Metal-Nonmetal Transition in Metal Solutions - Progress of ...

Supplement of the Progress of Theoretical Physics, No. 72, 1982

125

Metal-Nonmetal Transition in Metal Solutions Y oshio NAKAMURA

Department of Chemistry, Faculty of Science Hokkaido University, Sapporo 060 (Received October 16, 1979; Revised August 24, 1981) A brief survey has been given on the present status of the study of the metal-nonmetal transition in liquid materials. Experimental data on compound-forming liquid alloys and metal-ammonia (amine) solutions are reviewed and some attempts are presented to interpret them theoretically.

§ 1.

Introduction

Metal-nonmetal (M-NM) transitions have been studied in various noncrystalline materials.o-s> Among them, liquids are probably the most disordered and hence the most complicated materials. There are several different liquid systems which show an M-NM transition with change of thermodynamic parameters such as composition, temperature or pressure. The systems studied so far can be classified as follows: (1) solutions of metals in molecular solvents such as alkali metals in liquid ammonia or amines, (2) solutions of metals in corresponding molten salts such as sodium in molten sodium chloride, (3) binary alloys in which the electronegativity of the constituents is largely different such as molten Mg-Bi alloys, (4) expanded liquid metals and compressed metal vapours such as supercritical fluid mercury. All these liquid systems can be described as dilute liquid metals or liquid metals diluted by non-metallic components including a vacuum in the case of (4). Though the basic understanding of liquid metals has now been well established,s>.•> physical characters of the transition region of these liquids are relatively poorly understood in spite of extensive amounts of experimental data. In this report we review briefly the experimental results of some typical systems of the classes (1) and (3) liquids and then present some attempts to explain them theoretically.

§ 2.

M-NM transition in compound-forming molten alloys

It has been known that some molten binary alloys show anomalous electric

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Y. Nakamura

properties at a certain stoichiometric composition. The electrical conductivity has a cusp-like minimum, two or three orders of magnitude smaller than that of the pure components, and the thermoelectric power changes its sign or has a maximum at a composition usually corresponding to that of an intermetallic compound in the solid state. Other physical properties such as the viscosity and the compressibility also show a cusp-like maximum. The alloys around this composition have often been referred to as liquid semiconductors. 5J,aJ It is thus obvious that we have a metal-semiconductor transition, when we change the composition of these alloy systems towards the compound-forming region. Compound-forming molten alloys generally consist of components in which the respective electronegativity values are largely different. These alloy systems are classified into two groups: (i) alloys containing alkali or alkaliearth metals as an electropositive component and (ii) alloys containing chalcogen elements (Te, Se and S) as an electronegative component.

2. 1.

Alloys containing alkali or alkali-earth metals

A typical example of liquids of this type is liquid Mg-Bi which shows a metal-semiconductor transition around the composition Mg8 Bi 2 • This system is of particular interest because it is posible to follow continuously the transition from metallic behaviour to semiconductor behaviour by changing the alloy composition. Experimental results for the electrical conductivity and thermo8000

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Fig. 1. Electrical properties of liquid Mg-Bi. (a) Electrical conductivity at 900"C and (b) thermoelectric power at lOOOC above the liquidus temperature (Enderby and Collings 7l).

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Metal-Nonmetal Transition in Metal Solutions

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electric powern of this system are shown in Fig. 1. The thermodynamic data8> also indicate that a drastic change in the bonding characteristics takes place as we proceed from pure liquid Bi to pure liquid Mg. It has also been reported that liquid Li-Bi 9 > and Li-Pb 10> show similar electric properties, though experimental work on alloys of this group is still limited. Perhaps the most striking example of the class (3) liquids may be liquid Cs-Au.w It has been suggested from the measurement of electrolysis 12> that the bonding at the composition CsAu is totally ionic. Hence liquid Cs-Au has much resemblance to metal-molten salt mixtures of the class (2) which show an MNM transition as well.18), u>

2. 2.

Alloys containing chalcogen elements

These include liquid Ag-Te, Cu-Te, In-Te, Tl-Te, Tl-Se and others 5>,a> and represent the most widely studied systems among the class (3) liquids. Figure 2 shows as a typical example the electrical conductivity, thermoelectric power15>,ta> and Hall coefficienen of liquid Tl-Te as a function of composition. We see a sharp minimum in conductivity and a maximum in the Hall coefficient (negative) at Tl 2Te (XT1 = 2/3), where the thermoelectric power changes its sign. At this composition the entropy of mixing is very small 18> and the heat of mixing 18 >~ 2 o> has a deep minimum, as shown in Fig. 3. It is suggested 16>,ts> that the partial ionicity in the chemical bondings due to electron transfer from Tl to Te may be responsible for this large negative excess heat and the 1000

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Electrical properties of liquid Tl-Te. (a) Electrical conductivity and (b) thermoelectric power at 500"C (Nakamura and Shimoji' 6 >) and (c) Hall coefficient at 535"C (Enderby and Simmons">).


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reduction of the entropy of mixing. Recent measurements of the electrotransport21l in liquid Tl-Te show that Tl drifts towards the cathode and Te towards the anode, which is consistent with the model given above. An equally plausible model is to assume the existence of fairly well-defined molecular species such as Tl 2Tem,tsl, 22 l and to treat the system as consisting of such molecules and excess Tl and Te. This may be referred to as the pseudobinary alloy model. This model is also capable to explain the thermodynamic data18l as well as some of the electrical properties 23 l' 24 l as discussed later.

2. 3.

Pseudogaps

In view of the large difference in the nature of various compound-forming alloys, it is obvious that any single model cannot describe the essential features of all these systems which show a metal-semiconductor transition around the compound-forming region. Here, we present some models which are based upon the concept of the pseudogap developed generally for amorphous semiconductors by Mott and others.n Figure 4 shows a sketch of the density of states for non-crystalline materials.") The transition from the semiconducting state to the metallic state may be represented by the delocalisation of the electronic states at the Fermi level with increasing the amount of metallic components. The factor g defined as (1)

1s frequently used as a measure of the transition, where N (EF) is the density of states at the Fermi level and N 0 (EF) is that for the free electron case. Table I gives the results of an application of this idea to various molten ternary compound with the composition ABC2 (C: chalcogen) ."5l The transi-


Metal-Nonmetal Transition zn Metal Solutions

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N Since no change of sign of the Hall coefficient was observed at the "p-n transition" region,m this simple interpretation was found to be inconsistent with the conventional concept. Later, the Hall coefficient of the disordered materials has been studied theoreticall/ 6>,m and it is supposed that this objection is not a fatal one. 8 > Ender5y and Simmonsm proposed a quite different model which is illustrated in Fig. 5 (a). They supposed that the Hall coefficient rather than the thermoelectric power is a useful measure of both the sign and the density of the charge carriers. They considered that liquid Tl-Te consists of Tl 2Te mole-


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(a) .¥(E)

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Fig. 5. Density of states in liquid Tl-Te. (a) Enderby-Simmons model, and (b), (c) Faber model. (From Ref. 3) .)

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cules and excess Tl or Te atoms. The valence electrons in the molecules are mostly localised and form a band which is separated in energy from the conduction band. The valence electrons of the unbounded atoms will occupy the conduction band and can be treated in terms of the nearly free electron theory; this explains the peak of the Hall coefficient at the stoichiometric composition (Fig. 2 (c)). In order to explain the change of sign of the thermoelectric power, Enderby and Simmonsm postulated an energy-dependent scattering by a virtual bound state. The electrical conductivity in the metallic region of this system is discussed by Schaich and Ashcrofes> in terms of the Faber-Ziman theory, considering the scattering of nearly free electrons by Tl 2Te molecules and free Tl or Te ions. In order to avoid some difficulty in interpreting the change of sign of the thermoelectric power, Faber8 > modified the model of Enderby and Simmons, as illustrated in Figs. 5 (b) and (c). Faber suggests a modification of the conduction band with addition of excess Tl or Te atoms; an overlapping band is formed from two of non-localised states based upon the 5s and 5p valence electrons of the excess Te on the one hand, and the 6s and 6p valence electrons of the excess Tl atoms. These bands located at an energy somewhat below the conduction band resemble the impurity bands in heavily doped crystalline semiconductors. Both of the "impurity bands" due to the excess atoms are partially filled, but the position of the Fermi level is different with respect to the second peak in the impurity bands. This model is capable of explaining the change of sign of the thermoelectric power, since the thermoelectric power in the metallic region near the transition can be written as :n s~ -2n" k"T { dlnN (E)}

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131

alloys besides liquid Tl-Te and also to non-chalcogen alloys such as liquid Mg-Bi. We suppose, however, that these models are still speculative and the direct experimental evidence for the existence of the pseudogap and its modification with non-stoichiometric doping in large amount is now highly required. On the other hand, in order to carry out theoretical calculation of the band structures of liquid alloys such as Tl-Te, the experimental knowledge of local configuration of the constituent atoms is indispensable. In later papers Cutler28 )· 29) has put forwards the estimation of the electronic structure of liquid Tl-Te: He supposed the formation of molecules of the chain-like structure Tl(Te) n-Tl in the Te-rich side and a superposition of tight-binding bands due to Tl and Tl 2Te in the Tl-rich side. However, no detailed structural study has not yet been carried out to test the proposed model. The measurements of X-ray diffraction in liquid Tl-Te by Waseda and Tamaki 30 ) and Waseda et al. 3 u reveal that the total structure factors are practically invariant with composition between pure Tl and Tl 2Te and that the structure of Tl 2Te is little affected with increasing temperature up to 900°C. From these results they doubt the existence of a molecular-like compound at the stoichiometric composition Tl 2Te, though no definite structural model has not yet been presented.3u It should be noted that a totally different approach to the metal-semiconductor transition in compound-forming alloys has been proposed by Cohen and his coworkers 32 ) on the basis of the effective medium theory. In this model, liquid Tl-Te alloys are supposed to be formed of liquid Te containing Tl 2Te clusters for the composition XT1 ).

Li-CH,NH2 (methylamine) solutions

The electrical conductivity and thermoelectric power of liquid Li-CH,NH/ 4 > are shown in Fig. 10. This system shows an M-NM transition at XLi=0.15, a much higher metal concentration than in liquid Li-NH,. The Knight shift


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and 1/T1 in this system 45 > are also shown as a function of metal concentration in Fig. 11. Recently, Edwards et al. 46 > reported the relaxation time of ESR of this system. All of these data given in Fig. 12 strongly indicate that the delocalisation of electron states occurs above XLi~0.15. The correlation time of the electron-nuclear interaction changes by about two orders of magnitude between XLi = 0.14 and 0.20.' 5>·•6 >

3. 3.

Mott transition and concentration fluctuation

In Matt's original work•n.•s> it is argued that the screening of the long range Coulomb interaction of electron and ion (hole) in an array of oneelectron atoms is responsible for the existence of the insulating state. Thus, the M-NM transition occurs when

(4) where (5) and K.t is the static dielectric constant of the background and nc is the critical electron concentration. Modifications of the Matt criterion have been made by various authors. The easiest way is to introduce an effective dielectric constant of the medium instead of K.t. which is a sort of the average of the static and optical dielectric constants 49 > and has been applied to some metalammonia and metal-methylamine solutions88 >.••>.so> to obtain reasonable values for n 0 • Berggren and LindeW 0 evaluated the values of the effective Bohr radius by improving the atomic wave function and the screening function. Recently, Edwards and Sienko52 > proposed to determine values of aa* from the experimental ionisation energy (Eexp) of the localised state (donor) via aa* = e2/2K.tEexp rather than the dielectric properties of the host materials. They showed that the critical concentration observed in various host materials (ranging from 1018 to 1022cm- 8 ) , including liquid Li-CH8-NH2 , is given by the Matt criterion (Eq. (4)), if an appropriate value of aa* is adopted. Hubbard has discussed the problem as a single site correlation and attributed the M-NM transition to overlapping bands known as the Hubbard bands. 0 ' 58 > The lower band corresponds to singly occupied centres (donor) and the upper one to doubly occupied centres. The M-NM transition occurs when a half-full band splits into two bands. The criterion of this transition is similar to Matt's criterion!> We suppose that liquid Li-CH3 NH 2 undergoes this sort of transition. The experimental data indicate that the upper and lower Hubbard bands completely overlap at high electron densities (XL;>0.20) and the bands begins to open up in intermediate concentrations (0.20>XLi >0.14). These may also be seen in the curve of the relaxation time of NMR given in Fig. 12. It should be noted, however, that the edges of the two Hubbard bands may be modified considerably due to disorder. States at


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Y. Nakamura

the Fermi level are expected to be localised m the Anderson sense, 54 > and then the Hubbard gap may be replaced by a pseudogap. In such a case, the temperature dependence of the electrical conductivity and thermoelectric power can be explained in terms of excitation of electrons at the Fermi level to the mobility edge of the upper Hubbard band. This model was tested for the nonmetallic (semiconducting) region of liquid Li-CH 8 NH2 and a reasonable agreement with experiment was obtained.••> The same model was applied tentatively to liquid Li-NH 8 •88 > In view of the recent data of the electrical conductitivity and thermoelectric power of liquid K-NH 855 > we suppose that this simple application of the pseudogap model is not appropriate for the semiconducting region of metal-ammonia solutions in general. It appears that the effect of molecular entities which may vary with temperature should be taken into account in the concentration range where the peak of the temperature coefficient of the electrical conductivity is located (XLi ~0.03 for Li-NH 8 ) . An alternative explanation of the M-NM transition in metal ammonia solutions has been put forwards by Jortner and Cohen. 56 >. 5n In their model the transition arises from the existence of microscopic inhomogeneity in solutions and the conduction in the transition region has been interpreted on the basis of the semiclassical percolation theory. The assumption of this large scale inhomogeneity is based upon the fact that the concentration fluctuation deduced from thermodynamic data 58 > exhibits a considerable peak at the transition concentrations for Na-NH 8 and Li-NHs, in which a liquid-liquid phase separation occurs below the respective critical temperatures. The small angle neutron scattering measurements for liquid Li-ND8 also show the existence of clusters near the critical consolute temperature. 59 > The correlation length of the concentration fluctuation deduced from the neutron scattering data is """40A at 3°C above the critical temperature. 59 > If such large clusters due to the concentration fluctuation are non-metallic, they will affect the electronic properties of the system. We note also that the effect of tunneling of electrons through the non-metallic clusters becomes important, depending upon the size and the barrier-height of the clusters. 60 > We think as a conclusion that the M-NM transition in metal-ammonia or -amine solutions is of a Matt-Hubbard type, if large scale inhomogeneities are absent. The M-NM transition in Li-CH 8 NH2 is certainly of this type in view of the thermodynamic evidence.•u Metal-ammonia solutions at higher temperatures may also undergo this sort of transition. When we approach the critical consolute temperature of phase separation in these solutions, we do have large fluctuations of concentration.*> Even in such a case, we have to examine the validity of the proposed semiclassical percolation theory.m A *> Recently, a new two·phase region has been reported above the well-known critical point of Na-NHa solutions."> Further investigations will be needed to verify the existence of such an inhomogeneity range.


Metal-Nonmetal Transition zn Metal Solutions

137

proposed phase diagram for the Li-NH8 system is illustrated schematically in Fig. 13, which is a modification of the original sketch of Jortner and Cohen. 5n Further investigation is highly required to substantiate the proposal.

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§ 4.

Conclusion

We see that a metal-nonmetal transition takes place when we add a metallic component to various non-metallic solvents: molecular liquids (NH8 , CH8 NH2 ) , ionic liquids (CsAu, NaCl) or semiconducting liquids (Tl 2Te, Mg8 Bi2 ) , and when the solubility of metals is high enough, the resulting solutions show properties of liquid metals containing nearly free electrons. One of the most important characteristics of metal-nonmetal solutions is that the structure of the solutions changes drastically, when we proceed from the non-metallic state to the metallic state. The dissolved metals change their environment and hence the electronic structure of the system, since the structural relaxation occurs very easily in the liquid state. Therefore, we are always confronted with a problem to enquire, with adding the metals, what are the existing structures of the liquids under investigation. The change of structures may be expressed in terms of solvation, formation of complexes or special chemical bondings. This is in contrast with the case of heavily doped crystalline semiconductors, in which the structure of the host materials is unchanged with increasing the amount of dopants. The second difficulty is that the characteristics of the non-metallic side of the metal-semiconductor transition in the liquid state are still not clear; our understanding of liquid semiconductors is very limited in comparison with those of liquid metals, molecular liquids or ionic liquids. In spite of (and perphaps because of) these diffi~ulties, the problem of the metal-nonmetal transition in various liquids attracts us very much in both


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Y. Nakamura

the physical and chemical points of view. And we hope that investigations of these liquid systems will furnish us a deeper insight into the characters of materials involved in this transition phenomenon.

Acknowledgements The author wishes to thank Professor M. Shimoji for valuable discussions. References 1) N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971). 2) N. F. Mott, Metal-Insulator Transition (Taylor-Francis, London, 1974). 3) T. E. Faber, Introduction to the Theory of Liquid. Metals (Cambridge Univ. Press, London, 1972). 4) M. Shimoji, Liquid Metals (Academic Press, London, 1977). 5) M. Cutler, Liquid Semiconductors (Academic Press, New York, 1977). 6) ]. E. Enderby, Amorphous and Liquid Semiconductors, ed. J. Tauc (Plenum Press, London, 1974), Chap. 7. 7) J. E. Enderby and E. W. Collings, J. Non-Cryst. Solids 4 (1970), 161. 8) R. Hultgren, R. Orr, P. D. Anderson and K. K. Kelley, Selected Values of Thermodynamic Properties of Metals and Alloys (J. Wiley, New York, 1963). 9) G. Steinleitner, W. Freyland and F. Hensel, Ber. Bunsenges. 79 (1975), 1186. 10) V. T. Nguyen and J. E. Enderby, Phil. Mag. 35 (1977), 1013. 11) H. Hoshino, R. W. Schmutzler and F. Hensel, Phys. Letters A51 (1975), 7. R. W. Schmutzler, H. Hoshino, R. Fischer and F. Hensel, Ber. Bunsenges. 80 (1976), 107. F. Hensel, Adv. in Phys. 28 (1979), 555. 12) K. D. Kruger and R. W. Schmutzler. Ber. Bunsenges. 80 (1976), 816. 13) N. H. Nachtrieb, Adv. Chern. Phys. 31 (1975), 465. 14) D. A. Greenwood and V. K. Ratti, ]. of Phys. F: Metal Phys. 2 (1972), 289. 15) M. Cutler and C. E. Mallon, Phys. Rev. 144 (1966), 642. 16) Y. Nakamura and M. Shimoji, Trans. Faraday Soc. 65 (1969), 1509. 17) ]. E. Enderby and J. C. Simmons, Phil. Mag. 20 (1969), 125. 18) Y. Nakamura and M. Shimoji, Trans. Faraday Soc. 67 (1971), 1270. 19) T. Maekawa, T. Yokokawa and K. Niwa, ]. Chern. Thermodyn. 3 (1971), 143. 20) J. Terpilowski and E. Zaleska, Rcz. Chim. 37 (1963), 193. 21) Y. Kitazawa, Y. Nakamura and M. Shimoji, to be published. 22) A. B. Bhatia and W. H. Hargrove, Phys. Rev. B10 (1974), 3186. 23) W. Schaich and N. W. Ashcroft, Phys. Letters A31 (1970), 174. 24) V. K. Ratti and A. B. Bhatia, J. of Phys. F: Metal Phys. 5 (1975), 893. 25) Y. Ninomiya, Y. Nakamura and M. Shimoji, J. Non-Cryst. Solids 17 (1975), 231. 26) T. Matsubara and T. Kaneyoshi, Prog. Theor. Phys. 40 (1968), 1257. 27) L. Friedman, J. Non-Cryst. Solids 6 (1971), 329. 28) M. Cutler, Phil. Mag. 24 (1971), 381. 29) M. Cutler. Phil. Mag. 33 (1976), 559. 30) Y. Waseda and S. Tamaki, Z. Naturforsch. 30a (1975), 1655. 31) Y. Waseda, Y. Tsuchiya and S. Tamaki, Z. Phys. B32 (1979), 253. 32) M. H. Cohen and J. Sak, J. Non-Cryst. Solids 8-10 (1972), 696. M. H. Cohen and J. Jortner, Amorphous and Liquid Semiconductors, ed. J. Stuke and W. Brenig (Taylor-Francis, London, 1974), p. 167. 33) R. J. Hodgkinson, J. of Phys. C: Solid State Phys. 7 (1974), L9.


Metal-Nonmetal Transition in Metal Solutions 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49) 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61)

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K. Kai, S. Ukai and K. Suzuki, J. Phys. Soc. Japan 48 (1980), 2048. H. Ruppersberg and H. Egger, ]. Chern. Phys. 63 (1975), 4095. J. C. Thompson, Electron in Liquid Ammonia (Clarendon Press, Oxford, 1976). J. Jortner, ]. Chern. Phys. 30 (1959), 839. M. Hirasawa, Y. Nakamura and M. Shimoji, Ber. Bunsenges. 82 (1978), 815. R. D. Nasby and J. C. Thompson, J. Chern. Phys. 53 (1970), 109. U. Even, R. D. Swenumson and ]. C. Thompson, Can. ]. Chern. 55 (1977), 2240. N. W. Ashcroft and G. Russakoff, Phys. Rev. A1 (1970), 83. Y. Nakamura, Y. Horie and M. Shimoji, J. Chern. Soc. Faraday Trans. I 70 (1974), 1376. A. N. Garroway and R. M. Cotts, Electrons in Fluids, ed. ]. Jortner and N. R. Kestner (Springer-Verlag, Berlin, 1973), p. 213. Y. Nakamura, M. Hirasawa, M. Niibe, Y. Kitazawa and M. Shimoji, Phys. Letters 79A (1980), 131; ]. de Phys. CS (1980), 32. T. Toma, Y. Nakamura and M. Shimoji, Phil. Mag. 33 (1976), 181. Y. Nakamura, T. Toma, M. Shimoji and S. Shimokawa, Phys. Letters GOA (1977), 373. P. P. Edwards, J. R. Buntaine and M. ]. Sienko, Phys. Rev. B19 (1979), 5835. P. P. Edwards, Phys. Chern. Liq. 10 (1980), 189. N. F. Mott, Proc. Phys. Soc. A62 (1949), 416. N. F. Mott, Phil. Mag. 19 (1961), 287. ]. H. Simpson, Proc. Roy. Soc. A197 (1949), 269. G. Lepoutre and ]. P. Lelieur, Metal-Ammonia Solutions, ed. J. ]. Lagowski and M. ]. Sienko (Butterworths, London, 1970), p. 247. K. F. Berggren and G. Lindell, Solid State Comm. 13 (1973), 1589. K. F. Berggren, F. Martino and G. Lindell, Phys. Rev. B9 (1974), 4096. P. P. Edwards and M. ]. Sienko, Phys. Rev. B17 (1978), 2575. J. Hubbard, Proc. Roy. Soc. A276 (1964), 238; A277 (1964), 237. F. Yonezawa, M. Watabe, M. Nakamura and Y. Ishida, Phys. Rev. B10 (1974), 2322. M. Niibe, Y. Nakamura and M. Shimoji, ]. Phys. Chern. in press. J. Jortner and M. H. Cohe.n, ]. Chern. Phys. 58 (1973), 5170. ]. Jortner and M. H. Cohen, Phys. Rev. B13 (1976), 1548. K. Ichikawa and ]. C. Thompson, ]. Chern. Phys. 59 (1973), 1680. P. Chi ex, Phys. Letters A48 (1974), 493; ]. Phys. Chern. 79 (1975), 2891. N. F. Mott, J. Phys. Chern. 79 (1975), 2915. V. Steinberg, A. V oronel, D. Linsky and U. Schindewolf, Phys. Rev. Letters 45 (1980), 1338.
