Excitons Boidnd to Ionized Impurities Therefore wur experimental data testify that the properties of solvated electrons in strong alkaline methanol and strong alkaline water are different. The main differences aire tile f o l l o ~ n g : the extinction coefficient of e,- in methanol is appreciably less than in water and the recombination of e,- in alkaline water occurs faster than in alkaPine methanol. These differences are gradually removed
3771
with the increase of water content in the water-methanol mixtures.
Acknowledgments. We wish to express our indebtedness to A. Kulakov and A. Konopeshko for careful Linac operation and Dr. P. Glazunov and E. Shirshov for maintenance of electronic equipment.
Excitons Bsun to Ionized Impurities in Inorganic Crystals lkQIllQSS
Laboratoirre de Spectroscopie at d’Optique du Corps Solide.’ lnstitut de Physique, Strasbourg, France (Received April 21, 1972)
Pekeris’ method for helium atom generalized recently by Frost for the three-particle systems has been developed extensively to apply to excitons bound to ionized impurities in inorganic crystals. Haken’s exciton potential, where the dielectric constant between the two particles is a function of the interparticle distance, the optical and static dielectric constants, the electron and hole effective masses me* and mh*, ,respectively, and the longitudinal vibrational frequency of the lattice, has been generalized for the three particles. This potential shows the important effect of the polarizability. A considerable long recursion jrelation of 57 terms has been derived. In exciton-ionized-donor complexes, the binding energy of the system is a function of the mass ratio rne*/mh*. This complex has been studied for real systems such as CMS, ZnO, CuCl, CuBr, CUI, ZnSe, ZnTe, CdTe, S i c 6H, TlCl, and TlBr. The agreement with experiment is better than that obtained by the previous authors where the polarizability has been neglected. The calculations have also been carried out for exciton-ionized acceptor. In this case the results are given in terms of the mass ratio mh*/me*. For known inorganic crystals, this mass ratio is usually high, and consequently it is highly improbable to find such a stable complex for these crystals.
Introduction In inorganic crystals, experimental evidence2 has shown the existence of excitons bound to ionized donors, It is of interest to carry out some exact calculations for these complexes. The method given by Pekeris3 for the helium atom and generalized recently by F r ~ s t for ~ - the ~ threeparticle system has been developed further. Haken’s exciton potential7 in which the effect of the polarizability is included has been used in these calculations. For excitonionized-donor conqdexes the results are given in terms of the mass ratio F = mr*/mh* where me* and mh* are the effective masses of the electron and the hole, respectively. I n the case of the exciton-ionized-acceptor complexes the binding energies are function of the ratio 6, = mh*/me*. Comparison with experiment is also carried out for some real systems.
Farm of the Potentia As described by I4aken,7 the dielectric constant K(i-23) between the hole and the electron of a delocalized exciton iis tl function2 of the distance (i-23) separating the two particles, of their effective masses, of the optical ( K O )and the static (K,)dielectric constants, and of the longitudirial vibrational frequency ( w ) of the lattice. As atomic units in terms of a. certain effective dielectric constant K e r f are usually adopted, the generalized Haken’s poten-
tiaP for any two particles i and J of effective masses mi* and mj* in a crystal can be written in the following form
with
For the complex exciton-ionized donor given in Figure 1, (1) Research group associated with the Centre Nationale de ta Recherche Scientifique (CNRS), France. (2) For full review on the experimental evidence of exciton-ionizeddonor complex, see the references given in S. G. Elkomoss, Phys. Rev. 5,4, 3411 (1971). (3) C. L. Pekeris, Phys. Rev., 112, 1649 (1958). (4) A. A. Frost, J. Chem. Phys., 41,478 (1964). (5) A. A . Frost, M. Inokuti. and J . P. Lowe, J. Chem. Phys., 41, 482 (1964) (6) A. A. Frost, D. K . Harris, and J. D. Scargle, J. Chem. Phys., 41, 489 (1964). (7) H . Haken, Z. Naturforsch. A , 9, 228 (1954); “Halbleiter Probieme,” Vol. I, W. Shottky, Ed. Vieweg, Braunschweig, 1954, p 72: “Halbleiter Probleme,” Vol. II, W. Shottky, Ed., Vieweg, Braunschweig, 1955, p 1; J. Phys. Radium., 17, 826 (1956); J . Chin?. Phys., 55, 643 (1958): J. Phys. Chem. Solids, 8, 166 (1959). ~
The Journal ot Physical Chemistry, Vol. 76, No. 25, 1972
S.G. EIkomoss
3772
Figure 2. Exciton-ionized-acceptor complex.
Figure 1. Exciton-ionized-donor complex,
where Xl2, XIS, and X23 are the coefficients of the terms 1/r12, l/rls, and 1 / r 2 3 given by eq 1. For the complex exciton-ionized acceptor of Figure 2, the potential energy is given by eq 3 where A12 and A13 are of opposite signs to those for the exciton-ionized-donor complex of Figure 1. Due to the difficulties that may occur in solving the problems of exciton complexes using the general potential of eq 1, 2, and 3, mean values of Atj's should be considered. Knowing the wave function \k of the system, one can write
K & i 2 / e 2 M and Me4/K,ff2h2can be considered for length and energy, respectively with M = h = e2/Keff = 1. The exciton binding energy E, of eq 7 is simply 0.5 au. Notice that the atomic units corresponding to eq 5 , 6, and I are different from each other. For these different cases corresponding to the complexes given in Figures 1 and 2, the nonrelativistic Schrodinger equation takes one of the following three forms depending on the choice of the above atomic units
C~er'\k+ 6 x x ~ h J z \ +k 2(1 + G,)(E - V N = o exciton a
'The mean values ir,are denoted-by A, p , and V , respectively. The values of X, p , and v depend on the fundamental constants me*, mh*, K,, KO,and w. The binding energy of the complex given in Figure 1 can be calculated in terms of the neutral donor binding energy ED
ED= -me"e4/2K~2h2
(5)
The dielectric constant KD = K ( r d = Kerf and is evaluated using X = 1. In this case the atomic units Keffh2/ e2me* and me*e4/h2Kefr2 will be adopted for length and energy, respectively, and the units me* = h = 1 and e2/ Keff = 1 will be used. The binding energy ED is simply 1,hen equal to 0.5 au. For the exciton -ionized-acceptor complex of Figure 2, the binding energies are calculated in terms of the neutral acceptor binding energy EA EA
= -~nh*e~/ZK~~h~
(6)
The dielectric constant KA = K(r13) = Keff for the system of Figure 2 and is evaluated using p = -1. In this case the atomic units Keffh2/e2mh*and mh*e4/h2 Keff2are usually adopted for length and energy, respectively, with mh* .= k --. e2/Keff = 1 The binding energy EA of eq 6 is then simply equal to 0.5 au. In most of the inorganic crystals where the exciton-ionized-donor complexes have been observed, the binding energy ED of the neutral donor is not well determined. The binding energies of such complex are usually given in terms of the exciton binding energy E x E, = -Me4/2K,2h2
(7)
where M is the exciton reduced mass and K, = Keff = K ( r 2 3 ) , evaluated from iv = 1. In this case the atomic units The Journal ot Physicai Chemistry, Vol. 76,No. 25, 7972
I
(10)
'
For the systems where the calculations are carried out in terms of ED, eq 8 is used. In this case the electron effective mass me* is considered to be unity and the mass ratio u is equal to l/mh* au. For those cases in which the calculations are to be expressed in terms of E A , it is eq 9 that has to be solved. I n this case the atomic units are those that correspond to mh* = 1 and the mass ratio 6a is equal to l / m e * au. For the systems where the computations are given in terms of E,, eq 10 has to be used. In this case the exciton reduced mass M is considered bo be unity and the mass ratio 6, i s equal to me*/mh*. Note that the atomic units used in the three cases are different. In eq 8, 9, and 10, the potential V is given by eq 4 and 3 transformed into the appropriate atomic units, .Vet2 is the Laplacian for the electron i, c hI2 is that for the hole ,j, and E is the total energy of the system expressed in terms of the atomic units used.
Method of Solution Consider eq 8 corresponding to Figure 1. Use the classical method of Hylleraas,8 and introduce the perimetric coordinates u, u, and w. These coordinates2 depend on three variational parameters a , @, and y . Put = e-(1/2)(u+UiU)F(~,~,~~)
(11)
with m
~ ( uU ,, W ) =
C A ( I ,m, ~ ) L ~ ( u ) L , ( u ) L ~ ( w ) l,m,n
(12)
where L J , L,, and L , denote, respectively, the normalized Laguerre polynomials of order 1, m, and n. Using the different relations between these polynomials and their derivatives, one obtains2 a considerable long 57-term recursion (8)
E. A. Hylleraas, Z. Phys., 54,347 (1929).
Excitons Bound tu Ionized Impurities
3773
TABLE I
h=1 h h h h A h h
v v v v v v v v v v
p=1 p = 0.9
=1 =l =l = 1.05 = 1.15 =1
p p
= 0.8
= 0.7 p = 1 p = l p=l p=l p = 1.05 p = 1.9
= 1.05 h = 1.05 h=2
= l = l = l = l = l = l = 1.1 = 1.1 = 1.1 = l
h = l
h=l h=1 h=f h = 1.1 h=1 h=1 h = 1.05 h=2
Computations and Results The recursion relation takes the form of the eigenvalue problem H+t(P+uQ)=O (13) 'The procedure of computation is given in detail in ref 2 . The results of e in the determinants of eq 13 converge to four decimals. The maximum order attained for these determinants is 50. The first part of this work is to show whether the effect of the polarizability expressed in terms of h , . p , and v has an important effect or not. Computations were performed in double precision for different cases with different values of h, p, and v. Some of these cases correspond to the values of A, p , and v shown in Table I. For these cases the values of E / E D are represented in Figure 3 as a function of the mass ratio = l/rnh* au.. The values of 01 = 0.72, p = 1.5, and y = 0.55 that correspond to the minimization of energy for the case h = p = v = 1 have.been considered the same for all the curves given in Figure 3 with different values of h, p , and v and for different values of u . This approximation has been adopted to Save considerably the computer time involved in such elaborate long computations. The results A
= 0.95 = 0.85 = 0.75 = 0.65 p = 1
u=l
p=l
v v
v v v v
p=l p = 0.95
u
v v
p = 2 p = 3
h=3
irelation between the coefficients A( 1. rn, n). The solution of this recursion relation gives the energy ratio E / E D as a ffunction of l/mh* for different values of h, p , and v.
p p p p
= 1 = 1 =i = 1 = 1.05 = 1.2 = 1.1 = l =l
are strongly dependent on the values o f A, p 9 and V. This means that the variation in the dielectric constant due to the polarizability between the three particles makes an important contribution. The intersections between the curves E / E D = f(u) and E / E D = 1 give critical values ac for the mass ratios. The systems are stable for a Iac and unstable otherwise. From Figure 3 one can notice that the values of uc are also a function of 1,p, and v. The exponential part2 of the wave function 11 can be written in terms of 7-12, 7-13,and i-23 as follows \k
~
e-f(ar,z
-b w
t cr23)
F(u, u, w )
(14)
where a, b, and c are given by the following expressions
+ +
a = (1/2)(--a 0 y) b = (1/2Xa - P + 7 ) c = (1/2)(a+P - 7 )
(15)
Taking 01 = 0.72, p = 1.5, and y = 0.55, the corresponding values of a, b, and c are 0.665, 0.115, and 0.835, respectively, all positive numbers. These values of a, b, and c show that the repulsion between the hole and the donor has a smaller effect than the attractions along the directions of r12 and 7-23. Another important feature is the positive sign of the i-13 term in the exponential of eq 14. This sign has a significant physical meaning in that it explains the repulsive forces between the hole of the exciton and the donor. The two negative signs of ria and 7-23 in the exponential represent, of course, the attractive forces between the electron and the donor, as well as between the two exciton particles, respectively. The wave function 14 still converges since the value of b is much smaller than. that of a or c. This has been demonstrated by studying the integrals of eq 4 used for calculating the values of h, p, and u which are necessary for comparing experiment and theory. Notice that the coefficients a, b, and c of the exponential part of the wave function given in eq 14 as well as the energy 6 and consequently the totall energy E of the system are all determined from the long-recursion relation of eq 13.
Comparison with Experiment
i 0.2
63
0.4
0.5
0.6
0.7
,
0.8
-
0.9
l/m:
Figure 3. Plots of E / & and V.
vs. l/mh*for different values of A, p ,
To carry out the comparison between theory and experiment for a specific inorganic crystal, the corresponding mean values h, p , and v given in eq 4 have to be calculated. For these computations one needs to know the wave function q. The calculations concerning the wave function 14 are quite elaborate. For simplifications only the exponential part of this eq 14 has been used to evaluate the integrals of eq 4 and consequently to calculate the values of h, p, and Y. Using elliptical coordinates for integration, the values of A, p , and v for a given crystal are The Journal of Physical Chemistry, Vol. 76, No. 25, 7972
S. G . Elkomoss
3774
TABLE II: Fundamental Constants and the Computed Values of ( E ZnTe, CdTe, SIC SH, TICI, and TlBr
- fU)/fD for Exciton-Ionized-Donor
h = 1 , p = 1 1 . 0 0 1 7 , v = 1.06568 5.24d 5a 9.2c
CdS
Q.205n
ZnO
0.29e
1.8d
ZnSe
0.1d
0.6d
ZnTe
0.096h
0.6i
CdTe
0.096d,1
0.68
SIC 6H
O.Zn,q,
TlCl
0.5%'
2.7233
TIBr
0.28P
0.72P
0.7a,b
Complex in CdS, ZnO, ZnSe,
0.207
306
3.524
4.59d
0.1611
580
6.4
5.79
0.167
253
1.158
6.7k
0.16
206
0.875
7.13m
0.1412
168
1.827
X = l , p = l . O 0 0 1 7 , v = 1.021 9.66O 10.03O
6.7O
0.1
967O
3.007
h = l , p = l . O 0 3 3 8 , v - 1.21155 37.6P
5.1P
0.1915
174P
25.446
5.4p
0.3889
116P
6.73
h = 1 , p = 1.0012, v = 1.0697 11
8.5e
h = 1, p = 1.00057, v
-- 1.03242
9.2f
h = 1, p = 1.00044, v
-- 1.02763
10.38j
= 1, p = 1.00038, v = 1.02792
l.!jnJ
10.6m
3.5n
h = 1, p = 1.0051, v = 1.15727 35.1P
aReference 9 and J. 0. Dimmock, lnl. Conf. / / - V I Semiconduct. Compounds, 7967, 2 (1968); B. Segali, ibid., 327 (1968). b J J .J. Hopfield, J. Appl.
&vs. Suppl., 32, 2277 (1961). C R . E. ,Haisted, M. R. Lorenz, and B. Segall, J. Phys. Chem. Solids, 22, 109 (1961). See ref 10. eSee ref 1. f S,S. Mitra, J. PhYs. SOC. Jap. SUPPI., 21, 61 (1966). g M. Aven, D. T'. F Marple, and 6.Segali, J. A p l . Phys.'Suppl., 32, 226 (1961). R. L. Bowers and G. D, Mahan,
Phys. Rev., 185, 1073 (1969). ' D . T. F. Marple and M. Aven, lnt. Conf. l:V! Semiconduct. Compounds, 7967, 315 (1968). IS. Narita, H. Harada, and K. Nagasaka, J. Phys. Soc. Jap., 22,1176 (1967). k A . Nitsuishi in United States-Japanese Cooperative Seminar on Far-infrared Spectroscopy, Columbus, Ohio, 1965 (unpublished. I K . K. Kanazawa and F. C. Brown, Phys. Rev., 135, A1757 (1964). m D . De Nobel, Philips Res. Rep?., 14, 357 (1959): 14, 430 (1959); S. Yamada, J. Phys. SOC.Jap., 15, 1940 (1960). B. Ellis and T. S. Moss, Proc. Roy. SOC., 299, 383 (1967); H. J. Van Daal, W. F. Knippenberg a n d J . D. Wasscher, J. Phys. Chem. Solids, 24, 109 (1963). L. Patrick and W. J. Choyke, Westinghouse Scientific Paper No. 70-9d3OPGAP-P1, 1970 (unpublished). PSee ref 24, %,I*. 'me,,!.
given by the following expressions X = (Keff/Ks) .+ (1/2)Keff[(1/Ko)(l/KS)l{(SF)/[R(c
+ a1)3(C- 'd311
also a function of the energy E of the complex. This energy has been calculated using the determinant 13. In this case the computations have to be self-consistent as described in detail in ref2. For a particular material, different values of effective masses and dielectric constants are available in the literature. The computations corresponding to a particular substance are carried out for these different effective masses and dielectric constants. In Table 11, the fundamental constants and the computed values of ( E - & ) / E D for some of the cases that correspond to the best agreement with experiment in CdS, ZnO, ZnSe, ZnTe, CdTe, SIC 6H, TlC1, and TlBr are given. The problem of anisotropy for tlhe effective masses and the dielectric constants is eliminated by taking mean values for these constants using the formulas of Hopfield and Thomas.9
The stability of the exciton-ionized-donor complex calculated in Table I1 for these materials agrees with observat i o n ~ . The ~ ~ -value ~ ~ ( E - ED)/& = 3.524 x com(9) J. J. Hopfield and D. G. Thomas, Phys. Rev,. 122, 35 (1961). (10) U.C . Reynolds, C. W. Litton, and T. C. Collins, Phys. Status Solidi, 9, 645 (1965), (11) D. C. Reynolds, C. W. Litton, and T. C. Collins, Phys. Rev., 174, 8459 (1968). The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
Excitons Bound to Ionized Impurities
3775
TABLE 111: Fundamental Constants and the Computed Values of ( E
- E x ) / E X for Exciton-Ionized-Donor
Complex in CuCI, CuBr, and CUI
mh*
W
h = 0.98, p = 0.89, v = 1 GuCl
0.5a
13a,b
CuBr
0.415c.d
20C.d
7.43C.d
4.84e
0.03846
216c
0.146
0.02075
160
0.15
0.02075
150
0.13
h = "1.01,p = 0.925, v = 1 5.7d
4.4d
h = 1.002, p = 0.935, v = 1 c uI
0.415
20
4.8!je
6.Ze
a K . S. Song, Thesis, Strasbourg, 1967. OM. A. Khan, J. Phys. Chem. Solids, 31, 2309 (1970). CSee ref 25 d S . Lewonczuk, J. Ringeissen, and S. Nikitine, J. Phys., 32,941 (1971). e C . Carabatos, E. Prevot, and M. Leroy, C. R. Acad. Sci., 274, 707 (1972).
puted for CdS in Table I1 is in better agreement with experimenP (2.5 x than the best previous value (4.04 X calculat,ed recently by Suffczynski, et a1.,20,21 using Rotenberg and Stein's22 wave function. For ZnO, the computed value 6.4 >( is again in better agreement with experirnentl6gz3(9.61 X l o w 2 )than the previous best value 5.085 X 10-2 calculated recently by Suffczynski, et aL,21 using also Rotenberg and Stein's wave function. For the materials ZnSe, ZnTe, and CdTe, in spite of the observation of such a complex and which is confirmed by the calculations given in Table I1 for these crystals, the corresponding experimental values of ( E E D ) / E Dare not well determined. Taking v = 1.21155 and 1.15727 given respectively for TlCl and TlBr in Table 11, the corresponding exciton binding energies for these materials are 6.312 and 2.95 meV. These energies are in very good agreement wi.th the experimental values (11 f 2) and (6 r f r 1) meV given by Bachrach and Brown2* for TlCl and TlBr, respectively. For CuCl, CuBr, and CUI the neutral donor binding energy ED i:s not well determined. The experimental data are given in terms of the exciton binding energy E x . For these materials, the atomic units corresponding to Schrodinger eq 10 have been adopted. The self-consistent calculations have been carried out. The values of a, p, and y are determined from, the minimization of energy. For these three materials there are three different sets of values for A, k , and v as well as three different sets of values for a , p, and y.In Table 111, the fundamental constants and t,he computed values of ( E - E , ) / E x are given for these materials. For CuBr the effective masses given by Ringeissen, et ai., for CuCl have been considered. For CuCl and CuBr the computed values of ( E - & ) / E x are in good agreement with experiment 0.225926and 0.175,27-29 respectively. For CUI the experimental value of ( E - E x ) / E , is not well determined. But the computed value 0.13 for this material in Table 111 compared to the corresponding values in Cue1 and CuBr seems to be satisfactory. Equation 10 has also been considered for CdS and an agreement of the same order with experiment similar to that given in Table I1 has been obtained. For TlBr, Grabnerso observed recently two peaks, B1 and B2, which ha interpreted as bound excitons to unknown defects. The peak R1 may correspond to excitonionized-donor complex as predicted by the calculations given in Table IT for this material. The measurements30 are given in terms of the free exciton binding energy E,. Equation 10 has then to be considered and computations similar to those for CuCl, CuBr,. and CUI mentioned above
-A=-10
E /E,
____ a--to
1.30
6-0.667
1.25
+
; =l,O
/4=-?.0
6-0.929
--
D - 0 8 5 4 , 7-1.045
p--4.0 A = 0.843
-
I
3
=
1.05
= 0.948
j
,+
-
1.05 A. 0.95 ,L( -1.0 d-0.783 b = 0 . 8 W j r.-0.905
-
a =-1.0
1.20
1.15
1.10
1.05
1.00
0.95
0.90
,~
0.1
0.2
0.3
0.4
0.5
0.6
m;/q
Figure 4. Plots of E I E , vs. l/m,*
and I / .
for different values
of A , ,u,
(12) M. A. Gilleo, P. T. Bailey, and D, E. Hili, Phys. Rev., 174, 898 (1968). (13) E. Hal Boardus and H. B. Bebb, Phys. Rev., 176, 993 (1968). (14) D. C. Reynolds and T. C. Collins, Z. Naturforsch., 249, 1311 (1969). (15) D. C. Reynolds, "Electronic Structures in Solids," Plenum, New York, N. Y., (1969, p 110. (16) D. C. Reynolds, "Optical Properties of Solids." Plenum, New York, N. Y . , 1969, Chapter 10, p 239. (17) J. Conradi and R. R. Haering, Phys. Rev., 186, 1088 (1969). (18) D. E. Hill, Phys. Rev. 8, 1, 1863 (1970). (19) W. J. Choyke, Mater. Res. Bull., 4, S141 (1969) (20) M. Suffczynski and W. Gorzkowski, Acta Phys. Poi., A38, 441 (1970). (21) M. Suffczynski, W. Gorzkowski, and T. Skettrup, Internal Report of the Institute of Theoretical Physics, Warsaw University, and the Technical university of Denmark, Lyngby, 1970 (unpublished). (22) M. Rotenberg and J. Stein, Phys. Rev., 182, I (1969). (23) Y. S. Park, C. W. Litton, T. C. Collins, and D. C. Reynolds, Phys. Rev., 143, 512 (1966). (24) R. 2. Bachrach and F. C. Brown, Phys. Rev. B , 1, 818 (1970). (25) J. Ringeissen. Thesis, Strasbourg, 1967. (26) M. Certier, Thesis, Strasbourg, 1969. (27) A. Mysyrowicz, R, Heimburger, J, B. Grun, and S. Nikitine, C. R. Acad. Sci., 263, 1116 (1966). (28) C. Wecker, Thesis 30me Cycle, University Louis Pasteur, Strasbourg, Science Section, 1972. (29) T . Goto and M. Ueta, J. Phys. SOC. Jap., 22,488 (1967). (30) L. Grabner, to be submitted for publication. I wish to thank Dr. R. 2 . Bachrach who brought this reference to my attention. The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
Shubin, Sharanin, Pernikova, and Vinogradov
3778
are carried out. Using the fundamental constants given in Table I1 for this material, the three sets of values for A, p 9 and v obtained from the self-consistent computations are 1.05, 1, and I, respectively. The corresponding calculated value ( E - Ex)/EX= 0.53 is in good agreement with experiment O.6L3O For the exciton-ionized-acceptor complex the atomic units corresponding to Schrodinger eq 9 have been considered. The total energy E of the system is calculated in terms of the neutral acceptor binding energy EA. In this case the recursion relation of eq 13 takes the form
N’ -I-c(&P
f Qj =0
(181
The matrix H’is a function of A, p, and v. The values of X and p have opposite signs to those corresponding to the exciton-ionized-donor complex. Four different cases with different values of A, p, and u have been considered in Figure 4. In these calculations the values of a , p , and y corresponding to specific values of A, p, and u are determined from the minimization of the energy for this specific case. In Figure 4 the values of a , p, and y corresponding to the specific values of A, p , and v are given. This figure
shows again the important contribution of the polarizability expressed in terms of A, p, and v. As either the value of X or p or v changes, one gets different results for E / E A as function of l/m,* au. The intersections of EIEA = f(6a) with EIEa = 1 give critical mass ratios gaCbelow which the system is stable, otherwise it is unstable. In most known inorganic crystals the values of 6a are high and it is highly improbable that such a complex exist in such materials. The terminology of trapped hole and trapped electron in radiation chemistry could correspond to the same terminology of ionized donor and ionized acceptor, respectively, treated in this paper. It may be possible that similar treatment to that given in this paper leads to an explanation for the physical phenomena of trapped hole and trapped electron in radiation chemistry, particularly, in the radiation of glasses. Acknowledgments. The computations have been carried out a t the Computer Center of Cronenbourg-Strasbourg. I wish to thank Professor G. Monsonego, director of the Computer Center, and the personnel, in particular M. Gendner, for their considerable help.
Polaron Yields in Low-Temperature Pulse Radiolysis of Inert Aqueous Matrices V. N. Shubin,” Yu. I. Sharanin, T. E. Pernikova, and G. A. Vinogradov Institute of Electrochemistry of the Academy of Sciences of the
U.S.S.R., Moscow, U.S.S.R.
(Received April 27, 7972)
The competition between the recombination of activated electrons with parent positive ions and the formation of localized electrons (polarons) is considered. The equation obtained as the result of a physico-mathematical analysis gives a satisfactory explanation of the dependence of the polaron yield on temperature, experimentally observed in crystalline ice. The calculated value of the activation energy is equal to 0.12 eV, which suggests the localization process have an energy threshold. The possibility of polaron formation as result of dissociation of an excited state in alkaline glasses is postulated. This can be the reason for the anomalous dependence of G(e,- ) on temperature in 10 M alkaline solutions. Energetic characteristics of the water excited state were estimated. The calculated excitation potential is equal to 9.3 eV.
At present the existing theories of an electron in a local state ignore completely the kinetics of free electron transitions into localized state and the inverse processes. Nevertheless, it is possible that the quantitative characteristics (and, in particular, the polaron yield) are determined precisely by the competition between different kinds of electron stabilization processes (on a single molecule or on whole groups). Taking into account this approach, an attempt at a physico-mathematical analysis of such competition was carried out for frozen matrices. The interaction of ionizing radiation with substance leads to activation of molecular or atomic electrons into the conThe Journal of Physical Chemistry, Vol. 76, No. 25, 1972
duction band of a dielectric. Since, however, along with electron, there appears also a positive hole with which it will rapidly or slowly recombine, the kinetics of polaron formation in such a system has some specific features, as compared with the classical model of the ‘6excess” electron in crystal. Some of these features were pointed out in previous papers.l,2 If the number of the band electrons in unit volume is N V. N. Shubin, V. A. Zhrgunov, V. I. Zolotarevsky, and P. I. Dolin, Nature (London), 212,1002 (1966). V. M. Biakov, Yu. I . Sharanin, V. N. Shubin, Ber. Bunsenges. Phys. Chem., 271 (1971).
