# Volume of a dislocation network

Max Boleininger<EMAIL_ADDRESS>UK Atomic Energy Authority, Culham
Centre for Fusion Energy, Oxfordshire OX14 3DB, United Kingdom  Sergei L.
Dudarev<EMAIL_ADDRESS>UK Atomic Energy Authority, Culham Centre for
Fusion Energy, Oxfordshire OX14 3DB, United Kingdom  Daniel R. Mason
<EMAIL_ADDRESS>UK Atomic Energy Authority, Culham Centre for Fusion
Energy, Oxfordshire OX14 3DB, United Kingdom  Enrique Martínez
<EMAIL_ADDRESS>Department of Materials Science and Engineering, Clemson
University, Sirrine Hall, Clemson, SC 29634, USA

###### Abstract

We derive a simple analytical line integral expression for the relaxation
volume tensor of an arbitrary interconnected dislocation network. This
quantity determines the magnitude of dislocation contribution to the
dimensional changes and volumetric swelling of a material, and highlights the
fundamental dual role of dislocations as sources of internal strain as well as
carriers of plastic deformation. To illustrate applications of the method, we
compute the relaxation volume of a stacking fault tetrahedron, a defect
commonly occurring in fcc metals; the volume of an unusual tetrahedral
configuration formed by the $(a/2)\langle 111\rangle$ and $a\langle
001\rangle$ dislocations in a bcc metal; and estimate the relative
contribution of extended dislocations to the volume relaxation of heavily
irradiated tungsten.

###### pacs:

##  I Introduction

A material exposed to neutron or ion irradiation changes its volume [1, 2, 3,
4] and, in some cases, its anisotropic spatial dimensions [5]. If the exposure
of the material to a flux of energetic particles is low and the defects
produced by irradiation can be treated as a dilute gas of localised centres of
lattice distortion [6], the degree of volumetric expansion or contraction [7,
8, 1, 9] can be estimated from the relaxation volumes of individual defects
[10, 11, 12, 13]. These volumes can now be accurately computed using ab initio
methods [14, 15, 16, 17, 18].

However, a conceptual difficulty arises if the exposure to neutron or ion
irradiation exceeds approximately 0.1 dpa [19, 20]. In this high exposure
limit, defects form complex dense microstructures [21], with vacancies at
elevated temperatures coalescing into mesoscopic voids [1, 2, 3, 4]. The self-
interstitial defects cluster into dislocation loops, which then form rafts
[22, 23, 24], agglomerate into larger loops [22, 25], and eventually evolve
into an interconnected network of dislocation lines [19, 20].

The relaxation volume of a macroscopic void can be evaluated from its surface
stress or surface free energy [12]. In the absence of internal pressure from
helium or hydrogen gas filling the void, the relaxation volume of a void is
negative. As a result, vacancies and voids distributed in a material produce
negative lattice strain, readily observed using X-ray diffraction [8, 9, 12,
20]. At the same time, it is well known that materials swell, i.e. increase
their macroscopic volume, as a result of exposure to irradiation [1, 2, 3]. In
the limit of low radiation exposure, this can be readily explained by the fact
the relaxation volumes of self-interstitial defects, produced simultaneously
with vacancies during irradiation, are positive and significantly larger than
the (negative) relaxation volumes of vacancies [17, 18]. But in the limit of
relatively high exposure to irradiation this argument no longer applies, since
the majority of self-interstitial atom defects now agglomerate into large
dislocation loops and an extended network of interconnected dislocations [19,
20], effectively making them a part of the regular crystal lattice, and hence
barely detectable by electron microscopy or X-ray diffuse scattering. In fcc
metals, vacancies form stacking fault tetrahedra [26, 27, 28, 29], which are
dislocation configurations.

Extensive dislocation networks also form during severe plastic deformation of
materials, especially during rapid deformation [30, 31, 32], where the atomic
processes resemble those occurring in high-energy collision cascades initiated
by the incident high-energy ions or neutrons [33, 34].

To relate the complexity of dislocation structures and dislocation networks to
the macroscopic dimensional changes that they produce, and enable including
dislocations in the finite-element model formalism for the numerical treatment
of dimensional changes and radiation swelling of nuclear fusion reactor
components [10, 13], we propose a method for computing relaxation volume
tensors and relaxation volumes of arbitrary dislocation configurations, and
show how to evaluate the degree of volumetric expansion in a material
containing an interconnected network of dislocation lines.

##  II Relaxation volume of an isolated defect or a dislocation loop

We start by revisiting the concept of relaxation volume in linear elasticity
theory, casting it into a framework coherent with the established eigenstrain
[35] and dipole tensor [6] formalisms, and treating it from the perspective of
recent multi-scale applications [10, 13].

Consider a crystal occupying a finite volume in three-dimensional space,
subject to the traction-free surface boundary conditions. The relaxation
volume is defined as the change of the volume enclosed by the external surface
of the crystal, caused by the strain inside it [6, 36]. Hence, the relaxation
volume tensor $\mathbf{\Omega}$ is given by the surface integral of the
displacement field $u_{i}(\mathbf{x})$ over the volume boundary $S$ with
normal vector $n_{j}(\mathbf{x})$, namely

$\Omega_{ij}=\frac{1}{2}\int_{S}\left[u_{i}({\mathbf{x}})n_{j}({\mathbf{x}})+u_{j}({\mathbf{x}})n_{i}({\mathbf{x}})\right]\text{d}S.$
(1)

This quantity is a tensorial measure of the volume change to first order in
displacements [13], with the relaxation volume $\Omega$ given by the trace

$\Omega={\rm Tr}(\mathbf{\Omega})=\Omega_{ii},$ (2)

where summation over repeated indices is implied,
$\Omega_{ii}=\Omega_{11}+\Omega_{22}+\Omega_{33}$. Note that only the diagonal
elements of $\Omega_{ij}$ contribute to the relaxation volume, as shear
distortions conserve the volume of the material.

Applying the divergence theorem to (1), we arrive at [6, 36]

$\Omega_{ij}=\int_{V}\varepsilon_{ij}({\bf x})\,\text{d}^{3}x,$ (3)

where $\varepsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})$ is the strain tensor,
with subscripts after a comma denoting differentiation ($f_{,l}=\partial
f/\partial x_{l}$). The notion of the relaxation volume tensor enables
quantifying anisotropic dimensional changes, as its eigenvalues represent
variations of volume in the principal directions of the tensor.

We are interested in the case where the elastic strain is originating from the
defects inside the crystal. In Mura’s formalism [35], the defects are
described by a spatially-varying field of eigenstrain
$\varepsilon_{ij}^{*}({\bf x})$, which acts as a source of elastic
deformation. The strain tensor is thereby given by the sum of two terms

$\varepsilon_{ij}(\mathbf{x})=\varepsilon_{ij}^{\mathrm{el.}}(\mathbf{x})+\varepsilon_{ij}^{*}(\mathbf{x}),$
(4)

where the elastic strain $\varepsilon_{ij}^{\mathrm{el.}}({\bf x})$ also
includes the effect of boundary conditions. In the particular case where the
surfaces are free of tractions, the elastic strain tensor averages to zero
over the volume of the material [37, 38]. Consequently, the relaxation volume
tensor is given by the volume integral of the eigenstrain [13]:

$\Omega_{ij}=\int_{V}\varepsilon_{ij}^{*}(\mathbf{x})\,\text{d}^{3}x$ (5)

In applications, it is necessary to express the eigenstrain in terms of the
specific content of defects in the crystal. For a point $\mathbf{x}$ inside
the crystal, we can evaluate the eigenstrain from the body force $f^{*}_{i}$
by using the elastostatic equilibrium condition $\sigma_{ij,j}=-f_{i}$, viz.

$C_{ijkl}\varepsilon^{*}_{kl,j}(\mathbf{x})=-f^{*}_{i}(\mathbf{x}),$ (6)

where $C_{ijkl}$ is the stiffness tensor. The displacement field originating
from the body force is given by the volume integral

$\displaystyle u_{i}(\mathbf{x})$ $\displaystyle=\phantom{-}\int
G_{ij}(\mathbf{x}-\mathbf{x}^{\prime})f^{*}_{j}(\mathbf{x}^{\prime})\,\text{d}^{3}x^{\prime}$
(7) $\displaystyle=-\int
G_{ij,k}(\mathbf{x}-\mathbf{x}^{\prime})C_{jklm}\varepsilon^{*}_{lm}(\mathbf{x}^{\prime})\,\text{d}^{3}x^{\prime},$

where $G_{ij}(\mathbf{x}-\mathbf{x}^{\prime})$ is the elastic Green’s function
of the infinite body. Note that the above formula is not a solution to the
general boundary-value problem, as boundary conditions are not accounted for,
see Refs. [39, 40] for detail.

If the eigenstrain is associated with a defect localised in a region near
point $\mathbf{R}$ in the crystal, the asymptotic form of the displacement
field is found by expanding the elastic Green function to lowest order in
$\mathbf{x}^{\prime}$ at $\mathbf{R}$,

$u_{i}({\mathbf{x}})=-C_{jklm}\Omega_{lm}G_{ij,k}(\mathbf{x}-\mathbf{R}),$ (8)

where $\Omega_{lm}$ is the relaxation volume tensor given by Eq. (5). This
gives rise to the expression known from the dipole tensor formalism [6, 11],
namely

$u_{i}(\mathbf{x})=-P_{jk}G_{ij,k}(\mathbf{x}-\mathbf{R}),$ (9)

where the relaxation volume and elastic dipole tensor $P_{ij}$ of the defect
are related through

$\displaystyle\Omega_{ij}$ $\displaystyle=$ $\displaystyle S_{ijkl}P_{kl}$
(10) $\displaystyle P_{ij}$ $\displaystyle=$ $\displaystyle
C_{ijkl}\Omega_{kl},$ (11)

and $S_{ijkl}$ is the compliance tensor 111The compliance tensor is defined by
its relation to the stiffness tensor, namely
$C_{ijkl}S_{klmn}=\frac{1}{2}(\delta_{im}\delta_{jn}+\delta_{in}\delta_{jm}$)..
Owing to the symmetry properties of the stiffness and compliance tensors [42],
both the relaxation volume and elastic dipole tensors are symmetric. The
notion of elastic dipole tensor enables evaluating the energy of interaction
between the defect and an external slowly spatially varying strain field
$\varepsilon_{ij}^{\prime}({\mathbf{x}})$ to first order in the size of the
defect as $E({\mathbf{x}})=-P_{ij}\varepsilon_{ij}^{\prime}({\mathbf{x}})$,
see [43, 44]. For a detailed analysis of how elastic dipole tensors, formation
222Formation volumes of defects, explored for example in Ref. [47], differ
from relaxation volumes in that they do not solely refer to effects of lattice
distortions. A calculation of the formation volume of a defect is based on a
different definition, resulting in that the numerical values of formation
volumes of a vacancy and a self-interstitial atom defect are similar. and
relaxation volumes of small isolated defects can be evaluated from atomistic
simulations, we refer an interested reader to Refs. [15, 46, 11, 12, 17, 47,
18, 16].

To relate the relaxation volume tensor of a localised defect to Mura’s
eigenstrain, we define the density of relaxation volumes of defects [10]
similarly to the notion of charge density in electrodynamics, see e.g. Eq.
(28.1) of Ref. [48],

$\omega_{kl}({\bf x})=\sum_{a}\Omega^{(a)}_{kl}\delta({\bf
x}-{\mathbf{R}}_{a}),$ (12)

where summation is performed over the positions of individual defects
${\mathbf{R}}_{a}$, and $\delta(\mathbf{x})$ is the Dirac delta function
defined as $\delta({\bf x})=0$ for ${\bf x}\neq 0$ and $\int\delta({\bf
x})d^{3}x=1$. The relaxation volume tensor (5) can now be expressed as an
integral of $\omega_{kl}({\bf x})$ over the volume of the material
$\Omega_{kl}=\int_{V}\omega_{kl}({\bf x})\,\text{d}^{3}x$. Using (8) and (12),
the field of displacements generated by a spatial distribution of defects can
be evaluated as an integral of the density of relaxation volumes [35, 13]

$u_{i}(\mathbf{x})=-\int
G_{ij,k}(\mathbf{x}-\mathbf{x}^{\prime})C_{jklm}\omega_{lm}(\mathbf{x}^{\prime})\,\text{d}^{3}x^{\prime}.$
(13)

By comparing the above with Eq. (7), we arrive at the defect eigenstrain
theorem [13], which is the realisation that the density of relaxation volume
tensors is equivalent to the eigenstrain,

$\varepsilon^{*}_{ij}(\mathbf{x})\equiv\omega_{ij}(\mathbf{x}).$ (14)

Provided that the spatial distribution of crystal defects and their relaxation
volume tensors are known, the resulting macroscopic stresses can be evaluated
using conventional linear elasticity theory [10, 13]. This theorem highlights
the intrinsic multi-scale character of elastic forces originating from crystal
defects.

While the relaxation volume and dipole tensors of point defects and small
defect clusters can be readily obtained from atomistic or density functional
theory simulations [17, 47, 18], it is not immediately clear how to identify
the relaxation volume of dislocation-type defects, in particular when the
dislocations form a complex interconnected network. In what follows, we treat
the problem in the linear elasticity approximation and begin by exploring the
relaxation volume produced by an isolated dislocation loop, a prototypical
building block of a general dislocation network.

A dislocation curve is defined as the boundary of a cut surface inside the
crystal bulk, where the sides of the cut are displaced by the Burgers vector
$\mathbf{b}$ with respect to each another. As cut surfaces are generally open
surfaces, dislocation lines cannot have loose ends inside the crystal. Not
accounting for closures along free surfaces or grain boundaries, dislocation
lines must terminate either at dislocation junctions or by looping back onto
themselves. In the following, we refer to a dislocation network containing no
loose line segments as closed. Following the above definition, the density of
relaxation volumes or, equivalently, the eigenstrain of a dislocation loop, is
given by [49]

$\omega_{ij}(\mathbf{x})=\frac{1}{2}\int_{S}\left(b_{i}\text{d}S_{j}+b_{j}\text{d}S_{i}\right)\delta\left[\mathbf{x}-\mathbf{r}_{S}(v,w)\right],$
(15)

where

$\text{d}S_{n}=\left(\frac{\partial{\bf r}_{S}}{\partial
v}\times\frac{\partial{\bf r}_{S}}{\partial w}\right)_{n}\text{d}v\text{d}w$

defines the bounding surface of a dislocation loop in a parameterised form
through the vector function ${\bf r}_{S}(v,w)$ [50]. In the linear elasticity
approximation, the choice of a particular shape of the bounding surface ${\bf
r}_{S}(v,w)$ is immaterial, since strain and stress are unique and continuous
functions defined solely by the position of the dislocation line at the
perimeter of the bounding surface of the loop [51]. However, if one treats the
underlying discrete atomic crystal structure of the material, the variations
of the shape of the bounding surface can be detected and analysed, for example
by identifying the extrema of strain associated with the individual crowdion
defects constituting a dislocation loop [52, 53]. The fact that a perfect
dislocation loop is an assembly of crowdion point defects explains why, if we
neglect the core effects, the volume of a loop equals the number of defects
forming it, multiplied by the volume of an atom in the crystal lattice. We
refer an interested reader to Ref. [12] for a detailed analysis of scaling
relations for the relaxation volumes of larger defects, derived from atomistic
simulations.

Evaluating the volume integral (5) for the eigenstrain (15), we arrive at the
relaxation volume tensor of an isolated dislocation loop, see also equations
(27.11) and (27.12) by Landau and Lifshitz [51],

$\Omega_{ij}=\frac{1}{2}\left(b_{i}A_{j}+b_{j}A_{i}\right),$ (16)

where $A_{i}$ is the vector area of the loop. The vector area is defined
either as an integral over the bounding surface of the cut or, through the
Stokes theorem, as a line integral over the boundary $\Gamma$ of the cut [51],

$A_{i}=\int_{S}n_{i}({\bf
x})\,\text{d}S=\frac{1}{2}\oint_{\Gamma}\epsilon_{ikl}r_{k}\,\text{d}r_{l},$
(17)

where $n_{i}({\bf x})$ is the surface unit normal vector, $\epsilon_{ikl}$ is
the Levi-Civita tensor, and ${\bf r}=\\{r_{k}\\}$ is a coordinate of a point
on a dislocation line forming the perimeter of the dislocation loop. In vector
notation, equation (17) is [54]

${\bf A}=\frac{1}{2}\oint_{\Gamma}{\bf r}\times\text{d}{\bf r}.$ (18)

This definition is invariant with respect to the choice of the origin of the
Cartesian system of coordinates, since a translation of the origin by a
constant vector ${\bf r}_{0}$ adds to (18) a term

$\frac{1}{2}\oint_{\Gamma}{\bf r}_{0}\times\text{d}{\bf r}=\frac{1}{2}{\bf
r}_{0}\times\left(\oint_{\Gamma}\text{d}{\bf r}\right),$

which vanishes for a closed loop since ${\oint_{\Gamma}\text{d}\bf r}=0$.

The relaxation volume of a dislocation loop is given by the trace of (16) and
hence equals the scalar product of the Burgers vector of the loop and its
vector area, cf. Eq. (43) of Ref. [55] and Eq. (12) of Ref. [11]

$\Omega={\bf{b}}\cdot{\bf{A}}=\frac{1}{2}\oint_{\Gamma}{\bf b}\cdot({\bf
r}\times\text{d}{\bf r})=\frac{1}{2}\oint_{\Gamma}({\bf b}\times{\bf
r})\cdot\text{d}{\bf r}.$ (19)

The relaxation volume of an interstitial loop is positive $({\bf b}\cdot{\bf
A})>0$, whereas the relaxation volume of a vacancy loop is negative $({\bf
b}\cdot{\bf A})<0$. The relaxation volume of a loop formed by a pure shear-
type deformation is equal to zero $({\bf b}\cdot{\bf A})=0$.

The elastic dipole tensor of a dislocation loop can be readily found from the
relaxation volume tensor using equation (11) as

$P_{ij}=C_{ijkl}A_{k}b_{l}.$ (20)

In the above equation, there is no need to symmetrise the expression in the
right-hand side with respect to $i$ and $j$ since the symmetry properties of
$P_{ij}$ automatically follow from the symmetry of $C_{ijkl}$. Similarly,
there is no need to use the symmetrised form (16) in the right-hand side of
the equation.

As a slight digression, we note that the Burgers formula for the displacement
field of a curved dislocation line can be obtained by substituting (15) into
(13)

$u_{i}({\bf x})=-C_{klmn}b_{m}\int_{S}G_{ik,l}({\bf x}-{\bf
x}^{\prime})\,\text{d}S^{\prime}_{n},$ (21)

which is equivalent to Eq. (9.3) by Teodosiu [56], Eq. (4-6) by Anderson,
Hirth and Lothe [57], and Eq. (27.10) by Landau and Lifshitz [51]. In (21),
integration is performed over the bounding surface of a dislocation loop. The
differentiation of the elastic Green’s function in (21) is performed with
respect to coordinate ${\bf x}$, namely $G_{ik,l}({\bf x}-{\bf
x}^{\prime})={\partial}G_{ik}({\bf x}-{\bf x}^{\prime})/\partial x_{l}$, and
the sign convention in the definition of the Burgers vector in equation (21)
above is consistent with Refs. [51, 56, 55]. If one adopts the alternative
definition of the Burgers vector used in Ref. [57], the sign before equation
(21) must be amended from minus to plus. The observed physical quantities, for
example relaxation volumes, are independent of the Burgers vector sign
convention, and the volume of a self-interstitial dislocation loop is always
positive whereas the volume of a vacancy loop is negative.

A remarkable aspect of equations (15) and (19) is that, on the one hand, they
show that the fundamental origin of the positive (or negative) volume of a
dislocation loop is associated with the extra (or missing) material at the
bounding surface $S$ of the loop, illustrated by equation (15) and readily
identified in atomistic models [58, 59]. On the other hand, equation (19)
shows that the problem of evaluation of the volume of a dislocation loop can
be conveniently reduced to computing a contour integral along the dislocation
line at the perimeter of a dislocation loop. This result, replacing the
procedure of counting extra or missing atoms over the surface of a dislocation
loop by a line integration over its perimeter, enables generalising the
formalism developed above for an isolated loop to an arbitrary configuration
of entangled dislocations.

Equations (16), (19) and (20) show how to evaluate the relaxation volume and
elastic dipole tensors of an individual dislocation loop. The case of a
dislocation network, featuring complex configurations of dislocation lines and
dislocation junctions, requires a special consideration. The formalism for
evaluating the relaxation volume tensor of a dislocation network is given in
the next section of the paper.

##  III Relaxation volume of a dislocation network

Figure 1:  (a) Example of a closed dislocation network consisting of line
segments $G_{i}$ with Burgers vectors $\mathbf{b}(G_{i})$ (arrows). Triangles
indicate line sense. (b) The same network represented as a superposition of
closed loops $\Gamma_{i}$. Also given is the relation between area vectors
$A(\Gamma_{i})$ and $A(G_{i})$. (c) Closed loops are split into their
constitutive line segments, which are then grouped according to whether they
appear uniquely or not, into sets $S_{U}$ and $S_{N}$, respectively. The
decomposition is used to prove that all three representations yield the same
dipole tensor.

A dislocation network is defined by a set of dislocation junctions connected
by dislocation line segments $G_{n}$ with Burgers vectors ${\bf
b}(G_{n})\equiv b_{i}(G_{n})$. A closed dislocation network can be
equivalently represented as a superposition of closed dislocation contours
$\Gamma_{\alpha}$ with Burgers vectors $b_{i}(\Gamma_{\alpha})$ and area
vectors $A_{i}{(\Gamma_{\alpha})}$ by splitting the dislocation segments
according to the conservation rule of Burgers vector at junctions, analogous
to the Kirchhoff law of electric current flow [57]. The dipole tensor of the
network in the linear elasticity approximation is therefore given by the sum
of dipole tensors of the closed dislocation contours, namely

$P_{ij}=C_{ijkl}\sum_{\Gamma_{\alpha}}b_{k}{(\Gamma_{\alpha})}A_{l}{(\Gamma_{\alpha})}.$
(22)

While the above expression shows that the dipole tensor of a closed
dislocation network is unambiguously defined, it is in practice cumbersome to
partition the network into individual contours. We shall instead reformulate
the problem in terms of dislocation line segments by reversing the argument.

First, similarly to (17) and (18), we split the area vectors of closed loops
$\Gamma_{\alpha}$ into area vectors of their constituent line segments
$G_{n}$,

$A_{i}{(\Gamma_{\alpha})}=\sum_{G_{n}\in\Gamma_{\alpha}}A_{i}^{\alpha}{(G_{n})},$
(23)

where $A_{i}^{\alpha}{(G_{n})}=\pm A_{i}{(G_{n})}$ as the line sense of
segment $G_{n}$ may not match that of the loop $\Gamma_{\alpha}$, leading to
the dipole tensor

$P_{ij}=C_{ijkl}\sum_{\Gamma_{\alpha}}\sum_{G_{n}\in\Gamma_{\alpha}}b_{k}{(\Gamma_{\alpha})}A_{l}^{\alpha}{(G_{n})}.$
(24)

This decomposition is exact as integration is a linear operation, with the
caveat that the $A_{i}{(G_{n})}$ line integrals are not closed anymore.

Some new line segments were introduced in order to form the closed loop
representation, which are now being integrated over multiple times. Defining
$S_{U}$ as the set of line segments being integrated over once and $S_{N}$ as
the set of line segments being integrated over several times, Eq. (24) is
reordered as

$\displaystyle P_{ij}=$ $\displaystyle C_{ijkl}\sum_{G_{n}\in
S_{U}}b_{k}{(G_{n})}A_{l}{(G_{n})}$ (25) $\displaystyle+$ $\displaystyle
C_{ijkl}\sum_{G_{n}\in S_{N}}\sum_{\Gamma_{\alpha}\in
M(G_{n})}b_{k}{(\Gamma_{\alpha})}A_{l}^{\alpha}{(G_{n})},$

where summation over the set

$M(G_{n})=\\{\Gamma_{\alpha}\,|\,G_{n}\in\Gamma_{\alpha}\\}$ (26)

counts all appearances of segment $G_{n}\in S_{N}$. We refer to Fig. 1 for a
visual example of the decomposition.

Next, recalling that due to the conservation of the Burgers vector at
junctions

$\sum_{\Gamma_{\alpha}\in
M(G_{n})}b_{k}{(\Gamma_{\alpha})}A_{l}^{\alpha}({G_{n}})=b_{k}{(G_{n})}A_{l}({G_{n}}),$
(27)

as well as that the union of unique and non-uniquely appearing segments
contains all the line segments of the network exactly once, we arrive at the
central result:

$P_{ij}=C_{ijkl}\sum_{G_{n}}b_{k}{(G_{n})}A_{l}{(G_{n})}.$ (28)

Comparing Eq. (28) with Eq. (10), we find the relaxation volume tensor

$\Omega_{ij}=\frac{1}{2}\sum_{G_{n}}\big{[}b_{i}{(G_{n})}A_{j}{(G_{n})}+b_{j}{(G_{n})}A_{i}{(G_{n})}\big{]},$
(29)

the trace of which yields the relaxation volume of the network

$\Omega=\sum_{G_{n}}b_{i}{(G_{n})}A_{i}{(G_{n})}.$ (30)

Since the evaluation of areas $A_{i}{(G_{n})}$ can be reduced to the line
integrals along the dislocations, the dipole tensor, the relaxation volume
tensor, and the relaxation volume of a closed dislocation network can all be
obtained by the integration over the individual dislocation line segments.

In computational dislocation dynamics codes and post-processing algorithms for
detecting dislocations, dislocation lines are commonly represented by a set of
piecewise directionally-ordered linear segments [60], satisfying Kirchhoff’s
Burgers vector conservation law at dislocation junctions. Each linear segment
$C_{n}$ is defined by its Burgers vector $b_{i}^{(n)}$, the starting point
$p_{i}^{(n)}$ and the end point $q_{i}^{(n)}$. Noting that an element of the
vector area associated with an individual straight segment equals

${\bf A}^{(n)}=\frac{1}{4}\left({\bf q}^{(n)}+{\bf
p}^{(n)}\right)\times\left({\bf q}^{(n)}-{\bf p}^{(n)}\right)=\frac{1}{2}{\bf
p}^{(n)}\times{\bf q}^{(n)},$

we can simplify the expression for the dipole tensor of a dislocation network
(28) as

$P_{ij}=\frac{1}{2}C_{ijkl}\epsilon_{luv}\sum_{n}b_{k}^{(n)}p_{u}^{(n)}q_{v}^{(n)}.$
(31)

The relaxation volume tensor of a dislocation network described by straight
dislocation line segments is now

$\Omega_{ij}=\frac{1}{4}\sum_{n}\left[b_{i}^{(n)}\epsilon_{juv}+b_{j}^{(n)}\epsilon_{iuv}\right]p_{u}^{(n)}q_{v}^{(n)},$
(32)

with the corresponding total relaxation volume given by

$\displaystyle\Omega$
$\displaystyle=\frac{1}{2}\epsilon_{iuv}\sum_{n}b_{i}^{(n)}p_{u}^{(n)}q_{v}^{(n)}$
(33) $\displaystyle=\frac{1}{2}\sum_{n}{\bf b}^{(n)}\cdot({\bf
p}^{(n)}\times{\bf q}^{(n)}),$

where ${\bf p}^{(n)}$ and ${\bf q}^{(n)}$ are the coordinates of the start and
end points of straight dislocation segments. For a set of arbitrary curved
dislocation segments used in dislocation dynamics simulations [61, 62, 60, 63,
49], the relaxation volume can be written as a sum of line integrals along the
dislocation segments, linking dislocation junctions

$\displaystyle\Omega$
$\displaystyle=\frac{1}{2}\epsilon_{iuv}\sum_{n}b_{i}^{(n)}\int_{(n)}r_{u}\text{d}r_{v}$
(34) $\displaystyle=\frac{1}{2}\sum_{n}\int_{(n)}({\bf b}^{(n)}\times{\bf
r})\cdot{\text{d}\bf r}$ $\displaystyle=\frac{1}{2}\sum_{n}{\bf
b}^{(n)}\cdot\int_{(n)}({\bf r}\times\text{d}\bf r),$

where the direction of line integration along each curved segment follows the
direction of the corresponding dislocation, consistent with the law of
conservation of the Burgers vector at the dislocation junctions. Equation (34)
is one of the central results of our study, and for an individual dislocation
loop it reduces to (19).

To show that expression (34) is invariant with respect to an arbitrary
translation of the Cartesian system of coordinates, we note that such a
translation amounts to adding an extra term

$\displaystyle\frac{1}{2}\sum_{n}{\bf b}^{(n)}\cdot\left({\bf
r}_{0}\times\int_{(n)}\text{d}\bf r\right)$ (35) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s}{\bf b}^{(s)}\cdot\left({\bf
r}_{0}\times\left[{\bf r}^{(s)}_{2}-{\bf r}^{(s)}_{1}\right]\right),$

where ${\bf r}_{0}$ is the translation vector, and summation over $s$ is
performed over the full arbitrarily curved segments of the dislocation network
linking the dislocation junctions, with ${\bf r}^{(s)}_{2}$ and ${\bf
r}^{(s)}_{1}$ being the coordinates of the two junctions at the start and the
end of a full segment $s$. Since the choice of the start and end points of a
full segment reflects the sense of direction of integration along the segment,
each junction enters the sum in (35) with a plus or minus sign. This sign
rule, depending on whether a segment enters or leaves the junction, in
combination with the Burgers vector conservation law at the junctions, ensures
that the various terms cancel each other and the sum in (35) vanishes,
confirming the translational invariance of expression (34).

##  IV Periodic boundary conditions

Figure 2: Three example closures of an extended dislocation network consisting
of two dislocation dipoles in a box with periodic boundary conditions. The
network is closed by augmenting the dislocation lines with extra segments,
showed by dotted blue lines, that connect the dislocation line ends, with line
sense and Burgers vector chosen in accordance with the Burgers vector
conservation law at junctions. As the closure lines cancel with themselves
upon the repetition of the motif along the periodic cell vector $\mathbf{c}$,
all the examples of closure are equivalent.

In computational studies of complex dislocation networks, periodic boundary
conditions are commonly employed to represent systems of much larger size than
the size of the actual simulation cell. While any dislocation network
contained in a periodic cell must also be closed by definition, the treatment
of periodic boundary conditions requires an additional discussion.

First, networks crossing periodic boundaries must be correctly treated as
closed. Depending on the implementation of periodic boundary conditions, line
segments crossing a periodic cell boundary may be split and displaced by the
corresponding periodic cell vector, seemingly appearing as loose ends. In
order to avoid errors, the area vectors must be computed using an unbroken
(unwrapped) representation of the network. This is either achieved by
computing the distance vector between any two neighbouring points on a curve
modulo the periodic cell vectors [64], or by explicitly translating the
segments broken by a periodic boundary by the corresponding cell vector.

Second, the simulation box may contain networks that span the entire periodic
box along one or more directions, regardless of how they are translated. In an
unwrapped representation, such a network appears to contain terminated lines
that only become connected if the network is repeated along the periodic cell
directions. The simplest example of such a network is a pair of dislocation
lines, collinear to one of the periodic cell translation vectors, with
identical Burgers vectors and opposing line sense. In what follows, we refer
to such a network as extended. Extended networks have well-defined relaxation
volumes, provided that we connect the loose segments in such a way that the
network can be represented by a superposition of closed loops.

Specifically how should an extended network be closed? As it turns out, a
specific recipe for doing this is immaterial, provided that some simple rules
are followed. Consider a network extended along a periodic translation vector
$\mathbf{c}$, containing a set of $m$ curved segments terminating at points
$\mathbf{R}_{i}$ that are connected through translations by $\pm\mathbf{c}$.
We introduce a new set of dislocation lines, the closure, linking the loose
ends of line segments in a manner such that the Burgers vector conservation
law is fulfilled at every junction. Some examples of closure are shown in Fig.
2. The vector area of the closed network is invariant with respect to the
choice of closure since each closure segment cancels itself out with its own
periodic image. The most trivial choice of closure is to pick a single loose
end of a dislocation segment and let all other ends of loose segments connect
to it, keeping the line sense and Burgers vector unchanged, see Fig. 2
(right). The contribution of the closure to the relaxation volume is then
given by:

$\Omega^{\mathrm{(c)}}=\frac{1}{2}\sum_{i=2}^{m}s_{i}\mathbf{b}^{(i)}\cdot\left(\int_{(i,1)}\mathbf{r}\times\text{d}\mathbf{r}+\int_{(1,i)}(\mathbf{r}-\mathbf{c})\times\text{d}\mathbf{r}\right),$
(36)

where integrals over $(i,j)$ signify curve integral starting at
$\mathbf{R}_{i}$ and ending at $\mathbf{R}_{j}$, and $s_{i}$ is either $-1$ or
$1$ depending on the line sense of the loose segment terminating at
$\mathbf{R}_{i}$. After some manipulation, we arrive at a simple expression

$\Omega^{\mathrm{(c)}}=\frac{1}{2}\sum_{i=2}^{m}s_{i}\mathbf{b}^{(i)}\cdot\left[\mathbf{c}\times\left(\mathbf{R}_{i}-\mathbf{R}_{1}\right)\right].$
(37)

This closure correction needs to be included in the formula for the relaxation
volume given by equation (34) if a dislocation configuration happens to
contain loose ends because of periodicity, to ensure consistency with the
treatment of a fully closed dislocation network. The relaxation volume and
dipole tensors are then computed in exactly the same way as for a closed
network. In practice, we found the term given by (37) to be of comparable
magnitude to the relaxation volume of the network itself, and as such it
cannot be neglected.

##  V Applications

###  V.1 The relaxation volume of a stacking fault tetrahedron

Stacking fault tetrahedra (SFT), first observed in quenched gold by Silcox and
Hirsch [65], are thought to form through the condensation of individual
vacancies into a platelet that subsequently collapses into a loop of intrinsic
fault [66, 67]. Possible alternative reaction pathways, leading to the
formation of SFTs and likely dominating the dynamics of defects in the
mesoscopic limit, were extensively analyzed and reviewed in Ref. [68]. The
faulted loop is able to transform into an SFT by means of dissociation and
glide of the dislocation segments bounding the loop. As these processes do not
involve mass transfer, in the limit where dislocations are treated as elastic
discontinuities and core effects are neglected, the relaxation volume of the
SFT configuration is not expected to change during the transformation. The
individual components of the dipole and relaxation volume tensors may however
change.

Let the faulted Frank loop be of vacancy type with
$\mathbf{b}=\frac{1}{3}[\overline{111}]$. The loop lies in the plane
perpendicular to the Burgers vector, bounded by dislocations with
$\left\langle 110\right\rangle$ line directions and $\\{112\\}$ slip planes
forming an equilateral triangle with side length $l$, see Figure 3. We
introduce a reaction coordinate $\lambda$, parameterising the transformation
of the faulted loop at $\lambda=1$ to an SFT at $\lambda=0$, with
$1>\lambda>0$ describing the intermediate stages of the transformation
process. Intermediate configurations are approximated by a tetrahedron
truncated at a height $h^{\prime}=(1-\lambda)h$ along the direction of
$\mathbf{b}$, see Figure 3 illustrating the notations. Here, $h$ is the height
of a perfect SFT.

Figure 3: (a) Arrangement of the faulted Frank loop and stacking fault
tetrahedron corner points in the one-eighth of an fcc unit cell with the edge
length of $a/2$. Points labeled by Greek letters are at midpoints of
tetrahedral faces opposite to points labeled by the respective Roman letters.
(b-d) Transformation of the faulted Frank loop to a stacking fault tetrahedron
parameterised by the reaction coordinate $\lambda$. Burgers vectors are
described using the Thompson vector notation.

Using the parameterisation of the SFT transformation detailed in Appendix A,
we find the relaxation volume tensor from expression (32)

$\mathbf{\Omega}(\lambda)=-\frac{al^{2}}{12}\begin{bmatrix}1&\lambda^{2}&\lambda^{2}\\\
\lambda^{2}&1&\lambda^{2}\\\ \lambda^{2}&\lambda^{2}&1\end{bmatrix},$ (38)

with the relaxation volume $\Omega(\lambda)=\Omega_{ii}=-al^{2}/4$, where $a$
is the fcc lattice constant. The principal values of the relaxation volume
tensor are

$\mathbf{\Omega}_{\mathrm{p.v.}}=-\frac{al^{2}}{12}\left(1-\lambda^{2},1-\lambda^{2},1+2\lambda^{2}\right),$
(39)

indicating that an SFT ($\lambda=0$) generates isotropic volume contraction,
whereas a Frank dislocation loop ($\lambda=1$) gives rise to anisotropic
volume contraction. While the principal values of the relaxation volume tensor
vary as functions of $\lambda$, the total relaxation volume of the SFT is the
same as the relaxation volume of the Frank loop.

Using the stiffness tensor for a cubic crystal [42]333No summation over
repeated indices in the first term here.

$\displaystyle C_{ijkl}$
$\displaystyle=\left(c_{11}-c_{12}-2c_{44}\right)\delta_{ij}\delta_{jk}\delta_{kl}$
(40)
$\displaystyle\phantom{=}+c_{12}\delta_{ij}\delta_{kl}+c_{44}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right),$

we arrive at an expression for the elastic dipole tensor of an SFT using
either equation (11) or equation (31), namely

$\mathbf{P}(\lambda)=-\frac{al^{2}}{12}\begin{bmatrix}c_{11}+2c_{12}&2c_{44}\lambda^{2}&2c_{44}\lambda^{2}\\\
2c_{44}\lambda^{2}&c_{11}+2c_{12}&2c_{44}\lambda^{2}\\\
2c_{44}\lambda^{2}&2c_{44}\lambda^{2}&c_{11}+2c_{12}\end{bmatrix}.$ (41)

The number of vacancies $N_{\mathrm{v}}$ forming the SFT can be found by
dividing its relaxation volume by the volume of a missing atom in an fcc
lattice $-a^{3}/4$, resulting in $N_{\mathrm{v}}=l^{2}/a^{2}$. The relaxation
volume of an SFT, expressed in terms of the number of vacancies that it
contains, is

$\Omega=-\frac{1}{4}N_{\mathrm{v}}a^{3}.$ (42)

For comparison, the relaxation volume of a spherical void in an fcc metal,
containing the same number of vacancies, is [12]

$\Omega_{\rm
void}=-\left(\frac{3}{2}\right)^{5/3}\pi^{1/3}\left(\frac{1-\nu}{1+\nu}\right)\frac{sa^{2}}{\mu}N_{\mathrm{v}}^{2/3},$
(43)

where $s$ is the surface stress, $\mu$ is the shear modulus and $\nu$ is the
Poisson ratio. Equations (42) and (43) suggest that in the macroscopic limit
$N_{\mathrm{v}}\gg 1$, an SFT exhibits a much greater degree of elastic
relaxation than a void, in agreement with atomistic simulations [68]. This
does not imply, however, that a void is energetically more stable than an SFT
since, like for point defects [70, 71], it is often the core energy rather
than the elastic energy that dominates the total self-energy of a defect
configuration.

###  V.2 $\mathbf{\langle 111\rangle/\langle 100\rangle}$ bcc tetrahedron

Collision cascades in body-centered cubic (bcc) metals occasionally produce
unusually complex dislocation structures, which are topologically different
from the commonly occurring individual dislocation loops. An example of such a
topologically unusual dislocation configuration formed in a collision cascade
is given in Figure 10 of Ref. [72]. Similarly to an SFT, the structure
involves four dislocation junctions but, as opposed to an SFT, now
dislocations with two fundamentally different Burgers vector types are
involved in the formation of this structure. These two Burgers vectors,
schematically noted in Figure 4, are of the $(a/2)\langle 111\rangle$ and
$a\langle 100\rangle$ types, where $a$ is the bcc lattice constant.

Computing the scalar triple products for the six segments of the bcc
tetrahedron using the rule (33) produces values given in the last column of
Table 3, resulting in the total relaxation volume of the dislocation structure
in Figure 4

$\Omega=aL^{2}.$ (44)

The fact that this quantity is positive suggests that the dislocation
configuration depicted in Figure 4 has the net interstitial, rather than
vacancy, character.

The decomposition of the structure shown in Figure 4 into individual loops
highlights the subtlety associated with the evaluation of the volume of a
complex dislocation configuration. Indeed, applying vector calculus and
formula $({\bf b}\cdot{\bf A})$ for the volume of an individual loop (19), we
find that both $(a/2)\langle 111\rangle$ dislocation loops, shown in blue,
have the interstitial character and the same positive volumes of $aL^{2}$. The
$a\langle 100\rangle$ loop, shown in red, has vacancy character, its volume is
negative, and is equal to $-aL^{2}$. The sum of volumes of the three loops is
$aL^{2}$, in agreement with the result (44) obtained by a direct calculation
using equation (33), which notably does not require decomposing a dislocation
configuration into individual loops.

Figure 4: (a) Sketch of a tetrahedral defect structure formed by dislocations
with $(a/2)\langle 111\rangle$ (blue) and $a\langle 100\rangle$ (red) Burgers
vectors in a bcc metal. The configuration resembles a stacking fault
tetrahedron, but involves no stacking faults. The topology of this dislocation
configuration is not equivalent to that of a dislocation loop. Coordinates of
points A, B, C and D are given in Table 3. (b) Dislocation configuration shown
in (a) represented as an equivalent superposition of two $(a/2)\langle
111\rangle$ and one $a\langle 100\rangle$ loop.

###  V.3 An extended network of dislocations produced by irradiation

Metals exposed to irradiation by highly energetic particles, such as neutrons
in a nuclear reactor, develop microstructures often involving complex extended
dislocation network configurations. Whenever energetic particles scatter by
atoms in the crystal lattice, they may impart sufficient kinetic energy to
ballistically displace a large number of atoms from their lattice sites. The
displaced atoms eventually recrystallise, leaving behind defects of both
vacancy and interstitial type. With increasing exposure to radiation, the
density of radiation defects becomes large enough for defects to coalesce and
form complex interconnected dislocation networks. The dislocation network
transfers mass through the system under the influence of irradiation and
external stress, thereby contributing to irradiation-induced creep.

Figure 5: Dislocation networks extracted from simulated microstructure of
irradiated tungsten, showed in an unwrapped representation. In this example,
the dislocation microstructure is found to evolve from dispersed loops at low
dose, to extended dislocation networks at intermediate dose, to a large closed
interstitial loop at high dose. Networks with net interstitial and vacancy
content are colored red and blue, respectively. Structures (d) and (e) contain
extended networks, shown periodically repeated along $x$-direction (gray
lines), with segment end points marked by black dots. Dislocations of
$(a/2)\langle 111\rangle$ and $a\langle 100\rangle$ type are drawn with solid
and dotted lines, respectively.

We simulate the formation of an irradiated microstructure in single-crystal
bcc tungsten (W) using the molecular dynamics software package Lammps [73] and
the empirical interatomic potential developed in Ref. [74]. We start by
initialising a perfect bcc lattice in an orthogonal simulation cell with 220
unit cell repeats along the $[100]$, $[010]$, and $[001]$ cell vectors,
containing over 21.3 million atoms. Periodic boundary conditions are applied
in all three directions. We introduce 100 simultaneous cascades by choosing
100 random atoms, with a minimal distance of $15\text{\,}\mathrm{\text{Å}}$ to
each other, and assigning each atom a randomly oriented velocity corresponding
to the kinetic energy of $10\text{\,}\mathrm{keV}$. Following the method
commonly used in atomistic cascade simulation, we use an electronic stopping
model for atoms with kinetic energies above $10\text{\,}\mathrm{e}\mathrm{V}$
and an adaptive time step. For atoms with kinetic energy below the energy
corresponding to melting, we apply a damping term with the time-constant of
$15.6\text{\,}\mathrm{ps}$ to model electronic losses due to electron-phonon
coupling in the low velocity limit [75]. The system is propagated until the
temperature drops below $100\text{\,}\mathrm{K}$, which takes
$\sim$$10\text{\,}\mathrm{ps}$, at which point velocities are set to zero, and
atomic coordinates and box dimensions are relaxed using the Conjugate
Gradients method. The next round of cascades is initialised in the now damaged
microstructure, and this process is repeated until the desired radiation dose
is reached.

Using the damage model proposed in Ref. [76], we find that every new set of
cascades increments the radiation dose $\phi$ by the amount

$\Delta\phi=N_{\mathrm{c}}\frac{0.8T_{\mathrm{d}}}{2E_{\mathrm{d}}N}=$1.67\text{\times}{10}^{-4}\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$,$
(45)

where $N_{\mathrm{c}}$ is number of cascades per iteration,
$T_{\mathrm{d}}\approx$8\text{\,}\mathrm{keV}$$ is the cascade damage energy,
$E_{\mathrm{d}}=$90\text{\,}\mathrm{eV}$$ is the assumed threshold
displacement energy for tungsten, and $N$ is the total number of atoms in the
simulation cell. The dose is given in the dimensionless units of displacements
per atom (dpa), which is a standard measure of exposure of materials to
radiation used in the field of nuclear materials [76]. The dose of
$\phi=$1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$$ signifies that on average
every atom in the system has been a part of a Frenkel pair, consisting of a
vacancy and a self-interstitial defect, suggesting a significant accumulation
of radiation defects. We apply 6000 cascade iterations in order to reach the
dose of $1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$.

The resulting atomic configurations are analysed and their dislocation content
determined using the Dislocation Extraction Algorithm (DXA) [77] implemented
in the Ovito software [78]. The algorithm is able to identify the dislocation
line sense, its Burgers vector, the line coordinates, and the topology of the
dislocation network. Because the irradiated microstructure is vacancy-rich as
a large number of self-interstitial atom defects have coalesced into
dislocations loops and the dislocation network [19], some dislocation curves
identified by the DXA are disjoint and have loose ends. To ensure that all the
networks are closed, we restore the dislocation connectivity by iterating over
loose ends in the network and connecting them to other loose ends in the
neighbourhood.

Snapshots of the dislocation network formed at various doses are shown in
Figure 5. At low doses, the interstitial defects produced by cascades form
small spatially dispersed dislocation loops with $(a/2)\langle 111\rangle$ and
$a\langle 100\rangle$ Burgers vectors. As the dose increases, the interstitial
loops grow and coalesce, joining together to form more complex loops, which
eventually merge to form an extended network. At a higher dose
($\phi\sim$1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$$) the network breaks
apart, leaving behind a large interstitial loop of around
$60\text{\,}\mathrm{nm}$ diameter. We note that the entirety of the simulation
box is now filled with homogeneously distributed vacancy clusters, saturating
to the total vacancy content of $0.34\text{\,}\mathrm{\char 37\relax}$ over
the course of irradiation, see Figure 6c. Around $90\text{\,}\mathrm{\char
37\relax}$ of vacancy content is in the form of mono-vacancies, with the
remainder constituting sub-nanometer sized vacancy clusters. The vacancy
content was determined using the isosurface method presented in [79].

Figure 6: (a) The relaxation volume of the dislocation network and the total
change of volume of the simulation cell, expressed in the units of atomic
volume $\Omega_{0}$, shown as a function of dose. (b) Plot showing if the
dislocation network is extended or not as a function of dose. The
discontinuity in the computed relaxation volume occurs shortly after the
dislocation network becomes extended, suggesting that the change of the
relaxation volume is associated with the change in the topology of the
network. (c) Vacancy concentration, in the units of vacancies per lattice
site, shown as a function of dose.

In Figure 6a, we compare the change of volume of the simulation cell with the
total relaxation volume of the dislocation network computed using expressions
(34) and (36). The volume change is given in the units of bcc atomic volume
$\Omega_{0}=a^{3}/2$. In the dynamic steady state of the material forming in
the high dose limit [19, 20], the total box volume has increased by $\Delta
V/V_{0}=46\times 10^{3}/(2\times 220^{3})=$0.22\text{\,}\mathrm{\char
37\relax}$$. We note that the relaxation volume of the dislocation network is
at times higher than the total volume change of the simulation cell. This is
not a contradiction, as each of the many monovacancies contributes the
negative amount of $\Omega_{\mathrm{mono}}^{\mathrm{vac}}=-0.367\,\Omega_{0}$
to the total relaxation volume of the system [12, 17]. For the saturated
vacancy concentration of $0.34\text{\,}\mathrm{\char 37\relax}$, we obtain the
total relaxation volume of all the vacancies
$\Omega_{\mathrm{tot}}^{\mathrm{vac}}=-26.6\times 10^{3}\,\Omega_{0}$,
suggesting that the sum of the dislocation and vacancy relaxation volumes is
always smaller than the simulation cell volume change. The margin of error
stems from neglecting the non-linear relaxation volume effects [74] associated
for example with the core regions of dislocations.

The relaxation volume of the dislocation network exhibits a discontinuity at
$0.3\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ where it sharply drops by
$\Delta\Omega=-48.2\times 10^{3}\,\Omega_{0}$. This drop can be entirely
attributed to its transformation into an extended network. As seen in Figure
6b, the dislocation network first becomes simultaneously extended along the
$x$ and $y$ directions at $0.26\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$.
Shortly after, at $0.30\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$, the network
reorganises itself to being only extended in the $x$ direction. During this
process, it loses over half of its relaxation volume. Because the number of
atoms in the system is conserved, such a process can only occur if the crystal
lattice simultaneously undergoes a plastic transformation that increases the
number of lattice sites by an equal amount — this is possible because many
lattice sites in the simulation cell are unoccupied, populated by vacancies.
As the simulation cell volume does not exhibit a discontinuity, the drop in
the relaxation volume of the dislocation network must be fully counteracted by
the volume change arising from a corresponding plastic deformation of the cell
as a whole.

To illustrate this point, we estimate the change in the number of lattice
sites in the simulation cell. Given matrix $\mathbf{A}$, where the $j$-th
column is the $j$-th simulation cell vector, and matrix $\mathbf{B}$, where
the $j$-th column is the $j$-th primitive lattice vector, preserving the
continuity across periodic boundary conditions requires that

$\mathbf{A}=\mathbf{B}\mathbf{N},$ (46)

where $\mathbf{N}$ is a 3$\times$3 matrix of integers. Note that
$n_{\mathrm{lattice}}=n_{\mathrm{motif}}\det(\mathbf{N})$ is the number of
lattice sites, where $n_{\mathrm{motif}}=1$ for the primitive bcc unit cell.
For a bcc crystal, $\mathbf{B}$ takes on the form

$\mathbf{B}=\frac{a}{2}\left[\begin{array}[]{ccc}-1&\phantom{-}1&\phantom{-}1\\\
\phantom{-}1&-1&\phantom{-}1\\\
\phantom{-}1&\phantom{-}1&-1\end{array}\right].$ (47)

For the simulation cell given here before irradiation,
$A_{ij}=220a\delta_{ij}$, and therefore $N_{ij}=220(1-\delta_{ij})$.

Consider now the same system after it was exposed to irradiation, leading to
an elasto-plastic deformation, transforming the simulation cell to
$\mathbf{A}^{\prime}$. As the periodic boundary conditions still apply to the
transformed simulation cell, the continuity condition still applies in a
modified form

$\mathbf{A}^{\prime}=\mathbf{B}^{\prime}\mathbf{N}^{\prime},$ (48)

where $\mathbf{B}^{\prime}$ is the elastically distorted lattice cell and
$\mathbf{N}^{\prime}$ is the integer-valued matrix of cell repeats after
plastic deformation. The change in the number of lattice sites after elasto-
plastic deformation is then given by

$\Delta
n_{\mathrm{lattice}}=n_{\mathrm{motif}}\left[\det{(\mathbf{N}^{\prime})}-\det{(\mathbf{N})}\right].$
(49)

As the continuity condition still applies, $\mathbf{N}^{\prime}$ is
constrained to be integer-valued, therefore the smallest possible increase in
the number of lattice sites is $\Delta n_{\mathrm{lattice}}=48400$, which is
equivalent to the volume increase of the system by $\Delta V=48.4\times
10^{3}\,\Omega_{0}$. This volume change is very close to the magnitude of the
discontinuous change in the dislocation network volume of
$\Delta\Omega=-48.2\times 10^{3}\,\Omega_{0}$ found numerically at
$\phi=$0.3\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$$. This confirms that the
change of volume of the crystal involves a plastic deformation, mediated by a
structural reorganisation of the extended dislocation network. During
irradiation at low temperatures, where vacancies are immobile, this mechanism
is expected to play a significant role in changing the dimensions of the
crystal.

##  VI Conclusions

The question about the relaxation volume of an arbitrarily complex
interconnected network of dislocations naturally arises in the context of
macroscopic analysis of microstructures of heavily irradiated or heavily
deformed materials if one attempts to evaluate the effect of the dislocation
network on swelling or dimensional changes [10, 13]. While a formula for the
volume of an isolated dislocation loop is well known [55, 11], generalizing it
to the case of an arbitrary dislocation network has so far proved elusive. In
this study, we derived an analytical expression for the volume of an arbitrary
interconnected network of dislocations and showed that its volume can be
evaluated using piece-wise line integration along the arbitrarily curved
directionally ordered dislocation lines, linking the junctions of the network,
see Eqns. (33) and (34). We prove that this analytical expression for the
volume of the network is invariant with respect to the choice of the Cartesian
system of coordinates or the use of periodic boundary conditions, and
illustrate its applications using several representative examples of complex
dislocation structures.

###### Acknowledgements.

We are grateful to J. Marian, S. J. Zinkle, A. E. Sand, B. Uberuaga, A.
Stukowski and A. P. Sutton 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 by the RCUK Energy Programme, grant
No. EP/W006839/1. Enrique Martínez acknowledges support from the startup
package at Clemson University, and Sergei Dudarev acknowledges support from
the Centre for Non-Linear Studies at LANL. 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. The authors acknowledge
the use of the Cambridge Service for Data Driven Discovery (CSD3) and
associated support services provided by the University of Cambridge Research
Computing Services (www.csd3.cam.ac.uk) that assisted the completion of this
study.

##  Appendix A Parameterisation of the SFT formation process

Following Fig. 3, the corners of the one-eighth fcc unit cell $\mathbf{A}$,
$\mathbf{B}$, $\mathbf{C}$, and $\mathbf{D}$ define the coordinate system:

$\displaystyle\mathbf{A}$
$\displaystyle=\frac{a}{2}\left(\mathbf{\hat{x}}+\mathbf{\hat{z}}\right)$ (50)
$\displaystyle\mathbf{B}$
$\displaystyle=\frac{a}{2}\left(\mathbf{\hat{y}}+\mathbf{\hat{z}}\right)$
$\displaystyle\mathbf{C}$
$\displaystyle=\frac{a}{2}\left(\mathbf{\hat{x}}+\mathbf{\hat{y}}\right)$
$\displaystyle\mathbf{D}$ $\displaystyle=0,$

where $a$ is the fcc lattice constant. Points $\bm{\alpha}$, $\bm{\beta}$,
$\bm{\gamma}$, and $\bm{\delta}$ lie at midpoints of the tetrahedral faces
opposite the points $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$,
respectively:

$\displaystyle\bm{\alpha}$
$\displaystyle=\left(\mathbf{B}+\mathbf{C}+\mathbf{D}\right)/3$ (51)
$\displaystyle\bm{\beta}$
$\displaystyle=\left(\mathbf{C}+\mathbf{D}+\mathbf{A}\right)/3$
$\displaystyle\bm{\gamma}$
$\displaystyle=\left(\mathbf{D}+\mathbf{A}+\mathbf{B}\right)/3$
$\displaystyle\bm{\delta}$
$\displaystyle=\left(\mathbf{A}+\mathbf{B}+\mathbf{C}\right)/3.$

Tables 1-2 list the Burgers vectors $\mathbf{b}^{(n)}$, starting points
$\mathbf{p}^{(n)}$, and end points $\mathbf{q}^{(n)}$ of the piecewise linear
segments constituting the SFT dislocation structure. Note that the Burgers
vectors are given in the Thompson vector notation, such that

$\mathbf{b}=\bm{\delta\mathrm{D}}=\mathbf{D}-\bm{\delta}=\frac{1}{3}[\overline{111}].$
(52)

The Tables enable a simple computation of the dipole tensor, the relaxation
volume tensor, and the relaxation volume, following the general expressions
(31)-(33).

Table 1: Parameterisation of the faulted Frank loop ($\lambda=1$) in terms of piecewise linear segments. Burgers vector $\mathbf{b}^{(n)}$ ($a$) | start $\mathbf{p}^{(n)}$ ($\sqrt{2}l$) | end $\mathbf{q}^{(n)}$ ($\sqrt{2}l$)
---|---|---
$\bm{\delta\mathrm{D}}=\frac{1}{3}[\overline{111}]$ | $\mathbf{A}$ | $\mathbf{C}$
$\bm{\delta\mathrm{D}}=\frac{1}{3}[\overline{111}]$ | $\mathbf{C}$ | $\mathbf{B}$
$\bm{\delta\mathrm{D}}=\frac{1}{3}[\overline{111}]$ | $\mathbf{B}$ | $\mathbf{A}$
Table 2: Parameterisation of the Frank loop transforming to the SFT ($0\leq\lambda<1$) in terms of piecewise linear segments. Burgers vector $\mathbf{b}^{(n)}$ ($a$) | start $\mathbf{p}^{(n)}$ ($\sqrt{2}l$) | end $\mathbf{q}^{(n)}$ ($\sqrt{2}l$)
---|---|---
$\bm{\delta\beta}=\frac{1}{6}[0\overline{1}\overline{1}]$ | $\mathbf{A}$ | $\mathbf{C}$
$\bm{\delta\alpha}=\frac{1}{6}[\overline{1}0\overline{1}]$ | $\mathbf{C}$ | $\mathbf{B}$
$\bm{\delta\gamma}=\frac{1}{6}[\overline{1}\overline{1}0]$ | $\mathbf{B}$ | $\mathbf{A}$
$\bm{\gamma\beta}=\frac{1}{6}[10\overline{1}]$ | $\mathbf{D}+\lambda(\mathbf{A}-\mathbf{D})$ | $\mathbf{A}$
$\bm{\beta\alpha}=\frac{1}{6}[\overline{1}10]$ | $\mathbf{D}+\lambda(\mathbf{C}-\mathbf{D})$ | $\mathbf{C}$
$\bm{\alpha\gamma}=\frac{1}{6}[0\overline{1}1]$ | $\mathbf{D}+\lambda(\mathbf{B}-\mathbf{D})$ | $\mathbf{B}$
$\bm{\beta\mathrm{D}}=\frac{1}{6}[\overline{211}]$ | $\mathbf{D}+\lambda(\mathbf{A}-\mathbf{D})$ | $\mathbf{D}+\lambda(\mathbf{C}-\mathbf{D})$
$\bm{\alpha\mathrm{D}}=\frac{1}{6}[\overline{121}]$ | $\mathbf{D}+\lambda(\mathbf{C}-\mathbf{D})$ | $\mathbf{D}+\lambda(\mathbf{B}-\mathbf{D})$
$\bm{\gamma\mathrm{D}}=\frac{1}{6}[\overline{112}]$ | $\mathbf{D}+\lambda(\mathbf{B}-\mathbf{D})$ | $\mathbf{D}+\lambda(\mathbf{A}-\mathbf{D})$

##  Appendix B Parameterisation of a bcc tetrahedron

Fig. 4 shows the schematic structure of a tetrahedral dislocation structure
that can be formed in a bcc metal by the ${\bf b}=(a/2)\langle 111\rangle$ and
${\bf b}=a\langle 001\rangle$ dislocations. In principle, the dislocation
lines forming the edges of a bcc tetrahedron can be curved, see for example
Ref. [72], but for the purpose of illustrating the principle of how to
evaluate the volume of an unusual dislocation structure, we assume that all
the dislocation lines linking the dislocation junctions of the structure are
straight. The coordinates of the four junctions shown in Fig. 4 are

$\displaystyle\mathbf{A}$ $\displaystyle=(0,L,L)$ (53)
$\displaystyle\mathbf{B}$ $\displaystyle=(0,-L,L)$ $\displaystyle\mathbf{C}$
$\displaystyle=(L,0,0)$ $\displaystyle\mathbf{D}$ $\displaystyle=(-L,0,0).$

In these notations, vector $\mathbf{DC}$ is collinear with the $[100]$
crystallographic direction, whereas vector $\mathbf{BA}$ is collinear with the
$[001]$ direction. The Burgers vectors of the six dislocation segments forming
the bcc tetrahedron shown in Fig. 4, and the vector coordinates of the
segments themselves are given in Table 3. A direct examination of the
dislocation configuration shows that the Burgers vectors are conserved at
junctions, and that the $a\langle 001\rangle$ dislocations have a pure edge
character whereas the $(a/2)\langle 111\rangle$ dislocations have a mixed
character. The length of the $a\langle 001\rangle$ segments is $2L$ whereas
the length of the $(a/2)\langle 111\rangle$ segments is $\sqrt{3}L$.

Table 3: Parameterisation of the bcc tetrahedron shown in Fig. 4 in terms of piecewise linear dislocation segments. Burgers vector $\mathbf{b}^{(n)}$ ($a$) | start $\mathbf{p}^{(n)}$ | end $\mathbf{q}^{(n)}$ | $\Omega^{(n)}$
---|---|---|---
$[100]$ | $\mathbf{A}$ | $\mathbf{B}$ | $aL^{2}$
$\frac{1}{2}[1{\overline{1}}{\overline{1}}]$ | $\mathbf{D}$ | $\mathbf{A}$ | $0$
$\frac{1}{2}[111]$ | $\mathbf{C}$ | $\mathbf{A}$ | $0$
$[0\overline{1}0]$ | $\mathbf{C}$ | $\mathbf{D}$ | $0$
$\frac{1}{2}[1\overline{1}1]$ | $\mathbf{B}$ | $\mathbf{C}$ | $0$
$\frac{1}{2}[11\overline{1}]$ | $\mathbf{B}$ | $\mathbf{D}$ | $0$

The quantities referred to as $\Omega^{(n)}$ in Table 3 are the individual
terms in equations (33) and (34), computed in the Cartesian system of
coordinates where the positions of the junctions are given by (53). On their
own, quantities $\Omega^{(n)}$ are not invariant with respect to the
translations of the system of coordinates, but their sum
$\Omega=\sum_{n}\Omega^{(n)}$ is invariant with the respect to such
translations.

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