Approaches to SlowElectron Transport in
Condensed Matter
Mitio Inokuti

Physics Division

Argonne National Laboratory

9700 South Cass Avenue

Argonne, IL 604394843
The de Broglie wavelength l of an electron with kinetic
energy T = 150 eV is 0.1 nm, which is
comparable with distances between neighboring atoms in
an ordinary molecule and in usual condensed matter. In
general, l = 0.1 x
(150 eV / T ) ^{1/2}
nm becomes greater at lower T, and eventually
reaches 7.7 nm at thermal energy at room temperature.
Although this notion is well known for eight decades,
some of current calculations on transport of slow
electrons in condensed matter remain based upon the
classical trajectory of an electron. Examples include
trackstructure calculations extended to electrons of
low energies and analyses of electron recombination with
a geminate ion. Yet approaches are possible for carrying
out some calculations consistent with quantum mechanics.
Let us briefly survey several such possibilities.
The Feynman pathintegral method [1] is akin in
algorithm to a Monte Carlo simulation, and is
straightforward at least in principle to program on a
computer. Indeed, it has been applied for example to the
hydration of an excess thermal electron in water [24].
An obstacle in extending the treatment to electrons of
even eV's and higher lies in an extremely rapid increase
of the number of paths to be examined, resulting in a
prohibitive computation time. It is therefore desirable
to find some short cut, even as a sacrifice in accuracy,
perhaps somewhat similar in spirit to the
condensedhistory method in particletrack simulations.
The Wigner phasespace function method [57] is
attractive, because it provides an extension of the
distribution function in the sense of Boltzmann. Thus,
connections to the track length distribution, or the
degradation spectrum [8] of the familiar
electrontransport theory will be clear. A major
challenge in an application of this method is the
complexity of what corresponds to collision terms in the
Boltzmann equation. It involves amplitudes rather than
differential cross sections for individual collision
processes. Thus, the preparation of input data will have
to be far more extensive than in a classical case. One
way for dealing with this issue will be to start with a
potential for electronmolecule interactions as used in
the pathintegral method [9] and to generate amplitudes
on a computer as needed.
Some of the methods using electron orbital
functions also can be adapted to
electrontransport analysis in radiation physics. An
example is the theory of LEED (lowenergy electron
diffraction) [1012]. It focuses on the treatment of
multiple scattering of electrons of moderately high
energies by spatially fixed atomic lattices, and thus
need to be adapted to lower energies with some account
of atomic vibrations. In this respect, the method of Car
and Parrinello [13] and the END (electronnuclear
dynamics) [14] are highly suggestive and encouraging.
Concerning electron recombination with a geminate
ion in liquid, it is appropriate to view the
electronion system as being in a perturbed Rydberg
state interacting with many degrees of freedom of
surrounding molecules, in the spirit of the
multichannel quantumdefect theory [15]. Prototypes of
such a treatment are seen in studies on Rydberg states
of loosely bound dimers [16] and of large molecules
[17]. A similar idea about geminateion recombination
was indeed presented by Schiller [18], though technical
details remain to be developed further. A key point in
this line of thoughts is to distinguish electron
interactions with an ion into a short range and a long
range. Properties of shortrange interactions can be
deduced from existing knowledge about collisions and
spectra in the gas phase, while effects of longrange
interactions are intrinsically governed by
condensedphase properties. The goal of theory then is
to link the two elements in a coherent way.
Work was supported by the U. S. Department of Energy,
Office of Science, Nuclear Physics Division, under
Contract No. W31109Eng38.
References
1. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and
Path Integrals (McGrawHill, New York, 1965).
2. J. Schnitker and P. J. Rossky, J. Chem. Phys. 86,
3471 (1987).
3. J. Schnitker, et al., Phys. Rev. Lett. 60,
456 (1988).
4. B. J. Schwartz and P. J. Rossky, Phys. Rev. Lett. 72,
3282 (1994).
5. E. Wigner, Phys. Rev. 40, 749 (1932).
6. J. E. Moyal, Proc. Cambridge Philos. Soc. 45, 99
(1949).
7. W. B. Brittin and W. R. Chappell, Rev. Mod. Phys. 34,
620 (1962).
8. M. Kimura, M. Inokuti, and M. A. Dillon, in Advances
in Chemical Physics, edited by I. Prigogine and S. A.
Rice, 84, 193 (1993).
9. J. Schnitker and P. J. Rossky, J. Chem. Phys. 86,
3462 (1987).
10. J. B. Pendry, Low Energy Electron Diffraction: the
Theory and Its Application to Determination of Surface
Structure (Academic Press, London, 1974).
11. M. A. Van Hove and S. Y. Tong, Surface
Crystallography by Low Energy Electron Diffraction:
Theory, Computation and Structural Results
(SpringerVerlag, Berlin, 1979).
12. M. A. Van Hove, et al., LowEnergy Electron:
Experiment, Theory and Structure Determination
(SpringerVerlag, Berlin, 1986).
13. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471
(1985).
14. E. Deumens, et al., Rev. Mod. Phys. 66, 917
(1994).
15. U. Fano and A. R. P. Rau, Atomic Collisions and
Spectra (Academic Press, Orlando, 1986).
16. N. Y. Du and C. H. Greene, J. Chem. Phys. 90, 6347
(1989).
17. M. Thoss and W. Domke, J. Chem. Phys. 106, 3174
(1997).
18. R. Schiller, J. Chem. Phys. 92, 5527 (1990).
