J. Phys. Chem. i980, 84,1193-1196
fluctuations are not associated with metal transition?
-
1193
value at which we go from mean field to critical is precisely and identically determined by different techniques. Could you give a value for the v exponent? How close to T,can you really go in your experiment?
nonmetal
D. E. BOWEN.Our data indicate that concentration fluctuations are absent in Cs-NH,. As the MNM occurs in these solutims we conclude that the fluctuations are not directly associated with the MNM but with phase separation.
BOWEN.From the fo parameter we find that v changes from 1 for t > to -0.03 for e < I would emphasize that these values are not precise as they come from a theory that may not be applicable for the solutions. We can get to within 0.05 K of the critical temperature.
P. CHIEUX.I t is quite interesting to remark how the t
Are There Hot Carriers in Liquid Ar, Kr, and Xe? G. Ascarelli Department of Physlcs. Purdue Un/vers& West Lafayette, Indiana 47907 (Received July 17, 1079) Publlcafion costs assisted by the US. Department of Energy
It is shown that the saturation of the drift mobility observed in liquid Ar, Kr, Xe, and other insulating liquids is not a hot electron effect but can instead be understood as an effect due to shallow traps. Estimates of the binding energy of such traps in Ar indicate a value -90 meV. The effect of the addition of impurities can be understood as enhancing a Schottky ionization process.
Introduction From the early experiments of both Rice and co-workit became evident that e r ~ and ~ -Spear ~ and ~o-workers,~ the drift mobility of injected electrons in several liquids, particularly rare gases, is field dependent. The observed drift velocity initially increases with increasing electric field and finally appears to saturate. The shape of the drift velocity vs. applied field resembles that obtained for hot electrons in semiconductors and attempts have been made to explain the iexperimental resulb in a similar framework! More recenlly Schmidt and co-workers6made a careful study of the effects of several solutes on the field dependence of the drift mobility of electrons in liquid Ar, Kr, and Xe. Controlled impurities (N2,H2, CHI, (CH3)2, etc.) in concentratiions ranging from -0.1 to -1 atom % were dissolved in the liquid and it was observed that the saturation drift velocity increased with increasing impurity concentration. This work was extended by Kimura and Freeman who measured both the free ion yielde and the drift velocity' when the concentration of alkane impurities in liquid Xe vrlried from 1to 100 atom %. The increase of saturation velocities is again observed by these authors (in the case of ethylene up to 20% ethylene concentration) despite the fact that, the low field mobility decreases monotonically with increasing impurity concentration. Yoshino, Sowada, and Schmidt5proposed that this saturation of the drift mobility is due to inelastic collisions between the electrons and the atoms in the solution. It is the purpose of this note to point out that the hot electron explanation of the drift mobility saturation is inconsistent with experimental data. We do not know if the effect of shallow traps can also explain the near resonance which is1 observed in the drift velocity of electrons in several insulating l i q ~ i d s . ~ ? ' * ~ All existing ;measurements of drift velocity of electrons in insulating liquids are time-of-flight measurements. Therefore, if during its transit between two electrodes an electron is trapped either on an impurity or on a density
-
0022-3654/80/2084-1193$01 .OO/O
fluctuation, the time during which the electron is trapped does contribute to the drift velocity. The saturation drift velocities u, observed by all authors are small, in the sense that rnu,2/(2kT) varies between -0.17 in liquid Kr with 5 X 1020cmW3of methane atoms in pure liquid Xe. The corresponding ratio to 1.8 X in the case of hot electrons in Ges varies from 2.2 at 298 K to 17 at 77 K. The low drift velocities found in liquids give rise to the suspicion that the average electron energy is not much above thermal. Measurements by Rice ind co-workers2further indicate that the limiting drift velocity of electrons in liquid Ar decreases by about an order of magnitude when the temperature is increased from 85 to -150 K. The corresponding low field mobility can increase by up to a factor of 4. Again this is in contrast with observations in semiconductorsg(e.g., Ge or InSb) where an increase in mobility accompanies an increase of the saturation drift velocity. Finally the effect of addition of impurities is the most direct support of the contention that the saturation velocity does not arise from inelastic collisions. Yoshino, Sowada, and Schmidt5 measured the drift velocity of electrons in liquid Ar to which either 2.4 X cm-3 of H2 or 3 X 1019 of N2 had been added. The deviations of the observed drift velocities of electrons in the solutions from those observed in pure Ar have the same threshold when either N2 or H2 are used as the solute. Beyond the threshold, the very small differences between the effect of H2 and Nz impurities may be explainable in terms of the difference of the solute concentrations. If inelastic collisions with impurities were important, a drastic difference should have been observed between the case of N2 and H2impurities. The energy of the first vibrational1° state in H2 is 4395 cm-' above the ground state, while the first vibrational state in N2 is at 2359 6m-l. The corresponding rotational constantslo differ by a factor 30. Energy transfers to the recoil motion of the impurities is very different in the case of N2 and Hz. In all cases, if the 0 1980 American Chemical Society
1194
The Journal of Physical Chemistry, Voi. 84, No. 10, 1980
energy transfer per collision would determine both the threshold and the saturation drift velocity mentioned above they should be very different for N2and Hz.This is contrary to the experimental observations. The idea that the electrons collide with individual atoms of the fluid has difficulties. Since the electron is supposed to be in a conduction band about 0.25 eV below vacuum, it already reflects the interaction with a collection of atoms of the liquid. It is, therefore, contradictory to claim that as for electrons in a gas the electron can loose on the average a fraction m / M (M mass of the atom) of its kinetic energy. Only in a gas a steady state implies that the electron random energy must be (M/m)(mud2/2). Such an energy loss with an effective atom of mass M* = kT/c2,where c is the sound velocity, could be imagined to occur if the electron distribution would be Maxwellian with a temperature T, >> T and if acoustical phonons would be at the origin of the collisions? Such a mechanism would predict a drift velocity of the hot electrons proportional to the square root of the applied field. This is not o b ~ e r v e d .In ~ semiconductors the saturation of the drift velocity arises on account of interaction with optical phonons. Such phonons do not exist in the solid rare gases1"17 and are not expected in the fluid. At the most we may expect that the breakdown of k conservation law in the liquid would enhance the interaction with acoustic phonons, an effect that would tend to "cool" the electron distribution more than in the corresponding case of the solid and thus enhance the range where the drift velocity is proportional to the applied field. This is not ~bserved.~ In conclusion, despite the fact that high electric fields may appreciably increase the average electron energy, neither the observed saturation of the drift velocity nor its changes by the addition of impurities support such a conclusion.
Effect of Shallow Traps As mentioned above the presently available mobility data are obtained from the measurement of the time a carrier takes to cover a given distance. Deep traps that "permanently" remove carriers from the conduction process affect tba shape of the pulse describing the collected charge as a i%nction of time. They have been considered by various authors4J1J2and the experimental results reported in the literature have taken them into account. What has not been considered instead is the effect of shallow traps such that an electron is trapped and detrapped, on the average, several times (ai)by traps of type i during its transit from cathode to anode. The average time spent in a trap of type i is p;' where pi is the probability/unit time of detrapping an electron from such a trap. If we call t f the time during which the electron is free the mean transit time between anode and cathode is t = tf
* CniPi-1 i
(1)
If we call d the separation distance of the electrodes the apparent drift velocity v is
Calling P = u f / Ethe low electric field (tf >> Cin&') drift mobility where uf = d/t,, we get u =
PE
(3)
Ascarelli
I"
E-
" 2 '
l
o/
, 1 2 1
v,= 2.8 1O5crn/s
v
1G41 IO
, , , , , ,,,I
, , , , , ,,,I
, , , , , , ,,I
io3
IOZ
io4
, , , ,,, I
Io5
E (V/crn)
Figure 1. Comparison of the observed drift velocity in Ar, Kr, and Xe with the predictions of eq 4. The dfferent squares, circles, and triangles are from ref 4 and the meaning is explained therein. The broken line corresponds to the pure iiquki reported in ref 5. The full line is obtained from eq 4 wlth the Indicated drift mobilities and saturation velocities.
As long as the average distance traveled between trapping events is small compared with d, iii is proportional to d and the second term in the denominator of eq 3 is independent of d . Equation 3 predicts a dependence of the measured drift, velocity on electric field of the type
v=-
PE
1
+ AE
(4)
Data available in the literature4v6are compared with the results predicted from eq 4 in Figure 1. There is excellent agreement between the predicted and observed results in pure Ar, Kr, and Xe up to fields of -lo4 V/cm. In Ar for E > lo4 V/cm there is some discrepancy between the predicted and the observed field dependence of the drift velocity. This field corresponds also to the threshold for the effect of N2 and Hzimpurities reported in ref 5. We shall see below that the discrepancy can be completely explained by involving Schottky ionization of traps in a way that closely parallels earlier results for F centers in alkali h a l i d e ~ . ~ ~ J ~ From the form of eq 4 we clearly cannot recognize the effect of different types of traps on the field dependence of u. We shall therefore assume that it is sufficient to take into account only one type of traps so that
A = F1(fi/d)w
(5)
Hot Carriers in Liquid Ar, Kr, and Xe
The Journal of Physical Chemktty, Vol. 84, No. 70, 1980 1195
Since the same value of A describes equally well all the data of Miller et ala4for a given liquid at a given temperature (e.g., Ar at 85 K) for which 135 pm Id < 360 pm, we can give am upper limit to d / i i 100 pm. From eq 5 'we can now calculate p assuming d / i i = 100 pm. Assuming a constant density so that the binding energy E of the trap does not depend on temperature we expect to depend on temperature as
-
!3 = (1/r0)e+I'
(6) Unfortunately, we cannot obtain E and r o from the temperature dependence of p because c is a strong function of T due to density variations. We will have to estimate E at each temperature by making an educated guess for re In solid Ar at 4 K the highest phonon frequency is 2 X 10l2Hz.15 In the liquid16J7the characteristic frequencies are similar. Taking T~ = s we can estimate E as =90 s, the meV. If we had chosen d / t i = 10 p m and r0 = value of exp(c/kT) would have changed 100-fold. Despite this large change the calculated binding energy would only increase 40%, indicating the insensitivity of our estimated binding energy to the choice of parameters. The same estimates for Kr and Xe, taking d / i i = 100 pm, give binding energies of 125 and 180 meV near their respective boiling points (117 and 163 K). Given our choice of ( d / i i ) we can now estimate the density of the shallow traps. Their identification is, however, unknown. Detailed ballance imposes1*that the trapping probability per unit time per center ( a ) be related to by /?/a = ( 2 ~ r n K T / h ~ ) ~ / ~ e - ' / ~ ~ (7) from which a can be calculated. The trap density N can be calculatedl by remembering that ( d / i i = (l/ffN)uth, (l/ffN)Vth, wlhere uth is the random thermal velocity. In the case of liquid Ar this gives N 7 X 1013cm-3 a very reasonable value that can be compared with estimates of 2 X 1013cm-3 by Halpern et al.13 for the impurity concentration remaining after their extensive purification. From the value of a we can obtain the corresponding trapping cross sections: 1.4 X cm2,a value comparable with what is observed for donors in semiconductors.* These values correspond to lower limits of the trap density and upper limits of their cross section since only the product (ii/d')F1 is measured. Similarly upper limits of the trap density and lower limits of the binding energy can also be estimated by assuming that alii is of the order of the mean free path obtained from mobility measurements, Le., =lo0 A in the case of Ar. The resulting binding energies and trap densities are, respectively, E = 20 meV and 7 x io1' cm-:'.
-
Schottky Ionization of Trapped Electrons The discrepancy that exists between "pure" Ar and the prediction of eq 4 at high fields ( E > lo4 V/cm) is dramatically enhanced by addition of impurities.6 A t fields below lo4 V/cm the addition of 0.1% N2or H2 impurities to liquid Ar does not affect the dependence of the drift velocity on the applied field despite the fact that the impurity density is significantly larger than the density of preexisting traps. We are therefore led to conclude that the additional impurities neither create new traps nor significantly alter the binding energy of the electrons in the traps; they must,, however, contribute to the rate of change of the binding energy of the trapped electron with the applied field. It is well known13J4that in the case of hydrogen-like centers the potential barrier that must be overcome to thermally ionize a trapped carrier varies with field as Ell2.
I
I
t/
{ 0.2r c
OL--.--41 1.0
1.5 E'
X
I
1
2.0
2.5
3.0
16' (V/crn)"'
Figure 2. Deviation of the drift velocity of electrons in liquid Ar doped with -3 X I O 1 @cm3 of N, ( 4 )from that of the pure llquid ( vp) plotted as a function of the square root of the applied electric field. The straight line through the data is intended as a guide to the eye. Data from Figue 2 of ref 5.
As a result we expect that the probability of thermal ionization is 1 1 P = - c?xp(-~/KT) = - exp[-(co - C E 1 / 2 ) ] / k T (8) 70
70
where eo is the binding energy of the electron in the trap in the absence of field and C is a constant depending on the trap. The available experimental results5confirm such a field dependence of (Figure 2) since at high fields, in the saturation range, the drift velocity is proportional to P (eq 3). The addition of a small impurity concentration is1 not expected to change the binding energy of the shallow traps because impurities substitute only a few of the rare gas atoms of the liquid in the region swept by the trapped electron. However, the impurities will destroy the spherical symmetry of the surrounding of the trapped center and induce an electric dipole moment that enhances the interaction of the trap with the static applied electric field. These considerations explain why at low fields ( E 2: lo4 V/cm) a low concentration of impurities like H2or Nzdoes not alter the curve describing the variation of the (drift velocity with the applied field. A comparison of the effect of different impurities6 indicates that the polarizability of the molecules and their dimensions, rather than the vibrational and rotational states of the impurities, dominate the deviations of the field dependence of the drift velocity from that of the pure liquid. Thus when the same concentration of either Nz or H2 is added to Ar the curves describing the deviation of the drift velocity as a function of field from the pure liquid are the same despite the fact that all characteristic vibrational and rotational levels of H2 are appreciably larger than those of N2. The H-H and N-N distances are,21respectively, 0.746 and 1.097 A while the deviations of the dielectric constants K of the gas from unity are, respectively,21 254 X lo4 and 547' X lo4. The corresponding value for Ar is 517 X lo4. In this case effects due to size and polarizability of the impurities appear to compensate each other. A comparison of the effect of CHI and N2indicates that the deviation from the pure liquid dependence of the u vs. E curve has a threshold -200 V/cm in CH4 doped Ar while the threshold in N2 doped Ar is 2 X lo4 Vfcm. A comparison of (K- 1)of CHI, N2,and Ar indicate that they are, respectively,21944 X lo+, 548 X lo4, and 517 X lo4. The four H of CHI are on a sphere21 of 1.091-A radius, a value equal to the inter-nitrogen separation in N2. Here both the polarizability of the molecule and the perturbations associated with its size act in the same direction, i.e., to increase the difference between N2and CHI impurities. We would instead have predicted opposite
J. Pbys. Chem. 1980, 84, 1196-1199
1196
effects if we had considered inelastic losses to the rotational energy levels of N2and CHI since the rotational constant for the latter is about 2.6 times that of the former.10v22The energies of the vibrational states of N2and CHI, 2359 and 1526 cm-I, respectively,l0lz2would not suggest large differences of the rate of energy loss of electrons to these molecules. The data for ethane, propane, and butane impurities cannot be compared with those of CH4,N2,and Hz because there is not only a deviation of the u vs. E curve at high field but also a decrease of the low field mobility indicating a more complex behavior.
Conclusions An analysis of the dependence of the drift velocity of electrons in liquified rare gases suggests that the observed variation of the drift velocity with electric field is a result of the time spent by the electron in shallow traps rather than the effect of energy transfers during collisions. This conclusion is supported by the effect of addition of molecular impurities with widely different vibrational and rotational levels as well as by the sudden deviation of the expected field dependence of the drift velocity at given values of the electric field produced by the added impurities. This enhancement of the drift velocity is proportional to E l f 2 . Such a field dependence is to be expected from the decrease of the barrier that must be overcome for the thermal ionization of impurities. From the field dependence of the drift velocity observed by other authors4we obtained estimates of the trap density (e.g., 7 X lQ13 5 n 5 7 X 1017 in the case of Ar) and corresponding estimates of the binding energies of electrons in impurities (0.02 S 6 ,< 0.09 eV in the case of Ar). The trapping cross section of 85 K is estimated to be 1.4 X cm2, The origin of these traps remains, however,
undetermined and they could be either impurities or density fluctuations. Acknowledgment. The author thanks Professor J. W. MacKay for his numerous suggestions and for critically reading the manuscript. Support by the Department of Energy under contract no. DE-AC02-79ER10375.AOOO is gratefully acknowledged.
References and Notes H. Schnyders, S. A. Rice, and L. Meyer, phys. Rev., 150, 127 (1966). J. A. Jahnke, L. Meyer, and S. A. Rice, phys. Rev. A, 3, 734 (1971). J. A. Jahnke, N. A. W. Holzworth, and S. A. Rice, Pbys. Rev. A , 5, 463 (1972). L. S. Miller, S. Howe, and W. E. Spear, phys. Rev., 188, 871 (1968). K. Yoshino, U. Sowada, and W. F. Schmidt, Phys. Rev. A, 14,438 (1976); W. F. Schmidt, Can. J. Chem., 55, 2197 (1977). T. Kimura and G. R. Freeman, Can. J. Chem., 58, 749 (1978). T. Kimura and G. R. Freeman, Can. J. Chem., 56, 756 (1978). J. P. Dodelet and G. R. Freeman, Can. J. Chem., 55, 2264 (1977). E. M. Conweil, “High Field Transport in Semiconductors”, F. S e k , D. Turnbull, and H. Pnenrelch, Ed., Academic Press, New York, 1967, pp 16, 17. G. Herzberg, “Molecular Spectra and Molecular Structure”, Vol. I, Van Nostrand, New York, 1962. W. E. Spear and J. Mort, Roc. phys. Soc. London, 81, 130 (1963). K. Hetch, 2. Phys., 77, 235 (1932). R. N. Euwema and R. Smoluchowski, phys. Rev. A, 133, 1724 (1964). G. Splnolo and W. B. Fowler, Phys. Rev. A , 138, 661 (1965). D. N. Batchelder, M. F. Collins, B. C.G. Haywocd, and G. R. Sldey, J. Phys. C , 3, 249 (1970). V. Buontempo, S. Gunsolo, and P. Dore, phys. Rev. A, 10,913 (1974). P. A. Fleury, J. M. Worlock, and H. L. Carter, Phys. Rev. Lett., 30, 591 (1973). 0. Ascarelli and S. C. Brown, Phys. Rev., 120, 1615 (1960). B. Halpern, J. Lekner, S. A. Rice, and R. Gomer, Phys. Rev., 158, 351 (1967). H. A. Bethe and E. E. Salpeter, “Quantum Mechanics of One and Two Electron Atoms”, Academic Press, New York, 1957. “Handbook of Chemistry and Physics”, 47th ed, Chemical Rubber Co., Cleveland, Ohio, 1967. G. Herzberg, “Molecular Spectra and Molecular Structure”, Vol. 11, Van Nostrant-Reinhold, New York, 1945.
Similarities between Solvated Ground States and Gaseous Excited States of Alkali Atom Ron Catterall Department of Chemjstty and Applied Chemistry, University of Saiford, Saiford M5 4 W , England
and Peter P. Edwards” University Chemical Laboratoty, Cambridge CB2 IEW, England (Received July 17, 1979)
A close analogy is demonstrated between the optical and magnetic properties of alkali metal atoms solvated in a wide variety of media and those of excited alkali metal atoms in the gaseous phase. Developing this analogy one can show how the techniques of electron spin resonance (ESR) and optical spectroscopy, in conjunction with the trapping of excess electrons by solvated cations and anions, can be used to give a quantitative measure of ion-solvent interactions.
Introduction The dissolution of alkali metals in liquid ammonia results in the ionization of the ns valence electron from its parent ~ o r e , l a- ~process requiring some 2-4 eV if carried out on isolated alkali atoms in the gas phase.* The properties of extremely dilute metal solutions are then characteristic of an assembly of isolated fragments (Ma+and e,) in which case the solvated electron forfeits any parentage in the electronic states of the gas-phase alkali atoms.397bIn less polar solvents, situations also exist which
are intermediate between the completely ionized and high atomic character Here monomeric species, having the stoichiometry of solvated alkali atoms M,, retain the recognizable parentage of alkali atoms in the gas phase, but are subject to large perturbations arising from strong atom-solvent interaction^.^ Thus it is well recognized that the solvent exerts a pronounced effect on the electronic properties of these matrix-bound states.6,7 In this communication we attempt to address the problem of how the interaction between solvent and solute
0022-3654/80/2084-1196$01.00100 1980 American Chemical Society
