ON THE ORIGIN OF THE RED SHIFT OF THE RAMAN EXCITATION PROFILE FROM THE ABSORPTION SPECTRUM

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ON THE ORIGIN OF THE RED SHIFT OF THE RAMAN EXCITATION PROFILE FROM THE

ABSORPTION SPECTRUM

E. Mulazzi, R. Tubino, G. Dellepiane

To cite this version:

E. Mulazzi, R. Tubino, G. Dellepiane. ON THE ORIGIN OF THE RED SHIFT OF THE RAMAN EXCITATION PROFILE FROM THE ABSORPTION SPECTRUM. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-493-C6-495. �10.1051/jphyscol:19816144�. �jpa-00221209�

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JOURNAL DE PHYSIQUE

CoZZoque C 6 , suppZ6ment au n o 12, Tome 42, de'cembre 1981 page c6-493

ON THE O R I G I N OF THE RED S H I F T OF THE RAMAN EXCITATION PROFILE FROM THE ABSORPTION SPECTRUM

E . Mulazzi, R. ~ubino* and G . ~ e l l e ~ i a n e * "

I s t i t u t o d i Fisica de Z Z 'Universita', uia CeZoria 16, MiZano, ItaZy

.

Gruppo NazionaZe Struttura Materia del CNR, MiZano, ItaZy

.

* I s t i t u t o d i Chimica deZZe MacromoZecoZe del CNR, v i a Corti, 12, MiZano, ItaZy.

* * I s t i t u t o d i Chimica E'isica, Dipmtimento d i Chimica, v i a EZce d i Sotto, 8, Perugia, ItaZy.

Abstract.- A red s h i f t of the position of the Resonant Raman Scattering exci- t a t i o n p r o f i l e with respect t o the position of the maximum of the e l e c t r o n i c absorption band has been observed in a number of compounds. The variation of the inverse vibronic lifetime vn in the excited s t a t e w i t h the quantum number n i s believed to be the most relevant contribution t o the s h i f t of the Raman excitation p r o f i l e . Calculations based oh t h i s model f o r a molecule in a crys- t a l l i n e matrix a r e reported.

A detailed knowledge of the properties of the excited s t a t e s of a molecule in a crystal can be obtained from the analysis of the electronic absorption spectrum and Raman excitation p r o f i l e (REP). As a matter of f a c t , the e l e c t r o n i c absorption band shape and the resonant Raman cross section a r e determined by the electron vibrational coup1 ing and by the vibrational relaxation processes i n the excited e l e c t r o n i c s t a t e . While generally the REP closely mimicks the experimental absorption Spectrum, i n some ~ o m ~ o u n d s ~ ' ~ ' ~ the maximum of the REP i s considerably redshifted with respect t o the maximum of the absorption c o e f f i c i e n t . This e f f e c t cannot be explained within the existing theoretical approaches which a r e based on l i n e a r electron vibrational i n - teractions, harmonic potential function and a constant vibrational decay width v of the vibrational s t a t e s . In f a c t the commonly accepted approximation y=const,cannot be considered any longer valid f o r a molecule i n a s o l i d matrix, f o r which the impinging l a s e r beam creates many vibrational quanta i n the excited e l e c t r o n i c s t a t e , which ra- pidly decay through anharmonic interactions with the surrounding. The dependence of the vibronic lifetime on the vibrational quantum number has been quoted293 a s a pos- s i ble explanation f o r t h i S phenomenon. However no q u a n t i t a t i v e theoretical cal cula- tions have been reported so f a r . A simple theoretical model based on t h i s hypothesis i s worked out i n this report and calculations f o r a two s i n g l e t level diatomic molecule i n a crystal a r e presented. The model Hamiltonian f o r a diatomic molecule w i t h one vibrational normal mode interacting w i t h the l a t t i c e phonons of the host crystal i s :

+ + +

+ 1

+

where He=Eoaoa. + ~ , a > ~ ; He-ph=V(btb ) a l e l ; H =w(b b+ ?); E., a.,a,, a l a l ; E l represent the energies and the electron operators f o r t h e ground ph (0) and the excited electronic s t a t e (1); V i s the l i n e a r electron phonon coupling in the excited elec- t r o n i c s t a t e ; b, b

+

a r e the vibrational operators; w i s t h e frequency of the vibration- Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816144

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C6-494 JOURNAL DE PHYSIQUE

a1 normal mode; Hint represents a l l the anharmonic interactions between the molecule normal mode with the phonons of the host crystal and determines the relaxation proper- t i e s of the vibrational s t a t e s . We assume f o r simplicity t h a t H i n t i s the same i n the ground and i n the excited s t a t e . We note t h a t t h e introduction of Hint influences the time evolution of the vibrational operators b and b+ both in the absorption and in the Raman s c a t t e r i n g processes, thus introducing a variable decay width yn i n both the expressions. W i t h i n the approximation we j u s t sketched the e l e c t r o n i c absorption c o e f f i c i e n t can be written as

where n i s the l i g h t beam frequency, N(n) i s the r e f r a c t i v e index and I ( n ) , the op- t i c a l response function, i s

In Eq.(3) S(T)=;;Z (Zn(w)+l) i s the Huang Rhys f a c t o r , n ( ~ ) is the occupation number V2

f o r the vibrations, no i s the zero phonon frequency, and y n i s the vibrational decay width of t h e nth vibrational s t a t e . We note t h a t ^ydepends on n a s a r e s u l t of the anharmonic i n t e r a c t i o n s of H i n t i n the evolution of b and .'6 In Eq.(3) we have taken i n t o account t h a i the nth term i s determined by a process in which ( n - j ) vibrational quanta a r e created i n the excited s t a t e and j vibrational quanta a r e destroyed i n the ground electronic s t a t e .

By considering t h e approach of t h e secondary emission developped i n Ref.(4), the resonant Raman cross section f o r the k - t h Stokes processes, can be written a s

f o r simplicity in Eq.(4) we have neglected the contribution t o t h e Raman cross section coming from t h e terms i n which V 2 and t h e occupation number ( i n f r o n t t o t h e square modulus) appear with a power higher than k

.

In Fig.1 we show t h e r e s u l t s of the calculations f o r t h e absorption and the REP f o r the f i r s t and second order processes by considering n,=17600 c s - l ; V=700 cm-';

-

¹ ² 3

w.215 cm", .,,(cm )=5 + n + 0.35n a t 300°K.

The follorling comments can be made: i ) a s i g n i f i c a n t red s h i f t between absorption and t h e f i r s t and second order REP i s observed ( ~ 1 7 0 nm); i i ) the i n t e n s i t y of the

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f i r s t overtone i s much lower than the fundamental one. Calculation performed using a d i f f e r e n t functional dependence of y on n show t h a t a steeper variation of y with n y i e l d s lower overtone i n t e n s i t y , larger red s h i f t and larger half width of the absorpt-

ion band. Indeed t h i s e f f e c t has been experimentally observed i n Ref.(l) where R E P ' S and v i s i b l e absorption spectra of I2 i n d i f f e r e n t solutions a r e reported.

The model we have proposed here f o r the origin of the red s h i f t of the REP should be applied t o the case of Br2 in solid argon2 and t o tile molecular center I I I Ref .3.

Unfortunately i n both cases the absorption band shape i s not resolved and therefore the lack of t h i s important experimental information does not allow us t o perform ex- a c t calculations f o r these cases. Moreover good q u a l i t y absorption spectra f o r both cases would make i t possible t o give a more precise i n t e r p r e t a t i o n of the dependence of on n i n terms of r e a l i s t i c anharmonic interactions between the vibrational mode and the crystal phonons. We would l i k e t o point out t h a t i n an actual case of I2 in argon matrix, the Debye frequency of 75 cm" of the host crystal implies anharmonic interaction of order higher than the t h i r d one. As a consequence of t h i s f a c t a non l i n e a r dependence of y on n should be expected.

REFERENCES

Fig.1

-

Calculated absorption response function (Eq.3) and f i r s t and second order REP (Eqs.4-5), using the parameters given i n the t e x t .

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P.R. Fenstermacher and R.M. Cal lender, 0 p E c s Communications

2,

181 (1974).

2. J.M. Friedman, D.L. Rousseau and V,E. Bondybey, Phys. Rev. Letters 37, 1610 (1976) _-- 3. T.P. Martin, Phys. Rev. B 11, 875 (1975).

4. V.V. Hizhnyakov and I. T e h E r , Phys. S t a t . Solidi

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755 (1967).