Relaxation in the 3T1u state of F centres in CaO

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Relaxation in the 3T1u state of F centres in CaO

J. Cibert, P. Edel, Y. Merle d’Aubigné, R. Romestain

To cite this version:

J. Cibert, P. Edel, Y. Merle d’Aubigné, R. Romestain. Relaxation in the 3T1u state of F centres in CaO. Journal de Physique, 1979, 40 (12), pp.1149-1160. �10.1051/jphys:0197900400120114900�.

�jpa-00209202�

Relaxation in the 3T1u ^{state of F centres in CaO}

J. Cibert, ^P. Edel, ^{Y. Merle} d’Aubigné ^{and R. Romestain}

Laboratoire de Spectrométrie Physique (*), Université Scientifique ^et Médicale de Grenoble,

B.P. 53X, 38041 Grenoble Cedex, ^France

(Reçu ^{le 1 er} juin 1979, accepté ^{le 29 août} 1979)

Résumé.

Nous avons détecté optiquement la relaxation dans l’état excité 3T1u ^{du centre F dans CaO en utili-}

sant des hyperfréquences ^{en bande X} (9 GHz). ^A 1,7 K, ^on observe deux composantes : ^une composante ^lente égale à la durée de vie radiative (3,7 ms), due à la variation de la population ^{totale du niveau} 3T1u ; ^{une compo-}

sante rapide, due à des transitions d’un puits Jahn-Teller à l’autre. On montre que les composantes tétragonales

et les composantes trigonales ^des déformations induites par les phonons doivent être prises ^en compte. ^{Ces tran-} sitions dépendent beaucoup de la valeur des contraintes internes statiques. ^En utilisant la répartition ^de contraintes, déterminée dans un travail précédent, ^on peut reconstruire par ^un calcul numérique ^les signaux expérimentaux.

Abstract.

We have studied by optical detection the relaxation in the 3T1u ^{excited state of F centres in CaO}

using microwaves in the X band. At 1.7 K both a slow and a fast recovery ^are observed. The slow component, equal ^to the radiative lifetime (3.7 ms) ^{is due to a} variation of the total population ^{of the} 3T1u level. The fast component ^{is due to} transitions from one Jahn-Teller well to another. Evidence is given ^{that both the} tetragonal

and the trigonal components ^{of the} phonon induced strain field have to be considered. All these relaxations _depend

very much on the static internal strains. Numerical calculations using the distribution of strains determined in previous work fit the experimental ^data.

Classification Physics ^Abstracts

76.30M - 78.55

76.30M201378.55

Introduction.

The spectroscopic behaviour of the F centre in CaO is now well understood : optical

spectra ^{have been studied} by ^{Henderson et al.} [1, 2], optically ^detected magnetic ^resonance (ODMR) by

Edel et al. [3]. ^{Uniaxial stress} experiments [4, 5] ^show

the effect of strain on either the zero phonon ^{line or}

the ODMR lines, ^{confirm the} inhomogeneous ^nature

of these lines, ^and give ^some information concerning

the distribution of internal strains [6]. References to

previous ^{work are} given in several _papers [7, 8].

The excited state is both an orbital and a spin triplet 3Tlu. ^{It is} strongly coupled ^to Eg tetragonal ^modes

of vibration and undergoes ^a static Jahn-Teller effect

(JTE) [9]. The static nature of the JTE can be seen by

ODMR [3] : ^three equivalent tetragonal spectra ^are observed, ^{each of these} spectra corresponding ^{to one}

of the Jahn-Teller wells. The ground vibronic level is still a Tlu triplet [9]. ^{Inside this} triplet, matrix elements of off-diagonal electronic operators (when expressed

in the real basis 1 X >, 1 Y), 1 Z » ^{such as} spin-orbit interaction, dipole-dipole interaction and coupling ^to

the trigonal T2g ^{strains are strongly reduced [9]. This}

is not the case for the effect of the tetragonal Eg ^strains

eo, ee which ^is represented ^{in the} real basis by operators

(*) Laboratory associated with the Centre National de la Recher- che Scientifique.

having only diagonal matrix elements. As ^a result of the coupling ^to internal strains of Eg ^{symmetry, the}

orbital degeneracy ^{of the} 3Tlu ^ground vibronic level is lifted as shown in figure 1. In the relaxation effects

we will see that both the tetragonal ^{and the} trigonal

components ^{of the} time-dependent-strain ^{induced by}

the phonon ^{field have to be taken into account.}

Fig. ^1.

^{Level scheme of the} 3T 1 u multiplet. ^The magnetic ^field

is aligned along ^{the z} cubic axis [001]. The six ODMR lines are

refered to by ^the six numbers 1 to 6. The numbers in parenthesis give the relative deexcitation rates.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197900400120114900

We first present experimental ^{results on relaxation}

in the triplet 3T 1 u ^{level (sections 1 and 2). Then we}

show that from the results of the static studies it is

possible ^{to derive a} dynamical spin Hamiltonian (3)

which explains ^{well the} dynamical behaviour in high magnetic ^{field and at low} temperature (4).

1. Experimental ^methods.

Contrary ^{to a conven-}

tional Tl relaxation time measurement, ^we do not monitor the population difference between the two sublevels which are perturbed by the microwaves.

The dynamic behaviour of the populations ^{or, at} least, of some linear combinations of them, is obtained by monitoring ^the intensity of the emitted light ^{of a} given polarization ^after having ^{switched on or off the reso-}

nant microwaves [10]. Very ^little change ^{has to be}

made to the ordinary ^ODMR apparatus ^{to observe} relaxation. We have worked at X band (about ⁹ GHz)

using ^{a Varian} electromagnet (typically ^0.3 T) ; ^the microwaves are switched by ^{a PIN diode. The} sample

is cooled down to pumped liquid ^helium temperature

(typically ^1.7 K). ^{The F centre} luminescence is excited

by ^a mercury lamp. ^The light emitted in the whole fluorescence band is collected through ^a polarizer ^on

a photomultiplier (EMI ^{9558 or RCA} 6217). ^The output ^{current is} recorded, ^{via a} preamplifier ^with adjustable bandwidth and a multichannel analyzer having ^a time resolution of _{0.2 gs.} Signal averaging

is performed ^on typically ^{104 microwave} pulses.

2. Experimental ^results.

^{At fixed} temperature the signal ^is expected ^to depend ^{on :}

-

the pair of levels which is perturbed by ^the microwaves, i.e. which of the six ODMR lines has been selected,

-

the direction of the magnetic ^{field : we have}

worked with B parallel ^{to the} [001] axis,

-

the polarization with which the emitted light ^is

observed. In our case, we monitored signals ^{in a and}

in n polarizations, i.e. linear polarizations respectively perpendicular ^and parallel ^{to the} magnetic ^field.

The results at 1.6 K are reported ⁱⁿ figures ^{2 and} 3, schematically ^{for all cases and with} complete ^details

for some of them. When the microwaves are switched

on or off, ^{one observes a} signal which contains both ^a fast and a slow component (Fig. ^{2a). The} general

appearance of this signal ^{is the same} for all the low field lines (lines 1, 2, 3 ⁱⁿ figure 2a) ^but depends strongly

on the polarization ^{of the} monitoring light. ^{For the} high ^field lines, ^the signal ^{has the} opposite sign. ^We

will describe the signal observed when switching ^off

the microwaves.

2.1 FAST COMPONENT.

Depending ^{on the line} being saturated and the polarization ^{of the} monitoring light, ^{this fast} component ^{can or cannot be fitted to a} single exponential (Fig. ^{3). The} signal ^to noise ratio

was not good enough ^to try to extract more than one

exponential ^{from the} data. When the signal ^can

Fig. ^2a.

^Schematic diagram of the relaxation signal.

Fig. ^2b.

Experimental ^{results on ODMR line 6} (see figure ^{1 for}

the identification of the line).

reasonably ^{be fitted to an} exponential law (e.g. Fig. ^3a

in n polarization and the end of the signal ^{in 6} pola- rization), ^{the time constant} is found to be about 20 gs at 1.7 K. This rapid variation is opposite ⁱⁿ sign ^{on the}

two polarizations ^{of the} fluorescence and it involves

a change in the distribution of the populations among the different sublevels emitting light of different

polarizations.

2.2 SLOW COMPONENT.

It is exponential ^{with a}

time constant equal ^{to the whole} 3Tlu level lifetime

(i.e. ^3.7 ms) ^{as measured} by simply switching ^{off the}

excitation light. ^The sign ^{is the same on both} polari-

zations and here it involves a change in the total

population ^{of the} 3Tlu ^level.

At 4.2 K the rapid components ^{of all} signals (n ^or

Q whatever the line considered) ^{show a} quasi expo- nential behaviour with a time constant of about 4.5 _JlS, the signs ^{of the} signals being ^{the same as at 1.6 K.}

3. Theory.

^{3.1 RATE} EQUATIONS.

To com-

pute ^the relaxation signal ^{we have to consider the}

different mechanisms allowing transitions between the various levels. We are here interested in the evolution of the populations of the nine sublevels of the 3Tlu

level and we will have to consider four types ^{of tran-}

sitions, ^as illustrated in figure ^{4 :}

Fig. ^3.

Detail of the rapid component ^{of the} relaxation signal.

3a) Signal ^{for the centre} parallel ^{to the} magnetic ^field (line 6).

3b) ^and 3c) Signal ^{for the centres} perpendicular ^{to the} magnetic

field. The magnetic ^{field was} slightly misaligned ^{from the} [001]

direction in order to distinguish ^{the two lines that} would be other- wise superimposed.

-

non radiative transitions between two sublevels of the 3Tlu manifold with a rate R,j ^{for the} transitions from the sublevel j ^{to the sublevel i. The mechanisms}

of these internal transitions will be detailed later in this section,

-

transitions from one of these sublevels to the

ground ^state 1 Aig. ^{We will assume that they are purely}

radiative with a rate A; i and that there is no other deexcitation mechanism for the 3Tlu ^{level. Their}

Fig. ^4.

Definition of the different transition rates.

relative values are recalled on figure ¹ by ^{the numbers} in parenthesis,

-

transitions from the IT 1 u ^{level to the sublevel i}

of the 3Tlu ^{manifold. These} feeding transitions have

an unknown rate Pi. ^{We will see} however, ^{that since}

the internal transitions inside the 3Tlu ^{manifold are}

faster than the deexcitation rates A i, ^{the values of} Pi

have only ^a slight ^{effect on the final} populations ^and knowledge ^{of them is not} necessary for interpretation

of the signals,

-

microwave induced transitions with a rate

Wik

Wki ^{which is} obviously different from zero

only when i and k are the two sublevels between which microwave transitions are induced.

We assume that the conditions of weak optical pumping ^are satisfied. The ground ^level population

is then practically ^left unchanged ^{and the} populations

ni of the sublevels of the 3Tlu ^{manifold are controlled}

by ^{the nine} following linear differential equations :

The time dependence of n; ^{is then} given by ^{the sum}

of nine exponential functions :

The nine time constants r. depend only ^{on the tran-}

sition rates Rjj, ^{A ;, Wik and not on the feeding rates} Pi ^as long ^{as the} approximation ^{of weak} pumping

remains valid whereas the amplitudes B. and Bm depend ^{also on the} initial conditions.

3.2 INTERNAL TRANSITION RATES.

These tran- sitions between the different sublevels of the 3T 1 u

manifold are due to the coupling ^{of the F centre to}

the phonon ^{field. Here we} give only ^an outline of the calculation. More details are given ^{in the} appendix.

Considering ^{the low} temperatures ^{at which the} experiments have been performed, ^{we will assume}

that we have only direct transitions between the sub- levels : either direct transitions between spin ^sublevels

of a given Jahn-Teller well or transitions between sublevels of different Jahn-Teller wells.

We will make use of two simplifying features of the

problem : ^the phonon ^{bath can} be considered to a

very good approximation ^as being isotropic ^{and the} phonon wavelengths ^are large.

(i) ^The isotropy ^{of the} phonon ^bath is demonstrated

by ^the results of the measurements on sound velocity by ^Son and Bartels [llJ : vt

^{5 x 103} m/s ^and

vj 1 = 8.3 ^x 103 m/s ^for transverse and longitudinal polarizations respectively, ^{whatever the} direction of

propagation.

(ii) ^The long wavelength approximation (Van ^Vleck [12]) is usual when dealing ^with phonons ^{of low} energy,

i.e. with a wavelength ^much larger ^{than the} interatomic

separation (here ^a typical ^value of the random splitting

between the three orbital states is 3 cm -1, leading ^to

a phonon wavelength ^{of 500} Â). ^This approximation

states that the time dependent ^strain may be described

by ^the usual strain tensor. The coupling ^{to the} phonon

strain field is therefore described by the usual orbit lattice coupling Hamiltonian [4, 5] :

The exact definition of the different operators ^and of the strain components ^are given ^{in the} appendix,

ea, ee and ee’ e,, el and e, ^are linear combinations of the strain components that transform respectively

like the Ai (totally symmetric), ^E (tetragonal) ^and T2 (trigonal) representations of the cubic _group.

From stress experiments [4, 5], ^{we know that :} (i) Vi is small and _anyway we will see that the

term V, ea la ^is ineffective in spin lattice relaxation.

(ii) ^{The value of} V2 ^{is 4.5 x 104 cm-1.}

(iii) V3 ^{is not} measurable : it is strongly ^reduced by Jahn-Teller effect ; ^{we can} assign ^{5 x 102 cm-1}

as an upper limit of the reduced term V3 R, ^{where R} is the Ham reduction factor [9].

We can write the Hamiltonian JCoL ^{in a condensed}

form :

where F is the index of the representation (Al’ ^{E or} T2) ^and y the component of F (y = 0 or e, if F = E).

It is shown in the appendix ^{that the} transition rate

Rji ^{from the sublevel i to the} sublevel j ^is given by :

a is a numerical factor involving ^{the sound} velocity

and the density ^{of CaO. Its value is 0.15} ± ^0.04 when all energies ^are expressed ^{in cm -1.}

Using expression (1) ^{we can now determine the}

various relaxation paths ^{inside the} 3Tlu ^level.

3 . 2.1 Transition inside a given ^well.

^{We can use}

the values ( 1 1 0 Ty 1 j) ^that have been measured by

Le Si Dang et ^al. [5] when 1 i > and 1 j > belong ^{to the}

same well : these matrix elements are all smaller than 3 cm-1 and lead to transition rates smaller than 0.2 s -1 : these rates are much smaller than the deexci- tation ^rate and we will neglect ^{them as} they ^would

lead to relaxation times of the order of seconds, ^much longer ^{than the} measured times.

3.2.2 Transition between diffèrent ^wells.

^We

have only ^{to consider the} tetragonal ^{and the} trigonal

components of the strain field : the electronic operator involved in the symmetric term VI ea la being propor- tional to the unity ^{matrix, this term cannot induce}

any transition.

3.2.2. x Relaxation induced by ^the tetragonal

time-dependent-strain.

^{We look at the} transitions that can be induced by ^{the term} v2(ee le + eE lE). ^If

the vibronic eigenstates ^{are the} states 1 X > >, 1 Y», 1 Z )>), ^{we see that a} tetragonal time-dependent-strain,

being diagonal ^{in this basis, cannot induce a transition}

between these states. If such a transition exists, ^it

can only ^{be due to a} mixing [13] ^{of the orbital states.}

Such a mixing ^can be induced by any off-diagonal

operator. ^{The first case we can} think of is a mixing by ^the spin-orbit coupling ^{Âl. S. Let us} look for instance if a transition is possible ^{between the} states Z, +1 )) and ) Y, 0 )) (+ ^{1 and 0 stand for the} MS spin ^quantum number) ; ^{we have to} calculate :

where are the perturbed ^{states :}

Combining eqs. (2) ^and (3) ^one _{gets :}

where R is the Ham reduction factor [9].

We can see that, though tetragonal ^{strains are not}

affected by ^{the JTE, we need an} off-diagonal operator

to provide ^a mixing ^{of the states and the} effect of this

mixing operator ^is quenched by ^{the JTE, so} that, ^as

expected, ^the probability ^of tunnelling ^{between two}

wells has a factor proportional ^{to the} square of the

overlap ^{between the} wavefunctions of the two wells.

The various transitions made possible by spin-orbit mixing ^are given ⁱⁿ figure ^{5. On the same} figure ^we

recall the polarization selection rules : we can see

that this spin-orbit ^driven process (that ^{we will call}

now the Eg ^{process) connects} levels which emit light

of the same polarization. ^{So we can} group the sub- levels into two sets :

-

a group of levels emitting ^Q light, ^i.e. light polarized ^{in the} plane perpendicular ^{to the} magnetic field,

-

a group of levels emitting ^{either n} light ^or nothing.

A similar grouping ^{of levels was made} by ^{Bill and}

Silsbee in analyzing reorientation of 0- in CaF2 [14].

Other interactions can also provide ^the mixing :

the most important ^{are the} dipole-dipole coupling

which preserves the selection rules of the spin-orbit coupling, ^{and the} coupling ^{to the} random, static, trigonal ^strains.

3.2.2. Relaxation induced by ^the trigonal ^time- dependent-strain.

^{We now look at the} transitions that can be induced by ^{the term}

Fig. ^5u.

Relaxation rates induced by ^the spin ^orbit coupling

and the tetragonal component ^{of the} phonon strain field (Eg ^pro-

cess). ^E has the value

(see text).

Fig. ^5b.

Relaxation rates induced by ^the trigonal component of the phonon strain field (T2g ^{process). T has the value}

(see text). For both figures, ^{the random} splitting between the three orbital states is not represented.

We just replace ^the general operator Ory ^{of eq. ( 1 ) >}

by V3 lç, V3 111, V3 1, ^{and we get} transitions without

spin flip (Fig. ^{5), with a} transition rate :

if 1 ) and 1 j ) ^{have the same} spin ^state.

Here no mixing ^is necessary, due to the fact that the trigonal ^{strains are} off-diagonal operators. ^Their effect is also reduced by ^{JTE so that we see that both}

processes, the Eg ^{process and this process (which we}

will call T2g ^{process) are reduced by R, the Ham}

reduction factor, through ^{either the} mixing operator

or the strain operator.

There are two very important différences between the two processes :

-

the Eg process groups the sublevels into two sets characterized by ^their polarization (7c ^or u) ^selec- tion rules whereas the T2g process groups the sub- levels into three sets characterized by ^their spin quan- tum number,

-

the energy dependence ^{of the} Eg ^{process is given}

Fig. ^6.

Dependence of the relaxation rates on the phonon energy.

6a) T2g ^{process (1.5 K). 6b)} T2g ^{process (4.2 K : the} scale has been divided by ^{a factor 30). 6c)} Eg ^{process (1.5 K). 6d)} Eg ^process

(4.2 ^{K :} the scale has been divided by ^{a factor} 3). The functions

are AE’[exp(AEIkT) - 1J-1 ^{for the} T2g ^{process and}

for the Eg ^process.

These two dependences ^{are shown in} figure ^{6. We}

see that for the Eg ^{process, at very low phonon ener-} gies ^the transition rate does not fall to zero : the increase in the mixing ^{of the orbital states which}

varies as (Ei - Ej)’ ^{balances the}

dependence given by ^the coupling ^{to the} phonon

bath. We will come back to this important point ⁱⁿ

the final discussion. We have also to note that in our case we cannot make use of the high température approximation ^{since kT} (- ^1.2 cm-’) is smaller than but comparable ^{to the} phonon energy (- ³ cm-’).

One _may also think of relaxation _processes induced

by mixing with excited vibronic states. In fact, _contrary

to what happens ^{in the static case where} second order terms can be more important ^than first order ones,

higher ^{order terms} in relaxation _processes between different wells are always ^affected by ^{the Ham reduc-}

tion factor and are therefore negligible compared ^to

the first order terms.

4. Relaxation model.

4. 1 GENERAL CONSIDE- RATIONS.

Having found the different theoretical

expressions for the transition rates, ^one would in

principle ^{just have to} introduce the numerical values into the nine evolution equations ^and solve them. In fact we know only ^{the order of} magnitude ^{of the} spin

orbit coupling ^{constant and an} upper limit for V3.

We will therefore first examine whether we can make

use of some simplifying ^features.

The first simplification ^{one can think of would be}

to consider that only ^{one of the two} Eg ^and T2g

processes is rapid ^with respect ^{to the lifetime. A}

qualitative discussion of the experimental ^results (Figs. 2, 3) will show that such a simplification ^{is not} possible. ^If only ^{one of the two} processes is rapid,

we can group the levels into two or three sets inside which relaxation is rapid : ^{three sets} corresponding

to the three spin ^{states for a} T2g ^{process or two n}

and J sets for an Eg ^{process, as shown in figure 5.}

When switching ^{on or off the} microwaves, ^a quasi-

Boltzmann equilibrium is reached within each set.

Quasi-Boltzmann equilibrium ^{means that a Boltz-}

mann equilibrium ^{is not reached} (but ^{it does not}

mean a Boltzmann equilibrium ^{at a} temperature different from the sample temperature). ^The departure

from Boltzmann equilibrium ^{is of the order of the}

ratio of the deexcitation rates to the relaxation rates, and these quasi-Boltzmann equilibria, ^{with or without} microwaves, ^are only slightly different. So, ^{when the} microwaves are switched off, the fast component ^of the relaxation, which under this hypothesis ^corres- ponds ^to relaxation inside a set, has ^a very small

amplitude. ^One expects ^{to detect} only ^{a slow} compo- nent which corresponds ^{to a} rearrangement ^{of the}

populations ^{between the different sets and has a time}

constant of the order of the lifetime. This is obviously

not the case : one can see in figure 2 that the fast and the slow components of the relaxation signal

have comparable amplitudes. ^{In other} words, ^{in order}

to observe a non negligible ^fast component ^{in the} relaxation signal, ^{one has to assume} the existence of

a fast relaxation path between the two sublevels involved in the microwave transition. Figure ^{5 shows}

that this can only ^{be the case if both} Eg ^and T2g

processes ^are rapid compared ^{to the lifetime.}

The shape ^{of the} rapid signal (independently ^{of its}

time scaling ^{since we can consider as constant the}

variation due to the slow component) ^{is then} governed by ^{the ratio of the} Eg ^and T2g processes, i.e. the ratio

V2 ÂIV3 DE, ^{where AE} represents ^an average energy

separation between levels which is due to intemal strains. We will now consider the two extreme cases

V2 ÀIV3 4E > 1 and V2 ÀIV3 ^{DE 1.}

4.2 AN OVERSIMPLIFIED MODEL.

We will first

assume V2 ÂIV3 ^{AE » 1, which means that the} Eg

process is much more efficient than the T2g ^process,

both being rapid compared ^{to the lifetime. We will}

therefore _{group the} different sublevels in the two n

and 6 sets defined before. The Eg process induces very rapid transitions between the sublevels inside each set. Transitions between the two sets are induced

by microwaves and by ^the T2g ^{process (Fig. 7). The}

Fig. ^7a.

^Level grouping ^{in the} oversimplified ^model (Eg ^process

very much efficient than the T2g ^{process, see text).}

Fig. ^7b.

Expected ^ODMR signal ⁱⁿ this model.

population distribution at equilibrium ^{within the} 3Tlu

manifold when microwaves are applied ^{is then the} following :

i) ^Due to the Eg ^process quasi-Boltzmann equili-

brium is achieved inside each of the two groups and this whatever the feeding ^{rates since the} Eg ^process

is much more efficient than the radiative transition

rates.

ii) ^{The ratio of the} populations ^{of the two levels} corresponding ^{to the resonance} is intermediate bet-

ween unity (case ^of complete saturation of the line)

and quasi-Boltzmann equilibrium (no ^microwave

power), depending ^{on the} efficiency ^{of the microwave} compared ^to the average effect of the T2g ^process.

iii) ^{The total} population ^{of the} 3Tlu ^{manifold is}

given by ^the ratio between the feeding ^rate (which

is given by ^the power of the exciting light) ^{and the}

effective deexcitation probability

relative populations ^{of the} sublevels in each n or 6

set remain in quasi ^thermal equilibrium. ^The T2g ^pro-

cesses give ^{rise to a} transfer of population ^{from one}

set to the other, restoring ^the quasi-Boltzmann equi-

librium _{among the} nine sublevels. The observed rate of _recovery is an average of the rates of transfer bet-

ween the various sublevels, ^and may be easily comput- ed by introducing ^{into the} equation of evolution the fact that quasi ^thermal equilibrium ^is always ^achieved

within each n or J set. The signal ^has opposite sign

when detected on Q or n polarization. ^{The variation}

in the populations of the sublevels results in a variation of !eff’

the _average lifetime. As the populating ^rate

is left unchanged, ^{there is a} readjustment ^{of the total} population ^{of the} 3Tlu ^{level ; this} readjustment ^is

seen by ^a variation of the total fluorescence, ^{with a}

time constant equal ^{to the} effective lifetime.

This qualitative description ^{is in rather} good agreement ^{with the} experimental ^results. Quantitati- vely ^{one has} just ^to introduce into the rate equations,

the persistence of Boltzmann equilibrium within each set : this leads to two differential equations ^which

can be easily solved in order to obtain an analytical expression ^{for the} expected relaxation signal.

The rates given ⁱⁿ (4) ^and (5) ^however depend strongly ^on the energy DE of the phonons involved,

and this frequency is related to the local (internal)

strains. When averaging ^{over the} distribution of internal strains one gets ^a relaxation signal ^{which is}

very highly ^non exponential whatever the polariza-

tion of the monitoring light ^and whatever the ODMR line which is saturated. These conclusions are not

experimentally verified : see e.g. the quasi exponential signal obtained in n polarization ⁱⁿ figure ^3a.

Note that the opposite assumption, V2 ÀIV3 ^{AE « 1} (i.e. T2g process much ^more rapid ^{than the} Eg ^process)

would lead to an inverted signal ^{on 7r} polarization

and has therefore to be rejected.

4.3 MORE REALISTIC MODEL.

We are therefore in the intermediate case : V2 ÀIV3 ^{AE - 1.} Though

the Eg ^{and the} T2g ^{processes are still more efficient}

than the lifetime, ^both have the same order of magni-

tude. In this case only ^{a numerical} computation ^is possible. ^{We have to consider a} statistical distribution of the static tetragonal ^strain components e. and e,.

Their effect is determined by ^the following ^{results :}

i) ^{From the} study ^{of the} lineshape ^{of both the zero} phonon ^{line and the ODMR lines, Le Si} Dang et ^al. [6 ]

were able to determine the str’ain distribution : they

find the distribution for V2 e. and V2 e, ^{is best} repre- sented by ^a Voigt profile ^{but can be} approximated by ^a Gaussian distribution with ^a standard deviation u = 3.1 cm-’.

ii) ^{The same studies} [6] have shown that when the

magnetic ^{field is} along ^{the z} axis, ^the shape ^{of the}

ODMR line is practically governed by ^the parameter

V2 ee alone. Our calculations have therefore been

performed by giving ^{to this} parameter ^{the value}

corresponding ^{to the centre} of the ODMR line, ^where the relaxation measurements have been done

(V2 e(J

^2.2 cm-’) letting V2 e, ^{run over} all its dis- tribution. The calculation is then performed ^{in the} following way for a given ^{value of} V2 ÂIV3 : ^the

relaxation signal ^is computed when the microwaves

are switched off, the initial conditions being ^deter-

mined by solving ^the relaxation equations ^at equili-

brium in _presence of microwaves. We then compute

the effective signal by summing ^over V2 e, ^and using

the strain distribution mentioned before. The same

procedure is followed for different values of the ratio

V2 ÂIV3 ^{to find the} best value in order to fit the shape

of the experimental signal. Once the fit is found at 1.6 K, ^the signal ^is computed ^{at 4.2 K and} compared

to the experimental signal, ^{which is} quasi exponential

in both n and J polarizations ^{with a time constant of}

about 4.5 ps. A first set of parameters

gave ^a good ^{fit at 1.6 K but} predicted ^{a time constant}

of 1.5 _{gs at} 4.2 K, ⁱⁿ poor agreement ^{with the} experi-

mental value. This suggests that the 1.6 K temperature

was underestimated because of heating ^{of the} crystal (about ^{50 mW of} light). ^{In fact one can measure the} temperature fairly accurately by looking ^{at the} magnetic ^circular polarization, which has been shown to vary ^as th (g,uB B12 kT) [3]. We have checked this law with different powers of excitation and have found that at the levels of light ^at which the micro-

waves experiments ^were performed the effective tem-

perature fitting ^{the th} (gllB B12 kTeff) ^{law was of the}

order of 2 K when the temperature of the helium bath measured by ^its vapor pressure ^was 1.6 K. A new

fit at 2 K then _gave

V2 ÂR

^{700 cm - 2 and} V3 R = 200 cm-l,

these values leading ^{to a time constant of 3.4} gs at 4.2 K. From the stress experiment [4] ^{one knows the}

value V2

^{4.5 x 104} cm-’ - ^{so we can deduce a}

value of the reduced spin-orbit coupling ^{constant :}

ÀR

1.5 x 10-2 cm-1.

Figure ^{8 shows the} comparison ^{of the} experimental

and computed signals ^{for the line 6}

1

Fig. ^8.

Comparison ^between experimental ^and computed signal

on line 6. 8a) Plot on linear intensity ^{scale : the o} represent ^the computed signal. ^{8b) Same} signal plotted ^on logarithmic scale ;

+ : experimental signal ; ^{o :} computed signal.

One sees that the calculation _agrees well with several

interesting points :

i) ^{In J} polarization, ^the signal ^{cannot be fitted to}

an exponential function, ^this signal reflecting ^an

average ^over the strain distribution.

ii) ^{In n} polarization, ^the signal is much closer to

a single exponential function ; ^the calculation shows indeed that the n signal ^is mainly ^{due to the centres}

for which the |Z ) ^and thé 1 X > or ) | Y ^{states are} nearly degenerate, ^thus selecting partially ^{the centres} by their local strain.

iii) ^{Still in n} polarization ^{one can see that the} signal

seems to be delayed ^with respect ^{to the} switching ^off

of the microwaves. When the microwaves are switched off, only ^the time derivatives of the populations ^of

the two sublevels directly ^{involved in the transition} (emitting ^Q polarized light ^or nothing, ^{in the transition} being considered) ^are different from zero. The popu- lations of the four levels emitting ⁿ light begin ^there-

fore their rearrangement ^{with zero time} derivative, showing ^{hence a} delay ^{in their} response ^to the switch-

ing-off of the microwaves.

iv) ^The agreement ^is good ^not only ^{for the} shape ^of

the signals ^{but also for the relative} amplitudes ^{of the n}

and J signals.

5. Discussion.

First we will _compare the set of parameters ^obtained by ^the relaxation experiments

with those deduced by ^{Le Si} Dang et ^al. [5] ^from

strain and ODMR measurements. The agreement ^is rather good for the reduced spin-orbit ^{constant :}

ÂR

1.5 x 10-2 cm-1 for this work compared ^to ÂR ^{3 x 10-’} cm-’ given by ^{Le Si} Dang et ^al.

The agreement ^{is somewhat worse if we} introduce the value of  given by ^{Le Si} Dang et al. : 1  1 ^{15 cm-’.}

This value would give ^{a ratio} Y3/Y2 N 4.5 whereas

one expects ^a ratio of order of unity ^{or less} [5] since

the Jahn-Teller coupling ^is predominantly ^to Eg

modes. Different possibilities may be raised to explain

these discrepancies.

i) ^The analysis ^{of Le Si} Dang et ^al. gives directly

an order of magnitude ^{of the reduced} spin ^orbit coupling ^{constant ÂR. The true} coupling ^{contant À}

is not determined directly but is deduced from a

general fit of various experimental ^{results to a} simpli-

fied model. This model assumes Jahn-Teller coupling

to only ^one pair of Eg ^{modes and one Huang and Rhys}

coupling parameter ^{S. This is} obviously ^{an oversim-} plification ^{and as the} reduction factor R varies _expo-

nentially ^{with the} parameter S, ^{it can lead to a} large

error in  = ÂRIR.

ii) ^{We have} only ^{taken into} account two processes :

an Eg ^{process due to the spin orbit coupling and a} T2g ^{process due to the trigonal component} of the time

dependent strain field. We have neglected ^another Eg

process where the mixing ^{is due to the} trigonal ^static strains, ^{which has the same} polarization ^selection

rules as the T2g ^{process. Taking into account this}

process would have led to three more integrations ^on

the distributions of the local trigonal ^strains. Quali- tatively, ^{as this} process has the same selection rules and the same order of magnitude (but ^{not the same}

energy dependence) ^{as the} T2g ^{process one can say}

that taking ^{into account this} process would lead to

a lower value of V3 ^{R, thus} lowering ^{the ratio} V3lV2.

All the relaxation _processes are interwell _processes, i.e. the only efficient relaxation rates are obtained by jumps between different Jahn-Teller wells. The relaxa- tion _process inside a given ^{well is} very inefficient.

This is qualitatively ^confirmed by Krap et ^al. [16].

Applying ^{in zero} magnetic ^{field a} [110] ^{stress which}

leaves a non degenerate ^Z well lowest, they ^{show that}

the relaxation time increases rapidly ^{with the} applied

stress. However, ^the measured relaxation time is much too fast. The origin ^{of this} discrepancy ^has probably ^to be searched in the experimental ^{methods :}

as a result of the high light level and the high ^micro-

wave power (20 W), ^the crystal may be warmer than the helium bath. It seems that a temperature ^{of the} order of 2.5-3 K could explain their results.

The last point ^{we want to} emphasize ^{concerns the} polarization selection rules. In two recent papers by Bontemps-Moreau et ^al. [15] ^and Krap et ^al. [16], ^it

was demonstrated that the relaxation rates without

spin flip (our T2g ^{process) were} predominant compared

with the rates with spin flip (our Eg ^{process), whereas}

our work takes both into account in comparable

manner. There are two comments to make on this

point :

i) ^{In our} experiment, although the measured value is predominantly ^{due to the} T2g ^{process, the} Eg ^pro-

cess leads to shorter signals ^{but with} very small

amplitudes ^{due to} the detection scheme.

ii) ^In figure ⁶ one can see that the Eg ^{process is}

favored at low phonon energies, ^{due to the}

dependence ^{whereas the} T 2g ^process is favoured at

high phonon energies, ^{due to the}

dependence. ^Our measurements are essentially govern- ed by weakly ^{strained centres whereas the two other}

measurements are interested in highly ^{stressed centres :}

these highly ^{stressed centres are obtained either} naturally by selecting ^{them in the} wings ^{of the zero} phonon ^line [15] ^or by applying ^{an external stress} [lb],

so that their results are not surprising.

Conclusion.

We have shown that the relaxation

’occurs via jumps between different Jahn-Teller wells.

Similar cases were already observed in other systems

exhibiting ^a static Jahn-Teller effect : the ’Ti ground

state of 0- in CaF2 [14] ^{and the 2E} ground ^state

of Cu" in double nitrate [18, 19]andinzincfluorosili-

cate [20]. ^{For the F centre in CaO the} situation is more

complex ^{due to the} higher multiplicity ^{of the} 3Tlu

electronic level, ^{but the} optical ^{detection scheme} gives

more detailed information. Two relaxation _processes

are clearly identified : one is induced by ^the trigonal

component ^{of the} phonon induced strain field (T2g process) ^{and does not} change ^the spin ^{state ; the other}

one, induced by ^the tetragonal component ^{of the} phonon induced strain field and made allowed by ^the spin-orbit coupling (Eg ^{process), may change the spin}

state but gives only very small amplitudes ^{in the relaxa-}

tion signal, ^{due to} the detection scheme. A two step- mechanism including ^{these two} processes may ^reverse the spin ^{in a} given Jahn-Teller well. It was also shown that these two processes ^occur with comparable probability : ^{for the F centres in a} region ^{of the} crystal

with large ^internal strains the T2g ^process predominates

whereas in a region of small internal strain the E.

process predominates. Quantitative analysis ^{of the} decay times leads ^to values of the parameters ^{ÀR and}

V3 ^{R which are in} good ^{order of} magnitude agreement with estimation deduced from static ^stress experiments.

Acknowledgments.

^{We want to thank} Pr. R. Buis-

son and Dr. Le Si Dang for fruitful discussions and M. Merlin for assistance in computing.

Appendix.

^We shall calculate the transition rates induced by ^the orbit lattice Hamiltonian :

The quantities ea, ee and ef.’ e,, e,, and e ^{are linear}

combinations of the usual strain tensor components.

They ^transform respectively ^{like the} representations A1, Eg ^and T2g of the cubic _group and are defined by :

The quantities 1, ^{are orbital} operators ^defined by

Ham [9] ^{and Le Si} Dang ^{et al.} [5] ^{and the} V,’s ^are coupling coefficients ; V2 ^{has been defined} by ^Ham [9]

but V3 ^is different from the coefficient V3 ^{of Ham}

in order to take the same normalisation for the diffe- rent combinations _{ei. We} write the Hamiltonian in a more condensed form :

where F stands for the different representations (F

A1, E, T 2) ^{and y for the different} components of a given representation (y

^{0 or a if F}

E).

We have to represent ^the coupling Hamiltonian in terms of phonon creation and annihilation operators.

We will use for this the expression (2) of Stevens [17]

which gives ^the displacement ^{u of an ion at the} posi-

tion r :

In this expression _o)pl kpl ap, ap ^and gp, ^are respectively

the frequency, the wavevector, the annihilation and creation operators ^{of the} phonon ^mode p whose

polarization ^{vector is} _cpp. ^{M is the mass of the} crystal.

With these notations, ^the Hamiltonian JeL, ^the eigen- statues 1 t/1L ) ^and the energy EL ^{of the} lattice, ^treated

as harmonic, ^are given by :

where np ^is the number of phonons ^{in the} pth ^mode.

Using eq. (2) ^we get ^the values for the different strain combinations. For instance :

More generally

The fr, ^{have obviously the same} transformation

properties ^{as the} _ery. ^The only ^non vanishing ^matrix

elements of ery ^{are then the} following :

We will also have to use the phonon density ^of

modes. For longitudinal phonons ^we write that the number dN of phonon modes with an energy hco

within d(hm) and characterized by ^a wavevector k within the solid angle ^{dQ is} given by :

where Y is the volume of the crystal and v, ^the longitu-

dinal sound velocity. ^{The same} relation is valid for the two transverse phonon density ^of modes, just replacing ^the longitudinal velocity ^VI by ^{the trans-}

verse velocity vt.

The probability Rji of observing ^a transition of the centre from astate 1 ) ^{to another} state Ii) ^with

emission of a phonon of energy Ei - Ej ^{is then given}

by ^the following expression :

The term in brackets [ ] ^is given by ^Fermi’s golden rule and is the probability for the simultaneous transi-

tions 1 i > --> 1 j > ^{of the centre} and 1 n, >--- > np ^{+ 1 ) of the phonon bath. As we are only} interested in the transitions of the centre, ^{we sum over} all possible ^{states of the} phonon ^{bath, each state} being weighted by ^the probability ^Pr (nI... _np ^...) that it is occupied.

Taking ^{into account} the fact that the phonon density of modes is isotropic, eq. (3) gives :

The integral ⁷ appearing ⁱⁿ (4) ^is easy ^to evaluate :

The constant Ar depends ^{on the} representation ^{T .}

But in this _case, as we are looking ^{at a} problem ^of spherical symmetry (due ^to the fact that the sound

velocity, and therefore also the phonon density ^of modes, ^is isotropic) ^{we can take for F} the represen- tations of the rotation _group. We will therefore have

a constant A’J)o ^which gives Arl ^{in our case and a}

constant A’J)2 ^which gives ^{in our case} AE+ T2. ^This

shows that the constant Ar ^{is the same for the E and}

the T2 representations. ^{As we} don’t need the A,1

constant (there ^{is no} relaxation due to the symmetric component ^{of the} phonon induced strain field) ^we have just ^{to evaluate A}

AE

AT2. ^For longitudinal phonons ^{one finds : A = 4} nk2/5. ^For transverse pho-

nons one finds A

6 nk’15 ^{for the sum of the two} polarizations.

If we now make the additional assumption ^that

the lattice is and remains in thermal equilibrium, ^we

can replace 1 (np ^{+ 1) Pr (ni * 1 .} np ...) by ^{the Bose-}

inii

Einstein factor e1iw/kTI(e1iw/kT - 1). ^One gets ^the proba-

bility of transition with emission of ^a phonon :

The same calculation can be made for a transition with absorption ^{of a} phonon just by replacing (np+ 1 )

by n » :

where

If we put ^{in a the} numerical values available for CaO,

one finds ^a

0.15 + 0.04 if all energies ^are given ⁱⁿ

cm cm-1 . 1

Expressions (5) ^and (6) ^can be condensed in _expres- sion (7) whatever the energy ordering ^{of the levels i} and j :

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