Dynamical Jahn-Teller effect in relaxed excited state of F-center

d. Phys.Chem.Solids, 1972,Vol. 33, pp. 1225-1233. PergamonPress. Printedin Great Britain

DYNAMICAL JAHN-TELLER EXCITED STATE

EFFECT IN OF F-CENTER

RELAXED

T. IlDA, K. KURATA and S. MURAMATSU Department of Physics, Faculty of Science, Osaka City University, Sumiyoshi-ku, Osaka, Japan (Received 2July 1971 ; in revised f o r m 20 September 1971 ) A b s t r a c t - A vibronic model is set up to study the strong orbital reduction of the 'relaxed' excited

state of the F-center; J a h n - T e l l e r coupling between electronic 2p states and E~ vibrational mode has been considered. Expressions for the magnetic-field-induced circular polarization and the changes of polarization and of lifetime due to an applied electric field are derived on the basis of a vibronic model. By comparison of a vibronic model with low temperature experiments on KCI, reasonable values are obtained for the parameters for the Jahn-Teller coupling, the energy separation between 2p and 2s states and the internal field introduced by Bogan and Fitchen. 1. INTRODUCTION

IT HAS been of interest to study the nature of the relaxed excited state of the F-center in alkali halides. Although the F-center has been studied for many decades, it is only very recently that a progressive understanding for this problem is emerging. Swank and Brown [1] have found that the radiative lifetime for the excited F-center is - 10-6 sec, as opposed to - 10-8 sec for allowed transitions in atom. Why the lifetime should be so long has been a serious question for several years. An acceptable explanation to this problem has been given by Fowler[2]. who has been able to predict the right order of magnitude of the decay time on the basis of a diffused 2p states model, where the 2s state has been ignored. However, the recent measurements of Bogan and Fitchen[3] have not been able to be explained by the diffused 2.o states model; they have studied the Stark effect by analyzing the electric-field-induced polarization and the radiative lifetime change of the F-center emission. They have proposed a model (it is called Bogan's model) for the relaxed excited state of the F-center to give a reasonable explanation to their experimental results, According to Bogan's model, the relaxed excited states are taken to be quasihydrogenic, but with the 2s state lying slightly below the 2p states. They

have assumed an odd-parity internal field which mixes the 2p states into the 2s state. The mixing turns these states into the highest 2p-like state with some 2s component, the intermediate doubly degenerate quasihydrogenie 2p states and the lowest 2s-like state with some 2p component from which the emission occurs. They also have estimated the energy separation between the mixed 2s-like and 2p-like states. A successful measurement which strongly supports Bogan's model has been reported by Fontana and Fitchen[4]. They have measured the magnetic circular polarization of the F-center emission and found a strong reduction of the orbital angular momentum of the quasihydrogenic 2p states by analyzing their observation on the basis of Bogan's model. They have not explicitly discussed the mechanism causing the strong orbital reduction while instead they have suggested that it might result from countercirculating current on neighboring ions, or Jahn-Teller effect, or both. The purpose of this work is to examine in detail the suggestion by Fontana and Fitchen regarding the vibronic properties of the relaxed excited state and to give a consistent explanation to the experiments at low temperatures. Dynamical Jahn-Teller effect will be taken

1225

1226

T. I I D A , K. K U R A T A and S. M U R A M A T S U

into account in addition to the odd-parity internal field. When the F-center is of cubic symmetry, a triplet orbital State (2p states) couples linearly to vibrations of both Eg and T2~ symmetry. Even if we neglect the mixing of 2s and 2,o states by the internal field, the problem in five normal coordinates and three electronic states is complicated. The complexity of the problem has prevented us "from finding solutions without making assumptions about relative strength of the coupling coefficients.* In this paper, we include only Eg mode in addition to the odd. panty mternal field, assuming the effect of Et, mode is more dominant than that of T~u mode. As is well known, Alu mode generally plays an important role in the problem about the relaxed excited state. We assume that after the surrounding ions have relaxed to a new equilibrium configuration appropriate to the excited states, in the space of Al~ mode, adiabatic potentials referred to 2s and 2p states have approximately the same curvatures and equilibrium positions except their minimum values. From the theoretical calculation by Wood and ()pik[6], this approximation is allowed for the relaxed excited state at low temperatures. On this assumption, the effect of A~u mode does not appear explicitly, even if we include it; this will be discussed in Section 4. The spin-orbit interaction will not be taken into account. According to the observations' by Fontana and Fitchen[4], the magnetic interaction is temperature independent in the range of i-3--4~ this implies that the effect of the spin-orbit interaction may be negligible. Absence of the spin-orbit interaction has been confirmed by the most recent studies of .

.

/

"

*Recently, O'Brien[5] has discussed the Jahn-Teller coupling of a triplet state to both Eg and T2u modes in the strong coupling approximation; in this approximation the orbital angular momentum of a triplet state vanishes to first order. The orbital reduction which we discuss in this paper remains finite and will not be explained by the strong coupling approximation.

Fontana[7]. This fact makes it easy for us to discuss the dynamical Jahn-Teller effect of the relaxed excited state of the F-center [8]. In the vibronic model, the state from which the F-center emission occurs is to be a vibronic state associated with 2s electronic one mixed with a vibronic triplet through the odd-parity internal field. The vibronic triplet arises from the 2p states coupled with Eg mode through the dynamical J a n h Teller interactions, being geometrically degenerate in the space of Eg mode. In Section 2, we formulate the fractional changes of the circular polarization, lifetime and polarization due to applied fields. Expressions for these quantities contain three parameters unknown a priori; they are so complicated that it is necessary to do numerical computations. In Section 3, comparing the numerical computations with the experimental results on the F-center in KCI [3, 4, 7] we determine the magnitudes of parameters consistently, and make some discussions about physical features of the results obtained. In Section 4, the approximations that have been made in the practical numerical computations are examined and some remarks are made. 2. FORMULATION

(a) Vibronic model We start with assuming a model Hamiltonian for the F-center distorted by the odd-parity internal field in the k-direction as H = H0(r; Q2, Q.~)+ ~, U~(r)Qi+ Vk(r) (1) /=2,

3

where r stands for the electronic coordinates; Q2 and Q3 are components of E,J mode which transform as z2--[(x2+y2)/2] and ( V ~ 2 ) (xZ--y2), respectively. H0(r; Q2, Q3) consists of the electronic Hamiltonian Hoe(r) and the lattice Hamiltonian H0 n for Erj mode in the harmonic approximation. The second term stands for the vibronic coupling. Vk(r) is the potential associated with the odd-parity

R E L A X E D E X C I T E D S T A T E OF F - C E N T E R

1227

internal field in the k direction, k being x, y o r Z.

In order to investigate the relaxed excited state, we need a vibronic Hamiltonian in the manifold of the excited electronic states. T h e y are taken to be quasihydrog~nic 2s, 2p~, 2p.~ and 2p, eigenfunctions of Hamiltonian Hoe; we denote these electronic bases by q~,,,, ~.~, "4). and (ibm. The wave function of

9

~bJx = Z aj~,,~,,284)',.~+ E b~'],,~,,,'~(;b~,

o 0

0

tl

(2)

0r i l l

where the coefficients a~,, and b~,,, are to be real and a is s u m m e d over x, y and z. The wave ( h x ,~2 function is normalized a s Zn(ajX)2--[- a.mXVjm = 1. ~,," is a harmonic oscillator wave function satisfying the vibrational equation

[Ho"+2

(3)

i~2,3

where Wi'~ is (~b,~lUi[~b~). ~,,~(Q) is an undistorted harmonic oscillator wave function because of W;z~= 0. Three states g,#, /j,,y and ZS,,z have different equilibrium positions but are degenerate geometrically; their energies are displaced by W2/2Mto 2, the J a h n - T e l l e r stabilization energy, w h e r e M and co are the effective mass of E:, mode and its angular frequency*" W an appropriate c o m m o n factor for the matrix elements

.

.

~(0

"

(4)

The submatrix H,~ and the vector x. are defined as

0 0 (n + ~)8,,,,,

j-th vibronic state for the excited state of the F - c e n t e r distorted, for example, in the x direction can be written in a general form

..

--

F (n--(~)(~,,,,, R,,,,(2s, x)v~ H,,,,,= IR''(x'- 2s)vx (n+8)8,,,,,

Lo

.ft,,, . . . f t , , ,

(n+~)8.,,

and

P.q Ib,, I Lb,,zJ

x,,= Ib,./,

where 8,,,,, is the Kronecher delta. Here we omit the zero point energy, since it only causes a constant energy shift. The energy of the vibronic levels are measured from the midpoint between the vibronic states ~0Zs~b2s and ~0~b~; the energy level of ~0x~b~ is shifted above the midpoint by 8ho~. vz is written for (qSzs] Vxl~b~)(hw) -1 = (~bx[ V~l~bz,)(hw) -1. It is reasonably assumed that Vk(r) has the symmetry represented by the matrix relations as <6._,sl v~14,A = <4,._,s/V~16~> = <6.,slVzl4,z> = vhto. The notation R,,,,(a, a') represents the overlap integral between the oscillator wave functions with different equilibrium positions in the E:, mode space:

R,,,,,(a,a') = f dQ ~,,~(Q)* ZS,,,~'(Q) . (5)

W i~ .

The coefficients a.~,, and bj~,,, are determined by the following infinite coupled equations: *Precisely speaking, the angular frequencies are slightly altered for respective electronic states but we assume that the values are same.

It contains a parameter S = (W2/2Mto2)/hto measuring the strength of the vibronic coupling[8]. If we could neglect the off-diagonal elements R,mv~ in equation (4) connecting different

1228

T. I I D A , K. K U R A T A

vibrational states, equation (4) could be easily solved analytically. Most recently, Bogan and Fitchen[4] have estimated the strength of the internal field from the measurement of the temperature dependence of the field-induced polarization. The estimated strength of the internal field seems to be comparable with the vibrational quantum hto. This fact requires us to solve the equation (4) without neglecting the off-diagonal elements; in the next section, we will solve equation (4) numerically under appropriate approximations. Solving the secular equation (4), we can obtain the lower lying mixed vibronic wave function corresponding to the relaxed excited state from which the Fcenter emission occurs; the wave function with energy E0~ is denoted as

and S. M U R A M A T S U

the magnetic field is applied in the z direction, this will cause the mixing of the states having the symmetry y (or x) into the lowest excited state for the F-center distorted in the x (or y) direction through the orbital Zeeman term. According to the ordinary perturbation theory, we obtain the modified vibronic wave function for the lowest excited state of each F-center as

9

[(k+8)hto--Eo~]} Ck~qb2,,

and

[(k + 71

771

The coefficients, ao% and b ~ , and Eo~ are to be given by numerical values. We have also many pairs of degenerate wave functions ~v~ and ~z~bz, which are unaffected by the odd-parity internal field and many mixed states ~b~~ withj r 0. The above discussion can be applied also to the case where the odd-parity internal field is along the direction y or z. At low temperatures the F-center emission occurs mostly from the lowest lying relaxed state to the ground state of the system.. The vibronic wave function corresponding to the ground state is simply written by the product of the quasihydrogenic ls state wave function and the harmonic oscillator wave function, whose equilibrium position is displaced from that corresponding to the 2s state in the Eg mode space. We now proceed to calculate the effects of the applied magnetic and electric fields on the emission from the relaxed excited state.

eo ]}

where H denotes the magnetic field strength applied in the z direction and /3 is the Bohr magneton. The lowest excited state of the F-center distorted in the z direction is not altered by the magnetic field applied along the z direction. In order to calculate A ( H ) for bulk crystal, it is necessary to take average of a corresponding quantity for a single F-center over the x , y and z directions. We assume that Fcenters distorted in the x , y and z directions are distributed in equal probabilities in a crystal, or in other words, that the lowest relaxed states from which emissions occur are lying in the range smaller than kT. We can obtain the expression for the circular polarization A (H) as

A(H) ----

N2+N,JZ~ L ~k+~----~--~-_~o~ j (6)

(b) Circular polarization We calculate the magnetic-field-induced circular polarization A ( H ) = (I+--I_)11. If

where N,~2 = E m (b0,n) 2 with c~ = x, y. We use the closure relation referring to the final state

RELAXED EXCITED STATE OF F-CENTER

~b,~ in calculating the squares of matrix elements. In this calculation, the effects of the magnetic field higher than first order are neglected. From the assumption with equal probabilities of distribution, the energy values of the lowest relaxed states, E0 ~, E0 ~ arid E0~, are expected to be nearly equal at low temperatures. This makes the mixing ratios of 2p and 2s states in F-centers distorted along the x, y and z directions nearly equal one another;, the magnitudes of N~ 2, N~ 2 and N~2 can be assumed to be nearly equal. Substituting explicit forms of Rmk(y,x ) and Rmk(X,y) into equation (6), we obtain E

2fill

'

~x, u A ( H ) = - - N~ exp ( - 3 S ) ~ ] (k+~)hoJ--Eo'

1229

~bo'~(F) = ~bo~--yF ~ ( [ ~ ao~nRk.(x, 2 s ) ] /

[ (k + ~)hto--Eo]} ~k%bx, where a isy or z and y = -- (Far/F) (4,281x1~,~); we take V/cm and eV as the units of electric field and energy. On the other hand, the vibronic wave function of the F-center distorted in the x direction is unaltered by the electric field, if we neglect the terms having coefficients aoZnbJZnbTmand ~o,h~a~j.~jm l~'v with j # 0. After straightforward calculation similar to those for the circular polarization, we can obtain the lifetime change (AT/To) and the polarization z3d' = ( I , - I• ) / (1, + I• ), where Ill and I• denote the intensities of the emission polarized parallel and perpendicular to the applied field:

k

(7)

(l/r) -- (l/r0) lh-o

(AT/TO) =

assumption, N x 2 ~ N v 2 = Nz 2 ( = N "z) and Eo~-==E0 v - ~ E 0 z ( = E 0 ) , and define the following[9]

where

we use

(1 --

N 2) (Ku 2 + Kz2) 3N 2 exp (--S)3~F 2

(8)

~ bx (ma!~"2 (3S) lt2(k3-m3)Lmak3-raa (3S) mz O,(k2,m3) \ k3 ] ] for a = x, rk"(S) =

/m |\1/2

~] bg , m)/'''a'l ma

,~z, 3 \ k 3 ! }

(-- 1)k~-n'~(3S)Xtz(k3-m~)Lm~k~-m~(3S) for a = y,

where Lnm(x)'s a r e Luguerre polynomials and subscripts 2 and 3 refer to the Q2 and Q3 modes. The factor exp (-- 3S) Ek E ~ , u {[Trk'~(S)]21[(k+8)hoJ-Eo]} brings about the small size of A ( H ) or the strong orbital reduction.

and

(c) Stark effect If an electric field F is applied, say in the direction x, it gives rise to the Stark mixing of the states of x symmetry into the lowest excited state qJ0~. For the F-center distorted along the y or z direction, we have the modified wave functions as follows:

K ~ (S) =

--

K~ 2 + K z 2 2Nz exp (--S)y~b-~,

(9)

where K,2(S) is the function of S defined by

ag,(mz,ma) { k~_,k~

l'.~-~""~ll2(--1)m2-kz

,.:,m~

\n~!rn3!]

• (s_),,-

(_,,m3

k3'

1230

T. I I D A , K. K U R A T A

[(~+k3+~)h~o--Eo"]

},

, (10)

and the effect of the electric field higher than the second order is neglected. 3. C O M P A R I S O N OF M O D E L W I T H E X P E R I M E N T S

In this section, to examine whether the vibronic model can give a reasonable explanation to the observed values for A ( H ) , (Arlzo) and AP, we will compare computer calculations based on the formulas derived in Section 2 with the low temperature experimental results in the case of KCI. According to the observations by Bogan and Fitchen[3] and by Fontana and Fitchen [4], the re.suits are the following: (a) the changes of the lifetime and polarization due to an applied electric field are isotropic with respect to the crystalline axes, (b) the decrease in the lifetime change (AT/To) and the increase in the polarization AP by an applied electric field are quadratic in the field strength, (c) the size of the circular polarization A(H) is small and temperature independent in the range 1"3-4.3~ (d) the circular polarization A(H) decreases linearly in an applied magnetic field, (e) the observed magnitudes of A ( H ) , (AT~To) and AP are listed in Table I. The experimental result (a) implies that FTable I. Values o f Starkeffect coeJ~cients (10 -6 (kV/ crn) -2) and circular polarization f o r KCl. H is magnetic field strength &PIPTM (AT/To)/F2

A(H)

10 -----3 ta~ --5-8 --+ 1.0 "~ -- (9--+1) X 10-SH t"~

(a) Ref. [3], (b) Ref. [7].

and S. M U R A M A T S U

centers distorted in the x, y and z directions are distributed in equal probabilities in a crystal; this is to justify the assumption that has been made in Section 2. Equation (7), the decrease in the circular polarization, is linear in an applied magnetic field in agreement with (d); (8) and (9), the decrease in the lifetime change and increase in the polarization, are quadratic in an applied electric field in accord with (b). The expressions (7), (8) and (9) depend on S, 8 and v as well as ~r, KyZKz 2 and N2; the latter quantities are functions of the mixing coefticients a's and b's and energy E0 which are obtained by solving secular equations such as (4). The quantity which we can determine directly from the experimental results is N ~, which represents the magnitudes of p character in the emitting level. Inserting the observed values of (AT/to) and AP in (8) and (9), we obtain N ~ --- 0-13. It is not possible to obtain the values of parameters S, ~ and v by simple comparison of calculations with experiments; instead, we determine the values of them so that they can consistently reproduce the experimental values of A ( H ) , (AT~To) and AP. Before going into details, we briefly show a procedure for determining the values of parameters S, 8 and v: (I) for suitably chosen sets of numerical values of S, ~ and v, secular equations such as (4) are diagonalized and N z and A ( H ) / H are computed, (I1) out of suitably chosen sets are picked up those which satisfy two conditions; (a) they give the values of energies such that Ed: ~ Eo u ~- Eo~ and N 2 ~ 0.13 and (b) they give the experimental value of A ( H ) / H 9-0 x 10-a (G) -~, (III) APIF 2 are computed for the set of parameters selected in (ll); by comparing these computed values with observed one, a single set of parameters is uniquely determined. First, we solve infinite secular equations, such as (4), corresponding to the x, y and z

RELAXED EXCITED STATE OF F-CENTER

directions of distortion. To our knowledge, however, such a problem has not yet been explored. For the moment, we numerically solve the secular equations with coupling to over ten vibrational stat~s included; each secular matrix reduces to a" size (4 x 10) • ( 4 • I0). Even if this truncation is not entirely justified, it will have no crucial effect of undetermining the vibronic model. The diagonalization of the truncated secfilar matrices is carried out by the Jacobi method; with the results, N 2 and A(H)/H are computed. To compute A(H)/H, an effective frequency of hto = 0.012 eV for Eg mode is used; there has been available no accurate value of the frequency while the effective frequency in KC! has been estimated by some authors. These values are listed in Table 2. In above computations, we take 660 sets of three parameters S, ~5 and [vl; S varies from 1-2 to 2-2 at interval of 0.2; Ivl from 0.7 to 0-5; 6 from 0-4 to 2-4. It would be redundant to tabulate all numerical results for the all sets. We will instead show the results enough for our discussions. From the results of the above computations for 660 sets of parameters, we pick up the sets of parameters satisfying the conditions (A) and (B): they are plotted in Fig. 1 as a function of T = 1/Ivl, where corresponding energy values are also plotted. For each set of parameters plotted in Fig. 1, we compute the quantity AP/F z of equation (9), assuming the value o f y ~ to be 88.36 (,~)2,. The computed values are plotted as a function of T in Fig. 2. The curve in Fig. 2 shows reasonable behavior; with increase in the strength of the internal field, the field-induced polarization AP approaches to zero. This is

1231

Table 2. Effective frequency hto in KCI [ 10]

Rusell and Klick

Konitzer and Markham

Gebhardt and Kuhnert

0-011 eV

0.012 eV

0-012 eV

2.4

s 50

2.0

---o/h~ I

40

I

1.6 6O

3o 2 o 2.0

0.8 LO

0.4

I

I 0.4

0.6

0.8

1,0

1.2

1.4

T

Fig. 1. Values of parameters S and ,5 consistent with the experimental results of A(H) and N2; corresponding energy values of the lowest state, Eo, are plotted. The vertical dotted line indicates the value of T consistent with the experimental result of the polarization.

/

20 N I

\

1,5

>

'~

I0

'0

<3

*The factor (Fat/F) is usually regarded to be order of unity for diffused centers; for very compact centers, it may be given approximately by the classical Lorentz expression as [I + (n 2 -- 1 )/3] 11~ where n is the refractive index. For KCI, n 2 is 2-13 and then the factor is to be the order of unity also in compact centers. We put the value of the factor (FeulF) to be unity for KCI. The value o f ~ is then given by the quantity 1(4~,lxl~=)l 2, which has been calculated as 88.36 (,~.)2 by Wood and Opik [6].

0

0.2

0.4

0

0.8

Fig. 2. Computed values of AP/F~ for the sets of parameters plotted in Fig: 1 are represented as a function of T, where y~ ~ 88-36(A) z is used. The horizontal line indicates the experimental result of the polarization.

1232

T. IIDA, K. KURATA and S. MURAMATSU

consistent with the phYSical insight that the internal field becomes stronger, its effect will outweigh that of applied electric field. The value of T which gives the experimental value AP/b-~ -~ 10-5 (kV/cm) -2 is determined to be -~ 0.57 in Fig. 2. Returning to Fig. I, we can find the magnitudes of S,~8 and Eo/hoJ corresponding to the value of T obtained above; they are S = 1-6, 8 = 1-9, and Eo/hco = -- 2.6 for T = 0-57. This set of values for S, 8 and Ivl really reproduces the experimental magnitudes of A ( H ) / H , (Azh'0) and AP. Let us now compare our estimation with some related quantities which are obtained from other sources. The quantity (S + 2~)hco gives the electronic energy separation between 2s and 2p states in the relaxed excited states in which the internal field and the vibronic coupling are ignored. Using the set of parameters of the above result and the assumed frequency hoJ = 0.012 eV, we obtain the value ( S + 2 ~ ) h o J = 0-065 eV. This value satisfactorily agrees with the theoretical estimate of the energy separation - 0.08 eV by Wood and Opik [6]. We mention about a reason for the relatively long lifetime of the relaxed excited state in the F-center. The lifetime will depend on the 2p-like character of the emitting state, which is given by N ~ in the vibronic model. This quantity is to correspond to o?/(1 + o ? ) in the paper of Bogan and Fitchen[3]; their estimation = 0.26 is about two times as large as our result. The smallness of this quantity is partly responsible for the long lifetime. The lifetime at low temperatures is given by 1/1" = ( 4 n E 3 e 3 / 3 h 4 c ~ ) N 2 l ( dpl s Ixl~b:~) 12 [11], where n is the refractive index at the emission energy E. Using the experimental value E = 1-24eV[3], N 2 = 0-13 obtained above and I
the experimental value 7 = 0.715 x 10-6 sec at 0~ We remark that two facts of fair agreement observed above seem to be favorable for the vibronic model. 4. DISCUSSIONS

We first examine if an approximation of truncating infinite secular equations such as (4) is good enough for comparing the vibronic model with experiments. There is, however, no precise mathematical test to determine how many vibrational basis functions for secular matrix we need to obtain a good approximation. The energy separation between the lowest vibronic level and the highest one that we have taken into account turns out to be -~ 0.08eV for the set of parameters S - ~ 1.6, ~ - 1 . 9 and T - 0 . 5 7 obtained above. Roughly speaking, the effective odd-parity internal field strength causing admixture of 2s and 2p electronic states is given by Ivlhoa exp (-S/2) -~ 0.01 eV, which is about 13 per cent of the energy separation --- 0.08 eV. Therefore, higher vibronic levels that we have neglected will have negligibly small effects on the lowest excited level (emitting level). To examine the above consideration numerically, we have extended the size of secular matrices such as (4) from ( 4 • 2 1 5 2 1 5 to ( 4 x 1 5 ) x (4 • 15), and calculated the circular polarization, the lifetime change and the polarization, retaining the values of parameters previously found with (4 x 10) x (4 x 10) secular equations. The value of the magnetic circular polarization calculated in the case of (4 x 15) x ( 4 x 15) differs only 4 per cent from the one in the case of ( 4 x 1 0 ) x ( 4 x 10), the values of the lifetime change and the polarization remain the same within three significant figures. This observation is an evidence thai the truncation does not lead to substantial error. To extend the secular matrix requires the following considerations and also brings about complexities. The vibronic states associated with higher vibrational excited

RELAXED EXCITED STATE OF F-CENTER

states may approach more closely to continuum states of a conduction band. Therefore, if we take into account more vibronic levels we would have to consider the effects of both higher electronic bound states and the continuum states of the conduction band at the same time. It should be an important problem to be investigated at an early date. Alo mode plays an important role in relaxing the surrounding ions of the F-center after the optical absorption takes place. Even if Alg mode is taken into account, we obtain the essentially same result as that without Axg mode since Aao mode introduces an additional orthogonality to a matrix element, on the assumption about Alo mode stated in Section 1. For example, a matrix element of an electronic operator rr, (qba~nC~(Alo)~m~ • (E~)(TrlchB~t~(A~9)~k~(Eg)}, reduces to a form <4~=1~'14,~><~ma( Eo) I~k~ ( Eo) >6,t. We have devoted ourselves much to clarify origins of the odd parity internal field without satisfying results. It may be possible that the internal field is caused by the vibronic interaction with Tlu mode. If T,u mode is explicitly included in addition to Eo mode, one has to solve the vibronic problem in which electronic 2s and 2p states are coupled with both Eo and T,, modes. The vibronic interactions bring about too many parameters to compare with experiments available for the present time. We have investigated the effect of Eo mode in the relaxed excited state of the Fcenter. The expressions including the vibronic interactions explicitly are obtained for A ( H ) , (At/z0) and AP. Comparing the model with experiments, we find that dynamical

1233

Jahn-Teller effect of Eo mode plays an important role to give rise to the strong orbital reduction. The vibronic model is essentially consistent with Bogan's model. The latter is a useful phenomenalistic one to describe various properties of the relaxed excited state. Our approach based on the vibronic model is a reasonable next step in the problem of treating the relaxed excited states. However, it would seem that development of experiments demands further refinement of the models. Acknowledgements-The authors are indebted to Profs. H. Watanabe and H. Ohkura for continual helpful discussions. They also acknowledge Professor D. B. Fitchen and Dr. M. P. Fontana for communications on their experiments prior to publication.

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Note added in proof: In Fig. 1, the quantity Eolh~ should read - Eo]hOJ.