Approaches to Slow-Electron Transport in Condensed
Matter
Mitio Inokuti
- Physics Division
- Argonne National Laboratory
- 9700 South Cass Avenue
- Argonne, IL 60439-4843
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
track-structure 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 path-integral 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 [2-4]. 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 condensed-history method in particle-track simulations.
The Wigner phase-space function method [5-7] 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
electron-transport 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 electron-molecule
interactions as used in the path-integral method [9] and to generate
amplitudes on a computer as needed.
Some of the methods using electron orbital functions
also can be adapted to electron-transport analysis in radiation
physics. An example is the theory of LEED (low-energy electron
diffraction) [10-12]. 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 (electron-nuclear
dynamics) [14] are highly suggestive and encouraging.
Concerning electron recombination with a geminate ion
in liquid, it is appropriate to view the electron-ion system
as being in a perturbed Rydberg state interacting with many degrees
of freedom of surrounding molecules, in the spirit of the multi-channel
quantum-defect 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 geminate-ion
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 short-range
interactions can be deduced from existing knowledge about collisions
and spectra in the gas phase, while effects of long-range interactions
are intrinsically governed by condensed-phase 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. W-31-109-Eng-38.
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