potential wells where E tr is often in the range of 1 Ä 2 eV and even higher (e.g. see
[41–44] and the references therein). In practice, the release of particles from such
traps is only possible either for relatively high target temperatures or over a long
time.
To describe hydrogen and/or helium transport in the material lattice on large
spatiotemporal scales, the reaction-diffusion (RD) model is often used. The RD
model deals with free (mobile interstitial) and trapped (usually immobile) hydrogen
and/or helium atoms. The free atoms can be trapped in and de-trapped from the
trapping sites. For the simplest case where we assume that there is only one kind of
the traps that have the density N tr and can trap only one atom, the RD equations
describing the free and immobile trapped atom densities, n f , and, n tr , can be written
as follows:
∂n f
∂t
¼ ∇ Á D∇n f
ð
ÞÀK tr n f N tr À n tr
ð
Þþν dtr n tr ,
ð3:7Þ
∂n tr
∂t
¼ K tr n f N tr À n tr
ð
ÞÀν dtr n tr ,
ð3:8Þ
where K tr is the trapping rate constant, ν dtr ¼ ν 0 exp (ÀE tr /T) is the de-trapping
frequency and ν 0 is the bounce frequency of the atom in the trapping potential well.
The rate constant K tr is often described in a diffusive approximation, assuming that
the free atom becomes trapped when it approaches the trapping site at a distance R tr .
Then, considering the diffusion of the free atom into the trapping site in the dilute
approximation (the average distance between the trapping sites is larger than R tr ), we
find K tr ¼ 4πDR tr .
Although the Eqs. 3.7 and 3.8 are rather simple, the RD models used in practice
quickly become very complex when other effects, such as large amount of the
trapping sites with different trapping energies [45, 46], trapping of few atoms at
one trapping site [47], stress-induced advection of the free atoms [15, 48], nonlinear
generation of new trapping sites and modification of the existing ones [49–51], etc.,
are added.
The boundary conditions, which are needed to close the second-order differential
Eq. (3.7) used for hydrogen and helium transport studies, can be different because of
different structure of the effective hydrogen and helium potentials at the surface, as
shown in Fig. 3.7. Since the free helium atoms are not bounded to the surface, one
can take zero density of the free helium atoms at the “surface” as the boundary
condition for Eq. (3.7). The situation with hydrogen, which can be bounded to the
surface, is more complex and depends on the target material. For example, for the
case of tungsten, hydrogen is desorbed from the surface as a molecule. As a result,
the corresponding boundary conditions should involve some surface recombination
rate coefficient, K
surf
rec , describing the conversion of atomic hydrogen into the molecular form. However, the available experimental data on K
surf
rec are very unreliable and
differ by orders of magnitude (e.g. see [42] and the references therein). Therefore, it
is often assumed (e.g. for the interpretation of the TDS data) that hydrogen transport
58
3 Plasma-Material Interactions in Magnetic Fusion Devices
[41–44] and the references therein). In practice, the release of particles from such
traps is only possible either for relatively high target temperatures or over a long
time.
To describe hydrogen and/or helium transport in the material lattice on large
spatiotemporal scales, the reaction-diffusion (RD) model is often used. The RD
model deals with free (mobile interstitial) and trapped (usually immobile) hydrogen
and/or helium atoms. The free atoms can be trapped in and de-trapped from the
trapping sites. For the simplest case where we assume that there is only one kind of
the traps that have the density N tr and can trap only one atom, the RD equations
describing the free and immobile trapped atom densities, n f , and, n tr , can be written
as follows:
∂n f
∂t
¼ ∇ Á D∇n f
ð
ÞÀK tr n f N tr À n tr
ð
Þþν dtr n tr ,
ð3:7Þ
∂n tr
∂t
¼ K tr n f N tr À n tr
ð
ÞÀν dtr n tr ,
ð3:8Þ
where K tr is the trapping rate constant, ν dtr ¼ ν 0 exp (ÀE tr /T) is the de-trapping
frequency and ν 0 is the bounce frequency of the atom in the trapping potential well.
The rate constant K tr is often described in a diffusive approximation, assuming that
the free atom becomes trapped when it approaches the trapping site at a distance R tr .
Then, considering the diffusion of the free atom into the trapping site in the dilute
approximation (the average distance between the trapping sites is larger than R tr ), we
find K tr ¼ 4πDR tr .
Although the Eqs. 3.7 and 3.8 are rather simple, the RD models used in practice
quickly become very complex when other effects, such as large amount of the
trapping sites with different trapping energies [45, 46], trapping of few atoms at
one trapping site [47], stress-induced advection of the free atoms [15, 48], nonlinear
generation of new trapping sites and modification of the existing ones [49–51], etc.,
are added.
The boundary conditions, which are needed to close the second-order differential
Eq. (3.7) used for hydrogen and helium transport studies, can be different because of
different structure of the effective hydrogen and helium potentials at the surface, as
shown in Fig. 3.7. Since the free helium atoms are not bounded to the surface, one
can take zero density of the free helium atoms at the “surface” as the boundary
condition for Eq. (3.7). The situation with hydrogen, which can be bounded to the
surface, is more complex and depends on the target material. For example, for the
case of tungsten, hydrogen is desorbed from the surface as a molecule. As a result,
the corresponding boundary conditions should involve some surface recombination
rate coefficient, K
surf
rec , describing the conversion of atomic hydrogen into the molecular form. However, the available experimental data on K
surf
rec are very unreliable and
differ by orders of magnitude (e.g. see [42] and the references therein). Therefore, it
is often assumed (e.g. for the interpretation of the TDS data) that hydrogen transport
58
3 Plasma-Material Interactions in Magnetic Fusion Devices