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One-phonon spectra in absorption and fluorescence spectra of molecular impurities: breakdown of mirror symmetry
Volume
15, number
1
CHEMICAL
PHYSICS
1.5July 1972
LElTERS
ONE-PNONON SPECTRA IN ABSORPTION AND FLUORESCENCE SPECTRA OF MOLECULAR IMPURITIES: BREAKDOWN OF MIRROR SYMMETRY
Rcceiicd
Conditions spectra
for breakdown
of the structural
and intensity
1972
mirror
coupling strength,
of quadratic coupling is expected to be minimal.
to the zero-phonon
tion
In recent years the problem
of electron-phonon
coupling in impurity doped molecular solid solutions attention
considerable
[l-5]
. Studies
of
the one-phonon spectra in the electronic absorption and luminescence spectra of different impurity molecules in the same host medium can provide informa; tion pertaining to the phonon density oFstates function of the host and the pseudolocalized vibrations of the impurity mo!ecules [I,?] . To our knowledge localized external modes have not been observed in the ‘impurity sumably
spectra
of mixed molecular
crystllls *, pre-
because the phonon density of states of the hosts are devoid of forbidden gaps [6] and because, in most instances, the impurity molecules were heavier than the host. It has been noted
that in certain mixed crystal sys-
‘: terns there is a marked :.
phonon distribution
difference
between
i_
;:: ,:.
the one-phonon
-
bandlvidth,
is giv-en for defor a given linear electron-
to and fluorescence
from
the lowest
excited
im-
purity state [7,S] . In the ietracene in p-terphenyl system the lack ofstructural mirror symmetry isvery apparent and in addition it is evident that the integrated one-phonon intensitiesin absorption and fluorescence are different [8]. In this paper we consider the problem of the breakdown of the mirror symmetry. Out of the treatment presented here emerges a criterion for determining when tl~e quadratic coupling contribution to the width of the zero-phonon band would be expected to be minimal for ;1 given electron--phonon coupling
strength”*. The determinsrion of’ the lifetimes of higher energy excited electronic impurity states (nonemitting) from zero-phonon bandwidth studies in absorption requires an estimate of the quadratic cocpling contribution to the bandwidth at low temperatures, as recently emphasized by Richards and Rice.
ab-
In this paper we shall assume that the cuusmall to ensure that it is meaningful to and internal motions. One of the msni.,.'.:'talk about external _. festations of the breakdown of this separation is the appear:. modes
between
the one-
curves in the low temperature
::: * We exclude the internal modes of the impurity but recognize that a coupline betwen the internal and external
‘. ...
relationship
sorption and fiuorescence impurity spectra - absorp-
1. Introduction
has received
symmetry
and fluorescence spectra of molecular impurities arc established. A criterion
in the absorption
termining when the contribution phonon
6 .4pril
exists.
** The electron-phonon coupling strength is normally defined in terms of the linear coupling streqth, cf. cq. (2). N’z follow this conwntion here. If
pling is suificiently
arm of difierent
zerc+phonon _/. ,A_
phonon
vibronic
structures
bands.
built on the various
the c%upling iss~id tokweakor
sUong,cf.cqs.(23a)and
(24~1). 147
Volume 15. number 1 2.
15 July
CHEMICAL PHYSICS LETTERS
1972
equal to
Theory
a = _B-lA 2.1. Excited and ground state phonott normal coordinates We consider an impurity molecule in a host crystal (medium), referred to as the system in what follows. The ground electronic state of the system is the state in which the impurity and host molecules are in their ground electronic states. An escited electronic state of the system has the impurity in an excited electronic state with the host molecules in their ground electionic states. The ground state potential energy is expressed in the usual way as
N is the number of normal modes and {Qj}fL 1~ the set of ground state phonon oscillator coordinates. In eq. (1) ii is understood cht each &is of orderN-1/2. & = 0 represents the ground state equilibrium configuration of the system. If the excited state potentjal energy is expanded in terms of the ground state coordinates we have in general that
= 7~‘.
(6)
(I’ is then the excited state oscilIator displacement
vector relative to the ground state configuration. Finally we have that Q” = 0” denotes the equilibrium con& guration of the excited state and
iv A;’
E, (0”) = El(O) -
$C Bjj 7
j=l
(7)
’
where /I’= T-l.4
.
@I
2.2. Zero- ami one-phonoir tratlsitiom In this paper we limit the discussion of zero- and one-phonon transitiorrs to the Iow temperature (r+O) limit. The one-phonon structure in impurity vibronic spectra of molecular systems usually involves frequencies greater than about 20 cm-: and thus T = 4.2’K would satisfy the low temperature knit. In our treatment we assume that localized modes are not present in the system
so that
each a,jaj)
is cf the
order
X-112
[1,2,9,10J.
Zero-phonon transition probability We use the Condon approximation throughout so that the vibronic absorption or emission cross sections are proportional to the squares of the appropriate lattice vibrational overlap integrals [2,9, IO] . Without loss of generaiity * one could consider the zero-phonon transition as occurring between the zero-point itztratnolecular levels of the two electronic states. The ground and excited state zero-phonon wavefunctions are, respectively (i)
The linear and off-diagonal contributions to eq. (2) arise because of the change in impurity-host interactions upon impurity excitation [5]. The force constant matrix, 8, is symmetric and we define the similarity transformation T-‘BT
= B’ .
(3
B’ is diagonal and its elements are the squares of the excited state phonon frequencies. By substituting eq. (3) into eq. (2) we determine, after some manipulation, that
b$$2”)= 7 cF*(QJ,
N
El@“) = E1(O") + iI2
B'ijQ;'
,
(4?
where the q’s are harmonic oscil!ator wavefunctions. The subscript on the P is the oscillator quantum num-
with Q”=Q’_O’=T-l(Q_& (I is
is8
the ground state bscillator displacement vector
(5)
* Within the framework of the adiabaticand harmonic appro.ximation applied to e.xternal and internal motions of the nuclei, cf. p. 161 of ref. [I].
Voiume 15, number
1
15 July 1972
CHEM CAL PHYSICS LETTERS
ber. The overIap integral 3 oO = (~~(Q)i~~(Q”)}
00)
can be transformed to a more useful form by either expanding the qO(Qx-) or the q,$Q;l> in terms of the dispIacements ai or a;. For the latter case we find, with eq. (61, that to order 1/AJ G@Q;> = q,(Q;;)[ *
1 - $@i21
” ’ 1’2pI(Q;)a;. -i-(yPk)
2-3’2&a;;2@;>,
(11)
where f3;f= Zw’,/tl, w; is the frequency of the kth exe cited state mode. It is not difficult to see that the on: Iy terms in $,(Q”) which contribute toJO are ones which correspond to states with even vibrational quantum number, E To order I /Nonly terms corresponding to n = 0 or 2 need be considered_ It can be shown that n = 7, terms of the type conttinino, am should be larger than those containing pl ((21->~~ ((2;) [14]*. One also can argue that the total contribution of the tz = 2 terms to Jo0 is small relative to the single 11= Q term. We assume for the moment that the latter con:ributi; toJO is negligible (cf. discussion). Thus
617) Comparison of eqs. (14) and (16) reveals that in our treatment S = S’. However PSdiscussed below their frequency dcpendences can be dffeerenr. W should emphasize that it is a simple matter to include
the terms in JO0 which have been neglected. This, however, will only be necessary when the one-phonort integrated intensities in absorption and fluorescence are different (cf. discussion). (ii) The one-phonon spectra (a)dbsvlptio >L I n tl ~3 1ow temperature limit all ground state oscillators are in their zero-point levels. A ore-phonon transition in absorption corresponds to creation of one excited state phonon. The onephonon spectra1 shape function is given by
(13 G 1,(Q”)
= q (Q;‘, 7
L”~(Q;I)
(191
and
The intensity of the zero-phonon temperature limit is proportional 1/w J & = j& exp(d’)
band in the Iow to J& and to order
,
(14)
FI (Q;‘) = q1 CQ;>- +,;,li2
where
[qo(Q;,
- -%$p CZl)
(15) :.-
In the same way, by expanding the ground state ::I osciliator wavefunctions in a series of the displace;., men&, aj, one finds that J& is also equal to k,’ J2 = .2 Jo0 ev(-8 , (161 ;: 00 $ * This paper will extend the methods here lo a consideration ::. ;> - of the zero-phonon ?.’ i_ :
In evaluation of the overlap integral C$/o(Q)l$lti$Q”>> =Jolo we use the following form for PI (Qj’)
band shift and bandwidth
terms.
and eq. (11) for the pO(Qi). Again terms in eq. (19) with odd w do not contribute toJ,,(,-,. If we neglect then = 2 terms in eq. (19) we find that JgLtil = $3+;” j& exp(-S’> _ The total intensity of the one-phonon proportional to
(22)
spectrum is
(9-3a)
pi: F>, “.
I49
In view of the f&t tht
the phonon
density
of states?
*, in molecular
crystals represents a pseudo@aI continuum between 0 and WD, the Debye frequency, we may represent eq. (233) as
(23b) where&‘(o) = (2~/fi)~‘~.‘(ti) p’(w) = g’*(o)g(w) is the one-phonon spectra1 distribution function for absorption.. (b) Fhtorescetlce.
Following
the method outlined of the onein fluorescence is proportional to
above one can show that the intensitjr phonon
spectrum
15 July 1972
CHEXlICAL PHYSICS LETTERS
Vohme 15, number 1
the ccntribution
to the width
of the zero-phonon
band due to the off-diagonal quadratic coupling
terms is expected to be minimal. One should note that, because there is no causal relationship between the magnitudes of the linear coupling and off-diagonal quadratic terms in eq. (2), one cannot assume that the weaker the linear electron-phonon coupling, the smaller the contribution of the latter terms to the zero-phonon bandwidth at a given temperature. Considerable attention has been giv’en to the problem
of determining
the phonon
density
of states
func-
tion of the host crystal from the optical fluorescence spectra of the impurity [ 12,131. The problem is one of estracting the linear coupling function, &w), from the spectral distribution function,p(a). In this regard the usefulness of comparing the one-phonon structures in the absorption and iluorescence spectra is apparent from the above discussion.
=J&
$ ~*(w)S(w)dw,
(24b)
d(z)
where = (Zo/fi)J!* a(w). Thus the spectral distribution function cence is si_mplyp(w) = $Z2(w)g(w).
in fluores-
3. Discussion
Lastiy we emphasize that the neglect of certain highe,r order terms in the derivations of expressions for JG~, I,$ and J!,-, is justified if the integrated intensities of the one-phonon spectra in absorption and fluorescence are not very different. For systems in which the difference is significant the electronphonon coupling pawneters S and S’ cannot be equal and the equations in the test for the above three terms must be augmented.
Since it is impiicit in our treatment that S = S’, comparison of eqs. (233) and (24a) shows that Jil = Jfo or, in other words, that the one-phonon intensities in absorption and fluorescence are equal. Structure in the one-phonon spectra can arise from structure in the density of states, gjw), or in the displacement functions a”(o) (fluorescence) and Q’?(W) (absorption). The displace;llent functions are characteristic of the impurity molecule. Differences between
References
the one-phonon structure in absorption and fluorescence must be due to differences in the frequency de-
[ 1] K.K. Rebane, Impurity spectm of solids (Plenum Press,
pendence
of the displacement
functions.
It is apparent
if the off-diagonal elements of the force constant matrix 5’ are zero, that th? transformation matrix T is the identity and thus thar Q(W) = a'(o). Therefore, if in a given system the orte-phonon structures in absorption and fluorescence are identical or nearly so, * In &terns with low impurity concxnbation and no localiged modesg(w) for the system is that for the host.
_4&nowledgement The author would like to thank Professor Robin M. Hochstrasser for making available to him, prior to publication, the paper referenced [5].
New York, 19iOj, and refeiences
therein.
121 G.J. Small. J. Chem. Phys. 52 (1970) 656. [ 31 R. Ostertag and H.C. Wolf, Phys. Stat. Sol. 31 (1969) 139. [4] J.L.. Richardsand S.A. Rice, J. Chem. Phys. 54 (1971) 2014. IS] R.M. Hochstrasser and P.N. Prnsad, J. Chem. Phys., t,J be published. [6] G.S. hwIey, Phys. Stat. Sol. 2d (1967) 347. [7] A.B. Zahlan, ed., in: Excitons, magnons and pJlonons (Czmbridge Univ. Press, London, 1968).
Volume
15, number
1
[8] G.J. Smli, J. Chem. Phys., to be pubished. [Y 1 5I.H.L. Prym, in: Phonons in perfect lattices
[IO]
CHEMICAL
nnd in Int-
tices wi:h point defects, ed. R.W.H. Stevenson (Oliwr and Boyd, London, 1966). R.H. Sikbee, in: Optical properties of solids, cds. S. Nudleman 2nd S.S. hlitn (Plenum Press, NW York, 1969).
PHYSICS
LETTERS
1s July 1972
[ 11) G.J. Small, in preprtmtion. [I?] G.F. Imbosch, W.5f. Yen, A.L.
Schalow, D.E. XlcCumber and M.D. Sturge, Phys. Rev. 133 (1964) A1029. 1131 W.M. Yen, W.C. Scott and A.L. Schalow, Phys. Rev. 136 (1964) A271.
;_, :; ._ ._., !;_ ,
,.
:,, . _. ..
I-
ISI
15, number
1
CHEMICAL
PHYSICS
1.5July 1972
LElTERS
ONE-PNONON SPECTRA IN ABSORPTION AND FLUORESCENCE SPECTRA OF MOLECULAR IMPURITIES: BREAKDOWN OF MIRROR SYMMETRY
Rcceiicd
Conditions spectra
for breakdown
of the structural
and intensity
1972
mirror
coupling strength,
of quadratic coupling is expected to be minimal.
to the zero-phonon
tion
In recent years the problem
of electron-phonon
coupling in impurity doped molecular solid solutions attention
considerable
[l-5]
. Studies
of
the one-phonon spectra in the electronic absorption and luminescence spectra of different impurity molecules in the same host medium can provide informa; tion pertaining to the phonon density oFstates function of the host and the pseudolocalized vibrations of the impurity mo!ecules [I,?] . To our knowledge localized external modes have not been observed in the ‘impurity sumably
spectra
of mixed molecular
crystllls *, pre-
because the phonon density of states of the hosts are devoid of forbidden gaps [6] and because, in most instances, the impurity molecules were heavier than the host. It has been noted
that in certain mixed crystal sys-
‘: terns there is a marked :.
phonon distribution
difference
between
i_
;:: ,:.
the one-phonon
-
bandlvidth,
is giv-en for defor a given linear electron-
to and fluorescence
from
the lowest
excited
im-
purity state [7,S] . In the ietracene in p-terphenyl system the lack ofstructural mirror symmetry isvery apparent and in addition it is evident that the integrated one-phonon intensitiesin absorption and fluorescence are different [8]. In this paper we consider the problem of the breakdown of the mirror symmetry. Out of the treatment presented here emerges a criterion for determining when tl~e quadratic coupling contribution to the width of the zero-phonon band would be expected to be minimal for ;1 given electron--phonon coupling
strength”*. The determinsrion of’ the lifetimes of higher energy excited electronic impurity states (nonemitting) from zero-phonon bandwidth studies in absorption requires an estimate of the quadratic cocpling contribution to the bandwidth at low temperatures, as recently emphasized by Richards and Rice.
ab-
In this paper we shall assume that the cuusmall to ensure that it is meaningful to and internal motions. One of the msni.,.'.:'talk about external _. festations of the breakdown of this separation is the appear:. modes
between
the one-
curves in the low temperature
::: * We exclude the internal modes of the impurity but recognize that a coupline betwen the internal and external
‘. ...
relationship
sorption and fiuorescence impurity spectra - absorp-
1. Introduction
has received
symmetry
and fluorescence spectra of molecular impurities arc established. A criterion
in the absorption
termining when the contribution phonon
6 .4pril
exists.
** The electron-phonon coupling strength is normally defined in terms of the linear coupling streqth, cf. cq. (2). N’z follow this conwntion here. If
pling is suificiently
arm of difierent
zerc+phonon _/. ,A_
phonon
vibronic
structures
bands.
built on the various
the c%upling iss~id tokweakor
sUong,cf.cqs.(23a)and
(24~1). 147
Volume 15. number 1 2.
15 July
CHEMICAL PHYSICS LETTERS
1972
equal to
Theory
a = _B-lA 2.1. Excited and ground state phonott normal coordinates We consider an impurity molecule in a host crystal (medium), referred to as the system in what follows. The ground electronic state of the system is the state in which the impurity and host molecules are in their ground electronic states. An escited electronic state of the system has the impurity in an excited electronic state with the host molecules in their ground electionic states. The ground state potential energy is expressed in the usual way as
N is the number of normal modes and {Qj}fL 1~ the set of ground state phonon oscillator coordinates. In eq. (1) ii is understood cht each &is of orderN-1/2. & = 0 represents the ground state equilibrium configuration of the system. If the excited state potentjal energy is expanded in terms of the ground state coordinates we have in general that
= 7~‘.
(6)
(I’ is then the excited state oscilIator displacement
vector relative to the ground state configuration. Finally we have that Q” = 0” denotes the equilibrium con& guration of the excited state and
iv A;’
E, (0”) = El(O) -
$C Bjj 7
j=l
(7)
’
where /I’= T-l.4
.
@I
2.2. Zero- ami one-phonoir tratlsitiom In this paper we limit the discussion of zero- and one-phonon transitiorrs to the Iow temperature (r+O) limit. The one-phonon structure in impurity vibronic spectra of molecular systems usually involves frequencies greater than about 20 cm-: and thus T = 4.2’K would satisfy the low temperature knit. In our treatment we assume that localized modes are not present in the system
so that
each a,jaj)
is cf the
order
X-112
[1,2,9,10J.
Zero-phonon transition probability We use the Condon approximation throughout so that the vibronic absorption or emission cross sections are proportional to the squares of the appropriate lattice vibrational overlap integrals [2,9, IO] . Without loss of generaiity * one could consider the zero-phonon transition as occurring between the zero-point itztratnolecular levels of the two electronic states. The ground and excited state zero-phonon wavefunctions are, respectively (i)
The linear and off-diagonal contributions to eq. (2) arise because of the change in impurity-host interactions upon impurity excitation [5]. The force constant matrix, 8, is symmetric and we define the similarity transformation T-‘BT
= B’ .
(3
B’ is diagonal and its elements are the squares of the excited state phonon frequencies. By substituting eq. (3) into eq. (2) we determine, after some manipulation, that
b$$2”)= 7 cF*(QJ,
N
El@“) = E1(O") + iI2
B'ijQ;'
,
(4?
where the q’s are harmonic oscil!ator wavefunctions. The subscript on the P is the oscillator quantum num-
with Q”=Q’_O’=T-l(Q_& (I is
is8
the ground state bscillator displacement vector
(5)
* Within the framework of the adiabaticand harmonic appro.ximation applied to e.xternal and internal motions of the nuclei, cf. p. 161 of ref. [I].
Voiume 15, number
1
15 July 1972
CHEM CAL PHYSICS LETTERS
ber. The overIap integral 3 oO = (~~(Q)i~~(Q”)}
00)
can be transformed to a more useful form by either expanding the qO(Qx-) or the q,$Q;l> in terms of the dispIacements ai or a;. For the latter case we find, with eq. (61, that to order 1/AJ G@Q;> = q,(Q;;)[ *
1 - $@i21
” ’ 1’2pI(Q;)a;. -i-(yPk)
2-3’2&a;;2@;>,
(11)
where f3;f= Zw’,/tl, w; is the frequency of the kth exe cited state mode. It is not difficult to see that the on: Iy terms in $,(Q”) which contribute toJO are ones which correspond to states with even vibrational quantum number, E To order I /Nonly terms corresponding to n = 0 or 2 need be considered_ It can be shown that n = 7, terms of the type conttinino, am should be larger than those containing pl ((21->~~ ((2;) [14]*. One also can argue that the total contribution of the tz = 2 terms to Jo0 is small relative to the single 11= Q term. We assume for the moment that the latter con:ributi; toJO is negligible (cf. discussion). Thus
617) Comparison of eqs. (14) and (16) reveals that in our treatment S = S’. However PSdiscussed below their frequency dcpendences can be dffeerenr. W should emphasize that it is a simple matter to include
the terms in JO0 which have been neglected. This, however, will only be necessary when the one-phonort integrated intensities in absorption and fluorescence are different (cf. discussion). (ii) The one-phonon spectra (a)dbsvlptio >L I n tl ~3 1ow temperature limit all ground state oscillators are in their zero-point levels. A ore-phonon transition in absorption corresponds to creation of one excited state phonon. The onephonon spectra1 shape function is given by
(13 G 1,(Q”)
= q (Q;‘, 7
L”~(Q;I)
(191
and
The intensity of the zero-phonon temperature limit is proportional 1/w J & = j& exp(d’)
band in the Iow to J& and to order
,
(14)
FI (Q;‘) = q1 CQ;>- +,;,li2
where
[qo(Q;,
- -%$p CZl)
(15) :.-
In the same way, by expanding the ground state ::I osciliator wavefunctions in a series of the displace;., men&, aj, one finds that J& is also equal to k,’ J2 = .2 Jo0 ev(-8 , (161 ;: 00 $ * This paper will extend the methods here lo a consideration ::. ;> - of the zero-phonon ?.’ i_ :
In evaluation of the overlap integral C$/o(Q)l$lti$Q”>> =Jolo we use the following form for PI (Qj’)
band shift and bandwidth
terms.
and eq. (11) for the pO(Qi). Again terms in eq. (19) with odd w do not contribute toJ,,(,-,. If we neglect then = 2 terms in eq. (19) we find that JgLtil = $3+;” j& exp(-S’> _ The total intensity of the one-phonon proportional to
(22)
spectrum is
(9-3a)
pi: F>, “.
I49
In view of the f&t tht
the phonon
density
of states?
*, in molecular
crystals represents a pseudo@aI continuum between 0 and WD, the Debye frequency, we may represent eq. (233) as
(23b) where&‘(o) = (2~/fi)~‘~.‘(ti) p’(w) = g’*(o)g(w) is the one-phonon spectra1 distribution function for absorption.. (b) Fhtorescetlce.
Following
the method outlined of the onein fluorescence is proportional to
above one can show that the intensitjr phonon
spectrum
15 July 1972
CHEXlICAL PHYSICS LETTERS
Vohme 15, number 1
the ccntribution
to the width
of the zero-phonon
band due to the off-diagonal quadratic coupling
terms is expected to be minimal. One should note that, because there is no causal relationship between the magnitudes of the linear coupling and off-diagonal quadratic terms in eq. (2), one cannot assume that the weaker the linear electron-phonon coupling, the smaller the contribution of the latter terms to the zero-phonon bandwidth at a given temperature. Considerable attention has been giv’en to the problem
of determining
the phonon
density
of states
func-
tion of the host crystal from the optical fluorescence spectra of the impurity [ 12,131. The problem is one of estracting the linear coupling function, &w), from the spectral distribution function,p(a). In this regard the usefulness of comparing the one-phonon structures in the absorption and iluorescence spectra is apparent from the above discussion.
=J&
$ ~*(w)S(w)dw,
(24b)
d(z)
where = (Zo/fi)J!* a(w). Thus the spectral distribution function cence is si_mplyp(w) = $Z2(w)g(w).
in fluores-
3. Discussion
Lastiy we emphasize that the neglect of certain highe,r order terms in the derivations of expressions for JG~, I,$ and J!,-, is justified if the integrated intensities of the one-phonon spectra in absorption and fluorescence are not very different. For systems in which the difference is significant the electronphonon coupling pawneters S and S’ cannot be equal and the equations in the test for the above three terms must be augmented.
Since it is impiicit in our treatment that S = S’, comparison of eqs. (233) and (24a) shows that Jil = Jfo or, in other words, that the one-phonon intensities in absorption and fluorescence are equal. Structure in the one-phonon spectra can arise from structure in the density of states, gjw), or in the displacement functions a”(o) (fluorescence) and Q’?(W) (absorption). The displace;llent functions are characteristic of the impurity molecule. Differences between
References
the one-phonon structure in absorption and fluorescence must be due to differences in the frequency de-
[ 1] K.K. Rebane, Impurity spectm of solids (Plenum Press,
pendence
of the displacement
functions.
It is apparent
if the off-diagonal elements of the force constant matrix 5’ are zero, that th? transformation matrix T is the identity and thus thar Q(W) = a'(o). Therefore, if in a given system the orte-phonon structures in absorption and fluorescence are identical or nearly so, * In &terns with low impurity concxnbation and no localiged modesg(w) for the system is that for the host.
_4&nowledgement The author would like to thank Professor Robin M. Hochstrasser for making available to him, prior to publication, the paper referenced [5].
New York, 19iOj, and refeiences
therein.
121 G.J. Small. J. Chem. Phys. 52 (1970) 656. [ 31 R. Ostertag and H.C. Wolf, Phys. Stat. Sol. 31 (1969) 139. [4] J.L.. Richardsand S.A. Rice, J. Chem. Phys. 54 (1971) 2014. IS] R.M. Hochstrasser and P.N. Prnsad, J. Chem. Phys., t,J be published. [6] G.S. hwIey, Phys. Stat. Sol. 2d (1967) 347. [7] A.B. Zahlan, ed., in: Excitons, magnons and pJlonons (Czmbridge Univ. Press, London, 1968).
Volume
15, number
1
[8] G.J. Smli, J. Chem. Phys., to be pubished. [Y 1 5I.H.L. Prym, in: Phonons in perfect lattices
[IO]
CHEMICAL
nnd in Int-
tices wi:h point defects, ed. R.W.H. Stevenson (Oliwr and Boyd, London, 1966). R.H. Sikbee, in: Optical properties of solids, cds. S. Nudleman 2nd S.S. hlitn (Plenum Press, NW York, 1969).
PHYSICS
LETTERS
1s July 1972
[ 11) G.J. Small, in preprtmtion. [I?] G.F. Imbosch, W.5f. Yen, A.L.
Schalow, D.E. XlcCumber and M.D. Sturge, Phys. Rev. 133 (1964) A1029. 1131 W.M. Yen, W.C. Scott and A.L. Schalow, Phys. Rev. 136 (1964) A271.
;_, :; ._ ._., !;_ ,
,.
:,, . _. ..
I-
ISI