# Intense $\gamma$-photon and high-energy electron production by neutron
irradiation: effect of nuclear excitation on transport of defects

Luca Reali<EMAIL_ADDRESS>CCFE, United Kingdom Atomic Energy Authority,
Culham Science Centre, Oxfordshire OX14 3DB, UK  Mark R. Gilbert
<EMAIL_ADDRESS>CCFE, United Kingdom Atomic Energy Authority, Culham
Science Centre, Oxfordshire OX14 3DB, UK  Max Boleininger
<EMAIL_ADDRESS>CCFE, United Kingdom Atomic Energy Authority, Culham
Science Centre, Oxfordshire OX14 3DB, UK  Sergei L. Dudarev
<EMAIL_ADDRESS>CCFE, United Kingdom Atomic Energy Authority, Culham
Science Centre, Oxfordshire OX14 3DB, UK

###### Abstract

Effects of neutron irradiation on materials are often interpreted in terms of
atomic recoils, initiated by neutron impacts and producing crystal lattice
defects. We find that, in addition, there is a remarkable two-step process,
strongly pronounced in heavy elements, involving the generation of energetic
$\gamma$-photons in non-elastic collisions of neutrons with atomic nuclei
followed by the production of high-energy electrons through the scattering of
$\gamma$-photons by the atomic electrons. This two-step scattering creates a
non-equilibrium steady-state population of high-energy electrons in the bulk
of the material, which bombard atoms and stimulate vacancy diffusion,
resulting in a variety of driven athermal microstructural reactions including
the enhanced recombination of radiation defects. We find that tungsten
converts the energy of fusion or fission neutrons into a flux of
$\gamma$-radiation and subsequently high-energy electrons with the conversion
efficiency approaching 99%, explaining the low defect content observed at
moderate temperatures.

##  I Introduction

Neutron irradiation alters the microstructure of a material by producing
atomic scale defects. These defects interact and coalesce, forming extended
dislocation networks and voids. This ultimately results in macroscopic
swelling and dimensional changes, influencing the engineering design of
reactor components, similarly to the effect of thermo-mechanical and
electromagnetic loads. Populations of defects act as microscopic sources for
the macroscopic fields of stress and strain [1]. These populations evolve,
driven by the combined effects of stochastic generation of defects by neutron
and ion impacts, elastic interaction between the defects, thermal migration
and athermal relaxation. The resulting pattern of microstructural evolution
depends on the irradiation dose rate, temperature, and stress. The goal of
nuclear fission or fusion power generation is to exploit the energy of the
neutrons, produced by the decay of actinide nuclear fuel or by the deuterium-
tritium fusion reactions in plasma, and it is this energy that is ultimately
deposited in the coolant, structural materials or the tritium-generating
fusion reactor blanket.

One way this energy deposition occurs is through elastic collisions of
neutrons with the atomic nuclei. Primary knock-on atoms (PKA) initiate
displacement cascades that melt the material on atomic length and time scales,
leading to what is known as neutron heating. Another energy deposition
mechanism involves inelastic collisions of neutrons with the nuclei, either in
the form of neutron capture reactions at low neutron energies or nuclei break-
up reactions at higher (MeV) energies. These and other nuclear reactions
excite the internal degrees of freedom of atomic nuclei and give rise to the
generation of energetic photons through the subsequent nuclear de-excitation.
This process of absorption of these energetic photons is known as
$\gamma$-heating. It has long been realised that $\gamma$-heating depends
strongly on the atomic weight of an element and can be the dominant heating
process, more intense than heating through atomic recoils, for example by a
factor of ten in niobium [2]. It is also known that $\gamma$-photons can
produce high concentration of radiation defects in covalent and ionic crystals
even if they do not displace atoms from their lattice sites [3].

Exposure to neutrons is characterised by the magnitude of the total neutron
flux and its energy spectrum. The $\gamma$-photon flux, and in turn the high-
energy electron flux that the $\gamma$-photons produce in a material, also
have their own energy spectra. The energy spectra of all the three types of
particles determine the likelihood of various scattering events via the energy
dependence of the corresponding cross sections.

Electron irradiation on its own has been extensively used in the studies of
production and evolution of defects in transmission electron microscopy (TEM)
[4, 5, 6, 7]. Below, we show that in the nuclear environments — whether
fission or fusion — both neutron and electron irradiation are always present
simultaneously. Notably, there are no $\gamma$-photons and high-energy
electrons generated during ion irradiation, a fact that is not usually
highlighted when comparing neutron and ion irradiation experiments [8, 9].

Interaction between high-energy electrons and atoms in a crystal can generate
new defects and drive their subsequent motion [10, 5]. The maximum recoil
energy $E_{\textnormal{R}}^{\textnormal{max}}$ that an atom of mass $M$ can
receive in a collision with an electron of kinetic energy
$E_{\textnormal{el}}$ and mass $m$ is [11, 10]

$E_{\textnormal{R}}^{\textnormal{max}}=\frac{2(E_{\textnormal{el}}+2mc^{2})}{Mc^{2}}E_{\textnormal{el}},$
(1)

where $c$ is the speed of light. For instance, if $E_{\textnormal{el}}=3$ MeV
then in tungsten $E_{\textnormal{R}}^{\textnormal{max}}=140.8$ eV, which is
nearly twice the threshold displacement energy (TDE) [12, 13], the minimum
energy required to generate a stable vacancy-interstitial pair. If
$E_{\textnormal{el}}=500$ keV, then
$E_{\textnormal{R}}^{\textnormal{max}}=8.9$ eV, which is about five times the
vacancy migration energy in pure elemental W [14]. Vacancy migration in W,
driven by electron impacts, is observed under a 500 keV electron beam in a
transmission electron microscope even at cryogenic temperatures [5].
Similarly, athermal vacancy diffusion was reported in Zr illuminated by a 800
keV electron beam at 150 K [7], and in Pb exposed to a beam of 390 keV
electrons at 58 K [15]. The accelerated diffusion of vacancies stimulates
rapid microstructural evolution, involving the shrinkage of interstitial-type
dislocation loops [5] and growth of vacancy-type loops [7]. Electron
irradiation also stimulates the motion of defect clusters at low temperatures
in a variety of alloys and steels [16] and even in covalent materials like SiC
[17].

Athermal vacancy migration, induced by electron impacts, was modelled using
molecular dynamics (MD) by Xu et al. in W [12] and Satoh et al. in Fe and Cu
[18]. They found that the minimum recoil energy for displacing a vacancy was
directionally anisotropic, and strongly varied as a function of distance from
a vacancy. Satoh et al. [18] quantified the probability of a driven vacancy
hop as a function of recoil energy $E_{\textnormal{R}}$, the distance between
an atom impacted by an electron and a vacancy, and temperature. Even at high
$E_{\textnormal{R}}$, approaching 20-100 times the vacancy migration energy,
the probability of an electron impact on a nearest neighbour atom resulting in
a vacancy hop remains close to 0.3-0.5. This probability is higher at 300 K
than at 20 K, particularly at low $E_{\textnormal{R}}$.

Below, we detail the generation of $\gamma$-photons in a reactor environment
and its contribution to nuclear heating in various materials. We explain how
the photon spectra are derived from the neutron spectra, and how the high-
energy electron spectra are related to the $\gamma$-photon spectra. Then, we
simulate and analyse the effect of collisions between the high-energy
electrons and atoms in a lattice, and quantify a relationship between atomic
recoils and vacancy diffusion. Finally, we combine all the above steps and
evaluate the rates of neutron-$\gamma$-electron-stimulated vacancy migration
in irradiated materials.

##  II  $\gamma$-photons produced by exposure to fission and fusion neutrons

There are various sources of $\gamma$-photons (or $\gamma$-rays or
$\gamma$-radiation) in a nuclear reactor environment. In addition to
$\gamma$-photons, which are often produced by nuclei decay, it is also
possible to produce X-rays or, equivalently, the Röntgen photons, which are
are commonly generated through the de-excitation of atomic electrons [19].
These two forms of electromagnetic radiation are fundamentally similar and
there is no universally agreed differentiation between X-rays and
$\gamma$-photons. X-rays are defined as photons with energies below 100 keV, a
value that, as we shall see later, is conveniently close to the cut-off energy
threshold below which the energy of the electrons produced by the absorption
of photons is too low to influence the dynamics of atoms. The $\gamma$-photons
have the energies in the range from 100 keV to 10 MeV [20]. In this study, we
investigate the $\gamma$-photons produced by the relaxation of atomic nuclei
from their excited high energy configurations.

(a)  (b)

Figure 1: Mean free paths of neutrons (a) and photons (b) in Be, Fe and W as a
function of energy, defined as the reciprocal of atomic density times the
total cross section. The effect of atomic number on the position and character
of nuclear resonances on the energy axis is clearly visible. In (b), the
dominant scattering processes are highlighted. At $\sim$ MeV energies,
neutrons likely travel between 2 to 4 cm before experiencing mostly elastic
scatterings; $\gamma$-photons in W and Fe likely travel for 1 to 2 cm, before
undergoing a Compton scattering event. The lower panels show the ratio between
the absorption and total cross sections, highlighting that neutrons have a
much lower likelihood of being absorbed than photons, despite their mean free
paths being comparable.

In fusion, the energy released in D-T reactions is carried by neutrons and
$\alpha$-particles, formed in these events. Other types of fusion reactions
are also possible, but the international fusion research is focused primarily
on the D-T route. $\gamma$-photons with energies of 16.75 MeV and 13.5 MeV are
occasionally produced in D-T reactions, and this involves the formation of a
5He particle instead of the dominant $\alpha$-particle (4He). The branching
ratio for the 5He channel is very low compared to the main channel. Inertial
confinement fusion and accelerator-based experiments show that the branching
ratio for the $\gamma$-photon production directly from D-T reactions is in the
range from $10^{-4}$ to $5\times 10^{-5}$ [21]. Thus, this flux of
$\gamma$-photons originating directly from the D-T reactions in the fusion
plasma is negligible.

Meanwhile, the $\alpha$-particles produced in the D-T reactions do not have
the sufficient energy to induce any of the nuclear reactions in the plasma-
facing materials that could lead to $\gamma$-photon emission. The threshold
minimum energies for reactions such as ($\alpha$,n), in other words the
$\alpha$-absorption followed by neutron emission, which could lead to excited
isomeric states and thus $\gamma$-photons production analysed below, are
typically above the maximum 3.5 MeV energy of $\alpha$-particles produced in
D-T collisions, even for metals lighter than tungsten [22, 23, 24]. The direct
absorption reactions ($\alpha$,$\gamma$) resulting in the photon production,
are extremely rare [25]. Therefore, the only significant source of photons in
a fusion reactor are the neutrons produced by the D-T reactions themselves, or
those created by neutron multiplications, interacting with the nuclei in the
reactor materials.

In fission, the situation is more complex. Only a relatively small proportion
of the $\sim$200 MeV energy released in the fission of 235U is carried by
neutrons – an average of around 5 MeV per fission event, whereas the vast
majority ($\sim$ 80%) is carried by the fission fragments themselves [20].
Additionally, prompt $\gamma$-photons, emitted within $10^{-14}$ s of a
fission event, typically carry around 8 MeV, while the decay of fission
fragments releases around 19 MeV via $\beta$ decays and 7 MeV through delayed
$\gamma$-decay. The exact energy of individual $\gamma$-photons and $\beta$
particles depends on the nature of the fission fragments, which vary from one
fission event to another, and their decays. However, the overall intensity of
$\gamma$-photon production is significant; for example, around 8 prompt
$\gamma$-photons are produced per 235U fission [26], and there are of the
order of $3\times 10^{19}$ fission events per second per GW of fission power,
assuming 200 MeV of energy release per a fission event, leading to $\sim
2.4\times 10^{20}$ $\gamma$-photons per GW per second.

Such a large flux of photons could be problematic to structural components,
were it not for the relatively short penetration range of $\gamma$-rays
compared to neutrons. For example, the $\gamma$-photons produced by the decay
of 60Co have energies in the range from 1.1 to 1.4 MeV. Their absorption mean
free path in stainless steel is 1.6 cm and they would lose more than 90% of
their intensity within 3 cm [27]. Hence, despite the large $\gamma$-flux in
the fuel channels themselves and for experimental material samples placed in
core locations, fission does not generate an appreciable external flux of
$\gamma$-photons. This results in the same conclusion for fission as for
fusion above, namely that it can be assumed, as we will in the remainder of
this paper, that the dominant and the only source of photons in a bulk
structural material in a nuclear reactor is that originating from within the
material itself due to the nuclear reactions triggered by the incident
neutrons.

Fig. 1 shows the mean free paths of neutrons and photons in Be, Fe and W. This
defines the length scale between the subsequent collision events. The
penetration depth of the two types of particles also depends on the
probability of them being absorbed or scattered/re-emitted, see the lower
panels in Fig. 1. The relative probability of absorption can be estimated from
the ratio between the absorption ($\sigma_{\textnormal{abs}}$) and total
($\sigma_{\textnormal{tot}}$) cross sections. The values were taken from the
TENDL-2021 [28] and the XCOM [29] libraries for neutrons and photons,
respectively. For $\sigma_{\textnormal{abs}}$ of photons, we considered the
photoelectric effect and pair production. We find that in the $\gtrsim$ keV
energy range, the mean free paths of neutrons and photons are comparable, but
the photons are far more likely to be absorbed.

Unlike photons, neutrons undergo multiple scattering and propagate through
materials over much longer distances typically of the order of 10 cm and, to a
first approximation, neutron flux can be considered constant in the bulk of
materials on atomic length-scales; near interfaces between materials, neutron
fluxes can vary more rapidly [30], but we do not consider that case here. Fig.
2 compares the neutron spectra predicted for materials in the first wall,
plasma-facing environment of a fusion reactor to that expected at typical core
locations in the High-Flux Reactor (HFR) fission experimental facility in
Petten, Netherlands. These spectra were obtained using the MCNP transport code
[31, 32] and account for the reactor geometries – in the case of HFR for a
digital model of the physical reactor, and for fusion using a recent
conceptual digital design of DEMO, the next-step demonstration fusion reactor,
see [33, 34] for detail. The HFR spectrum was calculated specifically for a W
irradiation experiment, and this took into account the local experimental
environment as opposed to assuming a generic spectrum often quoted for
experimental facilities. Using this refined spectrum, analysis shows that
transmutation burn-up calculations with the FISPACT-II [35] inventory code
accurately predict the composition evolution of W samples placed in this
environment [30, 36].

Figure 2: HFR (fission) and EU-DEMO first wall (fusion) energy-differential
neutron spectra. These spectra were calculated through stochastic (Monte-
Carlo) simulations by MCNP [31], and are the volume-averaged flux as a
function of energy (originally in tallied histogram format, but converted to
differential fluxes here by dividing by bin width). They are obtained by
integrating (over angle) the energy-dependent angular flux of neutrons in a
region to give the density of particles, regardless of trajectory, at a point.
Over a distance element $ds$ this density can be thought of as the track
length density; MCNP estimates the average flux by summing track lengths,
which is a reliable approach in well-populated statistics [31]. The resulting
flux per source neutron are multiplied by the reactor power (converted to
number of neutrons) to give the flux-per-second values shown.

(a)  (b)

Figure 3: Source photon distributions computed using the SPECTRA-PKA code for
the neutron spectra shown in Fig. 2. $\gamma$-photon intensities in beryllium
are negligible and omitted, see Table 1. The curves give the energy-resolved
values of the source term $Q_{\textnormal{ph}}(E)$, defined by Eq. (10) and
referring to the generation of $\gamma$-photons per unit volume.

Photons are generated by various nuclear reactions involving neutrons, such as
inelastic scattering, neutron multiplication (n,2n) or neutron capture,
followed by direct $\gamma$ emission (n,$\gamma$). The fundamental origin of
$\gamma$-photon emission in all the cases is the same: when a neutron
interaction leaves a daughter reaction product or nuclide in an excited state,
often referred to as an isomeric state, or a metastable state if it is
sufficiently long-lived, photons are emitted to release energy and allow the
excited nuclei to transition through various energy levels to its lowest
energy ground state. A negligibly small amount of energy is also taken by the
recoiling nucleus as it decays [37]. While this ground state may also be
unstable on longer time scales, producing further $\gamma$-photons as part of
$\beta$-decay or $\alpha$-decay, these “delayed” photons are comparatively
rare compared to the “prompt” $\gamma$-photons considered in the present work.
The exact definition of prompt photons is not universally accepted, but in the
context of fission reactions, prompt $\gamma$-photons are considered to be
emitted within 10 fs of the initial reaction event [26, 38]. However, the
minimum half-life for an excited nuclide to be called “isomeric”, and thus not
an emitter of prompt $\gamma$-photons upon decay, can be anywhere from 1 ps to
1 µs, and is often taken to be around 1 ns [20, 39]. Such metastable nuclides
may persist for much longer – sometimes even for 100s or 1000s of years – but,
again, these infrequent strongly delayed $\gamma$-photons are not the subject
of analysis here.

Instead of using a transport code to calculate the prompt photon fluxes, which
is often done in complex nuclear geometries to evaluate the dose rates outside
of shielding to assess the safety of maintenance operations, we can instead
use an approach independent of geometry, to calculate the instantaneous source
term of photons in a region exposed to a known flux and energy spectrum of
neutrons. SPECTRA-PKA [40, 41] is a code developed to calculate the atomic
displacement source terms due to neutron irradiation, but it can also be used
to provide the equivalent $\gamma$-photon source terms using the available
nuclear library data of $\gamma$-photon emission cross sections as a function
of the incident neutron energy.

In what follows, we treat the photon sources and accounts for the attenuation
of photons in materials while at the same time evaluating the flux of high-
energy electrons generated as photons undergo interactions with atomic
electrons. This approach, starting from a neutron spectrum for a given
geometry, through a $\gamma$-photon source term and then $\gamma$-photon
attenuation and electron interactions in a material, is self-consistent and
avoids having to disentangle the local geometry attenuation from the pure
source spectrum of $\gamma$-photons.

Before using SPECTRA-PKA to define the flux and energy spectrum of these
local, prompt $\gamma$-photons, we first evaluate the neutron-induced energy
deposition, often called heating. To perform these calculations, we use the
FISPACT-II code, which can access the KERMA (Kinetic Energy Released per unit
MAss) cross sections, expressed in barns-eV units and included in nuclear
reaction data libraries. Nuclear heating is an integral measure of the energy
transferred to the material by neutron irradiation. By analysing the
constituent contributions to this heating, we can understand the relative
significance and microscopic effects resulting from the nuclear heating caused
by the $\gamma$-photons.

Table 1 summarizes FISPACT-II nuclear heating calculations for W, Fe and Be,
assuming exposure to the two neutron spectra shown in Fig. 2, and hence
enabling the comparison of fusion (DEMO) and fission (HFR) irradiation
environments. The total nuclear heating, given in W/g units, includes the
energy deposited as a result of elastic, inelastic and non-elastic (break-up)
nuclear reactions initiated by neutrons, as well as any locally deposited
energy from emitted secondary particles, including $\gamma$-photons. The table
also details the absolute and relative contributions to the total heating due
to the energy deposited by photons. The KERMA cross sections used here were
evaluated for the TENDL-2017 [42] nuclear data library by the NJOY processing
code [43, 44], assuming local absorption of the prompt photons and thus not
considering their transport and attenuation.

These FISPACT-II results represent the instantaneous heating rate at time
$t=0$, before any transmutation has taken place. The most striking observation
from these calculations is that for W and to a lesser extent for Fe, the two
technologically most significant fusion materials [45, 46, 47],
$\gamma$-photon heating entirely dominates the energy deposited in the
material exposed to neutron irradiation under both fission and fusion
conditions.

This result can be understood by considering that in high atomic mass elements
like W there is a greater scope for nuclei excited states because there are
more possible configurations of the nucleus in which to store energy. When the
nucleon number, i.e. the total number of protons and neutrons, is high there
is a significant and often dominant probability that the nuclear reaction
energy will be trapped in an excited isomeric state and subsequently converted
into $\gamma$-photons during isomeric transitions. In contrast, in simpler low
nucleon number materials like Be, there are fewer isomeric possibilities and
so the energy is more readily released directly in the form of kinetic energy
of the daughter products of nuclear reactions.

The results summarised in Table 1 demonstrate how important it is to consider
the effect of $\gamma$-photons on microstructural evolution, particularly in
the critical fusion and fission nuclear materials such as W and Fe [45, 46,
47], where the energy released during nuclear reactions is dominated by the
photons and not by atomic recoils. Table 1 shows that tungsten acts as a
highly efficient converter of the energy of fusion or fission neutrons into
electromagnetic $\gamma$-radiation, with the conversion efficiency approaching
99%.

Table 1: Nuclear heating, detailing the contribution of $\gamma$-photon emission to the energy deposited in materials exposed to fusion and fission neutrons.  Material | Heating (W/g)
---|---
| DEMO | HFR
| Total | Photon | Fraction | Total | Photon | Fraction
W | 2.73 | 2.65 | 97.0% | 10.48 | 10.36 | 98.9%
Fe | 1.68 | 1.19 | 70.8% | 2.00 | 1.63 | 81.5%
Be | 3.65 | $3\times 10^{-3}$ | 0.1% | 2.15 | $8\times 10^{-7}$ | 4$\times 10^{-5}$%

(a)  (b)

Figure 4: Various nuclear reaction channel contributions to the
$\gamma$-photon generation in W. Left panel: spectrum of photons generated by
fusion neutrons, right panel: spectrum of photons produced by fission
neutrons. Fusion and fission neutron spectra are taken from Fig. 2. The curves
give the energy-resolved values of the source term $Q_{\textnormal{ph}}(E)$,
defined by Eq. (10) and referring to the generation of $\gamma$-photons by an
individual atomic nucleus.

Using the neutron spectra shown in Fig. 2 as input for SPECTRA-PKA
calculations, we find the $\gamma$-photon flux distributions for Fe and W
shown in Fig. 3. Beryllium is omitted from this figure as it generates a
negligible flux of $\gamma$-photons, in agreement with the data given in Table
1. These distributions will be used later in the paper to evaluate the flux of
high-energy electrons produced by the $\gamma$-photons in the material, and at
this point it is instructive to highlight the fundamental origin of spectra of
$\gamma$-photons. To illustrate the principle, in Fig. 4 we plot the dominant
channel contributions for W under fusion (DEMO) and fission (HFR) conditions.
In both cases we see an overwhelming dominance of the neutron capture
(n,$\gamma$) reactions, particularly at higher $>$MeV $\gamma$-photon
energies.

In what follows, we assume that the $\gamma$-photons emitted by the excited
nuclei, and the high-energy electrons that these $\gamma$-photons produce in a
material, are generated isotropically and their flux has no directional
dependence. The neutron flux in a nuclear reactor inevitably has some angular
anisotropy because, for example, the 14.1 MeV neutrons produced in a fusion
plasma initially travel radially away from the plasma, whereas the lower
energy neutrons originating from multiple scattering events in the surrounding
structure have a strong back-scattered component. Any directional dependence
of the neutron flux is lost at low energies due to multiple scattering.
Similarly, in fission, there is a directional dependence of the fast (above 1
MeV) neutron flux emitted from the fuel pins, which impinges on the
surrounding structural materials, but the high thermal component shown in Fig.
2 is directionally more isotropic.

The prompt $\gamma$-photons are generated in materials primarily from the
neutron capture (n, $\gamma$) or inelastic scattering reactions, and can have
angular anisotropy associated with the initial direction of the neutron
momentum transferred to the compound nucleus [48], as observed in inelastic
scattering experiments on iron [49]. $\gamma$-radiation intensities have been
found to be correlated with the direction of motion of fission fragments [50].
However, since neutron capture and inelastic scattering dominate the
generation of $\gamma$-photons in structural materials, as illustrated in Fig.
4, and since these reactions typically occur at MeV energies and below,
especially in the case of neutron capture, most of the directional dependence
on the neutron flux will have been lost at the point of $\gamma$-photon
production due to the multiple scattering of neutrons. It is therefore
reasonable to assume the angular isotropy of the flux of $\gamma$-photons. The
flux of electrons $\phi_{\textnormal{el}}({\bf n},E)$, produced by the
scattering of $\gamma$-quanta, is also isotropic, and in the absence of
directional variables is solely a function of the kinetic energy of the
electrons.

##  III Energy spectra of $\gamma$-photons and high-energy electrons

A quantity central to the treatment of rates of scattering, atomic recoils,
and reactions in solids, is the flux of high-energy particles initiating the
respective scattering and reaction events. This flux, denoted by $\phi({\bf
n},{\bf r},E)$, equals the number of particles or photons with kinetic energy
$E$ that cross a unit area in the direction of unit vector ${\bf n}$ per unit
time in the vicinity of point ${\bf r}$. This flux can be computed by
multiplying the density $f({\bf n},{\bf r},E)$ of particles with kinetic
energy $E$, and moving in direction ${\bf n}$, by their velocity $v$.

For example, if $\phi_{\textnormal{el}}({\bf n},{\bf r},E)$ is the flux of
high energy electrons, the rate of scattering of electrons by an atom located
at ${\bf r}$ is

$\frac{\text{d}^{2}\sigma({\bf n},E\rightarrow{\bf
n}^{\prime},E^{\prime})}{\text{d}o^{\prime}\text{d}E^{\prime}}\phi_{\textnormal{el}}({\bf
n},{\bf r},E),$ (2)

where $\text{d}^{2}\sigma/\text{d}o^{\prime}dE^{\prime}$ is the differential
cross section of scattering of electrons into an element of solid angle
$do^{\prime}$ corresponding to direction ${\bf n}^{\prime}$, and an energy
interval $dE^{\prime}$. If the flux of electrons is mono-energetic then
integrating (2) over all the directions of scattering ${\bf n}^{\prime}$ and
energies $E^{\prime}$, as well over the directions of incidence ${\bf n}$, we
find the total rate of collisions of electrons with an atom situated at ${\bf
r}$, namely

$\nu_{\textnormal{tot}}(E,{\bf
r})=\sigma_{\textnormal{tot}}(E)\int\text{d}o\,\phi_{\textnormal{el}}({\bf
n},{\bf r},E).$ (3)

The total cross section of scattering

$\sigma_{\textnormal{tot}}(E)=\int\text{d}o^{\prime}\int\text{d}E^{\prime}\,{\text{d}^{2}\sigma({\bf
n},E\rightarrow{\bf
n}^{\prime},E^{\prime})\over\text{d}o^{\prime}\text{d}E^{\prime}}$ (4)

is independent of ${\bf n}$ due to the rotational invariance of the process of
scattering, requiring that the differential cross section depends only on the
angle between vectors ${\bf n}$ and ${\bf n}^{\prime}$ [51, 52].

In the treatment of threshold atomic recoil events below, we explore
quantities similar to (3), and often involving integration over a selected
range of solid angles corresponding to a reaction, for example a hop of an
atom from an occupied to a vacant lattice site. Such a hop occurs only if the
direction of the recoil is favourable and is able to initiate a transition
along a trajectory that crosses the energy barrier for the reaction.

The flux of neutrons, electrons or photons satisfies the Boltzmann transport
equation [53, 54], extensively used in the theory of radiative transfer [55],
namely

${\bf n}{\partial\over\partial{\bf r}}\phi({\bf n},{\bf
r},E)=I_{\textnormal{coll}}[\phi({\bf n},{\bf r},E)]+Q({\bf n},{\bf r},E).$
(5)

In this equation, $I_{\textnormal{coll}}[\phi({\bf n},{\bf r},E)]$ is the so-
called collision term that describes the effect of scattering by atoms or
nuclei on the propagation of particles through the material, and $Q({\bf
n},{\bf r},E)$ is the source term accounting for the generation of particles.
For example, in the Boltzmann transport equation for high-energy electrons,
$Q({\bf n},{\bf r},E)$ describes the generation of electrons by the
$\gamma$-photons, whereas in the transport equation for $\gamma$-photons the
source term refers to the generation of photons by the relaxation of excited
states of atomic nuclei [56].

For the $\gamma$-photons generated by neutrons, the source term has the form

$\displaystyle Q_{\textnormal{ph}}({\bf n}_{\gamma},{\bf r},E_{\gamma})$
$\displaystyle=$ $\displaystyle
n_{0}\int\text{d}o^{\prime}\text{d}o\int\text{d}E^{\prime}\text{d}E{\text{d}^{2}\sigma_{\textnormal{n}\gamma}({\bf
n}^{\prime},E^{\prime}\rightarrow{\bf n},E;{\bf
n}_{\gamma},E_{\gamma})\over\text{d}o^{\prime}\text{d}E^{\prime}}$
$\displaystyle\times\phi_{\textnormal{n}}({\bf n}^{\prime},{\bf
r},E^{\prime}),$

where $n_{0}$ is the number density of atomic nuclei,
$\phi_{\textnormal{n}}({\bf n}^{\prime},{\bf r},E^{\prime})$ is the flux of
neutrons interacting with the nuclei, and

${\text{d}^{2}\sigma_{\textnormal{n}\gamma}({\bf
n}^{\prime},E^{\prime}\rightarrow{\bf n},E;{\bf
n}_{\gamma},E_{\gamma})\over\text{d}o^{\prime}\text{d}E^{\prime}}$ (7)

is the differential cross section of scattering of a neutron by an atomic
nucleus, with the neutron changing its direction of propagation and energy
from $({\bf n}^{\prime},E^{\prime})$ to $({\bf n},E)$, accompanied by the
production of a $\gamma$-photon with energy $E_{\gamma}$ travelling in the
direction ${\bf n}_{\gamma}$. The integration of (III) over directions ${\bf
n}_{\gamma}$ and energies $E_{\gamma}$ gives the total number of
$\gamma$-photons generated in a unit volume of the material per unit time.
This quantity can also be computed numerically. For example, the curves shown
in Figs. 3 and 4 were computed using Monte Carlo MCNP simulations. They are
equivalent to integrating Eq. (III) over directions ${\bf n}_{\gamma}$ but not
energies $E_{\gamma}$, yielding the source term in a form differential with
respect to the energy of photons $E_{\gamma}$.

Eq. (5) itself in some cases can be solved analytically [55, 53], but more
often its solutions are found numerically using Monte Carlo methods [57, 54],
see Ref. [58] for more detail.

The collision term in the right-hand side of the Boltzmann transport equation
(5) has the form [53, 54]

$\displaystyle I_{\textnormal{coll}}[\phi({\bf n},{\bf
r},E)]=-n_{0}\sigma_{\textnormal{tot}}(E)\phi({\bf n},{\bf r},E)$ (8)
$\displaystyle+$ $\displaystyle
n_{0}\int\text{d}o^{\prime}\int\text{d}E^{\prime}{\text{d}^{2}\sigma({\bf
n}^{\prime},E^{\prime}\rightarrow{\bf
n},E)\over\text{d}o^{\prime}\text{d}E^{\prime}}\phi({\bf n}^{\prime},{\bf
r},E^{\prime}),$

where $n_{0}$ is the number density of atoms or, equivalently, atomic nuclei
in the material, and integration over $do^{\prime}$ and $dE^{\prime}$ is
performed over the solid angle and energy of particles scattered by these
atoms or nuclei. The first, negative, part of the collision term in (8)
describes the differential rate of loss of flux from an element of phase space
$({\bf n},E)$, whereas the second part of the collision term accounts for the
rate of scattering into this element of phase space. If absorption is the
dominant channel of scattering, only the first, negative, term needs to be
retained in the right-hand side of (8).

When the above equations are applied to the treatment of transport of high-
energy electrons, velocity ${\bf v}=v{\bf n}$ of an electron is related to its
momentum ${\bf p}$ through the relativistic kinematic formula ${\bf p}=m{\bf
v}/\sqrt{1-{\bf v}^{2}/c^{2}}$. The kinetic energy of an electron is
$E=c\sqrt{p^{2}+m^{2}c^{2}}-mc^{2}$ [59], where $p=|{\bf p}|$ and $c$ is the
speed of light.

###  III.1 Photons: iterative solution of the transport equation

If the source of $\gamma$-photons is directionally isotropic and the rate of
spatial variation of the field of photons is negligible in comparison with all
the other length scales involved, we can neglect the derivative with respect
to spatial coordinates in the left-hand side of (5), and arrive at

$I_{\textnormal{coll}}[\phi_{\textnormal{ph}}(E)]+Q_{\textnormal{ph}}(E)=0.$
(9)

The angularly isotropic photon flux and the source term, shown in Figs. 3 and
4, are

$\displaystyle\phi_{\textnormal{ph}}({\bf n},E)$ $\displaystyle=$
$\displaystyle{1\over 4\pi}\phi_{\textnormal{ph}}(E),$ $\displaystyle
Q_{\textnormal{ph}}({\bf n},E)$ $\displaystyle=$ $\displaystyle{1\over
4\pi}Q_{\textnormal{ph}}(E).$ (10)

In this approximation, the collision term (8) transforms to

$\displaystyle
I_{\textnormal{coll}}[\phi_{\textnormal{ph}}(E)]=-n_{0}\sigma_{\textnormal{tot}}(E)\phi_{\textnormal{ph}}(E)$
(11) $\displaystyle+$ $\displaystyle
n_{0}\int\text{d}E^{\prime}\phi_{\textnormal{ph}}(E^{\prime})\int\text{d}o^{\prime}{\text{d}^{2}\sigma({\bf
n}^{\prime},E^{\prime}\rightarrow{\bf
n},E)\over\text{d}o^{\prime}\text{d}E^{\prime}}.$

By denoting the kernel of the integral over the solid angle
$\text{d}o^{\prime}$ in (11) by

$K(E,E^{\prime})=\int\text{d}o^{\prime}{\text{d}^{2}\sigma({\bf
n}^{\prime},E^{\prime}\rightarrow{\bf
n},E)\over\text{d}o^{\prime}\text{d}E^{\prime}},$ (12)

we arrive at a closed equation for the energy spectrum of $\gamma$-photons

$\displaystyle
Q_{\textnormal{ph}}(E)-n_{0}\sigma_{\textnormal{tot}}(E)\phi_{\textnormal{ph}}(E)$
(13) $\displaystyle+$ $\displaystyle
n_{0}\int\text{d}E^{\prime}\,K(E,E^{\prime})\phi_{\textnormal{ph}}(E^{\prime})=0.$

This equation can be readily solved by iteration. This involves representing
the energy spectrum in the form of a series, where each term refers to the
number of scattering events undergone by an energetic $\gamma$-photon in a
material

$\phi_{\textnormal{ph}}(E)=\phi_{\textnormal{ph}}^{(0)}(E)+\phi_{\textnormal{ph}}^{(1)}(E)+\phi_{\textnormal{ph}}^{(2)}(E)+\dots$
(14)

The zero-order term in the series describes the flux of photons directly
emitted by the atomic nuclei

$\phi_{\textnormal{ph}}^{(0)}(E)={Q_{\textnormal{ph}}(E)\over
n_{0}\sigma_{\textnormal{tot}}(E)},$ (15)

and the subsequent terms

$\phi_{\textnormal{ph}}^{(i)}(E)={1\over\sigma_{\textnormal{tot}}(E)}\int\text{d}E^{\prime}\,K(E,E^{\prime})\phi^{(i-1)}(E^{\prime})$
(16)

refer to the contributions to the energy spectrum of $\gamma$-photons from
trajectories involving $i$ events of Compton scattering of photons by
conduction and inner-shell atomic electrons. Expression for the kernel
$K(E,E^{\prime})$ computed using (12) has the form

$\displaystyle K(E,E^{\prime})=\begin{cases}\frac{\pi
r_{\textnormal{c}}^{2}mc^{2}}{E^{\prime
2}}\bigg{[}\frac{E}{E^{\prime}}+\frac{E^{\prime}}{E}-1\\\
+\left(\frac{mc^{2}}{E^{\prime}}-\frac{mc^{2}}{E}+1\right)^{2}\bigg{]},&\parbox[t]{260.17464pt}{
$\frac{E^{\prime}}{1+\frac{2E^{\prime}}{mc^{2}}}<E<E^{\prime}$}\\\
0,&\text{otherwise}\end{cases}$ (17)

where $r_{\textnormal{c}}=2.8179$ fm is the classical electron radius. The
above formula results from inserting into Eq. (12) the Klein-Nishina cross
section

$\frac{\text{d}\sigma}{\text{d}o}=\frac{r_{\textnormal{c}}^{2}}{2}\left(\frac{E^{\prime}}{E}\right)^{2}\left[\frac{E^{\prime}}{E}+\frac{E}{E^{\prime}}-\sin^{2}\theta\right],$
(18)

where $\theta$ is the photon scattering angle.

The photon generation term $Q_{\textnormal{ph}}(E)$ in Eqs. (III) and (15),
expressed either in volumetric units cm-3s-1eV-1 in Fig. 3, or in atomic units
s-1atom-1eV-1 in Fig. 4, is obtained by folding the (n, $\gamma$) cross
section matrices with the energy-spectrum-resolved neutron flux. It is
calculated internally in transport codes such as MCNP, and can also be
explicitly obtained by inputting the cross sections generated by NJOY [44]
into the processing code SPECTRA-PKA [40]. The cross section in the
denominator of Eq. (15) and Eq. (16) depends on the photon energy. For photons
in the energy range from about 10 keV to 10 MeV,
$\sigma_{\textnormal{tot}}(E)$ is dominated by photoelectric absorption (PE,
the lower end of the energy spectrum), the Compton scattering (CS, the middle
part of the spectrum), or by the electron-positron pair production (PP, the
upper end of the spectrum). The total cross section can be written as a sum

$\sigma_{\textnormal{tot}}(E)=\sigma_{\textnormal{PE}}(E)+\sigma_{\textnormal{CS}}(E)+\sigma_{\textnormal{PP}}(E),$
(19)

where the numerical values for individual terms are available from tables [60]
or databases [29]. Of the three processes included in (19), only the Compton
scattering does not lead to the absorption of photons, and hence is the only
process that contributes to the scattering kernel (12).

Eq. (12) assumes that electrons in a material act as independent centres of
scattering of photons, regardless of them being free or bound in the inner
electronic shells of atoms. This approximation is justified if we are
interested in the photon energies higher than approximately 100 keV, see
Appendix A. Fig. 1 shows plots of the mean distance between photon scattering
events $[n_{0}\sigma_{\textnormal{tot}}(E)]^{-1}$ that, in the limit where
absorption is dominant, determine the characteristic attenuation distance of a
photon flux emitted by a localised source.

The inverse total scattering cross section is the most important scaling
factor in the representation of the photon flux as a series in the number of
scattering events (15) and (16). In the interval of energies where the Compton
scattering dominates, photons may lose nearly half of their energy in a single
scattering event, depending on the angle between directions ${\bf n}$ and
${\bf n}^{\prime}$. We also note that the above calculations assume that the
$\gamma$-photons are generated in the bulk of the material and that the local
flux of photons is proportional to the local flux of neutrons. If in addition
there is flux of photons from an external source, it will provide an extra
contribution to the various scattering events, including the generation of
high-energy electrons considered in the next section.

###  III.2 High-energy electrons: the continuous slowing down approximation

The PE, CS and PP processes, giving rise to the absorption or scattering of
$\gamma$-photons, result in the generation of high energy electrons. Similarly
to how nonelastic nuclear reactions give rise to the generation of
$\gamma$-photons described by the source term $Q_{\textnormal{ph}}(E)$ in
(III), the PE, CS and PP processes are responsible for the _electron_
generation term $Q_{\textnormal{el}}(E)$ in Eq. (5). This phenomenon of
production of high-energy electrons by $\gamma$-photons is well recognised,
for example the authors of Ref. [61] note the equivalence of $\gamma$-photon
and electron irradiation. The source term describing scattering of
$\gamma$-photons by atoms serves as a starting point in a calculation of the
flux of high-energy electrons.

If the flux of electrons is directionally isotropic and its variation on the
spatial scale of the problem is negligible, we write

$\phi_{\textnormal{el}}({\bf n},E)={1\over 4\pi}\phi_{\textnormal{el}}(E).$
(20)

The only variable characterising the flux of electrons is their energy $E$,
and now the collision term (8) describes electron energy losses, which can be
treated in the continuous slowing down approximation [54, 53, 62]. In this
approximation, assuming that the flux of electrons is independent of spatial
coordinates, the Boltzmann transport equation acquires the form

$0={\partial\over\partial
E}[\overline{\varepsilon}(E)\phi_{\textnormal{el}}(E)]+Q_{\textnormal{el}}(E),$
(21)

where $\overline{\varepsilon}(E)$ is the average rate of energy losses of an
electron with energy $E$, and $Q_{\textnormal{el}}(E)$ is the rate of
generation of electrons with energy $E$ by the $\gamma$-photons. The average
rate of energy losses per unit distance travelled $\overline{\varepsilon}(E)$
is related to the range $R(E)$ of electrons in the material via [63, 53, 54,
64, 65]

$R(E)=\int\limits_{0}^{E}{\text{d}E^{\prime}\over\overline{\varepsilon}(E^{\prime})}.$
(22)

Equation (21) can be readily solved, and its solution has the form

$\phi_{\textnormal{el}}(E)={1\over{\overline{\varepsilon}(E)}}\int\limits_{E}^{\infty}Q_{\textnormal{el}}(E^{\prime})\text{d}E^{\prime}.$
(23)

This analytical expression describes a directionally isotropic distribution of
high energy electrons (20) that energetic $\gamma$-photons generate in the
bulk of the material. In practice the upper limit of integration in (23) is
determined by the energy span of the source function $Q_{\textnormal{el}}(E)$,
whereas numerical values of function $\overline{\varepsilon}(E)$ are available
in literature [66]. The flux of electrons $\phi_{\textnormal{el}}(E)$ can also
be computed numerically using Monte Carlo simulations [57, 54], and below we
show that results of such simulations compare favourably with the analytical
result (23).

The electron generation term has the form

$Q_{\textnormal{el}}(E)=Q_{\textnormal{PE}}(E)+Q_{\textnormal{CS}}(E)+Q_{\textnormal{PP}}(E).$
(24)

At high energies well above the keV range, we neglect the electron binding
energy effects and write

$Q_{\textnormal{PE}}(E)=n_{0}\sigma_{\textnormal{PE}}(E)\phi_{\textnormal{ph}}(E).$
(25)

The Compton scattering contribution can be evaluated by noting that the energy
balance dictates that the electron energy equals the difference between the
photon energy before and after the event,
$E=E_{\textnormal{ph}}-E^{\prime}_{\textnormal{ph}}$. Therefore it follows
from Eq. (17) that the energy-differential Compton cross section involving an
electron receiving recoil energy $E$ is

$\displaystyle\frac{\text{d}\sigma}{\text{d}E}=\begin{cases}\frac{\pi
r_{\textnormal{c}}^{2}mc^{2}}{E_{\textnormal{ph}}^{2}}\bigg{[}\frac{E_{\textnormal{ph}}}{E_{\textnormal{ph}}-E}-\frac{E}{E_{\textnormal{ph}}}\\\
+\left(\frac{mc^{2}}{E_{\textnormal{ph}}}-\frac{mc^{2}}{E_{\textnormal{ph}}-E}+1\right)^{2}\bigg{]},&\parbox[t]{260.17464pt}{$0<E<\frac{2E_{\textnormal{ph}}^{2}}{2E_{\textnormal{ph}}+mc^{2}}$}\\\
0,&\text{otherwise}\end{cases}$ (26)

If the number density of electrons is $n_{\textnormal{el}}$, the resulting
electron generation term is

$Q_{\textnormal{CS}}(E)=n_{\textnormal{el}}\int\phi_{\textnormal{ph}}(E_{\textnormal{ph}})\frac{\text{d}\sigma}{\text{d}E}(E_{\textnormal{ph}},E)\text{d}E_{\textnormal{ph}}.$
(27)

If the photon flux is represented by a discrete $N$-point set on a grid of
photon energies $E_{i}$, we can write the energy-differential flux of
electrons in the form of a discrete sum of Dirac delta functions

$\phi_{\textnormal{ph}}(E_{\textnormal{ph}})=\sum_{i=1}^{N}\Phi_{\textnormal{ph}}^{(i)}\delta(E_{\textnormal{ph}}-E_{i}).$
(28)

This then simplifies Eq. (27) to

$Q_{\textnormal{CS}}(E)=n_{\textnormal{el}}\sum_{i=1}^{N}\Phi_{\textnormal{ph}}^{(i)}\frac{\text{d}\sigma}{\text{d}E}(E_{i},E).$
(29)

Finally, high-energy electrons can be generated through the production of
electron-positron pairs, resulting from the conversion of photons with energy
greater than $2mc^{2}\sim 1.022$ MeV into electrons and positrons. The sum of
kinetic energies of an electron and a positron is then
$E_{\textnormal{ph}}-2mc^{2}$. Assuming that this energy is split equally
between the two particles, we find that
$E=\frac{1}{2}(E_{\textnormal{ph}}-2mc^{2})$ and

$Q_{\textnormal{PP}}(E)=n_{0}\sigma_{\textnormal{PP}}(E_{\textnormal{ph}})\phi_{\textnormal{ph}}(E_{\textnormal{ph}}).$
(30)

Monte Carlo MCNP simulations were used for validating the approximations
involved in Eq. (15) and Eq. (16) for $\gamma$-photons and Eq. (23) for high-
energy electrons. The curves computed independently using the above equations
and MCNP simulations are shown in Fig. 5. The DEMO spectrum of neutrons from
Fig. 2 was used as a spatially homogeneous incident flux of neutrons in a
$20\times 20\times 20$ cm3 cube of tungsten, with periodic boundary conditions
(PBCs) applied along $x$ and $y$, and reflecting planes bounding the cell in
$z$. Neutrons, photons and electrons were tallied in the box to produce the
MCNP results shown in Fig. 5. Using tallies on a grid of 8000 1 cm3 voxels, we
confirmed that there was no detectable spatial variation of the spectra and
thus the results are representative of an effectively infinite bulk sample of
W. The total neutron flux of the input source spectrum was $5.04\times
10^{14}$ cm-2s-1. The neutron flux calculated by MCNP was normalised to the
same value. This resulting tallied neutron spectrum is slightly different from
the input spectrum, shown in the top panel of the figure because, as opposed
to the current simulation, the input neutron spectrum was produced in
simulation assuming a thin W layer in the first wall of a full-reactor DEMO
involving also other materials. Because of the high computational cost of
simulating the photon and particularly electron transport in MCNP, it was not
feasible to perform the N-P-E transport simulation using the full DEMO
geometry. The MCNP-generated neutron flux was used for evaluating
$\phi_{\textnormal{ph}}(E)$ through Eqs. (14), (15) and (16), and then for
comparing the outcome to the MCNP $\gamma$-photons result. Similarly, the
MCNP-generated photon flux represented input for high-energy electron flux
calculations, which was then compared to the MCNP result. It is worth noting
that the MCNP calculation with full photon and electron transport for this
relatively simple system (single cube with no complex surfaces or long
transport paths) and a relatively modest 108 source neutrons took
approximately 2 days using 8 processors, whereas the calculation required of
the order of seconds on a laptop.

Figure 5: Top: input DEMO spectrum shown together with the neutron spectrum it
generates in MCNP under periodic boundary conditions. Central and bottom
panels: MCNP photon and electron spectra compared to the
$\phi_{\textnormal{ph}}(E)$ and $\phi_{\textnormal{el}}(E)$ spectra computed
using equations given in the text. The abscissa is the same for the three
panels. Note that the two neutron spectra are different as one is
representative of a thin first-wall W layer in a mixed-material DEMO reactor
and the other is the result of that same spectrum being propagated through
pure bulk W. The photon and electron spectra are consistent with the MCNP bulk
W simulation and thus the equivalence between the MCNP and calculated curves
validates the methodology for the latter.

Neutron flux used for evaluating the generation rate of $\gamma$-photons and
high-energy electrons in Fig. 5 refers to what were to be expected if
scattering occurred in pure W. In Fig. 6 and Table 2 we plot the curves and
provide numerical data for the photon and electron spectra developing in bulk
Fe and W exposed to the neutron spectra shown in Fig. 2, calculated as
explained in Sec. III. In both fission and fusion scenarios the photon and
electron spectra are qualitatively similar. The intensity of photon fluxes is
comparable to the neutron fluxes. Notably, the neutrons generate
$\gamma$-photon spectra with the characteristic energy of $\sim 1-1.5$ MeV,
and electrons spectra with the characteristic energy of $\sim 0.5-1$ MeV,
which are skewed towards high energies.

The characteristic scale of electron energies that we see in Fig. 6 is the
same as the energy range of electrons used in transmission electron microscope
experiments [5] where it was found that the flux of electrons was able to
drive microstructural evolution of the material exposed to an electron beam.
In W, the photon and electron fluxes computed for the HFR input spectrum are
about 4 times higher than those computed for the DEMO neutron spectrum.

(a)  (b)  (c)  (d)

Figure 6: Energy-resolved differential neutron, $\gamma$-photon and electron spectra in the bulk of neutron-irradiated Fe (a, b) and W (c, d) corresponding to fusion (DEMO) and fission (HFR) conditions. Insets show the energy-differential electron fluxes on a linear scale. These are all maximum at approximately 200 keV, but even at $\sim 2$ MeV they still retain $\sim 10$ % of the peak intensity. Photon and electron spectra were calculated according to the theory presented in Sec. III. Table 2: Total neutron ($\Phi_{n}$), $\gamma$-photon ($\Phi_{\textnormal{ph}}$) and electron ($\Phi_{\textnormal{el}}$) fluxes in W and Fe and in DEMO and HFR conditions, calculated as presented in Sec. III, and median particle energy (MNE, MPE and MEE respectively), calculated as the energy below and above which lie the two halves of the total flux. | $\Phi_{\textnormal{n}}$ | MNE |  $\Phi_{\textnormal{ph}}$ in W | MPE in W |  $\Phi_{\textnormal{ph}}$ in Fe | MPE in Fe |  $\Phi_{\textnormal{el}}$ in W | MEE in W |  $\Phi_{\textnormal{el}}$ in Fe | MEE in Fe
---|---|---|---|---|---|---|---|---|---|---
| [cm-2s-1] | [keV] | [cm-2s-1] | [keV] | [cm-2s-1] | [keV] | [cm-2s-1] | [keV] | [cm-2s-1] | [keV]
DEMO | $5.04\times 10^{14}$ | 295 | $2.74\times 10^{14}$ | 1,410 | $1.78\times 10^{14}$ | 891 | $7.57\times 10^{12}$ | 589 | $2.36\times 10^{12}$ | 934
HFR | $6.83\times 10^{14}$ | 129 | $1.09\times 10^{15}$ | 1,480 | $2.04\times 10^{14}$ | 1,230 | $3.03\times 10^{13}$ | 589 | $3.22\times 10^{12}$ | 1,070

##  IV Atomic recoils produced by high-energy electrons

In the treatment of scattering of energetic particles by atoms in a material,
involving either the Boltzmann transport equation [53, 54] or Monte Carlo
simulations [57], the fact that scattering involves not only the change of
momenta of the incident particles but also atomic recoils, is often not
recognised. Experimental observations and theoretical analysis [67, 68, 69, 5]
show that even the relatively low energy electrons produce atomic recoils with
energies in the electron-volt range, comparable with the energy barrier for
vacancy migration in metals [70].

High-energy atomic recoils can be formally treated as multi-phonon excitation
events, whereas at low energies it is often sufficient to retain only the
lowest order single-phonon terms when computing the scattering structure
factor [71]. Irrespectively of the recoil energy, electron-atom recoils can be
treated as electron-phonon interaction events, resulting in the eventual
dissipation of the energy of high-energy electrons into the heat bath of
thermal atomic vibrations.

The treatment of relativistic collision kinematics involving an electron with
mass $m$ and an atom with mass $M$ gives a relation between the kinetic energy
of the recoil atom $E_{\textnormal{R}}$ and the angle of scattering $\theta$,
defined in the centre of mass frame [59]

$E_{\textnormal{R}}={Mc^{2}E_{\textnormal{el}}(E_{\textnormal{el}}+2mc^{2})\over(m+M)^{2}c^{4}+2Mc^{2}E_{\textnormal{el}}}(1-\cos\theta),$
(31)

where $E_{\textnormal{el}}$ is the kinetic energy of the fast electron,
${E_{\textnormal{el}}=mc^{2}/\sqrt{1-v^{2}/c^{2}}-mc^{2}}$, and
$\cos\theta=({\bf n}\cdot{\bf n}^{\prime})$. In the limit $m\ll M$, Eq. (31)
can be simplified as [6, 72, 56]

$E_{\textnormal{R}}={E_{\textnormal{el}}(E_{\textnormal{el}}+2mc^{2})\over
Mc^{2}}(1-\cos\theta).$ (32)

The maximum amount of energy is transferred to an atom in a collision where
the electron is scattered exactly backwards $\theta=\pi$. Taking
$\cos\theta=-1$, from Eq. (32) we find

$E_{\textnormal{R}}^{\textnormal{max}}(E_{\textnormal{el}})=2E_{\textnormal{el}}{E_{\textnormal{el}}+2mc^{2}\over
Mc^{2}},$ (33)

which agrees with Eq. (1). Estimates based on this formula suggest that
backscattering of electrons with kinetic energies in the MeV range can readily
generate atomic recoils with energies many times the magnitude of the
potential barrier for vacancy migration in metals.

The characteristic scale of angle $\theta$ in (31) depends on the differential
cross section of scattering. Elastic scattering of relativistic electrons by
atoms is well described by the screened Coulomb Rutherford cross section [6,
73, 62]

$\left(\frac{\text{d}\sigma}{\text{d}o}\right)=\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\left(\frac{1-\beta^{2}}{\beta^{4}}\right)\frac{1}{(1+\kappa-\cos\theta)^{2}},$
(34)

where $\beta^{2}=v^{2}/c^{2}=1-(1+E/mc^{2})^{-2}$, $Z$ is the nuclear charge,
and $\kappa$ is the screening parameter [62] inversely proportional to the
effective size of the atom,

$\kappa={\hbar^{2}\over 2p^{2}a^{2}_{\textnormal{TF}}},\>\kappa\ll 1$ (35)

where $p$ is the relativistic momentum of the incident electron
${p=mv/\sqrt{1-v^{2}/c^{2}}}$, and ${a_{\textnormal{TF}}=0.885a_{B}/Z^{1/3}}$
is the Thomas-Fermi atomic radius. Here, we use the atomic system of units
where the Bohr radius is ${a_{B}={\hbar^{2}/me^{2}}=0.52918}$ Å and
$e^{2}/a_{B}=27.2116$ eV.

The original Rutherford formula does not treat the effect of screening of the
electrostatic potential of the nucleus nor the effects of electron spin. The
Mott cross section takes the latter into account by solving the Dirac rather
than the Schrödinger equation [74]. The Mott cross section is usually
expressed in terms of the Rutherford cross section as

$\left(\frac{\text{d}\sigma_{\textnormal{M}}}{\text{d}o}\right)=\left(\frac{\text{d}\sigma_{\textnormal{R}}}{\text{d}o}\right)R_{\textnormal{M}},$
(36)

where factor $R_{M}$ for elements with $Z>20$ is found numerically [6]. Lijian
et al. [75] proposed a polynomial interpolation for the unscreened Mott cross
section

$R_{\textnormal{M}}=\sum_{j=0}^{4}a_{j}(Z,\beta)(1-\cos\theta)^{j/2}$ (37)

where

$a_{j}(Z,\beta)=\sum_{k=1}^{6}b_{k,j}(Z)(\beta-\bar{\beta})^{k-1}.$ (38)

and $\bar{\beta}=0.7181287$. The numerical values of the coefficients
$b_{k,j}$ can be found in tables [73, 75].

Since the scattering potential is radially symmetric, in the expression for an
element of the solid angle $do$ it is sufficient to retain only the polar
component ${\text{d}o=2\pi\sin\theta\text{d}\theta}$, where $\theta$ is the
angle of scattering of the electron. In the treatment of generation and
induced motion of lattice defects, it is convenient to work with the kinetic
energy of atomic recoils, related to $\theta$ by

$E_{\textnormal{R}}(\theta)=E_{\textnormal{R}}^{\textnormal{max}}\sin^{2}\frac{\theta}{2},$
(39)

which follows from Eqs. (32) and (33). Using the chain rule, we define a
quantity

$\left(\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}\right)=\left[2\pi\sin\theta\left(\frac{\text{d}\sigma}{\text{d}o}\right)\right]_{\theta=\theta(E_{\textnormal{R}})}\left|\frac{\text{d}\theta}{\text{d}E_{\textnormal{R}}}\right|,$
(40)

where

$\theta(E_{\textnormal{R}})=2\sin^{-1}\sqrt{\frac{E_{\textnormal{R}}}{E_{\textnormal{R}}^{\textnormal{max}}}},$
$\left|\frac{\text{d}\theta}{\text{d}E_{\textnormal{R}}}\right|=\frac{1}{\sqrt{E_{\textnormal{R}}(E_{\textnormal{R}}^{\textnormal{max}}-E_{\textnormal{R}})}}$

Similarly to how $\left({\text{d}\sigma}/{\text{d}\theta}\right)$ gives the
probability of an electron being scattered by angle $\theta$,
$\left({\text{d}\sigma}/{\text{d}E_{\textnormal{R}}}\right)$ is proportional
to the probability of a target atom receiving recoil energy
$E_{\textnormal{R}}$.

Using Eqs. (34) and (36), and noting that

$\frac{\sin\theta(E_{\textnormal{R}})}{\sqrt{E_{\textnormal{R}}(E_{\textnormal{R}}^{\textnormal{max}}-E_{\textnormal{R}})}}=\frac{2}{E_{\textnormal{R}}^{\textnormal{max}}},$

we simplify Eq. (40) as

$\left(\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}\right)=2\pi\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\left(\frac{1-\beta^{2}}{\beta^{4}}\right)\frac{2E_{\textnormal{R}}^{\textnormal{max}}}{(\kappa
E_{\textnormal{R}}^{\textnormal{max}}+2E_{\textnormal{R}})^{2}}$ (41)

if scattering is described by the screened relativistic Rutherford cross
section. Alternatively,

$\left(\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}\right)=2\pi
R_{M}\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\left(\frac{1-\beta^{2}}{\beta^{4}}\right)\frac{E_{\textnormal{R}}^{\textnormal{max}}}{2E_{\textnormal{R}}^{2}}$
(42)

if scattering is described by the Mott formula. In Fig. 7 we compare the two
expressions at two different electron energies. The Mott cross section is more
accurate, but computing it requires a large number of coefficients; the
Rutherford cross section may be more appropriate since the error that it
introduces is small given all the other approximations involved in the
analysis.

Figure 7: Differential cross sections for a tungsten atom to receive recoil
energy $E_{\textnormal{R}}$ after a collision with an electron with kinetic of
511 keV or 2 MeV, respectively. The screened relativistic Rutherford cross
section is compared to the Mott cross section. Both terminate at the maximum
recoil energy given by Eq. (1). The inset shows a magnified view of the curve
with the vertical axis shown on a linear scale.

To validate the method, we now apply Eq. (40) to the evaluation of the rate at
which high-energy electrons generate Frenkel pairs (FP) once the transferred
recoil energy $E_{\textnormal{R}}$ surpasses a certain threshold atomic
displacement energy barrier [4, 10]. This amounts to estimating the number of
stable defects $N_{\textnormal{d}}(E_{\textnormal{R}})$ remaining after a
collision event. Using the data derived from MD simulations, Yang and Olsson
[56] modified the arc-dpa (athermal recombination-corrected displacement per
atom) model [13] and found that the number of stable defects increases
linearly as a function of $E_{\textnormal{R}}$ above a certain minimum energy
$E_{\textnormal{d}}^{\textnormal{min}}$. A “defect production differential
cross section” can be defined by combining the energy-differential cross
section given by Eq. (40) and the Yang-Olsson expression

$N_{\textnormal{d}}(E_{\textnormal{R}})=\begin{cases}0,&E_{\textnormal{R}}<E_{\textnormal{d}}^{\textnormal{min}}\\\
\frac{0.8E_{\textnormal{R}}}{2E_{\textnormal{d}}^{\textnormal{avr}}},&E_{\textnormal{d}}^{\textnormal{min}}<E_{\textnormal{R}}<\frac{2E_{\textnormal{d}}^{\textnormal{avr}}}{0.8}\\\
\frac{0.8E_{\textnormal{R}}}{2E_{\textnormal{d}}^{\textnormal{avr}}}\xi(E_{\textnormal{R}}),&E_{\textnormal{R}}>\frac{2E_{\textnormal{d}}^{\textnormal{avr}}}{0.8}\end{cases}$
(43)

where

$\xi(E_{\textnormal{R}})=(1-c_{\textnormal{arcdpa}})\left(\frac{E_{\textnormal{R}}}{2E_{\textnormal{d}}^{\textnormal{avr}}/0.8}\right)^{b_{\textnormal{arcdpa}}}+c_{\textnormal{arcdpa}}.$

We then obtain the number of stable defects as a function of recoil energy.
The four parameters in Eq. (43) can be obtained by MD simulations, and thus
depend on the chosen interatomic potential. For the W potential used in this
work, after Ref. [76], we obtained the values:
${E_{\textnormal{d}}^{\textnormal{min}}=47}$ eV,
${E_{\textnormal{d}}^{\textnormal{min}}=106}$ eV,
${b_{\textnormal{arcdpa}}=-0.80}$, ${c_{\textnormal{arcdpa}}=0.23}$. We note
that in Eq. (43) we assume that the recoil energy is equal to the damage
energy. We are, in other words, neglecting electronic losses because during
recoils of energy 10—100 eV only about 5% of the energy is lost to electronic
stopping [56].

Since in a single collision any amount of energy
${0<E_{\textnormal{R}}<E_{\textnormal{R}}^{\textnormal{max}}}$ can be
transferred, the defect production cross section for a single atom is given by
the integral

$\sigma_{FP}=\int\limits_{E_{\textnormal{d}}^{\textnormal{min}}}^{E_{\textnormal{R}}^{\textnormal{max}}}\left(\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}\right)N_{\textnormal{d}}(E_{\textnormal{R}})\
\text{d}E_{\textnormal{R}}.$ (44)

The Frenkel pair production rate per unit volume is proportional to the
electron flux $\phi_{\textnormal{el}}$ and the target atomic density $n_{0}$,

$\sim\phi_{\textnormal{el}}n_{0}\sigma_{\textnormal{FP}}.$

This appears to suggest that the concentration of Frenkel pairs increases
linearly with time, which is approximately valid only at the very early stages
of irradiation, where the defects are isolated and do not coalesce or
recombine because of elastic interactions, and only if the temperature of the
material is sufficiently low so that the thermal diffusion of defects is not
activated.

Maury et al. [4] measured the variation of electrical resistivity due to the
formation of Frenkel pairs in electron-irradiated W at low cryogenic
temperatures below 7 K. Assuming that the resistivity increase is proportional
to the concentration of Frenkel pairs, in Fig. 8 we compare the data by Maury
et al. with the results derived from Eq. (44), finding good agreement for the
onset of damage — corresponding in our model to a change of slope at $\sim
1560$ keV — as well as for the shape of the curve with respect to the
experimental points.

Figure 8: If the maximum recoil energy given by Eq. (1) exceeds the threshold
displacement energy, electrons start creating Frenkel pairs that in turn
notably increase the electric resistivity of the material [4]. In this figure,
we compare the solid line, providing an estimate for the effective cross
section of production of Frenkel pairs in W, see text for detail, with
experimental results by Maury et al. [4].

##  V Molecular dynamics simulations

Molecular dynamics simulations of W and Fe were performed using LAMMPS [77]
and empirical potentials developed by Mason et al. [76] and Gordon et al.
[78], respectively. A single vacancy was created at the centre of a bcc
simulation cell containing 1024 atoms fully relaxed under periodic boundary
conditions. The structure was thermalised at a target temperature using a
Langevin thermostat with the damping constant of 15.7 ps for W and 0.84 ps for
Fe [79]. Electron collisions were simulated by adding a randomly oriented
momentum vector $\mathbf{P}$, corresponding to a given amount of energy
transferred to the atom by an electron, to the instantaneous thermal momentum
of the atom. The magnitude of $P$ was chosen in the way that the energy that
the struck atom would have gained in the absence of thermal motion was

$E_{\textnormal{R}}=\frac{P^{2}}{2M}.$

In what follows, we shall demonstrate how thermal vibrations of atoms
influence the spectrum of recoil energies caused by high-energy electrons. To
see why thermal motion is relevant, let us find the distribution of recoil
energies $f_{\textnormal{R}}(E)$ of atoms that, if at rest, would recoil with
$E_{\textnormal{R}}$. By adding the transferred momentum to the thermal
momentum, see Refs. [67, 68] and Appendix B for detail, we find that the
distribution of recoil energies is well approximated by a Gaussian centred at
$E_{\textnormal{R}}$ and with variance
${\sigma^{2}=2(k_{\textnormal{B}}T)E_{\textnormal{R}}}$, namely

$f_{\textnormal{R}}(E)=\frac{1}{\sqrt{4\pi
k_{\textnormal{B}}TE_{\textnormal{R}}}}\exp\left[-\frac{(E-E_{\textnormal{R}})^{2}}{4k_{\textnormal{B}}TE_{\textnormal{R}}}\right].$
(45)

In the above equation, $k_{\textnormal{B}}$ is the Boltzmann constant. This
expression is valid in the limit $k_{\textnormal{B}}T\ll E_{\textnormal{R}}$,
applicable to the treatment of collisions of atoms with high-energy electrons
considered here. We note that although the thermal energy is very small
compared to the recoil energy, it has a major effect on the shape of spectrum
of atomic recoils [67, 68]. This remarkable manifestation of the Doppler
effect is illustrated in Fig. 9.

Figure 9: Probability distribution for the energy of recoils $x$ computed from
Eq. (45) assuming that electrons collide with an atom undergoing thermal
vibrations at a lattice site. The recoil energy transferred in a collision
would be exactly equal to $E_{\textnormal{R}}=5$ eV if the atom were at rest.
Even if the thermal energy is as low as about 0.05 eV, the broadening of the
recoil spectrum is close to 2 eV.

Simulations were carried out for temperatures $T=0$, 300, 600, and 900 K in W,
and 0, 150, 300, and 450 K in Fe. In relation to Fe, we note that two other
potentials, by Mendelev et al. [80] and Malerba et al. [81], frequently
produce a spurious “split vacancy” defect in near-threshold collisions, see
Appendix C. This configuration was very rarely found in the simulations
performed using the potential by Gordon et al. [78].

We note that although the treatment of collisions with $\sim$ MeV electrons
requires using relativistic mechanics [59], the velocities of atoms receiving
recoils are in the non-relativistic $10^{3}$ m/s range. 50,000 collisions were
simulated at each recoil energy over the energy interval extending up to 10
eV. Depending on the relative orientation of thermal and transferred momenta,
the energy of the recoil atom after a collision can be higher or lower than
$E_{\textnormal{R}}$, as shown in Fig. 9. This is significant, as an electron
may collide with an atom moving thermally towards a vacancy, away from it, or
at an angle. The total duration of the simulation less than 1 µs is such that
pure thermally activated hops have a negligible probability. Even at 900 K, a
vacancy hops in tungsten on the timescale of milliseconds, and hence MD
simulations performed in this study describe only the mode of diffusion
resulting from atomic recoils generated by high-energy electron impacts.

Since collisions with electrons delivering recoil energies in the eV range are
quite rare, we simulated them starting each time from a different thermalised
configuration. A computational loop was set up as follows: (i) a simulation
was run at a constant temperature for 0.310 ps for W and for 0.220 ps for Fe;
(ii) the resulting atomic configuration was saved at the end of each run;
(iii) a randomly oriented momentum $\mathbf{P}$ was added to the thermal
momentum of an atom in the neighbourhood of a vacancy; (iv) MD simulation was
run for 1 ns to give the vacancy a chance to hop to a neighbouring site; (v)
simulation was restarted from a configuration saved at step (ii). The time
between samplings of initial thermalised configurations was about 12 times the
inverse Debye frequency
${\omega_{\textnormal{D}}=k_{\textnormal{B}}T_{\textnormal{D}}/\hbar}$, where
$T_{\textnormal{D}}$ is the Debye temperature, equal to 310 K for W and 410 K
for Fe [82], giving $\omega_{\textnormal{D}}=4.06\times 10^{13}$ s-1 and
$\omega_{\textnormal{D}}=5.37\times 10^{13}$ s-1, respectively. The sampling
frequency ensured that a broad spectrum of initial conditions was explored. As
expected, we found that the $x$, $y$ and $z$ components of initial velocities
of atoms followed three independent Maxwell-Boltzmann distributions
corresponding to the thermalisation temperature.

By recording whether the vacancy had moved after each kick, we found the
fraction of successful hops as a function of recoil energy at different
temperatures, shown in Fig. 10. A reader will appreciate that, owing to the
Doppler effect of Eq. (45), vacancies in Fe or W at a temperature of about
15-25% of the melting point have a finite probability of hopping even when the
neighbouring atom recoils with up to about half of the minimum energy that is
required to initiate a vacancy hop at 0 K. However, the effect of temperature
is no longer significant if the recoil energy is above approximately twice
this minimum energy.

(a)  (b)

Figure 10: Fraction of atomic recoils resulting in a vacancy hop increases
with the amount of energy transferred in a collision, as illustrated by the
plots for Fe (a) and W (b). At 0 K, there is a sharp threshold for the
reaction. At a finite temperature, thresholds are smooth and vacancy hops
occur even at very low recoil energies.

##  VI Vacancy diffusion driven by high-energy electrons

We now evaluate the effect of high-energy electron impacts on vacancy
diffusion. In a pure material, the thermal part of the vacancy diffusion
coefficient $D$ is [83, 84]

$D_{\textnormal{th}}=\frac{N_{\textnormal{1NN}}}{6}d^{2}\nu_{\textnormal{th}}\exp\left[-\frac{E_{\textnormal{a}}}{k_{\textnormal{B}}T}\right],$
(46)

where $N_{\textnormal{1NN}}$ is the number of first nearest neighbour (1NN)
atoms, $d$ is the hop distance (if $a$ is the lattice parameter then
$d=a\sqrt{3}/2$ for bcc and $d=a\sqrt{2}/2$ for fcc crystals),
$\nu_{\textnormal{th}}={\omega_{\textnormal{D}}/2\pi}$ is the thermal attempt
frequency, and $E_{\textnormal{a}}$ is the vacancy migration energy.

In the presence of impurities, the diffusion coefficient can be estimated
following Refs. [85, 86]. For example, in Fe and W containing carbon at a
concentration exceeding the concentration of vacancies, we find

$D_{\textnormal{th}}^{\textnormal{C}}=D_{\textnormal{th}}\frac{[V]}{[V_{\textnormal{tot}}]},$
(47)

where $[V]$ is the vacancy concentration and
$[V_{\textnormal{tot}}]=[V]+[VC]+[VC_{2}]+...$ is the total concentration of
isolated vacancies and carbon-vacancy complexes VCn. A single vacancy in Fe
and W can trap between one and four C atoms, with the VC2 complex being the
most stable [85, 87, 86]. Using the binding energies $E_{n}^{\textnormal{b}}$
calculated using density functional theory for Fe [85] and for W [86] we find

$[VC_{n}]=[V][C]^{n}\exp\left(\frac{E_{n}^{\textnormal{b}}}{k_{\textnormal{B}}T}\right)$

resulting in

$D_{\textnormal{th}}^{\textnormal{C}}=\frac{D_{\textnormal{th}}}{1+\sum_{n=1}^{4}[C]^{n}\exp\left(\frac{E_{n}^{\textnormal{b}}}{k_{\textnormal{B}}T}\right)}.$
(48)

If atoms are bombarded by high-energy electrons, diffusion is always
accelerated, irrespectively of the presence or absence of impurities, by the
transfer of kinematic momentum to atoms from electrons. A collision of a high-
energy electron with an atom close to a vacancy can stimulate an atomic hop
or, equivalently, a vacancy hop to an adjacent lattice site. The threshold
energy for a hop depends on the crystallographic direction of the impact and
the distance between the atom receiving the impact and a vacancy.

To estimate the contribution of electron impacts to vacancy diffusion, we
first need to identify the atoms around a vacancy that could contribute to
diffusion through random recoils requiring the least amount of energy to
initiate a successful vacancy hop. We simulated, at $T=0$ K, the atomic
processes initiated by atomic impacts using the interatomic potentials for
tungsten by Mason [76] and Marinica [88], and the interatomic potential for
iron by Gordon [78]. We considered initial impacts in the $\langle
111\rangle$, $\langle 100\rangle$ and $\langle 110\rangle$ directions
involving the first, second and third nearest neighbours of a vacancy, and
also atoms further away from it.

Table 3: Minimum recoil energies resulting in a vacancy hopping to a nearest neighbour position. The values, given in eV units, were computed for $T=0$ using two different potentials for W and one for Fe. The atoms receiving the recoils were assumed to be in the first, second, and third NN position in relation to the vacancy. Recoil energy thresholds are visualised for the Mason potential for bcc W [76]. |  | W [76] | W [88] | Fe [78]
---|---|---|---|---
Direction | distance | minimum energy [eV]
$\langle 111\rangle$ | 1${}^{\text{st}}$ | 1.80 | 2.23 | 0.83
2${}^{\text{nd}}$ | 2.19 | 3.72 | 1.25
3${}^{\text{rd}}$ | 3.03 | 5.81 | 1.99
4${}^{\text{th}}$ | 3.97 | 8.08 | 2.87
5${}^{\text{th}}$ | 4.99 | 10.3 | 3.78
6${}^{\text{th}}$ | 6.06 | 12.5 | 4.70
7${}^{\text{th}}$ | 7.19 | 14.3 | 5.59
$\langle 100\rangle$ | 1${}^{\text{st}}$ | 10.1 | 10.1 | 4.76
2${}^{\text{nd}}$ | 19.6 | 19.1 | 8.46
3${}^{\text{rd}}$ | 30.1 | 28.2 | 12.7
$\langle 110\rangle$ | 1${}^{\text{st}}$ | 43.3 | 32.9 | 17.0

Table 3 shows that electron-induced diffusion in W and Fe is dominated by the
events involving high-energy electrons colliding with atoms situated along the
$\langle 111\rangle$ crystallographic directions. In bcc metals there are
$N_{\langle 111\rangle}=8$ atoms nearest to a vacancy. The electron collision
contribution to the vacancy diffusion coefficient is proportional to the
weighted sum of frequencies $\nu_{\langle 111\rangle}^{(k)}$ with which one of
the atoms at a $k$-th position along a $\langle 111\rangle$ atomic string
causes a vacancy to perform a hop following an event of interaction with a
high-energy electron

$D_{\textnormal{el}}=\frac{N_{\langle 111\rangle}}{6}d^{2}\left(\nu_{\langle
111\rangle}^{(1)}+\nu_{\langle 111\rangle}^{(2)}+\ldots\right),$ (49)

where $d=a\sqrt{3}/2$ is the nearest neighbour distance between sites in bcc
lattice. This provides a lower bound for $D_{\textnormal{el}}$ as it neglects
the events where the struck atom is not situated along a $\langle 111\rangle$
atomic string.

To evaluate $\nu_{\langle 111\rangle}^{(k)}$, we note that a vacancy hop to a
nearest lattice site involves the following sequence of events

  1. 1.

an atom near a vacancy is struck by an electron with sufficiently high energy
$E_{\textnormal{el}}$ to initiate a recoil in the eV energy range;

  2. 2.

the atom recoils in a specific direction with kinetic energy
$E_{\textnormal{R}}$ transferred in the collision with the electron;

  3. 3.

one of the atoms in the vicinity of a vacancy, not necessarily the atom
impacted by the electron, crosses the potential barrier and moves into a
stable energy minimum position at the initially vacant lattice site.

Recalling Eq. (3), the _total_ frequency of collisions between an atom and
electrons depends on the integrated over the solid angle electron flux
$\phi_{\textnormal{el}}(E_{\textnormal{el}})$ defined by (20) and the total
cross section of scattering of electrons by an atom
$\sigma_{\textnormal{tot}}(E_{\textnormal{el}})$,

$\nu_{\textnormal{tot}}(E_{\textnormal{el}})=\phi_{\textnormal{el}}(E_{\textnormal{el}})\sigma_{\textnormal{tot}}(E_{\textnormal{el}}).$
(50)

To account for points 1 and 2 above, we need the frequency _distribution_ of
atoms as a function of recoil energy $E_{\textnormal{R}}$, resulting from
collisions with electrons over the entire spectrum of electron energies. The
recoil energy-resolved frequency of impact events for an atom at a lattice
site is

$\displaystyle\nu(E_{\textnormal{R}})$ $\displaystyle=$
$\displaystyle\int\text{d}o\int\text{d}o^{\prime}\int\text{d}E^{\prime}{\text{d}\sigma({\bf
n}^{\prime}\rightarrow{\bf n})\over\text{d}o}\phi_{\textnormal{el}}({\bf
n}^{\prime},E^{\prime})$ (51) $\displaystyle\times$
$\displaystyle\delta\left\\{E_{\textnormal{R}}-{1\over
2}E_{\textnormal{R}}^{\textnormal{max}}(E^{\prime})[1-\cos({\bf n}\cdot{\bf
n}^{\prime})]\right\\},$

where $E_{\textnormal{R}}^{\textnormal{max}}(E)$ is given by Eq. (33), and
$\delta(x)$ is the Dirac delta function. This equation involves only the
elastic cross section of scattering of electrons by an atom because inelastic
scattering of high-energy electrons primarily contributes to energy losses but
not to atomic recoils [89]. Integrating (51) over $E_{\textnormal{R}}$, we
recover expression (50).

If the flux of electrons bombarding the atoms is isotropic, see Eq. (20), in
Eq. (51) we integrate over the directions of ${\bf n}^{\prime}$ and arrive at

$\displaystyle\nu(E_{\textnormal{R}})$ $\displaystyle=$ $\displaystyle
2\pi\int\limits_{0}^{\pi}\text{d}\theta\sin\theta{\text{d}\sigma\over\text{d}\theta}\int\limits_{0}^{\infty}\text{d}E\,\phi_{\textnormal{el}}(E)$
(52) $\displaystyle\times$
$\displaystyle\delta\left\\{E_{\textnormal{R}}-{1\over
2}E_{\textnormal{R}}^{\textnormal{max}}(E)[1-\cos\theta]\right\\}.$

Since the angle of scattering $\theta$ enters Eq. (52) only as an argument of
$\cos\theta$, we change the variable of integration to $\xi=1-\cos\theta$ and
write

$\displaystyle\nu(E_{\textnormal{R}})$ $\displaystyle=$ $\displaystyle
2\pi\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\int\limits_{0}^{2}\text{d}\xi\int\limits_{0}^{\infty}\text{d}E\,\phi_{\textnormal{el}}(E)\left(\frac{1-\beta^{2}}{\beta^{4}}\right)$
(53) $\displaystyle\times$
$\displaystyle\frac{1}{(\kappa+\xi)^{2}}\delta\left[E_{\textnormal{R}}-{\xi\over
2}E_{\textnormal{R}}^{\textnormal{max}}(E)\right],$

where the cross section of scattering is given by the screened Rutherford
expression (34). Integrating over $\xi$, we find

$\displaystyle\nu(E_{\textnormal{R}})=2\pi\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\int\limits_{0}^{\infty}\text{d}E\,\phi_{\textnormal{el}}(E)\left(\frac{1-\beta^{2}}{\beta^{4}}\right)$
(54) $\displaystyle\times{2E^{\textnormal{max}}(E)\over\left[\kappa
E_{\textnormal{R}}^{\textnormal{max}}(E)+2E_{\textnormal{R}}\right]^{2}}\Theta\left[E_{\textnormal{R}}^{\textnormal{max}}(E)-E_{\textnormal{R}}\right],$

where $\Theta(x)$ is the Heaviside function, $\Theta(x)=1$ for $x>0$ and
$\Theta(x)=0$ for $x<0$.

For a mono-energetic flux of electrons with kinetic energy ${\cal E}$, the
expression for the flux has the form
${\phi_{\textnormal{el}}(E)=\Phi_{0}\delta(E-{\cal E})}$ and the distribution
of the frequency of atomic recoils with respect to the energy of recoils is

$\displaystyle\nu(E_{\textnormal{R}})$ $\displaystyle=$ $\displaystyle
2\pi\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\Phi_{0}{(1+{\cal
E}/mc^{2})^{2}\over[(1+{\cal E}/mc^{2})^{2}-1]^{2}}$ (55)
$\displaystyle\times$
$\displaystyle{2E_{\textnormal{R}}^{\textnormal{max}}({\cal
E})\over\left[\kappa E_{\textnormal{R}}^{\textnormal{max}}({\cal
E})+2E_{\textnormal{R}}\right]^{2}},$

for $E_{\textnormal{R}}<E_{\textnormal{R}}^{\textnormal{max}}({\cal E})$.
There are no recoils with energies higher than
$E_{\textnormal{R}}^{\textnormal{max}}({\cal E})=2{\cal E}({\cal
E}+2mc^{2})/Mc^{2}$.

Figure 11: Representative examples of trajectories found in events involving
collisions of electrons with atoms in a 1NN shell of a vacancy in W. The
[322], $[114]$ and $[22\bar{1}]$ trajectories start at about 11∘, 35∘ and 55∘
off the $[111]$ crystallographic direction, respectively, and require recoil
energies of 2, 5 and 9 eV to enable the atom to successfully complete a hop
into a vacant site (marked as $V$).

(a)  (b)  (c)  (d)

Figure 12: Fraction of successful vacancy hops in W for 0 K (a), 300 K (b),
600 K (c), 900 K (d) if the kicked atom is one of the 1${}^{\text{st}}$,
2${}^{\text{nd}}$, 3${}^{\text{rd}}$ or 4${}^{\text{th}}$ atoms along the
$\langle 111\rangle$ directions. Discrete data points, with error bars
indicating one standard deviation, are derived from MD simulations (potential
[76]), whereas the fitting curves are given by Eq. (VI). Fitting all the data
simultaneously yielded a single pair $\alpha=0.3504$,
$k_{\textnormal{B}}T_{0}=2.038$ eV (with values of $E_{\textnormal{a}}^{0}$ in
Eq. (58) from Table 3). Eq. (VI) fits all the data points with $R^{2}=99.7$ %.

The same result can be obtained from the energy-differential cross sections,
Eq. (41) for the screened Rutherford interaction and Eq. (42) for the Mott
scattering, resulting in

$\nu(E_{\textnormal{R}})=\int\limits_{0}^{\infty}\text{d}E_{\textnormal{el}}\phi_{\textnormal{el}}\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}.$
(56)

In the above equations, $\nu(E_{\textnormal{R}})$ is expressed in units
s-1eV-1 whereas the flux of electrons is expressed in the cm-2s-1eV-1 units.

We now address point 3 above by quantifying the probability of the vacancy
hopping to a neighbouring lattice site if a collision with a high-energy
electron transfers energy $E_{\textnormal{R}}$ to one of the neighbouring
atoms. This probability, which in what follows we refer to as
$J(E_{\textnormal{R}},T)$, depends on the recoil energy and temperature.

Since the spectrum of recoil energies given by Eq. (55) is broad, vacancy
hopping events are expected to be dominated by the electron impacts generating
recoil energies comparable or greater than the vacancy migration energy. This
energy scale is many times the energy of thermal motion of atoms, and hence
the statistics of transitions stimulated by electron impacts are expected to
be different from the statistics of thermally activated events, detailed in
Ref. [90].

The treatment of thermally activated transitions involves computing the rate
of a many-body atomic system crossing a potential barrier associated with a
reaction where an atom hops to a neighbouring vacant site [91, 92, 90]. For
thermally activated processes, this rate is exponentially small, and the
thermal motion of atoms at equilibrium positions is assumed to be unaffected
by the occurrence of transitions.

A typical recoil event stimulated by a high-energy electron collision with an
atom has the energy well above the thermal energy but still below the
threshold for the formation of a Frenkel pair. The motion of an atom receiving
the recoil and its neighbours remains confined to the vicinity of equilibrium
positions, although the amplitude of motion is significantly larger than the
amplitude of thermal vibrations. The initial energy of the recoil dissipates
by phonon radiative transfer. Occasionally, on the timescale of several Debye
oscillations, this energy gets partially projected onto the trajectory of a
many-body reaction involving a vacancy hopping to a neighbouring lattice site.

Providing an accurate quantitative estimate of $J(E_{\textnormal{R}},T)$ is
difficult because of the atomic many-body character of recoil-stimulated
vacancy hopping events. Trajectories of MD simulations shown in Fig. 11
suggest that the initial direction of an atomic recoil does not have a strong
effect on the outcome of a sequence of local interactions between the atoms
that eventually result in vacancy migration. Even the recoils at a large angle
to the direction of a vacancy hop appear to contribute significantly to the
vacancy hopping rate.

At $T=0$ K, function $J(E_{\textnormal{R}},0)$ vanishes for all the recoil
energies $E_{\textnormal{R}}<E_{\textnormal{a}}$, where $E_{\textnormal{a}}$
is the minimum recoil energy required to initiate a vacancy migration. At a
finite temperature, $J(E_{\textnormal{R}},T)$ is a monotonically increasing
function of $E_{\textnormal{R}}$, likely saturating at high recoil energies
$E_{\textnormal{R}}\gg E_{\textnormal{a}}$, provided that $E_{\textnormal{R}}$
is still well below the Frenkel pair threshold production energy. MD
simulations performed in this study and in Ref. [18] suggest that at high
recoil energies function $J(E_{\textnormal{R}},T)$ approaches a limit close to
0.27. Notably, this value is higher than the $1/8=0.125$ fraction of the solid
angle spanned by the directions from an atom in a corner of a cubic cell
towards a vacancy in its centre.

Temperature effects are expected to assist the recoil-stimulated vacancy
diffusion. Thermal velocity adds to the velocity of an atom derived from a
recoil event if both vectors are aligned towards a vacancy. Also, the
effective free-energy migration barrier is lowered by the vibrations of atoms
though the entropy effect [93]. A treatment similar to a transition rate
theory approach suggests a suitable functional form for
$J(E_{\textnormal{R}},T)$, see Appendix D for detail

$\displaystyle J(E_{\textnormal{R}},T)$
$\displaystyle=\alpha\Bigg{(}1+\frac{1}{2}\left(1-\sqrt{\frac{E_{\textnormal{a}}}{E_{\textnormal{R}}}}\right)\text{erf}\left[\frac{\sqrt{E_{\textnormal{R}}}-\sqrt{E_{\textnormal{a}}}}{\sqrt{k_{\textnormal{B}}T}}\right]$
$\displaystyle-\frac{1}{2}\left(1+\sqrt{\frac{E_{\textnormal{a}}}{E_{\textnormal{R}}}}\right)\text{erf}\left[\frac{\sqrt{E_{\textnormal{R}}}+\sqrt{E_{\textnormal{a}}}}{\sqrt{k_{\textnormal{B}}T}}\right]$
(57) $\displaystyle+\sqrt{\frac{k_{\textnormal{B}}T}{\pi
E_{\textnormal{R}}}}\text{sinh}\left[\frac{2\sqrt{E_{\textnormal{R}}E_{\textnormal{a}}}}{k_{\textnormal{B}}T}\right]\exp\left[-\frac{E_{\textnormal{a}}+E_{\textnormal{R}}}{k_{\textnormal{B}}T}\right]\Bigg{)},$

where $\alpha$ is a constant that can be determined from MD simulations. Eq.
(VI) was derived assuming that the activation energy is the same as in Eq.
(46). To account for the entropy term [93], we treat $E_{\textnormal{a}}$ as a
weakly temperature dependent quantity and define

$E_{\textnormal{a}}(T)=E_{\textnormal{a}}^{0}\left(1-\frac{T}{T_{0}}\right),$
(58)

where the constants are material-dependent with ${T_{0}\sim 10^{4}}$ K.
$E_{\textnormal{a}}^{0}$ is the minimum energy required for an atom to hop
into a vacant lattice site and it can be derived from a single MD simulation
for each of the atoms in the $\langle 111\rangle$ string of atoms, see Table
3. As for the choice of parameters $\alpha$ and $T_{0}$, we found that a
single set of two constants is sufficient to fit all of the data derived from
extensive MD simulations involving randomly kicking of atoms in the vicinity
of a vacancy, at one to four nearest neighbour distances in a $\langle
111\rangle$ direction. The MD data and the fits based on the analytical
formula Eq. (VI) are shown in Fig. 12.

For a given electron flux and resulting distribution of recoil energies, the
success rate is given by the product of Eq. (56) and Eq. (VI). Since both the
electron and recoil energies are given by continuous probability
distributions, the total success rate $\nu_{\langle 111\rangle}^{(k)}$ is
obtained by integrating over all the electron and recoil energies, yielding

$\nu_{\langle
111\rangle}^{(k)}(T)=\int\limits_{0}^{\infty}\text{d}E_{\textnormal{el}}\int\limits_{0}^{\infty}\text{d}E_{\textnormal{R}}\phi_{\textnormal{el}}(E_{\textnormal{el}})\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}J_{k}(E_{\textnormal{R}},T).$
(59)

Numerically, the electron flux is represented by a set of $N$ values
$\Phi_{\textnormal{el}}^{(i)}$ corresponding to discrete electron energies
$E_{i}$, and thus the above equation can be written as

$\phi_{\textnormal{el}}(E_{\textnormal{el}})=\sum_{n=1}^{N}\Phi_{\textnormal{el}}^{(i)}\delta(E_{\textnormal{el}}-E_{i}).$
(60)

Substituting this in Eq. (59) we find that

$\nu_{\langle
111\rangle}^{(k)}(T)=\sum_{i=1}^{N}\int\limits_{0}^{\infty}\Phi_{\textnormal{el}}^{(i)}\frac{\text{d}\sigma}{\text{d}E_{\textnormal{R}}}J_{k}(E_{\textnormal{R}},T)\text{d}E_{\textnormal{R}}.$
(61)

The $k$-th index of $J_{k}(E_{\textnormal{R}},T)$ indicates that the
corresponding energy threshold refers to the $k$-th atom along the $\langle
111\rangle$ direction. Inserting Eq. (61) into Eq. (49) enables one to
calculate $D_{\textnormal{el}}$, the high-energy electron recoil contribution
to the diffusion of vacancies in a material exposed to neutron irradiation.
The barriers listed in Table 3 were obtained for pure W and Fe. In the
presence of C atoms at the vacancy site, we expect them to be higher. We
denote by $D_{\textnormal{el}}^{\textnormal{C}}$ the diffusion coefficient
calculated after raising all the minimum hopping energies in Table 3.

##  VII Vacancy migration in W, stimulated by high-energy electrons

High-voltage TEM is known to be effective in stimulating vacancy migration in
W. We can apply Eq. (49) to evaluate the diffusion coefficient of vacancies
stimulated by a flux of $10^{21}$ cm-2s-1 electrons with kinetic energies 200
and 2000 keV [5], as shown in Fig. 13 together with $D_{\textnormal{th}}$ and
$D_{\textnormal{th}}^{\textnormal{C}}$. The estimate for
$D_{\textnormal{th}}^{\textnormal{C}}$ is valid in the limit $[V]\ll[C]$, i.e.
in the case where the accelerating voltage is not sufficient to produce
vacancies by electron collisions or, if it is, at the early stages of
irradiation. We assume the thermal activation energy of 1.66 eV, and binding
energies $E_{n}^{\textnormal{b}}$ of 1.93, 1.97, 1.87, 1.54 eV for $n=1,2,3,4$
respectively [86]. If the carbon concentration is close to 1 appm,
$D_{\textnormal{th}}^{\textnormal{C}}$ is dominated by the [VC] complexes in
the temperature interval between approximately 30 and 1500 K, yielding an
Arrhenius behaviour with a pre-exponential factor
${D_{0}^{\textnormal{C}}=\nu_{\textnormal{th}}a^{2}/[C]=6.55\times 10^{3}}$
cm2/s and an activation energy
${E_{\textnormal{a}}^{\textnormal{C}}=1.66+1.93=3.59}$ eV. On the basis of
this, when calculating $D_{\textnormal{th}}^{\textnormal{C}}$, we raised the
activation energies entering the four hopping frequencies $\nu_{\langle
111\rangle}^{(k)}$ by 1.79 eV to $E_{\textnormal{a}}=[3.59,3.98,4.82,5.76]$
eV.

Conventional thermally activated diffusion exhibits a very strong temperature
dependence [90], whereas the temperature dependence is much weaker for the
high-energy electron recoil contribution to the vacancy diffusion coefficient.
To compare thermal and electron-stimulated contributions to vacancy diffusion,
we define the notion of the effective equivalent temperature

$T_{\textnormal{el}}=\frac{E_{\textnormal{a}}}{k_{\textnormal{B}}\log(D_{0}/D_{\textnormal{el}})},$

where $D_{0}$, $E_{\textnormal{a}}$ and $D_{\textnormal{el}}$ depend on
whether C impurities are considered or not. At 0 K under electron irradiation,
vacancies appear to be as mobile as in pure tungsten at temperatures close to
$T_{\textnormal{el}}=635$ and 672 K for the two energies of the incident high-
energy electrons, respectively. This is fully consistent with experimental
observations showing that vacancies were highly mobile under electron
irradiation [5]. If 1 appm of C is also present in the material,
$T_{\textnormal{el}}^{\textnormal{C}}$ equals 966 K for the 2 MeV case whereas
the electron-driven diffusion is suppressed in the case of 200 keV electrons,
because $E_{\textnormal{R}}^{\textnormal{max}}=2.85$ eV is lower than the
lowest activation energy of 3.59 eV. However, due to the Doppler effect, even
if $T$ is as low as 300 K, the electron-stimulated vacancy diffusion is strong
enough to give $T_{\textnormal{el}}^{\textnormal{C}}=803$ K. These
calculations were performed using the Mott scattering cross section. The
Rutherford cross section yields results that are approximately 1.5 times lower
but qualitatively consistent with the above estimates.

Eq. (49) enables calculating the vacancy diffusion coefficient and the
associated $T_{\textnormal{el}}$ for a given electron spectrum. We performed
this calculation for the two spectra shown in Fig. 6. Table 4 compares the
computed values with the data for high-voltage TEM electron irradiation,
assuming the irradiation temperature of 0 K. Both fission and fusion neutron
spectra yield a similar effective temperature for the driven vacancy
migration, assuming pure tungsten, of about 135 ∘C. This temperature is on the
lower end of the reported temperature for stage III recovery of neutron-
induced damage in W, which is in the range 100—450 ∘C [94].
$D_{\textnormal{el}}$ in HFR is about four times larger than in DEMO. The
effective temperature is significantly higher, and is close to 400 ∘C, if the
effect of 1 appm of carbon is taken into account. However, we note that if
under prolonged irradiation the vacancy concentration increases enough so that
the fraction trapped by the carbon atoms becomes small, the effect of the
carbon-vacancy interaction becomes less noticeable.

Figure 13: Temperature dependence of electron-stimulated vacancy diffusion in
W, $D_{\textnormal{el}}$, Eq. (49), compared to thermal diffusion in an ultra-
pure, $D_{\textnormal{th}}$, Eq. (46), and in C-doped,
$D_{\textnormal{th}}^{\textnormal{C}}$, Eq. (47), conditions. The effect of
carbon is included in $D_{\textnormal{th}}^{\textnormal{C}}$ by increasing the
energies $E_{\textnormal{a}}^{(k)}$ in Eq. (61) so that the lowest energy
matches the C-doped thermal barrier of 3.59 eV. The analysis corresponds to a
TEM scenario, with beam energies of 200 keV and 2 MeV and electron flux of
$2\times 10^{21}$ cm-2s-1. An estimate (VII), corresponding to the limiting
case for $1000/T\rightarrow\infty$, is also shown.
$D_{\textnormal{th}}^{\textnormal{C}}$ for 200 keV electrons vanishes if
$T\rightarrow 0$ as $E_{\textnormal{R}}^{\textnormal{max}}=2.85$ eV falls
below the raised hopping barrier. Therefore, the electron-driven diffusion
observed in this case is entirely due to the Doppler effect.

Finally, we note that the photons generating the electron flux in a neutron-
irradiated material, in the analysis above are only those generated
_internally_ by the W nuclei. In the case of a relatively thin W layer like in
the current designs of the first wall of a breeding blanket, the _external_
photon flux is going to play the dominant part in determine the effective
vacancy diffusion temperature.

In W, for the electron energies above hundreds of keV, we find that the
correction factor $R_{\textnormal{M}}$ entering the Mott cross section, the
choice of the screening parameter $\kappa$ in the Rutherford cross section
and—most critically—the temperature of the material all have a limited effect
on vacancy diffusion. This suggests that it should be possible to derive a
simple analytical formula not requiring the numerical integration of Eq. (61).
To do this, we take the limit $\lim_{T\to 0}J(E_{\textnormal{R}},T)$ in Eq.
(VI) and arrive at

$J(E_{\textnormal{R}},0)=\alpha\left(1-\sqrt{\frac{E_{\textnormal{a}}}{E_{\textnormal{R}}}}\right)\Theta[E_{\textnormal{R}}-E_{\textnormal{a}}],$
(62)

noting that Eq. (62) sets a lower bound on $J(E_{\textnormal{R}},T)$. Then we
set $\kappa=0$ in Eq. (34), evaluate the integral in Eq. (61), insert the
result into Eq. (49) and find

$\displaystyle\widetilde{D}_{\textnormal{el}}=$
$\displaystyle\frac{\pi\alpha}{3}\left(\frac{Ze^{2}}{4\pi\varepsilon_{0}mc^{2}}\right)^{2}\Phi_{0}a^{2}\left(\frac{1-\beta^{2}}{\beta^{4}}\right)\times$
$\displaystyle\sum_{k=1}^{4}\Bigg{[}\frac{E_{\textnormal{R}}^{\textnormal{max}}}{E_{\textnormal{a}}^{(k)}}+2\sqrt{\frac{E_{\textnormal{a}}^{(k)}}{E_{\textnormal{R}}^{\textnormal{max}}}}-3\Bigg{]}.$
(63)

This equation is valid if
$E_{\textnormal{R}}^{\textnormal{max}}>E_{\textnormal{a}}^{(k)}$, whereas
$\widetilde{D}_{\textnormal{el}}=0$ otherwise.

We considered the mono-energetic electron beam case ($N=1$ in Eq. (61)) and we
considered only the first four atoms along the $\langle 111\rangle$ directions
as contributing to the recoil-stimulated diffusion. For W, the activation
energies are $E_{\textnormal{a}}=[1.80,2.19,3.03,3.97]$ eV, see Table 3. This
formula provides a simple but accurate estimate of the effective temperature
of a W sample in a TEM given the accelerating voltage and the electron flux.
The value yielded by Eq. (VII) is shown in Fig. 13 next to the corresponding
curve obtained by direct numerical integration. The difference with respect to
the right-most point of the curves, i.e. $T=1$ K, is a factor of about 1.5,
which is entirely due to the ratio between the Mott and the Rutherford cross
sections. Taking as an example a 2000 kV TEM scenario where the
$\Phi_{\textnormal{el}}=2\times 10^{21}$ cm-2s-1 and $T=300$ K, we find that
$D_{\textnormal{el}}=2.22\times 10^{-15}$ cm2/s if the estimate involves using
the Mott cross section and $1.52\times 10^{-15}$ cm2/s if the Rutherford cross
section is used. The corresponding value derived directly from Eq. (VII),
hence at 0 K, is in this case $1.46\times 10^{-15}$ cm2/s.

Table 4: Diffusion coefficient of vacancies in W resulting from exposure to high-energy electrons. Parameters for the high-voltage TEM (HVTEM) electron irradiation, the accelerating voltage of 2 MV and the lower and upper values of flux of electrons, were taken from Ref. [5]. The Mott cross section was used in the calculations. | $D_{\textnormal{el}}$ | $T_{\textnormal{el}}$ | $D_{\textnormal{el}}^{\textnormal{C}}$ | $T_{\textnormal{el}}^{\textnormal{C}}$
---|---|---|---|---
| [cm2/s] | [K] | [cm2/s] | [K]
DEMO | $4.88\times 10^{-24}$ | 396 | $1.84\times 10^{-24}$ | 657
HFR | $1.99\times 10^{-23}$ | 408 | $7.55\times 10^{-24}$ | 672
HVTEM | $2.15\times 10^{-15}$ | 672 | $1.20\times 10^{-15}$ | 966
$(2\times 10^{21}$ cm-2s${}^{-1})$
HVTEM | $5.38\times 10^{-18}$ | 556 | $3.00\times 10^{-18}$ | 848
$(5\times 10^{18}$ cm-2s${}^{-1})$

##  VIII Conclusions

Neutron irradiation deposits energy in materials through a variety of nuclear
interactions; these include elastic and inelastic collisions with atomic
nuclei and non-elastic interactions such as neutron capture or multiplication.
The resulting nuclei are left in internally excited states that emit
$\gamma$-photons as they decay. The fraction of energy of neutrons converted
into a flux of $\gamma$-photons strongly depends on the elemental content of
the material. For example, in W the proportion of kinetic energy of neutrons
converted into electromagnetic $\gamma$-radiation approaches 99 %, which is
higher than the light generation efficiency of LED devices, whereas in Be this
fraction is less than 1 %, assuming exposure to identical neutron spectra.

The energetic secondary $\gamma$-photons play a significant part in the
microstructural evolution of W exposed to neutron irradiation. In a two-step
process, neutrons generate $\gamma$-photons that in turn excite high-energy
electrons in the material. Starting from an energy-resolved neutron spectrum,
we compute the energy-resolved $\gamma$-photon and electron spectra. In W and
Fe, the DEMO fusion and HFR fission neutrons produce the $\gamma$-photon and
electron spectra that have energies in the range from hundreds of keV to
several MeV. These high-energy electrons have sufficient energy to stimulate
athermal migration of vacancies and other defects through relatively low
energy atomic recoils, which on their own do not produce radiation defects.
The frequency of vacancy hops stimulated by interaction with electrons depends
on the energy-differential atomic cross section and many-body atomic
transitions triggered by the recoils. A functional form for the athermal
diffusion coefficient is confirmed by molecular dynamics simulations. The
results highlight the role of the Doppler effect in lowering the threshold
energy for vacancy migration.

The diffusion coefficient for electron-stimulated diffusion in W is evaluated
assuming DEMO, HFR and high-voltage TEM irradiation exposure. In a neutron
field, even if W is assumed to be at very low temperature, vacancies are as
mobile as if, in the absence of impurities, the temperature were close to 400
K. This effective temperature is even higher if we take into account the
interaction of vacacies with impurities, for example carbon. Since the process
of electron-driven vacancy diffusion is entirely athermal, no cooling of the
material can deactivate this neutron-photon-electron mode of diffusion
stimulated by irradiation. Atomic recoils initiated by the high-energy
electrons, produced by the $\gamma$-quanta resulting from the neutron-
stimulated nuclear excitations, have a broad energy spectrum extending into
the several eV range even in tungsten, and hence the same mechanism likely
stimulates a variety of other reactions between the defects involving high
activation energies.

###### Acknowledgements.

The authors would like to thank M. Rieth, D. Terentyev, S. Kalcheva, and G.
Pintsuk for stimulating discussions. This work has been carried out within the
framework of the EUROfusion Consortium, funded by the European Union via the
Euratom Research and Training Programme (Grant Agreement No 101052200 —
EUROfusion) and from the EPSRC Energy Programme (grant number EP/W006839/1).
To obtain further information on the data and models underlying the paper
please contact<EMAIL_ADDRESS>Views and opinions expressed are
however those of the authors only and do not necessarily reflect those of the
European Union or the European Commission. Neither the European Union nor the
European Commission can be held responsible for them. We gratefully
acknowledge the provision of computing resources by the IRIS (STFC)
Consortium.

##  Appendix A The total cross section of Compton scattering

The kernel of Eq. (12) is the energy-differential cross section involving only
the Compton scattering of photons. This cross section describes scattering of
a photon with the initial energy $E^{\prime}$ into a state with any energy
between $E^{\prime}$ and $E^{\prime}/(1+2E^{\prime}/mc^{2})$. Therefore, the
total Compton scattering cross section in Eq. (19), tabulated in Ref. [60],
can also be found by integrating Eq. (12) over all the energies $E$ after
scattering

$\displaystyle\tilde{\sigma}_{\textnormal{CS}}(E^{\prime})=$
$\displaystyle\int\text{d}E\int\text{d}o^{\prime}{\text{d}^{2}\sigma({\bf
n}^{\prime},E^{\prime}\rightarrow{\bf
n},E)\over\text{d}o^{\prime}\text{d}E^{\prime}}$ (64) $\displaystyle=$
$\displaystyle\int\text{d}E\,K(E,E^{\prime}).$

By integrating Eq. (12) over the interval of energies
$E^{\prime}/(1+2E^{\prime}/mc^{2})<E<E^{\prime}$ we find that

$\displaystyle\tilde{\sigma}_{\textnormal{CS}}(\varepsilon)=$ $\displaystyle
2\pi
r_{\textnormal{c}}^{2}\bigg{[}\frac{2}{\varepsilon^{2}}+\frac{(\varepsilon+1)}{(2\varepsilon+1)^{2}}+\frac{\log(1+2\varepsilon)}{2\varepsilon}$
(65)
$\displaystyle-\frac{2(\varepsilon+1)}{\varepsilon^{3}}\text{tanh}^{-1}\left(\frac{\varepsilon}{\varepsilon+1}\right)\bigg{]},$

where $\varepsilon=E^{\prime}/mc^{2}$.

Comparison of Eq. (65) and $\sigma_{\textnormal{CS}}$ from [29] is given in
Fig. 14. Eq. (7) of Ref. [60] provides a different expression for Eq. (65),
which is also included in the figure. We see that for energies of interest of
$\sim$300 keV and above, the assumption that there are $Z/\omega$ independent
electrons, where $\omega$ is the atomic volume, is very accurate. At 100 keV
Eq. (65) overestimates the data from [29] by only 16 %.

Figure 14: Total Compton cross section of scattering for a single electron,
obtained by integrating the Klein-Nishina cross section (blue), is compared
with the data available in the XCOM database for W [29] (green) and analytical
expression from Ref. [60]. The assumption that even electrons in the inner
shells of atoms act as independent scattering centres is very accurate if the
photon energy exceeds $\sim$300 keV.

##  Appendix B Thermal broadening of recoil energy

Consider a collision between a high-energy electron and an atom. If the latter
is at rest, it recoils with velocity
$(v_{x,\textnormal{R}},v_{y,\textnormal{R}},v_{z,\textnormal{R}})$ and kinetic
energy
${E=E_{\textnormal{R}}=\frac{m}{2}(v_{x,\textnormal{R}}^{2}+v_{y,\textnormal{R}}^{2}+v_{z,\textnormal{R}}^{2})}$.
If the atom also had some initial thermal velocity
$(v_{x,T},v_{y,T},v_{z,T})$, the kinetic energy $E$ after the collision is

$E=\frac{m}{2}v^{2}=E_{T}+E_{\textnormal{mix}}+E_{\textnormal{R}},$ (66)

namely the sum of the thermal energy
${E_{T}=m/2(v_{x,T}^{2}+v_{y,T}^{2}+v_{z,T}^{2})}$, the athermal recoil energy
$E_{\textnormal{R}}$, and a mixing term

$E_{\textnormal{mix}}=m(v_{x,T}v_{x,\textnormal{R}}+v_{y,T}v_{y,\textnormal{R}}+v_{z,T}v_{z,\textnormal{R}}).$
(67)

At $T>0$, the atom may receive more or less energy than it would at absolute
zero, enabling collisions below the threshold to provide sufficient energy to
cause a transition. The distribution of energy $E$ depends on the distribution
of the individual terms, one of which is a constant $E_{\textnormal{R}}$.

$E_{T}$ follows the Maxwell-Boltzmann distribution for a particle of mass $m$
at temperature $T$. $E_{\textnormal{mix}}$ is the sum of three independent
terms, where each is a product of thermal velocity that obeys a 1D Maxwell-
Boltzmann distribution, i.e. a Gaussian distribution, and a term that is
uniformly distributed between two extrema $\pm\sqrt{2E_{\textnormal{R}}/m}$.
It can be shown that $E_{\textnormal{mix}}$ follows a Gaussian distribution
whose mean is zero and whose standard deviation is
$\sigma=\sqrt{2k_{\textnormal{B}}TE_{\textnormal{R}}}$. For the simulations
that we considered, $E_{\textnormal{R}}$ is of the order of 1-10 eV, whereas
$k_{\textnormal{B}}T$ is of the order of 0.03-0.08 eV. Hence, the distribution
of $E$ is primarily that of $E_{\textnormal{R}}+E_{\textnormal{mix}}$, i.e.
Eq. (45)

$f_{\textnormal{R}}(E)=\frac{1}{\sqrt{4\pi
k_{\textnormal{B}}TE_{\textnormal{R}}}}\exp\left[-\frac{(E-E_{\textnormal{R}})^{2}}{4k_{\textnormal{B}}TE_{\textnormal{R}}}\right],$
(68)

plus a correction given by the thermal contribution $E_{T}$, which is not
trivial to evaluate as $E_{\textnormal{mix}}$ and $E_{T}$ are not independent.
However, this correction is of second-order if $k_{\textnormal{B}}T\ll
E_{\textnormal{R}}$. We note that the effect of temperature on the kinetic
energy after the collision does not depend on the thermal energy
$k_{\textnormal{B}}T$, but rather on
$\sqrt{(k_{\textnormal{B}}T)E_{\textnormal{R}}}$, and therefore can be much
greater than what one might anticipate from some simple energy addition
argument [67, 68].

##  Appendix C Interatomic potential for Fe

The energy landscape that an atom experiences when hopping into a vacancy,
treated as a function of reaction coordinate, may have local minima. If this
happens, the atom can get stuck in one of these minima. We found that this
commonly occurs in Fe, but does not happen in W. The local minimum corresponds
to a “split vacancy” defect, where the Fe atom finds itself in a stable
position between two neighbouring lattice sites, which both can be identified
as vacancies. For this to happen, the atom must arrive at the midpoint between
the two lattice sites with a low kinetic energy. The depth of the local
minimum depends on the potential. As this local minimum is an artifact of
parameterization of the potential, we selected a potential where this minimum
is as shallow as possible. To this end, we tested three different potentials,
namely by Mendelev et al. [80], Malerba et al. [81], and Gordon et al. [78].

Consider the simple case where one of the first nearest neighbours of the
vacancy receives a kick precisely along the $\langle 111\rangle$ direction
towards the vacancy, at 0 K. If using the Mendelev potential, a split vacancy
remains stable after kicks in the recoil energy range from 0.9-1.05 eV, while
the hop is not successful if the recoil energy is 0.85 eV and successful if it
is above 1.1 eV. Similarly, the Malerba potential predicts a split vacancy for
the recoil energies between 1.05-1.15 eV, an unsuccessful hop at 1.0 eV and a
full hop at 1.2 eV. On the other hand, with the Gordon potential an attempt is
unsuccessful at 0 K if the recoil energy is 0.81 eV and is fully successful if
it is 0.83 eV, with a much more narrowly-defined activation energy of 0.82 eV.

The influence of the intermediate local minimum on the results is very strong.
Fig. 15 shows histograms of the displacement experienced by the 1NN between
the beginning and the end of 20,000 recoils in random directions with various
energies and at different temperatures. The two expected outcomes for the
displacement are either approximately null (if the attempt is unsuccessful),
or approximately $\sqrt{3}a/2\approx 2.48$ Å (if the attempt is successful),
with possibly a spread due to thermal vibrations. In the top row of Fig. 15,
produced using the potential by Mendelev et al., one can clearly appreciate
the presence of a third cluster of realisations, where the atom is stuck in
the middle of the hop. This undesirable third outcome is substantially more
likely than the full hop if the recoil energy is close to the vacancy
migration energy. On the other hand, the same simulations repeated with the
potential by Gordon et al. show that half-hops are very unlikely at 0 K and
never occurr at 300 K.

Fig. 16 shows the effect of the choice of potential on the fraction of
successful hops for the same potentials and for different temperatures. The
presence of half hops fundamentally alters the results for the Mendelev
potential, but has a negligible effect for the Gordon potential. For the
results, and for Fig. 10, the very few instances of half hops that were found
at 0 K were equally divided between successful and unsuccessful outcomes.

(a)  (b)  (c)  (d)  (e)  (f)  (g)  (h)

Figure 15: Histograms showing distributions of displacements of 1NN hops after
recoils in random directions simulated at 0 K and at 300 K. Displacement is
defined as the distance between the coordinate of the atom at the beginning
and the end of each realisation. The most likely outcome is that the atom
returns to the initial position. Successful hops form the distributions
centred at $\approx 2.48$ Å. The instances of formation of “split vacancy”
configurations are responsible for the distributions shown in the centre. The
split vacancy configuration forms much more often with the Mendelev potential
(a-d) than with the Gordon potential (e-h).

(a)  (b)  (c)  (d)  (e)  (f)

Figure 16: Fraction of successful hops in Fe plotted for the Mendelev (a-c)
and Gordon (d-f) potentials, similarly to Fig. 10. For the former potential,
the instances of half hop are comparable in the frequency of occurrence to
those of complete hops, and are dominant for the recoil energies close to the
top of the migration barrier. For the latter potential, the occurrence of half
hops is very unlikely and only detectable at 0 K.

##  Appendix D Probability function of successful hops: Eq. (VI)

In order to relate electron-induced recoils and vacancy diffusion, one has to
quantify the probability that a collision of an electron with a random atom in
a random direction leads to a vacancy hopping by one or more lattice spacing.
We assume that this function ${J(E_{\textnormal{R}})}$ is a function of the
recoil energy only. The electron flux $\phi_{\textnormal{el}}({\bf
n},E_{\textnormal{el}})$ is assumed to be isotropic and thus solely a function
of the electron energy because the $\gamma$-photons that are generated by
nonelastic neutron reactions are emitted in a random direction and have long
attenuation distance of the order of a cm before they generate high energy
electrons that in turn are excited in a random directions. We present a semi-
analytical model to predict $J(E_{\textnormal{R}})$, and verify the model by
MD simulations, which are also used to parameterise the analytical result.

Assuming the vacancy as a quasi-particle moving in a one-dimensional potential
energy landscape, we start from a one-dimensional Langevin equation for a
particle of mass $m$ and friction coefficient $\gamma$,

$m\frac{\partial^{2}x}{\partial t^{2}}=-\frac{\partial V(x)}{\partial
x}-\gamma\frac{\partial x}{\partial t}+\eta(x,t).$ (69)

The particle moves in a potential that we assume for simplicity to be

$V(x)=E_{\textnormal{a}}\sin^{2}\left(\frac{\pi x}{\lambda}\right),$ (70)

with the periodicity of $\lambda=\frac{\sqrt{3}}{2}a$ in a bcc crystal.
$\eta(x,t)$ is a stochastic force that satisfies the so-called white noise
conditions

$\left<\eta(x,t)\right>=0$ (71)
$\left<\eta(x,t)\eta(x^{\prime},t^{\prime})\right>=2k_{\textnormal{B}}T\gamma\delta(x-x^{\prime})\delta(t-t^{\prime}),$
(72)

i.e. it has zero mean and is uncorrelated in space and time.

The particle initially is at $x_{0}$ and has velocity $v_{0}$. At $t=0$, it
receives a recoil so that the velocity becomes

$v=v_{0}+v_{\textnormal{R}},$

where $v_{\textnormal{R}}=\sqrt{2E_{\textnormal{R}}/m}$ is defined at 0 K. We
assume that the influence of stochastic force and friction are both
negligible. The former gives rise to thermal fluctuations on the energy scale
about 10—100 times lower than $E_{\textnormal{R}}$. The latter is weak on the
time scale ${\sim 1}$ ns of the transition. For a vacancy to migrate, the
kinetic energy after the recoil must be above the barrier height

$\frac{1}{2}mv^{2}>E_{\textnormal{a}}-V(x_{0}).$

If the temperature is low, we assume that ${E_{\textnormal{a}}-V(x_{0})\approx
E_{\textnormal{a}}}$, leading to the condition

$v^{2}>\frac{2E_{\textnormal{a}}}{m}.$ (73)

To find the likelihood of this condition being met, we need to know the
probability distribution function (PDF) for $v^{2}$. In a 1D problem, the
thermal part is a Gaussian distribution $f_{v_{0}}(v)$ with zero mean and the
standard deviation of $\sigma=k_{\textnormal{B}}T/m$. Contribution from
collisions $v_{\textnormal{R}}$ is drawn from the uniform distribution

$f_{v_{\textnormal{R}}}(v)=\begin{cases}\displaystyle\frac{1}{2v_{\textnormal{R}}},&|v|<v_{\textnormal{R}}\\\
0,&\text{otherwise,}\end{cases}$

as this generates recoils with random orientations in 3D. The PDF for the sum
of the two is given by their convolution

$\displaystyle f_{v}(v)$
$\displaystyle=\int\limits_{-\infty}^{\infty}\text{d}v^{\prime}\,f_{v_{0}}(v-v^{\prime})f_{v_{\textnormal{R}}}(v^{\prime})$
$\displaystyle=\frac{1}{4v_{\textnormal{R}}}\left[\text{erf}\left(\frac{v+v_{\textnormal{R}}}{\sqrt{2}\sigma}\right)-\text{erf}\left(\frac{v-v_{\textnormal{R}}}{\sqrt{2}\sigma}\right)\right].$
(74)

The PDF of $v^{2}$ follows from Eq. (D) through a change of variable
$g(y)=y^{2}$ with $y=v^{2}$. From

$f_{y}(y)=f_{v}\left(g^{-1}(y)\right)\bigg{|}\frac{\text{d}\left(g^{-1}(y)\right)}{\text{d}y}\bigg{|},$
(75)

we find

$f_{v^{2}}(v^{2})=\frac{1}{4v_{\textnormal{R}}\sqrt{v^{2}}}\left[\text{erf}\left(\frac{\sqrt{v^{2}}+v_{\textnormal{R}}}{\sqrt{2}\sigma}\right)-\text{erf}\left(\frac{\sqrt{v^{2}}-v_{\textnormal{R}}}{\sqrt{2}\sigma}\right)\right].$
(76)

The probability of a successful hop $p(v^{2}>\frac{2E_{\textnormal{a}}}{m})$
follows as

$\displaystyle\qquad
p\left(v^{2}>\frac{2E_{\textnormal{a}}}{m}\right)=\frac{1}{2}\int\limits_{2E_{\textnormal{a}}/m}^{\infty}\text{d}s\,f_{v^{2}}(s)$
$\displaystyle=\frac{1}{2}+\frac{1}{4}\left(1-\sqrt{\frac{E_{\textnormal{a}}}{E_{\textnormal{R}}}}\right)\text{erf}\left[\frac{\sqrt{E_{\textnormal{R}}}-\sqrt{E_{\textnormal{a}}}}{\sqrt{k_{\textnormal{B}}T}}\right]$
$\displaystyle-\frac{1}{4}\left(1+\sqrt{\frac{E_{\textnormal{a}}}{E_{\textnormal{R}}}}\right)\text{erf}\left[\frac{\sqrt{E_{\textnormal{R}}}+\sqrt{E_{\textnormal{a}}}}{\sqrt{k_{\textnormal{B}}T}}\right]$
$\displaystyle+\sqrt{\frac{k_{\textnormal{B}}T}{4\pi
E_{\textnormal{R}}}}\text{sinh}\left[\frac{2\sqrt{E_{\textnormal{R}}E_{\textnormal{a}}}}{k_{\textnormal{B}}T}\right]\exp\left[-\frac{E_{\textnormal{a}}-E_{\textnormal{R}}}{k_{\textnormal{B}}T}\right],$
(77)

where the factor of $1/2$ in front of the integral indicates that the kick can
be given either towards or away from the vacancy. We assume that the 1D
approximation adequately describes a reaction along a one-dimensional reaction
pathway, and that the 3D atomic dynamics alters the result only though an
extra scaling factor that we call $\alpha$. This leads to the functional form
$J(E_{\textnormal{R}},T)$ for the transition probability given by Eq. (VI).

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