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).
Polaron Yields in Low-Temperature Pulse Radiolysis
3777
and the number of holes, accordingly, N+, the probability of their recombination is proportional to the product of these quantities P J - N + . It can be assumed with certainty that the electron will be trapped by the Coulombian field of the hole if it comes into its vicinity of radius
rB = e2/2ekT
(1)
The "thermal" electron itself having the velocity ij = (2kT/m,)1
during the time At is "'Iocalized" in the sphere of radius3 = DAt. The probability of partial or total overlapping of these spheres (which leads to recombination) is proportional to the products of the volumes
Since in the final analysis the dependence of G, on T is determined, it would be expedient to combine all the coefficients into one and write
W1 = 6(hT)-3/2
(3)
Apart from recornhination, there exists a possibility, first pointed out by Landaq4 that from an unstable band state the electron will transform into a specific local state such as polaron. Having carried out a physico-mathematical analysis, Pekar5 arrived a t the concept that the band electron of dielectrics will be autolocalized into the optimum polaron state as the result of monotonic deepening of the polarization potential well a t the expense of energy of the electron field. It i s not impossible, however, that near the top edge of the polaron band there should exist some intermediate region from which the electron, loosely bound with the medium, may be transferred, due to thermal or some other fluctuations, with equal probability either to the conduction band or to the polaron state. For this reason, the electron transition into the polaron state will be of a jumpwise (threshold) nature, i.e., the medium will constitute a stable trap for the electron only starting from a certain definite energy value of the polarization potential well E0.6 If Eo Is a trap of minimum depth in which electron can be stabilized and account is taken of the fact that the probability7 of the electron being captured by the trap is proportional to its depth (see Appendix I) and the number of the traps with an energy E , the Boltzmann factor, the following expression is valid mEe-E'k dE
( + f*)
= ae-EolkT(hT)21
(4)
Here w ~ ~the~probability ~ i s of the electron being captured by all traps of depth E' 2 Eo and a is the proportionality factor. Since it would be reasonable to assume Eo >> kT, expression 4 can kie restricted to the second term, so that
W2 = p(kT)e-Eo/kZT
Figure 1. Graphical solution of eq 8 from the experimental data of the authors and ref 9 and 11: 0, this paper, e,ref 11; 8 , ref 9. The straight line has been constructed using the leastsquares method. perature of a dielectric
(5)
The total probability of transition into the polaron state is proportional to the total number of electrons formed during ionization, so that the electron formation-decay equilibrium during the pulse action can be written as8
where N is the Avogadro value and Go the primary ionization yield. Using this expression we can obtain an equation for the radiation yield of detected polarons, determining a definite dependence of the polarone yield, G,, on the tem-
Combining all the unknown constant coefficients into one constant and writing out the dependence of the yield on temperature in an explicit form, we finally obtain Go
- GP GP
= constant x
T-5/2 &IkT
(8)
In Figure 1 this equation is solved by plotting it as In {[(Go - Gp)/Gp]T5/2] us. 1/T using the data of the authors as well as the results10 of other papers.gJ1 It can be seen that eq 8 agrees satisfactorily with experiment a t Eo = 0.12 eV *15%, which corresponds to the upper boundary of the librational (rotational) spectrum, equal to -0.125 eV.I2 This confirms the concept advanced earlier2314 that interaction with phonons has a determining effect on a number of properties of the electron local state in crystalline ice and liquid water. A radically new conclusion is drawn here, namely, that electron localization does not take place at all if the energy of the polarization well is less than the upper boundary of the phonon spectrum, i.e., the maximum energy of an optical phonon. If this is the case an electron can be ejected from the trap after an interaction with a phonon. It is convenient to believe an electron which relaxes with a positive hole is an excited state equivalent which has a half-life A t and an indeterminate energy A€. Then according to the principle of indeterminacy A L A t = h = APAx or A(meV2/2)At = A(rneP)Ax so that Ax = rL = PAt N 100 A at the temperature 273°K. L. 0.Landau, Phys. Z. Sowjetunion. 3, 664 (1933). S.I. Pekar, Issled. Elektron. Teor. Kristall., (1951). One can see that if an electron is autoiocalized during the process of monotonic deepening of the polarization potential well then the energy value €0 will be equal to zero simply. A. S. Davydav, "Kvantovaya mechanica," PhisMathGIS, Moscow,
1963. Radical OH does not form a Coulombian trap for electron and OHcan appear only if the time of an electron passing the hydroxyl sec, exceeds the time of the radical, equal to t = r O H / V 2 energy losses through dipole radiation (see Appendix i) t = (fi4c3/ e2a2 E 3 0 ~ - ) N IO-$ sec. Here €OH- is the ionization potential of OH- and a, the trap size. As one can see, the condition is not fulfilled and the OH- formation, because of an electron interaction with hydroxyl radical, does not take place. That is confirmed experimentaliy.Q I. A. Taub and K. Eiben, J. Chem. Phys., 42,2499 (1968). The values of G, obtained by different authors were normalized to the value of G, = 0.8 ion/100 eV at 263"K, obtained by direct electrochemical measurement^.'^ G. Nilsson, H. C. Christensen, i . Fenger, P, Pagsberg, and S. 0. Nieisen, Advan. Chem. Ser., No. 81, 71 (1968). I. E. Bertie and E. Whailey. J. Chem. Phys., 40, 1637 (1964). Yu. i. Sharanin, V. N. Shubin, and P. I, Doiin, Dokl. Acad. Nauk SSSR, 195,896 (1970). V. N. Shubin and S. A. Kabakchi. "Teoriya i metody radiatsionnoy khimii vody," "Nauka," Moscow, 1969. The Journal of Physical Chemistry, Vol. 76, No. 25, 7972
Shubin, Sharanin, Pernikova, and Vinogradov
3778
7
i
/OO
- 50
50
Assuming, after the authors of ref 19, this temperature to be the arbitrary zero of the temperature scale and taking into consideration all that has been said above about the dependence of E on T, let us transform expression 9 into the form (10)
Here G ( p )and GeXtare the yield of the electrons transferred from the discrete level to the polaron statezl and the primary yield of excited states (excitons), respectively. The value of the energy gap E(T), which is a function of the sample temperature, can be for convenience rewritten as E(T) = E300"~ A E T ~ K
i
Figure 2. Graphical solution of eq 12 from the experimental data of ref 19. T h e straight line has been constructed using the least-squares method. The dashed lines illustrate the change with temperature ot the first ( - - - - ) and t h e second*(-.-.-.-) terms of the equation.
Apart from ionization, radiation can lead to appearance of the excited states of water molecules. In such states an activated electron is on a discrete level located much deeper than the lower boundary of the conduction band. For this reason thermal dissociation of such exciton^'^ to form a band electron appears to be extremely problematic. Apparently, Frank15 was the first to suggest that thermal fluctuation can transfer a n excited electron into the polaron state (or eaq-). According to Frank, having absorbed a n energy quantum, the electron can (a) be thermally transferred into the polaron state with the probability - e - E I I R T , where E1 is the difference of energies of two levels, and (b) recombine with the hole radiating light with the probability independent of temperature. This leads to the expression for the quantum polaron yield (compare also ref 16)
As it, was first pointed out earlier,Z the residence of the polaron in the field of alkali metal cations resulted in appreciable (up to 13%, see ref 17) deepening of the polarization potential well. This affects the change in the value of Elcharacterizing the difference between the electron energies on a discrete exciton level, and in the polaron state. Moreover, the value of E l , as it follows from the results of numerous (see, e.g., ref 18) experimental ipvestigations, should be a function of temperature. As a consequence, one can see from analysis of expression 9, the observed polaron yield obtained upon cooling of the sample may increase. This phenomenon was experimentally observed by Ruxton, Cattell, and Dainton19 during pulse radiolysis of a I O M NaOH-KOM mixture in the temperature range 30077°K. With temperature decreasing from 300 to -200"K, the observed yield of esnlv- a t first increases smoothly from 3.3 to -5 ion/100 eV, and subsequently drops to revious values (-3) with further temperature decrease. ccording to ref 19, this drop correlates with complete disappearance at T *,= 135°K of translations of water molecules favoring spatial separation of chargesz0 during exciton dissociation. Due to steric difficulties arising in this case, the recombination phenomena begin to prevail in the competing processes, h results in a corresponding decreme of the detected of localized electrons. The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
(11)
Here E 3 0 0 0 ~is the gap width a t 300°K anid AETOK,the change in its value when the temperature drops from 300°K to a lower value. The value of A E ~ o ~ c aben determined using the data on the shift of EA,,, upon cooling of the ' K . f ~writing out sample,lg since A E T ~ K= 1.5 A E ~ l r ~ ~ ~ 1 " Then the dependence on temperature in a n explicit form, we obtain finally instead of (10) an equation that can be conveniently solved graphically
As it follows from the solution results (Figure 21, eq 12 agrees satisfactorily with experiment a t Gex, = 7. This value agrees closely with the value of the exciton yield estimated earlier in analysis of the anomalous behavior of the radiation-induced electrical conductivity of crystalline ice.2z From the slope of the straight line obtained, it is possible to estimate the energy characteristics of the dissociative transition exciton-polaron and the energy of the electron excitation to the exciton state. The value of the energy gap at 300"K, calculated from analysis of the experimental data, is 0.52 eV f 10%. Adding the calculated value of the energy gap to the polaron well depth a t this temperature and subtracting the quantity obtained from the ionization potential of a water m0lecule,~3we obtain Eext =
12.56 -
(E300"K+
l.jEh,,x300"K)E 12.56 - (0.52
-+ 2.74) = 9.30 eV
The calculated value closely coincides with the energy of one of excited levels of a water molecule given in ref 24 and is equal to 9.2 eV. Thus the results of the estimation carried out support the hypothesis that polaron is anion of an excited state (exciton).
Acknowledgment.
The authors wish to thank Professor
R. R. Dogonadze for discussion of this paper and valuable remarks. J. Frank, unpublished, cited in R. Livingstan, "Chemical Reactions Caused by ionizing Radiation," in coll. "Radiobiology," Moscow, 1955. N. F. Mott, Proc. Roy. SOC., 167, 384 (1938). S. A. Kabakchi, Khim. Vys. Energ., 5, 180 (1971). A. K. Pikaev, "Solvatirovannyy electron v radiatsionnoy khimii," "Nauka," Moscow, 1970. 6 .V. Buxton, F. G . R. Cattell, and F. 5. Dainton, Trans. Faraday SOC., 67, 687 (1971). E. Koldin, "Kinetics of Fast Reactions ~n Solution," "Nauka," Moscow, 'I 966. Less the yield of the electrons formed as the result of ionization and completely transformed into eaq- (see in more detail),'$ Le., G ( p , = G(e,'-)tota'- G(e,-)lOniz. Yu. I. Sharanin, V. NT Shubin, T. E. Pernikova. and V. i. Zolotarevsky, P. I. Dolin, Nature (London), 234, 15 (1971). W. C. Price, J. Chem. Phys., 4, 149 (1936). C. I . Hochanadel, "Comparative Effect of Radiation," M. Burton, eta/., Ed., Wiley, NewYork, N.Y., 1960, p 151.
Polarori Yields in Low-Temperature Pulse Radiolysis
Appendix I It is possible to estimate the probability of electron filling the traps of different depth which is equivalent to the determination of the problem of photon energy radiation during the electron transition from the zonal state to the lowest level of the trap. Suppose that the initial electron state is weakly bound with the trap. Then the radiation process can be calculated with the help of the first-order matrix: The equation fop:the matrix element will be 3,,1 =
-eJ’q2
A $ , d x i m
where i s the ekectromagnetic field potential and $1 and $2 are the wave functions of the initial and final states. Since these are steady states, $1 and +z include the time in the form of e-Eb i t and P - ’& z t so that the matrix element will be S&-.f = -27rlUL-r 6 (E1
3779
where a is a characteristic trap size and c, the light velocity. Consequently, the traps are not filled uniformly, but proportionally to their depth c.ubed. For a system with the Coulombian interaction a = (e2/ Aw) so that the obtained equation one can simplify to
P,, = (eLhc)w(wa/c)2= w / ( h ~ / e = ~ )~~/ ( 1 3 7 ) 3 The mean energy of the electron interaction with the induced polarization is equal to E = Y3E1, where E I is~ the polaron energy in the ground state. Then using the equation of the usual theory of the hydrogen-like atom
E
+’3E1,
=I 3
I w = nh one can estimate the electron frequence in the polaron state
- c2 - w)
where Evidently that the relation between the probabilities of a or a thermalized electron being captured by a proton, PH+, polarization trap, P,, will be and ~d i s the frequence of the electron oscillation into the trap with an energy E , and 6 the 6-function. Since the clipole radiation a t I = 1 is the most likely one, the probability of such transition can be written as
Then taking into account that the Legendre adjoint polynomial is equal to 1, and the integrand corresponds to the matrix unit of the dipole moment, the probability of transition in time unit will be
where E H is the ionization potential of hydrogen atom. It is clear that when the concentrations of “free” protons and polarization traps become equal the observed yield of eaqshould be essentially less then the primary ionization yield as it takes place in the cristalline In the case of liquid water the same effect was observed experimentally by Kenney and Walker.25 (25) G. A. Kenney and D. C.Walker, J. Chem. Phys., 50,4074 (1969).
The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
