Download Theory of the Diffusion of Hydrogen in Metals

8. Theory of the Diffusion of Hydrogen in Metals
K.W.Kehr
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8.1 Overview
Hydrogen in metals has a large mobility. At room temperature and below, its
mobility is many orders of magnitude larger than that of other interstitially
dissolved atoms, as has been pointed out in Chapter 12. The question arises of
why the mobility of hydrogen is so large. Is the interaction between hydrogen
and the host lattice weak compared to other interstitials, such that the potential
barriers between equilibrium sites of the hydrogen atoms are low ? Are other
mechanisms than thermally activated jumps over the potential barriers the
cause of the large mobility? Not much is known about the detailed interaction
of hydrogen with the host metal atoms. Here we shall concentrate on the
discussion of the possible mechanisms for diffusion of hydrogen atoms. Since
hydrogen atoms have a small mass compared to other interstitials, quantum
effects in the diffusion are likely to be observed for hydrogen, if they occur at all.
The three isotopes of hydrogen have large mass ratios, and in addition the
positive muon can be considered as a light isotope of the proton (cf. Chap. 13).
Hence isotope effects can be studied over a wide range, which is very important
for determining a distinction between different possible diffusion mechanisms.
First a survey of the different possible diffusion mechanisms of a light
interstitial in a host lattice will be given. Figure 8.1 gives a rough subdivision of
the different possible regimes as a function of temperature. At the lowest
temperatures, the interstitial must be delocalized in the form of a band state,
unless it is trapped by lattice defects. The propagation in the band state is
limited by the scattering on termal phonons or lattice defects. At some higher
temperature the interstitial will be localized at or about a specific interstitial
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K. W.Kehr
site. This process will be discussed in detail in Section 8.2.3. The elementary step
of the diffusion process is now a thermally activated jump from one to another
interstitial site. The particle might execute the jump by tunneling from one to
the other interstitial site or by hopping over the potential barrier. In the first
case, thermal activation is necessary to bring the energy levels of both sites to
the same height. In the second case, a higher activation energy is required to
overcome the barrier. Hence this process contributes at higher temperatures.
Finally, at the highest temperatures the interstitial will be mainly in states
above the potential barriers. It should perform a diffusion somewhat similar to
a dense gas or a liquid, where many collisions occur, provided in our case by the
thermally fluctuating host lattice. This regime will henceforth be called
"fluidlike diffusion". The boundaries separating the different regimes are not
sharply defined and one can consider subdivisions and modifications within
one regime. Also it is not certain that all regimes really occur in a given system.
It will be one of the aims of this chapter to assess the applicability of the
different mechanisms to the diffusion of hydrogen in metals.
In the regime of thermally activated tunneling transitions and over-barrier
jumps, a natural separation between the individual jump processes and the
diffusion in the lattice of interstitial sites appears. This separation is well
justified as long as the individual jump processes have a short duration
compared to the mean time of stay at an interstitial site, and are uncorrelated.
Under these conditions the contribution of the diffusive jumps to the lattice
diffusion can be evaluated by using master equations (cf. Chap. 10). We shall
direct our attention to the statistical-mechanical problem of the individual
jump processes.
It is instructive to compare the problem of interstitial diffusion with other
related mobility problems. The regime of band diffusion is encountered in the
electronic conductivity. In some metal oxides "small polarons" occur, i.e.,
electrons which are self-trapped within a local lattice deformation. They have a
mobility which is determined by the thermally activated tunneling processes
corresponding to the second regime. The theory of these processes was first
derived for small polarons (see Holstein [8.1, 2]). Paraelectric centers in crystals
can reorient themselves; molecules or molecular groups in molecular crystals
can perform rotational motions. The crystal potentials in which the paraelectric
centers or molecular groups move differ from system to system. Thus these
systems offer the opportunity of studying the different regimes, their limits, and
transitions between them induced by the variation of parameters such as the
temperature. All regimes including tunneling at lower temperatures and
fluidlike diffusion at higher temperatures are found.
So far, the diffusion processes of one hydrogen atom in a host lattice have
been considered. The diffusion of many dissolved hydrogen atoms is important
because of the large solubility of hydrogen in some transition metals. Not much
theoretical work has been done on the diffusion at higher hydrogen concentration, with the exception of critical phenomena. The hydrogen-metal
systems exhibit a critical slowing down of the hydrogen diffusion (AleJeld et aJ.
Theory of the Diffusion of Hydrogen in Metals
t99
[8.3]) at the critical point of the phase transition between the a: and a:' phases.
The connection between critical slowing down and the elastic interaction
leading to the phase transition has been discussed by Janssen [8.4]. Almost no
theoretical work has been done on the diffusion in the dense phases of
hydrogen in metals such as the p phase of Nb-H.
The next section contains a discussion of the interaction between hydrogen
and host metal atoms, and a discussion of the localization problem of hydrogen
atoms in metals. The following section describes the quantum mechanical rate
theory and discusses its applicability to experiments. In the fourth section the
classical rate theory is presented, together with its quantum mechanical
modification and the experimental observations. The limits of the description
by individual jump processes are discussed in Section 8.5.
The theory of the diffusion of light interstitials has been reviewed by
Sussma'ln [8.5], Stoneham [8.6], and Maksimov and Pankratov [8.7].
Additional summary comments have been given by Stoneham [8.8, 9].
8.2 Interaction of Hydrogen with Host Metal Atoms
8.2.1 Phenomenological Interaction Parameters
A knowledge of the interaction between the dissolved hydrogen atoms and the
host metal atoms is essential in order to discuss the diffusive processes of these
interstitials. However, the microscopic derivation of this interaction starting
from the interacting system of host metal ions, protons, and electrons seems to
be too difficult for the transition metals of interest. On the other hand, the
hydrogen-metal interaction produces observable effects which allow the determination of interaction parameters. Since only a few parameters are
determined in this way, the theory assumes a model character. The use of model
interactions is quite reasonable for the discussion of the basic diffusion
mechanisms, which is intended in this chapter. Of course, a derivation of
quantities such as the activation energy for diffusion in specific metals would
require an explicit investigation of the hydrogen-metal interaction.
In a microscopic derivation of the interaction of hydrogen with the host
metal, the screening of a proton at a n interstitial site has to be considered
(Friedel [8.10]). Friedel treats the screening of a proton in a free electron gas
and obtains a screened Coulomb potential in the linearized Thomas-Fermi
limit, and superimposed oscillations ("Friedel oscillations") in the linearized
self-consistent Hartree approximation. Popovic and Stolt [8.11] treat the
nonlinear screening of a proton in the electron gas and derive activation
energies for diffusion in AI and Mg. They have shown that nonlinear screening
is essential for protons in metals. The situation in transition metals with narrow
d-bands such as Pd is even more complicated. A qualitative discussion has been
given by Friedel [8.10], but no quantitative derivations have appeared in the
literature.