J . Phys. Chem. 1984,88, 3709-3711
qualitatively by the mobility value of (6 f 2) X cm2/(V s) which is obtained by extrapolation of the mobility of solvated electrons in liquid ammonia at T = 233 K5 to higher temperatures (Figure 1). The diffusional motion of localized electron similar to that of solvated ions can be excluded in this density range since the electron mobility is much higher than the mobility of cations which is obtained by extrapolating the measured mobilities to higher densities (Figure 1 ) . At still higher densities, however, the cavities should collapse giving rise to a further mobility edge: a transition from the
3709
localized state to the quasi-free state should be observable.
Acknowledgment. I thank my co-workers Drs. M. Wantschik, V. Giraud, and K. Bukowski and Mr. M. Heintze for their help in the experimental work. I also acknowledge critical comments by Professor Sir Nevi11 Mott. The realization of the high-pressure photocell was greatly facilitated by the expertise of the precision mechanic Mr. W. Baltz. Part of this work was supported by the Deutsche Forschungsgemeinschaft. Registry No. NH,, 7664-41-7; H 2 0 , 7732-18-5.
Mobillty of Excess Electrons in Methylamine Vapor and Electron Scattering Processes in Dense Polar Vapors J. C. Thompson,* U. Even, and D. K. Blanks Department of Physics, The University of Texas at Austin, Austin, Texas 7871 2 (Received: August 25, 1983; In Final Form: November 28, 1983)
The mobility I.L of electrons photoinjected into CH3NH2vapor was measured with a drift time-of-flight apparatus. At room ~ )70 cm2/(V.s) at the highest temperature the I.L values ranged from 1100 cm2/(V.s) at the lowest density (n = 6 X 10l8~ m - to density (n = 8 X 1019cm-’). The normalized mobility nw drops with increasing density as for other polar fluids. We use the Ioffe-Regel criterion for the onset of strong scattering. A simple criterion for evaluating the strengths of the electron-molecule interaction (suggested by Warman) permits us to determine a relation between dipole moment and strong scattering, valid for all polar fluids studied to date. Some conjectures about electron localization are offered.
Introduction There have been three recent reports of electron mobility p measurements in dense vapors of molecules with a substantial dipole We report here measurements on yet another-methylamine-as well as an analysis which shows how the dipole moment influences the onset of diffusive transport or “strong” scattering with increasing density n and, therefore, eventual localization. In the low-density limit each electron interacts with one gas molecule at a time. When only single scattering processes occur, the normalized mobility n p is independent of n. For an electron described by a Maxwell-Boltzmann distribution interacting with a polar molecular of moment p (in Debye) the result4 is n p / T’/2[(Vcms-K)-’] = 8.40 X 1OZo/p2.The low-density values of n p for the materials of present interest are rather close to this value. However, this is not true for polar gases with smaller values of p . There are severe problems with the Born approximation result5 for p > 2.5 D as well, as Christophorou and Christodoulides6 have shown for many gases. As the density is increased the normalized mobility begins to drop below the low-density value. Several factors can contribute to this decrease, which is common to nearly all gases studied to date. Once the density is high enough that electrons can interact simultaneously with two or more molecules, multiple scattering begins to be important.’ Structural effects then enter.s At still
higher densities localization often occur^,^*^ though a conduction band may form instead.*O Discussions of the localization process in dense materials with intrinsic conduction electrons, e.g., Hg vapor or P-doped Si, are often introduced in terms of the Ioffe-Regel criterion.’ ‘,12 Ioffe and Regel (IR) pointed out in 1960 that values of the mean free path A so small that kFA < 2a (where kF is the Fermi wave number) are impossible within the confines of a free electron model and the Boltzmann approximation. Kaplan and Kitteli3 stated a similar criterion somewhat earlier in analyzing M-NH3 solution conductivities. Whenever A is less than the electron de Broglie wavelength, A, strong scattering must be operating and the propagation-with-occasional-scatteringpicture appropriate to weak scattering must be abandoned. We have used the IR criterion, X = A, to establish the onset of strong scattering for excess electrons in dense polar vapors.’,9 This application departs from the usual one in several ways. As noted these are excess electrons, injected from a photocathode or some other source, and thermalized. The density is low so classical statistics apply. Therefore, the electron momenta are low and the de Broglie wavelength long. This has the apparently anomalous consequence that the scattering is considered to be “strong” when the mobility is as high as 100 cmz/(V.s) (at room temperature). Nevertheless, the IR criterion provides a consistent basis for describing the electron transport process as “nonfree” or “quasil~calized”.~~ For thermal electrons, described by a
(1) P. Krebs and M. Wantschik, J . Phys. Chem., 84, 1155 (1980); P. Krebs and M. Heintze, J. Chem. Phys., 76, 5484 (1982); V. Giraud and P. Krebs, Chem. Phys. Lett., 86, 85 (1982). (2) L. G. Christophorou, J. G. Carter, and D. V. Maxey, J . Chem. Phys.,
(7) T. F. O’Malley, J . Phys. B, 13, 1491 (1980). (8) M. Nishikawa and R. A. Holroyd, J . Chem. Phys., 77, 4769 (1982). (9) N. Geeand G. R. Freeman, J . Chem. Phys., 78, 1951 (1983); Can. J . Chem., 61, 1664 (1983). (10) S. S.-S. Huang and G. R. Freeman, J. Chem. Phys., 68,1355 (1978). (11) A. F. Ioffe and A. R. Regel, Prog. Semicond., 4, 237 (1960). (12) N. F. Mott, ”Metal Insulator Transitions”, Taylor and Francis, London, 1974. (13) .I. Kaplan and C. Kittel, J . Chem. Phys., 21, 1429 (1953). (14) J. Jortner and A. Gaathon, Can. J . Chem., 55, 1801 (1977).
76, 2653 (1982). (3) N. Gee and G. R. Freeman, Can. J . Chem., 60, 1034 (1982). (4) S. Altshuler, Phys. Rev., 107, 114 (1957). (5) Y. Itikawa, Phys. Repr., 46, 117 (1978). (6) L. G. Christophorou and A. A. Christodoulides, J . Phys. B, 2, 71 (1969).
0022-3654/84/2088-3709$01.50/0
0 1984 American Chemical Society
3710 The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 Maxwellian distribution, the IR criterion is satisfied at a mobility given by pIR= 2.8 X 104/T where p is in cm2/(V-s) and T is in K. In particular, one may note that n p ceases to be independent of n when p < pIR.The knee in a p n vs. n plot may not be precisely coincident with pIRbut the two are always close. We shall use pIRin what follows to define nIR, the density at which strong electron-molecule interactions become important. In the following sections the present experiments are described, the data on electrons in CH3NH2presented, and an analysis of the dependence of pIRand nIR on the molecular dipole moment discussed. Experimental Section A straightforward time-of-flight technique was used. An Ag photocathode was illuminated by a pulse from a Xe flashlamp or a dye laser. The photoelectrons are then pulled across a 1-mm gap by an applied potential. The mobility is then computed from the pulse length, potential, and electrode spacing. The signal was captured, averaged, and displayed by a Biomation 8 100 transient recorder controlled by a PDP-11/23 minicomputer. The interface/driver between the transient recorder and computer triggered the recorder, called the data to the adder, summed each sweep with the previous ones and initiated the next pulse. As all these functions were carried out by hardware, operation was possible at a 200-Hz rate using all 2048 channels of the transient recorder. The computer also was used to deconvolute the light pulse shape from the transit pulse. The observed transit times were in excess of 1 ps. Warman and SauerIs give thermalization times for electrons in NH3, at the pressures of these experiments, to be of order 10 ps. Thus one can be certain of measuring the transit time of thermal electrons at the low fields used. The values of Eln (field to density V ratio) varied from 0.12 to 0.21 townsend (1 Td = cm2/molecule), with most values near 0.15 Td for the data reported here. The major limitation in the experiment is geometrical. The electrode spacing and the field homogeneity cannot be determined with the same precision as the other quantities required in the mobility computation. In addition, any deconvolution process must be regarded as suspect because of the necessity of using a smoothing function to eliminate noise in the back Fourier transform.16 The latter problem was empirically eliminated by checking deconvoluted pulses against those produced by electrons injected by a IO-ns laser pulse. The shift in apparent transit time was less than 3%. Geometrical problems were less satisfactorily eliminated. The use of electrode spacings between 0.5 and 2.0 mm changed the apparent mobility by less than 8% a t low fields. However, systematic distortions of the pulse shape at higher fields lead us to suspect substantial field inhomogeneities. Shape changes do not affect values of p, which depends on duration. The present values of p for N H 3 compare well with those previously reported.'s2 For example, at 300 K and 1.2 X lOI9 cm-), the present result is 330 cm2/(V-s) while Krebs' gives 390 cm2/(V.s) and Christophorou2 370 cm2/(V.s). At all pressures Christophorou's data lie halfway between the present data and those of Krebs. These differences probably lie in the measurement of electrode separation. Data were obtained on both N H 3 and CH3NH2 a t 300 K. Figure 1 shows the dependence of the normalized mobility n p on the density. Despite the limited range of density covered, there is a clear knee in the curve (when plotted linearly) and the density n1R a t which the IR criterion is satisfied is marked with an arrow in Figure 1. Discussion The CH3NH2data differ from those reported for other polar fluids in ways which are consistent with the dipole moment p = 1.3 D. We discuss the low-density limit first. The present value (15) J. M. Warman and M. C. Sauer, J . Chem. Phys., 62, 1971 (1975). (16) M. Deutsch and J. Beniaminy, Reu. Sci. Instrum., 53, 90 (1982).
Thompson et al. loz3
, I i
,
,
I
"
'
-0201
1019
1020
IO"
1022
n(~rn-~)
Figure 1. Normalized mobilities np for several polar gases. The symbols V and A refer to the present data for NH3 and CH3NH2. The upper solid curve is for isobutene (ref 8) and the lower for H20 at 463 K (ref 1). The arrows mark the onset of strong scattering as determined from
the Ioffe-Regel criterion. t
I
Figure 2. The low-density value of np, (np),,, for a variety of polar gases as a function of dipole moment, p . In addition to the present work, data were drawn from ref 1, 2, 3, and 8. The points are identified in Table I. The line is taken from dipole scattering theory as described in the text.
of the low-density limit ( n p ) , is above those for H 2 0 (p = 1.9 D) and NH, (p = 1.5 D) yet well below those for larger, less polar molecules. Figure 2 collects some recent date of Freeman and co-workersg as well as those already mentioned. One sees that the gases with large dipole moments and compact molecules satisfy the Born formula rather well compared to the others. Larger, less symmetric, less polar molecules generally scatter less than expected [the measured (np), is larger than the computed] while propane (C,H,) scatters more [the measured (np), is smaller than the calculated]. The basis for this difference is not clear, though it must be noted that discrepancies with the Born approximation result are well-known for dipole ~ c a t t e r i n g . ~ . ~ The constancy of np at low densities has been taken as indicative of single scattering. An elementary argument suggests that p cc
The Journal of Physical Chemistry, Vol. 88, No. 17, 1984 3711
Mobility of Excess Electrons in Methylamine TABLE I no. la lb
formula
name
P, D
H*O
water
1.85
CH, OH N H;
methanol ammonia
1.69 1.47
IC
2 3a 3b 3c 4a 4b 5 6 7
8 9 10
11 12
(CH,),O
dimethyl ether
1.3
CH, NH, i-C,H, c, H6 C,H,-1 cis-C, Ha-2 i-C, H i-C,Hlz C,H,
methylamine isobutene propene 1-butene cis-2-bu tene isobutane isopentane propane
1.28 0.5 0.366 0.34 0.3 0.132 0.1 3 0.084
,
may be indicative of scattering by clusters containing x molecules. If the NH3 and H 2 0 mobilities are analyzed in that way (there being too few CH3NH2data) one finds x = 3 and 4, respectively. Long ago Jortner14 suggested 6 and 4 molecules were required for localization. Warmanl7 has suggested that strong scattering processes should become important when the potential energy of the electron in the dipole field is of the order of the thermal energy kT. If the electron-dipole separation is taken to be a fixed fraction of the intermolecular distance, then a simple expression for the density nkT at which this occurs can easily be found. There are then two independent estimates of the onset of strong scattering. One, nkT, derived from energetic arguments and the other, nIR, from transport properties. If electron-dipole interactions are indeed the dominant scattering mechanism then the two should be the same. If nkT is taken to be the same as nIR, one predicts Krebs' has already found the PI2dependence of nIR. The relation is not quantitative, but in Figure 3 one sees that the dependence o n p and Tis correct. Data are collected in Table I. The absence of a quantitative relation is not surprising inasmuch as both the IR and Warman criteria are simply statements that the distances and energies, respectively, are comparable. The almost complete consistency among the data of Figure 3 is striking when one recalls the inconsistencies with a simple scattering law in Figure 2. It should be noted that the gases for which ( n p ) olies above the Born approximation curve have values of nIR below the density of the knee in a n p vs. n graph. Values of n]R are slightly above the knee for the more polar gases. As may be seen in Figure 1, nIR occurs for n p values about 90% of
(w)o.
We have clearly established the dominant role of charge-dipole scattering in the first stages of the electron localization process in polar fluids. Gee and Freemang have reached similar conclusions. The role of the dipole in the final stages (solvation) has long been recognized.I8 At intermediate densities, fluctuations (17) J. M. Warman in "The Studv of Fast Processes and Transient Swies by Electron Pulse Radialysis", J. H. H, Baxendale and F. Busi, Eds., Reidel, Reid Dordrecht, 1982, p 520. (18) D. A. Copeland, N. R. Kestner, and J. Jortner, J . Chem. Phys., 53, 1189 (1970).
PIR,
(w)o,
1OZ1(V.cms)-' cmz/(V.s) 3.4 3.2 2.9 4.5 8.4 6.5 4.5
53 57 65 15 43 56 94 70 105 94
8.0 5.5 6.9 2000.0 66.0 4000.0 4500.0 6000.0 7500.0 160.0
80 85 73 70
80 60 225
T--T ~K , 5 25 493 433 364 650 5 00 300 400 26 8 3 00 340 317 370 390 3 84 444 125
nIR> lOI9
ref
cm-,
6 .O 5.3 4.8 6.0
17.0 11.0 . 4.3 10.0 5.2 6.4 22.6 68.0 45.0 70.0 94 .O 90.0
70.0
1 1
1 8 2 2 2 3 3 present work
8 8 8 8 8 8 7
t
t
Figure 3. The density at which strong scattering begins as determined from the mobility as a function of the ratio of dipole moment to T. The solid line indicates only the -'/* power law. The sources of data are the same as Figure 2 except that temperature-dependent results yield several points for a single gas, see Table I.
in density (clusters) or dipole orientation must also play a role in the localization p r o c e s ~ . ' In ~ ~this ~ ~regime dipolar effects are weakened by the tendency of adjacent dipoles to be aligned oppositely. Fluctuations and electron polarization might then be expected to play a more important role, and the polar fluids to resemble more the nonpolar fluids. Acknowledgment. This work was supported in part by the Robert A. Welch Foundation and the u.s. National Science Foundation. The data acquisition system was designed by J. P. Coose. J.C.T. acknowledges a stimulating conversation with J. M, Warman, R, B. Coffman assisted in data analysis. Registry - . No. CH?NH,, - 74-89-5. (19) C. E. Krohn, P. R. Antoniewicz, and J. C. Thompson, Surf. Sci., 101, 241 (1980). (20) P. R. Antoniewicz, G. T. Bennett, and J. C. Thompson, J . Chem. Phys., 77, 4573 (1982).
