

through

^{the resonant}

cavity through

^{slots cut} in its wall. Stress was

applied

^to

the

sample through

^the quartz ^rods

by

^{means of a}

lever system ^{which was set}

just

^{above the resonant}

cavity.

The direction of

applied

^{stress was}

always perpendicular

^to that of the

magnetic

field. The maxi-

mum pressure

applied

^{was about 8}

kg/mm2.

^The

sample,

^{the resonant}

cavity

^{and the lever} system ^were immersed into the

liquid

helium bath.

3.2 METHOD OF ANALYSIS OF STRESS EFFECTS. -

Our

samples

^were grown

by

^{W. C.}

Spicer

^Ltd

using

the arc fusion

technique. They

contained both F and F+ centres as a result of the reduction condition in the arc furnace

during

^the

growth

process. The

concentration

^{of F centres was}

generally high,

^so

that most of the luminescent centres were very close

to the surface of the

crystal receiving

^the

illuminating light.

^{Therefore we}

mainly

observed uniaxial stress effects on excited states of F centres localized near

the surface and ^not those in the bulk. This could be an

important

^{source of error if the stress} apparatus ^was

not

operating

under favorable conditions. In fact two measurements of stress

effects,

made with the same

applied

pressure ^on the same

sample

^but

by

^illumi-

nating successively

^{its two}

opposite faces,

gave results which differ

by

^{as much as 20}

%.

^To reduce this type of _error, it was decided to measure

directly

^stress

effects in that

part

^{of the}

crystal

from which the lumi-

nescence took

place.

^{Hence our}

experimental

proce- dure was as follows : stress effects on the EPR _spec- trum and the zero

phonon

^{line were} recorded simul-

taneously.

^{Then the} shift Av of the low _energy components ^{of the zero}

phonon

^line

[7]

^{was used}

as a calibration of

applied

^{stress. The}

advantage

of this method of

analysis

^is illustrated in

figures

^5a

and 5b. In

figure 5a,

^{the shift}

AHy

of the low field EPR line of the Y centre and the shift Av of the _{low energy} components ^{of the zero}

phonon

^{line are}

plotted against

^the pressure

F applied along

^the

[100]

^direction.

The low stress data is

bad, indicating

^{an error in the}

definition of the zero

applied

^{stress. Also the}

experi-

FIG. 5. ^- Illustration of the method of analysis ^{of stress effect.}

AHO (fl ⁼ Y, Z) ^is the shift of the low field line of the vibronic state 1 fi, 0, 0 », and Av the shift of the low energy components of the zero phonon line, with FII[IOO], H//[OO1], ^{T = 1.7 K}

(see text).

mental

points

^marked

by

^{arrows do not} fit with the others. In

figure 5b,

^{the same data are shown but}

by plotting AHy against

^{Av. The}

scattering

^of

experi-

mental

points

around the

straight

^{line is}

quite

^{small :}

the zero stress is well

defined,

^{and the}

points

^marked

by

^{arrows now} fit well with the others. The shift of the EPR line _per unit _pressure is related to the

slope

of the

straight

^line

by (Fig. 5b) :

where

fi

⁼

X,

^{Y or}

Z, and cE

^{is a} linear combination of the

compliance

coefficients

(see Appendix I).

The value of

V2’

^{= 4.5 x 104}

^{cm -’}

^obtained

by

Edel et al.

[7]

^{was confirmed}

by

^{our own} measurements.

As discussed in section

2,

the effect of a

[ 111 stress

on the vibronic

triplet 1 fui, 0, 0 »

^is

strongly ^reduced,

and no shift of the zero

phonon

line could be observed.

Thus for this direction of the

applied

^{stress one had}

to

rely

^on the calibration of the stress apparatus.

3.3 EXPERIMENTAL ^{RESULTS. - As}

previously reported [3]

six EPR lines were observed around g ⁼ 2 for an

arbitrary

orientation of the

magnetic

field.

They

result from the

superposition

^{of three}

equivalent

axial spectra ^whose symmetry ^{axes are}

along

^the

[100], [010]

^and

[001]

directions. Each of these spectra

corresponds

^{to the} relaxed excited states of the F centre in one of the three vibronic states

1 fi, 0,

⁰

».

^Under

applied

stress, the shifts of the two

lines of each EPR spectrum ^{were observed to be}

opposite, regardless

of the direction ^of the

magnetic

field H and of the pressure F. This

clearly

^{shows that}

the main effect of

applied

^{stress is to}

change

^{the fine}

structure

(through

^{the two}

parameters

^{D and E of the} standard

Spin

Hamiltonian of a S ⁼ 1

level),

^and

that the effect on

the g

^{tensor is}

negligible.

This is the

reason

why

^{the cross terms}

involving

both the strain and the

magnetic

^{field were not} considered in the effective Hamiltonian of table I.

a) 100

^{stress. - The EPR}

lineshape

^{did not}

change

much with the

application

^{of a}

[100]

^stress.

Only

^{a line}

broadening

^{of at most 20}

%

^{for stress} up to 8

kg/mm2

^{was observed. When the stress was}

released,

^{the linewidth went back to its}

original

^value.

The most

striking

^{feature was the}

change

ⁱⁿ

intensity

of the X lines

(see Fig. 6).

^With

increasing

pressure, the Y and Z lines shifted

linearly (see Fig. 5b),

^while

the X line shifted and

severely

decreased in

intensity vanishing

^{at about 4}

kg/mm2.

^{This can be}

explained

^as

follows. A

[100]

^stress

splits

^{the X} vibronic level from the Y and Z vibronic levels

by

_y ^{LI =}

3 V2

^F

_(see

^inset

2c

E

(

in

Fig. 6).

^{Since the X} vibronic level is shifted toward

higher

energy, its

intensity

will decrease as a result of thermalization inside the vibronic levels

[5, 10].

However due to internal

strain,

^the

intensity

^{of the X}

line does not fall

exponentially

^with

applied

^stress

FIG. 6. ^- Effect of [100] ^{stress on the low field} part ^{of the EPR} spectrum ^at 9.15 GHz with HII[OOI] ^{and T = 1.7 K.} Inset shows the combined effect of internal (b) and external [100] ^stress (d)

in the vibronic triplet 1 fi, 0, ⁰ ».

as

expected

^{from a} Boltzmann distribution. Instead the observed resonance

signal originates

^{from all}

the X centres for which

(d - ô) ;! kT,

where à is the

splitting

of the vibronic levels due ^to internal strain.

Typically

^one has kT ’" 1.2

cm-1 (1.7 K),

^{£5 ’" 6}

cm-1, (d - ô) -

^1.7

^cm-1

^{for F = 2}

kg/mm2.

^{As a conse-}

quence of this

effect,

the measured shift of the X line is not a linear function of the

applied

stress, and the information obtained on this line is _very

unprecise.

Making

^the measurement at

high

temperature, ^i.e.

kT >

(d - 8),

should reduce the effect of internal strain on the X centre. However at 4.2

K,

^the

signal

to noise ratio decreased for the whole EPR spectrum, and the _accuracy of the measurements remained very poor.

Three different

samples

^{were used for}

[100]

^stress

experiments.

^{In table II are}

given

^the

averaged

^shifts

of the low field lines of the

X,

^{Y and Z centres for the}

magnetic

^{field H}

parallel

^to

[001].

^{Due to the} poor accuracy of the X line measurements,

only

^two

linearly independent equations

^are obtained for the three parameters

A, %

^{and e.} Measurements with H

along

a

[011]

^direction

give

^no

independent

information. It has been shown elsewhere

[10]

^{that the additional}

information needed to determine these three _para- meters can be obtained from the

analysis

^{of the zero}

phonon

^{and EPR}

lineshapes.

^The

resulting

^{set of}

values for A, ^{% and C}

satisfying

both the data in table II and the

lineshape analysis

^is

given

^{in table III. It is}

found that the two

parameters

^{and C are much}

smaller than the

parameter

^%.

Uniaxial stress effects at zero

magnetic

field have been studied

by Krap et

^al.

[11].

^Their

parameters Gij defining

^the

spin-lattice coupling

^to

Eg

^{strain are}

related to our parameters, ^{? and} e,

by

and

Using

their data of

Gij,

^{one obtains}

TABLE Il

Effect of [100]

^{stress on the EPR} spectrum

A, % and C are the spin-lattice coupling parameters (see text),

and t) ⁼ 0

cm-1,

ⁱⁿ

good

agreement ^{with our measu-}

rements

(Table III).

b) [111] stress.

^-

Figure

⁷ shows the effect of a

f 111 ]

stress on the low field line of the Z centre for H

parallel

to

[112].

The line shifted and broadened

strongly

^even

for

relatively

^{low stress.} On the other hand, ^no effect could be observed on the zero

phonon

^{line which was}

recorded

simultaneously.

This indicates that there is no induced

Eg ^strain,

^{and that} the observed effect

on the EPR line is

only

^{due to}

T2g ^(or Alg)

^strains.

Variations in the

lineshape

^{made the} measurement

of the line shift somewhat

ambiguous.

^{Also as dis-}

cussed in section

3.2,

^{since no}

splitting

^{was observed}

for the zero

phonon

^{line one had to}

rely

^{on the cali-}

FIG. 7. ^- Effect of [111] ^{stress on the low} field line of the vibronic state Z, 0, 0 )) ^{at 9.15} GHz, with H//[112] ^{and T = 1.7 K.}

bration of the stress apparatus. ^This

might

^{result in}

a

large uncertainty

^{for the} actual value of the

applied

stress. For all these reasons, it is better to use the ^zero field data of

Krap et

^al.

[11]

^{to determine} the parame- ters 9) and e. These authors observed that the EPR line of the Y centre

split

^{into two} components ^under

an

applied

^stress

parallel

^to

[101]. They

assumed that the

spin-lattice coupling

^to

T2g

strain is _zero, i.e.

9) ⁼ 6 ⁼

0,

and attributed the observed

splitting

^to

a

misalignment

^{of their}

crystal.

^{We checked that,}

in order to

explain

the measured

splitting

^{in terms of}

coupling

^to

Eg

^strain

only,

^{one would have to assume}

TABLE III

Parameters

defining

^the

effective

Hamiltonian

The main contributions of the various orders of the perturbation calculation are given in columns 3 to 5. R, fa, ^..., fd ^are overlap integral

functions of the harmonie oscillator defined in appendix ^II and also shown in figure ² Perturbation calculation

0 ^Ref. [5], (b) ^Ref. [1 )], (C) ^Ref. [3], (d) ^Ref. [7], e) ^Ref. [11].

a

misalignment

of more than 10°. This is _very

large

^foi

a

crystal

^which

presents good

cleaved faces. Therefore

one has to consider the

coupling

^{of the}

spin

^{levels to}

T2g

^strain.

^Assuming

^{a maximum}

misalignment

^{of 5°.}

and

using

the data of

Krap

^{et al.}

[11],

^one gets

This result is not in contradiction with our own

measurements.

4. D1SCUSSIOII. -4. 1 MECHANISMS OF THE SPIN- LATTICE COUPLING. ^- We shall now discuss the

physical ^origin

of the various ^terms of the effective Hamiltonian which describe the

spin-lattice coupling (terms

^{7 to 11 in table}

I). They

^{do not} contain the

magnetic ^field,

^so

they

^{are related to} the variations of the fine structure with

applied

^{stress. These terms}

can be calculated

by

^the

perturbation technique

^used

by

^Ham

[4]

^to determine the fine structure constant D.

Details of such calculations are

given

ⁱⁿ

appendix

^II.

Here in order to make clearer the mechanisms of the

spin-lattice coupling

and without

entering

^into many

calculations,

^we shall discuss the case of strong Jahn- Teller

coupling

where the adiabatic

approximation

can be made.

The

unperturbed

Hamiltonian

JCo

^{includes the} linear

coupling

^to

Eg

modes of vibration. We assume

that the

coupling

^to

T2g

^{modes can be}

^neglected.

Then the

states 1 fi, n,,, n, » M )

defined in section 2

are

eigenstates

^of

Jeo.

^The

perturbation

Hamiltonian

Je1

^{is the sum of 4 terms :}

which describe

respectively

^the

spin

^Zeeman

effect,

the

dipole-dipole interaction,

^the

spin-orbit

interaction and the direct effect of strain in the electronic manifold

3T 1 u (1

⁼

1,

^{S =}

1).

^Detailed

expressions

^are

given

in

appendix

^{II. For zero}

applied

^stress and up ^to second-order

perturbation,

^{the fine structure}

(term

³

in table

I)

is described

by

^a

single

parameter ^{D :}

Ddd

is the contribution from the

dipole-dipole

^inter-

action

Jedd.

It appears in first order of the

perturbation

since

Jedd

^contains

diagonal

^orbital

operators. Dso

comes from the second order effect of the

spin-orbit

interaction

(Eq. (8)),

^{and is}

given by :

The

complete expression

^{of D}

(Eq. (14))

^{should include}

a third term

resulting

^from the second order effect of the

spin-orbit

interaction between the

3T lu

^and

1 T 1 u

^levels

[3].

^The

separation

between these levels

being large ( N

^{7 000 cm’}

1),

^{this term} is small and will be

neglected

^{in the} present

analysis.

When the Jahn-Teller

coupling

^is

large, fb

^{has the}

simple asymptotic

^value

1/3 SE [4].

Then the expres- sion

(15)

^{becomes :}

As illustrated in

figure

^{1, the}

quantity

³

EJT

^{is the}

separation

^{of the}

potential

energy surfaces at the

equilibrium position

^{in the lowest surface. Now}

from

(14)

^and

(16)

^{one sees that the}

spin-lattice coupling

may

proceed through

three different mecha- nisms :

(i) Effect of

^{strain on the}

dipole-dipole coupling.

Just as

Ddd,

this effect _appears in first order of the

perturbation.

(ii) Effect of

^{strain on the}

spin-orbit coupling.

This term appears in second order of the

perturbation

as :

(üi) Effect of

^{strain on} the vibronic levels. This effect is shown in

figure

^{1 for a strain} eo. The set of

, states X;

ne,

ne» and Y, né, n;»

^{are shifted}

by

! V2 eo,

^{and the set of}

states Z, n;, n; » by - V2

eo.

This is

equivalent

^{to a} variation of the Jahn-Teller energy

f1EJT = ! V2

^{eo, and of}

Dso by :

This term appears in third order of the

perturbation (second

order of the

spin-orbit coupling,

first order of the orbit-lattice

coupling).

The detailed calculations of these effects should take into account the fact that the fine structure is described

by

^{a tensor.}

They

^{are carried out} in appen- dix II for a Jahn-Teller

coupling

^of

arbitrary strength

i.e. without

making

^{the adiabatic}

approximation.

The result of the identification with the effective Hamiltonian is

given

^{in table III.}

fa, fb, f,,, and hi

^are

sums of harmonic oscillator

overlap integrals.

^Their

dependence

^{on the}

Huang

^and

Rhys

^factor

SE

^is

shown in

figure

^{2. Note that for}

SE >

^{1 one} gets

fc fd Z 1 P. ^{and Àr}

^are defined in _appen-

(3 SE)

²

*

dix II.

They

describe the variations of the

dipole- dipole

^and

spin-orbit

interactions with the components of the strain ^tensor which transform like the irredu- cible

representation

^{r. In table}

III,

^{it can be seen that} the mechanism

(i)

contributes to A, % ^and D, ^the mechanism

(ii)

^to all the five parameters .4, ..., 9, and the mechanism

(iii)

^to

only 3

^{and 9. In}

^addition,

another mechanism is found in 2) and 9, which invol-

ves first order of electron-lattice and

dipole-dipole

interactions.

4.2 ESTIMATE OF THE JAHN-TELLER _COUPLING, THE SPIN-ORBIT INTERACTION AND THE DIPOLE-DIPOLE INTER- ACTION. ^- To our

knowledge

there exist no direct

measurements or estimates of the

spin-orbit

^inter-

action

parameter Â

^{and the}

dipole-dipole

interaction parameter

PI

^{of F centres in CaO. From} theoretical calculations of the electronic states, Wood and Wilson

[12]

^found

SE

^{= 2.14 and}

nWE

^{= 266 cm-1.}

On the other

hand,

^the temperature

dependence

^of

the emission spectrum ^was studied in detail

by

^Hen-

derson et al.

[2].

These authors deduced the

Huang- Rhys

factor of the

coupling

^to all modes S ⁼ 5.7 from the

intensity

ratio of the zero

phonon

^{line and of}

the broad band.

Using

their data of half the Stokes shift Shco ⁼ 1 100

cm - 1,

and of the second moment of the broad band at very low temperatures

one obtains S ⁼ 6.45 and nw ⁼ 172

cm - 1.,

^nw

being

the mean energy of

coupled

^{modes. However}

analysis

of the temperature

dependence

of the second moment,

M2(T),

^{leads to hco = 274}

^{cm - 1.}

Another value of nw ⁼ 160

cm - 1

was obtained

by

Bates and Wood

[13]

in

interpreting

^the temperature

dependence

^{of the}

emission spectrum ^{in the} range 50 I)C-350 OC.

In the

following

^{we shall} try ^{to estimate}

SE, nWE’ À

and

PI by using

^{data in table III.}

Combining

^rela-

tions

(3.8)

^and

(3.9)

^{of table}

III,

^one gets :

An upper limit for

PF

^can be obtained

by assuming 1 PE I 1 Pl L

^{which is not} unreasonable for a

dipole- dipole

interaction.

Then PE

^{can be}

neglected

ⁱⁿ

equation (19)

since it will be shown below that

1 PI I

^0.1

cm - 1.

One obtains a system ^{of three} relations

(3. 1), (3. 3), (19)

^{for the four} parameters

SE, h(OE, Â, Pl.

^{Then this} system of relations is solved

by taking hcoe

^{as a free} parameter ^{in the} range 170- 270

cm - 1.

It is found that for

hcoe k 250 cm - 1,

^,

the obtained value of

EJT

⁼

SE hcoe

exceeds the

experimental

value of half the Stokes shift.

Taking h(OE

^{= 200}

cm -1,

^one obtains the

following

^{set of}

parameters :

Choice of another value of

liWE

^{in the} range 170- 250

cm -1

would

change SE by

^{less than 10}

%, and 1 À 1 by

^{less than 20}

%,

^while

PI

^remains

negative,

- 0.1

cm - 1 Pi ;

0. These results remain valid in a more detailed

analysis

^{in which}

PE

^{is not}

dropped

from

equation (19),

^but

kept

^{as a free}

parameter

^of the same order of

magnitude

^than

Pl.

Uncertainties on the values of A and C are so

large

that relations

(3.7)

^and

(3.9)

^{cannot be used to}

determine

ÂA

^and

ÂE.

Nevertheless

they

indicate that

ÂA and ÂE probably

^have

opposite signs. Assuming IÂFI- JÂJ I and IPFI- IP, 1,(F=A, E, T), and using

^{the set of} parameters

(20)

^one gets

which is not in contradiction with data in table III, and shows that mechanisms

(i)

^and

(ii)

^are

actually

effective.

By combining (3.10)

^and

(3.11)

^{one obtains}

_ 9) - Â 2 V3 fc/(1iWE)2. V3

is unknown but

expected

to be

equal

^{to or} smaller than

V2.

^{In such a case one} gets 6 - + % ²

cm -1,

^{to be}

compared

with the value of 0.85

cm-1

determined from data of

Krap et

^al.

[11].

Similar estimate can be made for terms 12 to 18 of the effective Hamiltonian

(Table I).

^It

gives

^{the same order}

of

magnitude

for all the

coefficients,

^{... xr} 1

cm -1.

In conclusion the present

analysis

showed that the dominant mechanism of

spin-lattice coupling

^{is that}

involving

the direct effect of strain on the vibronic levels

(mechanism (iii)

in section

4.1).

^It

gives

^rise

to terms such as

Use of the effective Hamiltonian allows a syste- matic

approach

^{of the}

problem

^to be made and no

importand

^{term of the}

spin-lattice coupling

is omitted.

Moreover table 1 is _very

helpful

for the discussion of the

broadening

^{of the zero}

phonon

^{and EPR}

lines

[10],

and of the

paramagnetic

relaxation. It is

interesting

^{to note that terms 7 to 11 involve}

diagonal

orbital operators, ^and thus allow

only

^relaxation

inside a

given

Jahn-Teller well. It can be

easily

^shown

that

they give

^{rise to} relaxation times of the order of

one second at 4.2 K. This is much

longer

^{than cha-}

racteristic times

(

²⁰

ys)

^observed

by

^{Cibert et al.}

[5]

and

Bontemps-Moreau et

^al.

[6].

^{These authors}

showed that the most effective mechanism for the

spin-lattice

relaxation should involve

tunnelling

pro-

cesses between Jahn-Teller wells.

Acknowledgments.

^{- We wish to thank M. Glas-}

beek for

making

his data avalaible to us

prior

^to

publication.

^{We are thankful to R.}

^Buisson,

^J.

^Cibert,

P. Edel and R. Romestain for _very

helpful

discussions.

The technical assistance of R.

Legras

^is

greatly

appre- ciated.

APPENDIX 1

SOME DEFINITIONS. ^-

Operator /.

^-

lTy

^operates

inside the

ground

^vibronic

triplet T 1 u (l’

⁼

1).

^For

r ⁼

A,,, ^1£

^{is the}

^{unity operator.}

^{For r =}

^Tlu,