Quadratic non-condon effect in optical spectra of impurity paramagnetic centers in dielectric crystals

Transcription

1 Journal of Physics: Conference Series Quadratic non-condon effect in optical spectra of impurity paramagnetic centers in dielectric crystals To cite this article: R Yu Yunusov and O V Solovyev J Phys: Conf Ser Related content - Electron-virational interactions in polyatomic molecules M D Frank-Kamenetski and A V Lukashin - Spin relaxation in Kondo lattices S I Belov, A S Kutuzov and B I Kochelaev - MODERN METHODS IN THE THEORY OF MANY-PHONON PROCESSES Yu E Perlin View the article online for updates and enhancements This content was downloaded from IP address on /9/8 at 5:

2 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 Quadratic non-condon effect in optical spectra of impurity paramagnetic centers in dielectric crystals R Yu Yunusov and O V Solovyev Kazan (Volga Region) Federal University, 48, Kremlevskaya Street 8, Kazan, Russia thor@chelnycom Astract Analytical expressions for the asorption and luminescence form functions of impurity paramagnetic centers in dielectric crystals at zero temperature are derived in the adiaatic approximation taking into account the quadratic non-condon effect It is proved that, if the optical transition is foridden due to symmetry selection rules, the non-condon asorption and luminescence spectra are not mirror symmetric and can contain a zero-phonon line, contrary to the case of linear non-condon effect Conditions under which the zero-phonon line is contained in the optical spectra of a symmetry-foridden transition are determined Introduction The purpose of this theoretical study is to derive and analyze formulas for calculating the form of virational optical spectra of weakly allowed transitions in impurity centers of dielectric crystals For a correct description of the spectra of weakly allowed transitions, it is necessary to go eyond the Condon approximation In [] we derived analytical expressions for non-condon asorption and luminescence spectra of impurity centers for the case of zero temperature taking into account linear non-condon effect In the present paper we consider linear and quadratic non-condon effects The statement of the prolem is determined y the necessity of interpreting the interconfigurational 4f n 4f n- 5d spectra of asorption and luminescence of heavy rare-earth ions (n > 7) for which transitions etween the ground state of the 4f n electronic configuration and the lowest-energy state of the 4f n- 5d electronic configuration are spin-foridden In this study, we use the following simplifying assumptions: () the host crystal is transparent in the spectral region under investigation; () the concentration of impurity optical centers is low, and hence the cooperative effects due to the interaction etween these centers can e ignored We consider the following Hamiltonian of the electron virational system []: H = He + Hint + Hvi () H e Here is the Hamiltonian of optical electrons of the impurity paramagnetic center, which includes the energy of interaction of the electrons with a static crystal field (the set of coordinates of the optical electrons of the impurity ion with respect to its nucleus is designated as r) The Hamiltonian of the virational susystem is considered in the harmonic approximation: H vi To whom any correspondence should e addressed Pulished under licence y Ltd

3 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 H vi = ( q / q ) h /, where q stands for the real dimensionless normal coordinates of the virational susystem (the set of normal coordinates is designated as q) and are the corresponding frequencies The mechanism of relaxation of the virational susystem is postulated, and the relaxation time is supposed to e short in comparison with the lifetime of excited electronic states We consider the electron-virational interaction Hamiltonian H int linear in the normal coordinates of the virational susystem: = v q, where v are the electronic Hermitian operators, which have dimension of energy Hint The non-condon asorption and luminescence generating functions This work is ased on the adiaatic approximation Let us assume that the eigenvalues and eigenfunctions of the Hamiltonian H e are known: He ψn() r = Enψn() r Let ψ n(, r q) (in what follows, we will also use the notation n >, ψ n ( r ) = n > ) e the nth solution of the electronic Schrödinger equation for a fixed configuration of the virational susystem: h r r () ( He + Hint + q ) ψn(, q) = Un( q) ψn (, q) Equation () will e solved using the perturation theory with the Hamiltonian taken as perturation (the Kuo Toyodzawa approach [3]) We find the electronic wave function ψ n(, r q) to the second order in the perturation: < c H n > < c H n > ψ ψ ψ ψ int int n(, r q) = n() r + () () c r n r c n En Ec c n ( En Ec) + < c H k >< k H n > < n H n >< c H n > int int int int () ψ c c n k n ( En Ek)( En Ec) ( En Ec) r (3) After sustituting the adiaatic potential U ( n q ) into the equation for the virational susystem and considering the latter to the first order in the perturation H int, we otain again the Hamiltonian of the system of independent harmonic oscillators with the same viration frequencies ut with the displaced n equilirium positions and denote it as H : vi U q q E q H n n( ) h / / = n h n / + vi, where the new equilirium positions are given y qn =< n v n > / h Then, we consider the transition from the electron state a to the electron state upon asorption of an electromagnetic radiation quantum y the impurity virational system Let d e the projection of the electric dipole moment of the impurity center onto the direction of the polarization of the photon (the expressions given elow can e easily generalized to the case of multipole radiation) The dependence of the asorption coefficient on the frequency is determined y the form function [] + (as) (as) a ( Ω ) = a ( )exp( Ω ), H int F I t i t dt (5) where I (as) a () t is the asorption generating function, which in the adiaatic approximation is represented y the expression [] (4)

4 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 (as) a a ( vi ) ( vi ) I ( t) = < a d > exp ith / h < d a> exp ith / h exp( iω t), (6) where a means temperature averaging over the states of the virational susystem in the electronic state a and Ω = h + / (7) a ( E Ea )/ ( qa q ) is the frequency of the zero-phonon transition We neglect the electron-virational interaction for the lower energy electronic state, involved in optical transition In practice, this approximation is fulfilled with good accuracy in the case of interconfigurational optical transitions, for example, 4f n 4f n- 5d transitions Therefore, we can set a>= a > and q a = Then, it is necessary to choose the approximation in which the wave function of the excited electronic state is calculated In the Condon approximation, the static electronic wave functions are considered at fixed virational coordinates ( q =, as the Kuo Toyodzawa approach implies) In this case, the matrix element of the operator d is factored outside the sign of averaging over the states of the virational susystem in equation (6) Setting >= > and introducing the notation < a d >= d a, the following expression for the Condon asorption generating function can e otained [4]: a a cond (as) (as) Ia ( t) = da exp iωat q + q exp( it) = da Ia ( t ) (8) Here we introduced the normalized Condon generating function of asorption (as) () t, which corresponds to the Condon form function of asorption F (as) a ( Ω ) normalized y π In the Condon approximation, the asorption and luminescence form functions are mirror symmetric []: I ( t) = I ( t) exp( iω t) (lum) (as) a a a and F (lum) (as) a a a I a ( Ω ) = F ( Ω Ω) For a spin-foridden or symmetry-foridden electronic transition a the matrix element d a is small or equal to zero, and it is necessary to go eyond the Condon approximation The foridden transition can ecome allowed through orrowing the intensity from allowed transitions via electronvirational interaction; this mechanism was proposed y Herzerg and Teller [5] Let us consider the dynamic wave function (3) for the electronic state and the corresponding matrix element of the electric dipole moment, containing terms linear and quadratic in the normal coordinates: (9) * a γ () < a d >= d + q + B q q, γ = < v c > d, ca () c ( E Ec) < c v >< v c > < c v k >< k v > B = d a + d ac c ( En Ec) c k ( E Ek)( E Ec) 3

5 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 < v >< c v ( E Ec) > We will mark the generating functions and form functions calculated within the estalished approximation y the index We sustituted () into equation (6) and the analogous equation for luminescence and calculated the temperature averages (for the case of zero temperature) over the states of the virational susystem, using the representation of the second quantization and the Feynman s operator calculus [4, 6] In all further formulas the symmetrized coefficients sym B = ( B are used, with the notation «sym» omitted for revity We otained the + B )/ following formulas for the generating functions: () () I ()/ t I () t = B exp( ± i( )) Re( ) exp( )(exp( ) ) + t m B q γ ± it ± i t m * + B B q exp( )(exp( ) )(exp( ) ) 3 q3 ± it ± i t i t m ± m 3 3 * γ it da γq it + exp( ± ) + m (exp( ± ) m ) () m m (3) + B q q (exp( ± i t) )(exp( ± i t) ) + B 4 In this formal expressions modulus concerns complex constants, ut not exponential functions of time; upper and lower signs are taken for asorption and luminescence, respectively The mirror symmetry of asorption and luminescence reaks due to the terms included in (3) with different signs for asorption and luminescence Physical cause of the asymmetry lies in the fact that transitions etween the same virational sulevels have different weights in asorption and luminescence spectra, eing temperature averaged If we put all coefficients B equal to zero, (3) transforms to the formulas for the linear non-condon effect, which we otained in [] If we put also coefficients γ equal to zero, we otain the Condon case (see (8)) Note, that the otained formulas (3) will e also valid if we take into consideration quadratic electron-virational interaction in the first order of the perturation theory for the electronic wave function; this will only alter the values of B constants (see ()) 3 The non-condon asorption and luminescence form functions Let s find the Fourier transforms of the otained generating functions I (as) a () t and I (lum) a () t (see (5), (3)) It is convenient to present the explicit formulas for the asorption and luminescence form functions in terms of convolution operators For the distriution A( ) determined for positive values of the argument, we define the convolution operators ρ A and ρ + A, which, y acting on a function ϑ( Ω ), give its convolutions with A( ) []: + ± ρ A [ ϑ( Ω )] = A( ) ϑ( Ω± ) d (4) ± * By definition, we assume that ρξ = ρ ± ξ * Similarly, we introduce the convolution operators with the distriutions of two and three frequencies: 4

6 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34// ρ A [ ϑ( Ω )] = d d A(, ) ϑ( Ω± ± ), (5) ± (, ) ± A(,, 3) Ω = d d d 3A 3 Ω± ± 3 m (6) ρ [ ϑ( )] (,, ) ϑ( ) The following equality can e proved for an aritrary function f ( x ), which can e expanded in the Maclaurin s series: m f A( )exp( ± itd ) Φ( t) exp( iω tdt ) = f( ρa ) Φ( t) exp( iωtdt ) (7) Analogous equalities involving the distriutions A(, ) and A(,, 3) can e written Utilizing (5), (8), (9), (7) one can write the following expression for the normalized Condon asorption and luminescence form functions: (as) F L d m (lum)( Ω ) = πexp( ( ) + ρ ) [ δ( ΩΩa)], where the upper and lower signs are taken for asorption and luminescence, respectively; integrals here and further are taken in the limits from zero to the maximum frequency of virations; spectral distriution L( ) is estalished as ( ) = δ( )/ L (8) L q (9) The integral L( ) d has the meaning of the Huang-Rhys parameter of the transition With the use of (5), (3), (7) we otained the following expression for the non-condon form functions of asorption (upper signs) and luminescence (lower signs) at zero temperature: (as) F ( (lum) Ω ) = ( ρ m A + ρ m P + ρ m * mρ m Q mρ m * + ρ m G mρ m M m ρ m * + ρ m P Q M η + ρ m f * m m m (as) a γ ρϕ ρn ρζ (lum) 4 + d + q ± + m + B q q + B )[ F ( Ω)] () Here the following distriutions were introduced (the distriutions f ( ) and ϕ( ), which correspond to the linear non-condon effect, have een estalished in []): A(, ) = B δ( ) δ( ), P(, ) γ B q δ( ) δ( ), * * = Q( ) = P(, ) d, * G(,, 3) = B B q ( 3 q3δ ) δ( ) δ( 3 ), 3 3 M(, ) = G(,, 3) d3, η( ) = M(, ) d, 5

7 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 * ϕ ( ) = γ qδ ( ), f ( ) = γ δ( ), N(, ) = B q ( qδ ) δ( ), 4 ζ ( ) = N(, ) d () 4 Computational formulas for the non-condon form functions For computational reasons it is convenient to express the sums over the normal coordinates, appearing in (7), (9), (), through the advanced Green s functions for the virational susystem In [] we introduced a complex function of the positive frequency with the indices corresponding to four electron states, i = 4, which are the eigenstates of the Hamiltonian H : i D 34 ( ) = Im G ( ) () a <, H int > < 3 H int 4 > * Here and further the sign of the imaginary part should e assigned to the advanced Green s function [7] for the Hermitian virational operators; the complex coefficients, ie, the matrix elements of the electronic operators, are factored outside the sign of the imaginary part The properties of symmetry of * the introduced function are ovious: D ( )= D ( )= D ( ) The required sums over the normal coordinates can e expressed through the D ( ) 34 functions with the use of the following equation, which can e proved for an aritrary function f ( ): e h v v f( ) δ ( ) D ( ) ( ) 34 f π < * >< 3 4 > = (3) For example, the distriution L( ) (9), which defines the Condon and shape, with the use of D ( ) (3) can e written as L( ) = π h Introducing B = C < 3 4 v >< 3 v 4 >, where coefficients C can e determined from the expression (), symmetrized upon indices,, we can rewrite the distriutions (), involved in the non-condon form-functions, in the following form A(, ) C C D ( ) D ( ), * = h π 8 d P C D ( ) ac * 34 (, ) = ( 3 4 c ), 4 c π E Ec h D 4 D ( ) D ( ) G(,, ) C C D ( ), = h * π 8 3 * dac dac' h cc, ' π E Ec E Ec' f( ) = Dcc ' ( ), ( )( ) 6

8 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 d Dc ( ) ϕ ( ) =, ac c π E Ec N D ( ) D ( ) = (4) 3 4 (, ) C 3 4 π 4 5 Group-theoretical analysis of the properties of the non-condon and Let G e the point group of symmetry of the Hamiltonian He The electron-virational interaction Hamiltonian Hint can e expressed in terms of symmetrized displacements QΓ γ of the ions of the crystal lattice, which are transformed according to the irreducile representations of the group G ( γ is the row of the irreducile representation Γ ) The electron functions of the level, containing the state, are also transformed according to a certain irreducile representation of the group G ; we will denote this representation as Γ (for revity, the level itself will also e denoted as Γ ) In the adiaatic approximation, we take into account only interaction with the virations that are adiaatic for this level These are the totally symmetric virations, which are transformed according to the identical representation Γ, and the virations transformed according to the representations Γ that are K not contained in Γ Γ Here K denotes the symmetric part of the direct product of the representation y itself if H e is the Hamiltonian of the even numer of electrons and the antisymmetric part if H e is the Hamiltonian of the odd numer of electrons Let s consider the otained non-condon form-functions of asorption and luminescence () for a symmetry-foridden transition a, for which the electric dipole moment matrix element vanishes in Condon approximation: d = In [] we showed that in this case the distriution ϕ( ) and the quantity a * γ q (in fact, the integral of ( ) ϕ with the opposite sign) are equal to zero Now we will show that for the symmetry-foridden transition a the distriution N(, ) (see (4)) is also equal to zero Sustituting the explicit expressions of the coefficients in (4) we otain C 3 4 d D ( ) D ( ) D ( ) D ( ) ac ck k k ck = + c k π ( E Ek)( E Ec) N(, ) d ac D ( ) Dc ( ) Dc ( ) D ( ) + c π ( E Ec) (5) Let us consider the second line in (5) Since the imaginary parts of the crystal Green s functions are diagonal in the indices of the irreducile representations and only interaction with the totally symmetric virations contriute to the matrix element < Hint >, the sum over the states c for these terms is limited y states of the same symmetry as the state Thus, the terms in the second line in (5) give zero contriution to the distriution N(, ) Now, consider the first line in (5) From similar reasoning it turns out that the sum over the states k is limited y states of the same symmetry as the state, and the sum over the states c for these terms is limited y states of the same symmetry as the state k Hence the terms in the first line in (5) also give zero contriution to the distriution N(, ) 7

9 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 The distriution ζ ( ) and the quantity B q q, eing the integrals of the distriution 4 N(, ), are also equal to zero for a symmetry-foridden transition a It can e seen that the remaining distriutions, involved in the non-condon form functions (), in general case do not vanish for a symmetry-foridden transition Given the aove, the formulas () for the non-condon form functions of a symmetry-foridden transition reduce to simpler expressions: F (as) (lum) m m m m m m m m m m (as) ( Ω ) = ( ρa + ρp + ρ * mρq mρ * + ρg mρm mρ * + ρ f B )[ (lum)( )] P Q M η + ρ + F Ω (6) As follows from (6), the form functions F (as) ( Ω ) and F (lum)( Ω ) of a symmetry-foridden transition, contrary to the case of linear non-condon effect [], are not mirror-symmetric, and can contain a zero-phonon line The only term in (6) that causes a zero-phonon line in the spectra is B, since all other terms give convolutions of the normalized Condon form function with some distriutions Let s determine the conditions under which the quantity B is nonzero for a symmetry-foridden transition We express the quantity B through the Green s functions of the virational susystem (see (), (3)): h d B = Im G ( ) d ( )( ) < > < > ac a c H int k, k Hint c k π E Ek E Ec h d ac a G ( ) c Hint, Hint c π E E c Im ( ) d < > < > (7) The second term in (7) vanishes similarly to the second line in the expression (5) for the N(, ) distriution Now let s determine the conditions under which the first term in (7) does not c vanish We should find such a state c ( Γ is not equivalent to Γ ) that satisfies the following conditions: () d a Γ Γ Γ c, ie, the mixing of the state c with the state can allow the transition a ; () In the expansion of the electron-virational interaction Hamiltonian H int over the symmetrized ionic displacements there is such a displacement Q Γ γ that K (a) Γ Γ Γ, ie, the viration is adiaatic for the level Γ (ut not totally symmetric); () Γ Γ Γ k and Γ Γ c Γ k, ie, interaction with this viration mixes oth states c and k and states k and Thus, the zero-phonon line can e allowed in the spectra of a symmetry-foridden transition via mechanism of a clearly quadratic nature mixing, induced y interaction with virations, of the states and c through the third state k ; if the aforementioned conditions cannot e satisfied simultaneously, the zero-phonon lines will e asent in spectra within the estalished approximations 6 Conclusions In the present paper we develop theory of quadratic non-condon effect in optical spectra of impurity ions at zero temperature within the adiaatic approximation This effect has not een considered earlier in literature The results otained can e used in further investigation of mechanisms of optical 8

10 International Conference on Resonances in Condensed Matter: Altshuler Journal of Physics: Conference Series 34 () 35 doi:88/ /34//35 spectra formation in the case of the Condon approximation violation, in modeling and interpretation of interconfigurational 4f n 4f n- 5d asorption and luminescence spectra of impurity rare earth ions, which is necessary for predicting characteristics of potential phosphors and scintillators in the vacuum ultraviolet region of the spectrum of electromagnetic radiation The strength of the quadratic non- Condon effect in comparison with the linear non-condon effect will e estimated in our future investigation of the LiYF 4 :Lu 3+ crystal optical spectra, corresponding to symmetry-foridden 4f 4 4f 3 5d transitions in the Lu 3+ ion Acknowledgements This work was supported y the RFBR Grant References [] Solovyev O V Phys Solid State 5 78 [] Perlin Yu E and Tsukerlat B S 974 Effects of Electronic Virational Interaction in the Optical Spectra of Impurity Paramagnetic Ions (Kishinev: Shtiintsa) [3] Kuo R and Toyodzawa Y 955 Prog Theor Phys 3 6 [4] Lax M 95 J Chem Phys 75 [5] Herzerg G and Teller E 933 Z Phys Chem At B 4 [6] Rickayzen G 957 Proc R Soc London, Ser A 4 48 [7] Böttger H 983 Principles of the Theory of Lattice Dynamics (Berlin: Akademie-Verlag) 9