P. Smejtek and M. Silver
3890
the zero point energy of KrHZ+ was not taken into consideration and therefore a lower value of 0.33 eV of the threshold was given.)23 The diagram of Figure 8d shows that H4f is unstable with respect to its decay into H3+ + H. The H4f level lies half-way between the levels of the reactants and products. H At high energies, the reaction Hz+ + H2 -* Ha+ shows stripping behavior. At low energies, the H3+ intensity is peaking around the CM point in the contour diagram which is explained by intermediate complex format i ~ n . ~However. " ~ ~ since there is no potential well for H4+ according to Figure 8d, we would rather explain the observed symmetry of product intensity around the CM
+
point by the impulsive isotropic scattering model as in the case of the Kr+ Hz reaction. (The level of H3+ H in Figure 8d was calculated assuming a proton affinity of EIz of 4.1 eV, since t(H2-H+) from ab initio calculations is 4.56 e V 3 9 ~ ~and 0 the zero point energy of H3+ is about 0.5 eV.)
+
+
.I. Durup and M.Durup, J. Chlm. Phys., 64, 386 (1967). M. T. Bowers, D. D. Eileman, and J. King, J. Chem. Phys.. SO, 4787 (1969). L . D. Doverspike and R. L. Champion, J. Chern. Phys.. 46, 4718 (1967). W. Kutzelnigg, R. Ahlrichs, I. Labib-lskander, and W. A. Bingel, Chem. Phys. Lett.. 49, 4965 (1967). I. G. Csizqadia, R. E. Kari, J. C. Polanyi, A. C. Roach, and M. A. Robb, J. Chern. Phys., 52, 6205 (1970).
t Electron Injection into Dense Methane, Carbon Monoxide, and Carb P. Smejtek and M. Silver* Department of Physics, University of North Carolina, Chapel Hill, North Carolina 27514
(Received May 19, 1972)
Publication costs assisted b y the University of North Carolina
We have been studying injection currents in order to understand the electronic transport properties in liquids and dense gases. In particular we use tunnel cathodes to inject hot electrons into c&co, and COz. We have measured the current us. voltage characteristics for various densities. In order to analyze our data we use a simple model for injection. From this model we can estimate the cross section for momentum exchange scattering and the energy relaxation time of the hot electrons. The density dependence of these parameters and their implication are discussed.
Introduction Despite the recent excellent research using drift techniques,Z the nature of the quasi-free electron in liquids remains obscure. Part of the difficulty is that we do not know the energy of the conducting state nor what cross section to use for momentum exchange scattering. In order to look into these questions we have been injecting electrons into dense gases. As will be seen, from such experiments one can determine the diffusion cross section and the lifetime of the injected hot electrons as a function of density. Since the relevant cross sections are known a t low densities, we can see what density dependent changes take place as one approaches liquid densities. Assuming no change in the scattering length, density can be expected to affect the cross sections through multiple scattering3 and through atom-atom correlation.4 The problem might be further complicated since the scattering length will likely be altered by the presence of near neighbors. In this paper we will show some preliminary experimental results which indicate that none of the simple models mentioned above is adequate. Basically, the experiment consists of injecting hot electrons into the media and measuring the current as a function of applied electric field. In order to interpret such experiments one adopts a model. We have chosen a simpie model in order to minimize computational difficulty in the analysis of our data. In the absence of more sophisticated theoretical guidance, we believe that this is a The Journal of Physical Chemistry, Vol. 76, No. 25, 1972
viable approach recognizing that this may lead to contro__ versy regarding the approximations and errors (which for our experimental conditions we estimate to be less than 50% from more exact calculations) in our derived parameters. Figure 1 shows a schematic representation of what may happen to an electron injected into a dense medium from a metallic electrode. The entering elect,ron may be back scattered into the electrode (process 1). In such an event this electron does not contribute to the current. The electron may undergo several momentum exchange scatterings and then give up part of its kinetic energy by exciting vibrations o r rotations of a host molecule. This may occur closer or further from the electrode than the position of the maximum in the potential. In the former case the electron has a barrier to overcome in order to be collected by the anode while in the latter the electron has a barrier to overcome in order to be returned to the cathode. These two possibilities are depicted by process 2 and 3 in Figure 1. I
Model for Injection In our model we assume that every electron undergoes many scattering events before i t is either collected or ther(1) Supported.by the Army Research Office. (2) H. S. W. Massey, E. H. S. Burhop, and H. B. Gilbody, "Electronic and ionic Impact Phenomena," Oxford University Press, London, 1969; D, E. Golden, N. F. Lane, A. Temkin, an3 E, Gerjuoy. Rev. Mod. Phys., 43, 642 (1971). (3) W. Legler, Phys. Lett. A . , 31, 129 (1970). (4) M. H. Cohen and J. Lekner, Phys. Rev., 158, 305 (1967)
Hot Electron Injection into Dense CH4, CO, and C 0 2
3891
ments). We have made more exact calculations and the errors introduced due to our approximations are less than 60% if the energy is 0.3 eV and electron mfp 5 A. For the case of electrons having an energy of about 1 eV, the field driven motion is small compared with the diffusive motion and further L)h can be considered to be approximately constant, independent of position. The current of hot electrons is jt,
= - &,antl/ax
(8)
One final set of approximations is made and these are related to the boundary conditions. The boundary conditions for the hot electrons is given in terms of the current balance
Figure 1. Schematic representation of t h e injection process. malized. If the electron is thermalized we assume that it undergoes many scattering events before it is collected either by the anode or cathode as the case may be. In this case the electron motion can be evaluated from the continuity ofcurrent equation v.1 = 0
-
+ ntl/T = 0
(3)
where the subscript h stands for hot and the subscript t stands for thermalized, and R is the density of carriers. Equations 2 and 3 follow from eq 1 because J = itl + it
(4)
and
iin
~7
---I),, -sf' iin
-I),7-!.d x
+ n,,p,,E(x)
(6)
(7 1 + n ,!JutE(x) where the U and p are the diffusion coefficients and the mobilities, respectively, E ( x ) = Ea - e/4cx2 is the positiondependent electric field, and & is the applied voltage divided by the electrode spacing. To further simplify the problem we assume that the diffusive motion of the hot electrons starts after the first mean free path, A, and that X is large enough so that the contribution of the image potential energy to the total energy ofthe hot electron may be neglected. This is the case if the energy of the hot electrons is 1 eV or more and if-X is greater than IO b; (these conditions are generally met in our experi.,j .,
==
The other parameters are p = p l E ~ , / D and t K I= t>/.it/.Zcf)t. There are two independent parameters a t each density which determine the current and these are the thermalization time r and momentum exchange scattering cross section ukl which is defined as I / M where N is the number density of the medium. They are obtained by comparing the experimental data with the predictions of the model. A typical fit between the experimentally observed current voltage characteristics and that obtained from our theoretical model i s shown in Figure 2. A small computer was used to make a least-squares fit to our data to obtain t,he values for r a n d u,]. Considering the approximations, the fit is good and the low density values for c,, and r agree with other experimenk2 Experimental Results
The expressions for the respective currents j h and J L are j,,
+
(1)
Because energy relaxation in these systems is expected to be very fast, we make our first approximation and that is that we are dealing with a two-state system; the hot electron state where the total energy of the electrons is essentially constant and is equal to some average energy the electrons have before thermalization, and the thermalized electron state where the electron has a kinetic energy equal to 3k7'/2. The thermalized electrons are produced only through the energy relaxation of the hot electrons. Our second approximation is that the energy relaxation can be characterized with a single characteristic time r. With these two approximations eq 1 may be written as a system of two coupled equations --"it
where 10 is the current available from the emitter, n h ( X ) is the density of hot electrons a t x = A, and v(X) = [ ( 2 t o / m ) [l e E , X / q + (e2/4tXco)]]1/2, where (0 is the average kinetic energy of the hot electrons measured above the barrier maximum a t x = x m . The boundary condition for the density of thermalized electrons is that nt is finite everywhere, including a t the electrodes. With these boundary conditions one can solve f u r the total current J collected by the anode and this is
As a source of electrons we use a tunnel cathode.5 We have previously reported6 results using these electrodes in helium, liquid argon, and cyclohexane as well as other liquids and gases. The gases used were obtained commercially and the impurity level was in COz, 50 ppm, in CH4, 100 ppm, and in CO, 1000 ppm. The electronics and t he experimental set up is shown in Figure 3. There are two independent circuits. One of the circuits is used for. the operation of the electron emitter. This circuit consists of a power supply with a potentiometer to bias the AI-Al203-Au diode, a (5) R. M. Handy, J. Appl. Phys.. 37,4620 (1966). (6) D. G. Onn and M. Silver, Phys. Rev. A , 3, 1773 (1971); M . Silver, P. Kumbhare. P. Smejtek. and D. G. Onn. J. Chem. Phys., 52, 5195 (1970); M. Silver, 0 . G. O m , and P. Srnejtek, J . Appl. Phys.. 40, 2222 (1969); P. Smejtek, M. Silver, and D. G. Onn, J Chem. Phys., in Dress. The Journal ol Physical Chenustry. Vol 76, No. 25. 1972
3892
P. Smejtek and M. Silver
10-1-
r
Io-' 1/13
)/io Corbon monoxide, density, IOzo ~ r n - ~ 0 188 x285
I 0-3
334 5 92 v 104 e 167
a 43.8 066.6 t775
c 10.4 10-4
1 density,
0
77.
-Methane,
I
i020crn-3
6
Figure 2. Comparison ,between theory and experiment for the current vs. Ea," in COz. Solid lines are theory and points are the experimental results. In order to obtain the agreement shown, it was necessary to make the thermalization t i m e and the cross section density dependent (these dependences are depicted in Figures 5 and 6 ) . DC POWER SUPPLY HARRISON 6201 B
.E,&
i0-'4F---7---I---;---rl
t
1
1
co
9
4
I
I
I ELECTROMETER KEITHLEY 602 \
Vcm2
Figure 4. Experimental current-voltage characteristics in meth.ane ( T = 200°K), carbon dioxide ( T = 318'K), and carbon monoxide ( J = 160'K). j is the current collected in gas, j o is the emitter current measured in vacuum, and €,IN is the ratio of t h e applied electric field and gas density. No theoretical comparison is shown.
DlGlTEC 251 D V M i
t i
I
SAMPLE CHAMBER
I k AiUMlNU
I
I I
r
I
I
!COVERED
I I
I
GOLD ILAYE
I
i
I
Figure 5. Momentum exchange scattering cross section vs. density as derived from theory using the experimental data s h o w n in Figure 4. Figure 3. Schematic diagram of the experimental setup and elec-
tronics.
Keithley 610B electrometer, and a digital voltmeter to monitor the diode film current and bias voltage. The electron emitter is operated under conditions which ensure a long lifetime and stability of the emitted current.? The second circuit is used for the measurement of the injected current (electrometer Keithley 602) as a function of the electric field applied between collector and emitter (gold layer). The Journal of Physical Chemistry, Vol. 76, NO. 25, 1972
The current is plotted us. the ratio of the applied field to the gas density. Our experimental results for the electron injection into methane, carbon dioxide, and carbon monoxide are shown in Figure 4. The drop of the current with density for a given E / N indicates the loss of injected hot electrons due to their thermalization within the range of the image barrier. This decrease of the current with density is (7) D. G . Onn, P. Smejtek, and M . Silver. to be submitted for publication.
Hot Electron lnjeclion into Dense CH4, GO, and GO2
__
r
10-10
T---
i 0-15
7 -
I
5& 10-16 sec
___
3893
b
.'0
I
IO+
\
b ' \ I------------\
0
\
I I
0
0
t 10-l~
N, ~ r n - ~ Figure 7. Comparison between the experimentally derived momentum exchange scattering cross section (circles) and that calculated from the theory: (a) Lekner,'O (b) l.egler.3
Figure 6. Therrnalization time vs. density as derived from theory using the experimental data shown in Figure 4 . The thermalization distance x g can be estimated from eq 11 assuming the average kinetic energy of injected electrons to be 1 eV.
clearly seen in CO and C&. In methane, however, we do not find that the current voltage characteristics drop as in CO and CO2. Rather, the current us. EIN is independent of density. This suggests that the thermalization is occurring well beyond the range of the image potential. Using these data we have calculated the density dependence of the cross section for the momentum exchange scattering up and the lifetime 7 , These are shown in Figure 5 and 6, respectively.
Discussion The experimental results presented are in accord with our simple model for injection. While the many simplifications used in the model might raise questions regarding how precise the derived scattering cross sections might be, the agreement between the model and the data do indicate that momentum exchange scattering and energy relaxation are very important in determining the current. When energy relaxation is fast the image potential also must be considered. This is generally true at very high density and is expected to be very important in the liquid phase. The derived Scattering cross sections and thermalization times show very interesting density dependences. Density dependences of the relevant cross sections were previously noted from drift experiments8 and from theory.9 Comparison between our experimental results and drift show that there is general accord, ie.,the cross section for momentum exchange scattering in C decreases with increasing pressure. 'The theories of L e k n e P and Davis11 which are single scattering theories but which include molecule-molecule correlation predict a density dependence of the momentum exchange scattering cross section. Legler3 has calculated a density dependence from a multiple scattering theory but he does not include mol ecule-molecule correlation. The predictions of the variation of up with density in CHI are shown in Figure 7, and the experimentally derived density dependence from. Figure 5 is repeated for clarity.
Neither of these t h e o r i e ~by ~ ?themselves ~~ seem to be satisfactory. A multiple scattering theory including moleculemolecule correlation might work in CH4. This i s presently being tried. That the momentum exchange cross sections in CO and COz are independent of density is somewhat reassuring. As Davis has pointed out11 the incoherent scattering might dominate in these anisotropic molecules. Since CHS is spherical the incoherent terms might be expected to be small. Incoherent scattering may also mute the effects of multiple scattering and again little or no density dependence might be expected. All of these points require further theoretical investigation. The density dependence of T indicates the opening of new inelastic channels or the widening of the present ones. That the hot electron lifetime might vary as the reciprocal of the gas density was expected and is apparent in COz and GI&. What the extra density dependence in CO is due to is not understood.12 Two possibilities come to mind. (1) The presence of nearest neighbors affects the selection rules for rotational and vibrational excitations. This would increase the inelastic cross section and thereby decrease 7 as the density is increased. (2) Clustering of molecules a t these densities provide new rotational and vibrational modes. These new channels of the energy loss would decrease electron lifetime faster than the reciprocal density. Both of these speculations suffer from the fact that they are not found in CHI and COz. Further investigations including wider density range are in progress to clarify these points. Many experiments on drift and photoinjection of electrons into liquids of these molecules are being performed. We believe that our experimental results show that at present there is no general rule one can follow regarding the prediction of what mean free paths OF lifetime one should use in the liquid phase from data obtained in the gas or from the isolated molecule. Consequently, we urge caution when interpreting data in the liquid phase or when applying models using isolated molecule parameters. (8) H . Lehning, Phys. Lett. A , 29, 719 (1969): H Lehning, ibid.. 28, 103 (1968). (9) 8. Kivel, Phys. Rev.. 116, 926 (1959): T. F. O'Malley. ibid.. 1363, 1020 (1963). (10) J. Lekner, Phys. Rev., 158, 130 (1967). (11) H. T. Davis, L. D. Schmidt, and R. M. Minday, Phys. Rev., 3, 1027 (1971), (12) We have previously observed a similar effect in dense Nr and H2: J. Chem. Phys., in press. The Journal of Physical Chemistry, Voi, 76, No. 25, 1972
