RAMAN SCATTERING IN RESONANCE WITH THE ABSORPTION BAND OF THE F CENTRE IN ALKALI HALIDES : BREAKDOWN OF THE SCATTERING SELECTION RULES DUE OF THE SPIN ORBIT INTERACTION

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RAMAN SCATTERING IN RESONANCE WITH THE

ABSORPTION BAND OF THE F CENTRE IN

ALKALI HALIDES : BREAKDOWN OF THE

SCATTERING SELECTION RULES DUE OF THE

SPIN ORBIT INTERACTION

E. Mulazzi, M. Bishop

To cite this version:

JOURNAL DE PHYSIQUE Colloque C7, supplément au n° 12, Tome 37, Décembre 1976, page C7-109

RAMAN SCATTERING IN RESONANCE WITH THE ABSORPTION BAND

OF THE F CENTRE IN ALKALI HALIDES : BREAKDOWN

OF THE SCATTERING SELECTION RULES DUE OF THE SPIN

ORBIT INTERACTION

E. MULAZZI f

Istituto di Fisica, Universita degli Studi di Milano Via Celoria 16, 20133 Milano, Italy, and Gruppo Nazionale di Struttura della Materia del CNR

M. F. BISHOP (+*)

Department of Physics, University of California. Irvine 92717, U. S. A.

Résumé. — Nous présentons la théorie et les évaluations numériques pour la section de la diffu-sion Raman en résonance avec les bandes d'absorption F des centres F dans les cristaux d'halogénure alcalin, caractérisés par un couplage d'interaction spin orbite X différent de zéro dans les états électro-niques excités. On montre que pour toutes les polarisations de la lumière incidente et diffusée les sections de la diffusion Raman pour les densités d'états phononiques ri", r j et rt sont toujours différentes de zéro. La théorie est appliquée aux deux cas particuliers des centres F dans les cristaux de K l et CsF.

Abstract. — We present the theory and the theoretical evaluations for the cross sections of the

polarized Raman scattering in resonance with the F absorption bands of the F centres in alkali halides, characterized by a spin orbit interaction coupling coefficient A different from zero in the excited electronic states. We show that for all the polarizations of the incident and scattered light the cross sections of the scattering processes determined by the densities of phonon states transforming according to the irreducible representations r^, r j , / £ are always different from zero. The theory is applayed to the particular cases of the F centres in CsF and in KI.

Introduction. — We report the results of the first

theoretical calculations that prove the breakdown of the usual selection rules [1] for first-order resonant Raman scattering induced by defects that are charac-terized by a spin degenerate ground state and dege-nerate excited state with strong spin orbit interaction. The present results extend those of a recent general theory [2] which predicts that selection rules for Raman scattering from defects depend not only on the

sym-metry of the defect site but also on the electronic structure of the defect [2]. Here we show that due to

the spin orbit interaction in the excited electronic state new selection rules are valid ; in Oh symmetry

these state that for all the polarizations of the incident and scattered light the cross sections of the scattering processes determined by the densities of phonon states

(f) Work supported in part by ONR contract number NOOO14-69-A-0200-9003.

(+) Work supported in part by AFOSR contract number 76-2887.

(*) Present address : Department of Physics, Purdue Uni-versity West Lafayette, Indiana 47907.

transforming according to the irreducible represen-tations rf, r$, r% are always different from zero.

The breakdown of the selection rules that we discuss here is important for a large variety of systems, in particular for all systems with impurity ions that have a degenerate ground state, transforming according to the irreducible representations of the double group of the local site symmetry. These include most of the 3d, 4f, and 5d group impurity ions in polar crystals [3].

In the present paper, however, in order to show clearly the breakdown of the usual selection rules and to illustrate the importance of electronic structure in determining the Raman scattering selection rules, we consider the scattering in resonance with the absorp-tion transiabsorp-tion « ls-2p » of the F centre in KI and CsF, taking into account the spin orbit interaction in the excited 2p state and the spin degeneracy in the

ground state Is. The local site symmetry is Oh in

both the cases. We have chosen the two cases as exam-ple for the present theory because the F centres that are characterized by spin orbit interaction in the 2p state have different properties for the electron phonon interactions in the excited electronic state in alkali

C7-110 E. MULAZZI, M. F. BlSHOP

halides and in cesium halides [4]. The two cases consi- dered are the best candidates in order to prove the breakdown of the usual selection rules because of the following reasons : (i) The 2p excited state of the F centres in both the crystals presents strong spin orbit interaction. (ii) The Raman spectra cannot have any contribution from the ground state vibronic levels scattering [5], since the electronic ground state is a

r'6f state and has only spin degeneracy. Then the experi- mental Raman scattering spectra in these cases are only determined by the scattering processes of the phonons created by the electron phonon interactions in the degenerate excited state. (iii) The laser line frequencies usually available are in resonance with the absorp- tion F bands considered. Recent experimental results reported in ref. [6] of Raman scattering in resonance with the F band in KI shows clearly the breakdown of the usual selection rules, due to the spin orbit interac- tion in the excited state. Since in that case the more important electron phonon interaction in the excited electronic state is that transforming according to the

r:

irreducible representation of the local site symmetry Oh [7], the breakdown is proved by the appearance of the T: density of phonon states in all the polarized spectra for every incident and scattered light polarizations.

Absorption band shape and resonant raman scat- tering. - Since the properties of the optical tran- sition absorption band are very important in deter- mining the Raman cross section intensity in reso- nance with this transition, we first evaluate the opti- cal absorption band for the systems considered. We begin with the F centre in CsF absorption band. This case is particularly interesting because the electronic absorption band due to the transition we consider exhibits structure due to the spin orbit (SO) interaction and to the dynamic Jahn-Teller (JT) effect via the electron-phonon (EP) interactions [8, 91. The SO inter- action determines the splitting of the excited elec- tronic state such that the new states transform accor- ding to r'i and

r,.

Therefore, the absorption band is a superposition of the two bands due to the electronic transitions from the ground state, transforming as r6+, to the excited electronic states

rg

and

ri.

The JT effect determines the structure of the band related to the electronic transition from the degenerate ground state

r6+

to the

ri

excited state [8, 91. In this excited state, the JT active EP interactions transform accor- ding to the

r:

and T: irreducible representations of the point group Oh. On the contrary the F band in KT has a structureless bandhape because the T': EP interaction is stronger than the SO interaction and than the JT active r': and r': EP interactions. Then the overall effect coming from the SO interaction and the JT active EP interactions appears only as giving contributions together with the

r:

EP interaction to the moments of the band.

In the both cases the model Hamiltonian for the excited electronic state is

where He is the electronic Hamiltonian in which the spin orbit interaction has already been taken into account, that is :

In this expression

where El and E2 are the electronic energies of the

I'i

and

r g

electronic states respectively ; E is the energy of these states, in the absence of the spin orbit interact ion, relative to the ground state energy, which we set equal to zero ; I is the coupling constant for the SO interaction ; a', ai are the electron operators, where i labels the six electronic basis vectors of the representations and T s . H, is the harmonic pho-

non Hamiltonian for the imperfect polar crystal,

and He, is the EP interaction Hamiltonian. Because

the EP interaction is localized about the impurity, it is more useful to analyze it in terms of irreducible representations Ty (y labels the degeneracy of r ) , where

r

= r:,

r:

and r': :

where h = 1 ; b+ and b are the phonon operators

labeled by the irreducible representations to which they belong, and h(ry) are the EP interaction matrices evaluated in the excited electronic state and transform- ing according to T:, ~ 3 f , and r:, and whose coupling coefficients are ~ ( r : ) , ~(r:), and c(T:) respecti- vely. These matrices are easily evaluated by using the equivalent operator technique [lo] and will be given in a more extended publication.

BREAKDOWN OF THE RAMAN SCATTERING SELECTION RULES C7-111

1 X2

dx-

o a ( ~ ) ~ / '

where p is the concentration of defects in the crystal, f is the electric dipole moment oscillator strength for the transition, and

Also,

where

The plot of the absorption band of the F centre in CsF versus frequency eq. (6) is given in figure 1, for

A , =0.61AI, A3=0.35111, and A, =0.451AI at

15 OK, with the experimental curve for comparison

191. We have used the value

o

= 0.5

1

A 1, which is

approximately the average of the phonon density spectrum of the pure crystal [12]. The plot of the F band in KI will be given elsewhere. The band shape

FIG. 1 . - Absorption band shape R(w) from eq. (5) with A1 =0.6 11I, A ) =0.35 112

1,

A s =0.45 112

1,

G = 0 . 5 ( 3 , \ a t

15 K with the experimental curve from ref. 191 for comparison.

E is the energy of the electronic excited state without the spin orbit interaction and in the static lattice. Here E = 0 and w is

given in units of ) t

1.

-

5

in this case is gaussian like and it has been evaluated by using the values of Ai as derived by the coupling coefficients and the densities of phonon states given in 171.

In order to evaluate the resonant Raman scattering for the F centre in CsF and KI, we use the theory given in 1131, by taking into consideration of the properties of the JT active EP interaction matrices in the excited states, which in this case anticommute at any time. In addition, we calculate the Raman cross section for different incident and scattered light polarizations. That is, we consider the different matrices of the pola- rized dipole moment, in the directions of the electric fields, for the transitions from the doubly degenerate ground state (r:) to either the four-fold degenerate

( F 8 ) or doubly degenerate

(rd)

excited states. It is possible to derive that for all the polarizations of the incident and scattered lights the cross sections of the scattering processes determined by the Raman active densities of phonon states, i. e. those transforming according to I':, r:, r:, are always different from zero.

The results for the

1

100

1

+ 1100

1

polarized first order resonant Raman scattering per unit solid angle

SZ and for 900 scattering geometry is given (at T w 0) by :

( a )

_-

_THEORY

0 - 1 1 2 E. MULAZZI, M. F. BISHOP

where

Here, wL is the laser frequency, c is the speed of light,

o is the Stokes frequency shift for the first order pro- cess,

p(r,?,

m) are the perturbed projected densities of phonon states transforming according to the irre- ducible representations

r,?

with v = 1, 3, 5 and the Sij(wL, w ) are related in a complicated way to the functions given in eq. (6a)-(6d) and will be given in a more extended publication.

Analogous results are obtained for the

I

110

I

-+

1

1 i0

1

and the

1

100

1

+ 1010

I

polarized Raman scatterings. Also, for these cases the differential Raman cross sec- tions are given in terms of the S i j ( o L , w ) and of all the three one phonon densities of states transforming as I':,

r:,

and

r:.

In order to estimate the impor- tance of the breakdown of the selection rules with respect to the usual selection rules, it is useful to eva- luate as a function of w , the intensity of the Raman cross sections for the separate processes induced by the densities of phonon states transforming according to r : ,

I

'

:

and

r:

respectively. In figures 2 and 3 we give the partial and the total Raman cross sections for the polarized scatterings in resonance with the F band in CsF, as explained in the captions. We use the same values for the Ai as in the calculation for the absorp- tion spectrum, figure 1.

From the figures we note that all the processes deter- mined by the r : ,

r:

and

r:

densities of phonon

. .

LASER FREQUENCY w,

Fro. 2.

-

Raman scattering intensities for the. polarized

1

100

1

+ 1100 ] first order process. The partial intensities for the process supported by the densities of phonon states trans- forming as T:,

rz

and

r:,

with the total intensity, are given. The parameters are the same as in figure 1, and w~ is the laser

frequency (eqs. (7) and (S)), in units of

I

IZ

I,

with E = 0.

states have always intensities different from zero in all the polarized scatterings. The absolute maxima of the intensities are for w , E 0.5

1

I

1

and a t this laser

frequency there are also the maxima of the intensities for the processes allowed by the usual selection rules,

i. e. the processes supported by the T : and

r:

den- sities of phonon states in the [loo] -, [loo] polarized scattering and the processes supported by the T : density of phonon states in the [110] + [ I ~ o ] polarized scattering. We note also from figure 2 that in the reso- nance range for the laser frequency w , <

-

2

1

I

1

and

o,

>

3

1

1

1

the intensity of the processes determined by the density of phonon states is larger than that due to the processes determined by the T : den- sity of states. The same is valid for the [I101 + [ I ~ o ] polarized scattering in the resonance region for the laser frequency w , > 3

1

1

1

(see fig. 3). Then the Raman spectrum in resonance with the F band consi- dered here, changes with the different resonant condi- tion of wL, not only in relation with the different contributions of the processes determined by the new selection rules, but also in relation with the different contributions of the processes allowed by the usual selection rules. Moreover by performing the same kind of evaluations one finds that the absolute maximum of the intensity for the processes in the [100] + [010] polarized scattering is at 0, 21 0.5

1

I

) and at this

laser frequency the more intense processes are those determined by the T : density of states. Furthermore also for this polarized scattering there is an impor- tant contribution coming from the phonon processes supported by the T : and

I

'

:

EP coupling interactions.

L A S E R FREQUENCY U L

FIG. 3. -The same as in figure 2 for the polarized

1

110

1-+1

170

1

BREAKDOWN OF THE RAMAN SCATTERING SELECTION RULES C7-113

By performing the same kind of evaluations for the tizations. The intensities ratio of the [loo] -t [OlO] Raman scattering in resonance with the F band in KI scattering with respect to the [loo] -, [loo] scatter- by using the EP coupling coefficients and the densities ing for the

r:

density of states peak at o = 97.7 crn-' of phonon given in 171, it is possible to derive that the is 0.16 when the laser frequency is o, = 6 000

a,

first order scattering process supported b y the

r:

while this ratio becomes 0.23 when the laser frequency density of phonon states appear for all the three pola- is o, = 63 28

A.

References

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[2] MULAZZI, E. and TERZI, N., Solid State Commun. 18 (1976) 721.

[3] ABRAGAM, A. and BLEANEY, B., Electron Paramagnetic

Resonance of Transition Ions (Clarendon, Oxford)

1970.

[4] HENRY, C. H. and SLICHTER, C. P. in Physics of Color

Centers edited b y Fowler W. B. (Academic Press

N. Y.) 1968.

[5] GUHA, S. and CHASE, L. L., Phys. Rev. B 12 (1975) 1658. [6] BUISSON, J. P., SADOC, A. and TAUREL, L., Proceedings of the Third International Conference of Light Scattering

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