Dislocation Structure and Mobility in Hcp Rare-Gas Solids: Quantum versus Classical

Abstract

We study the structural and mobility properties of edge dislocations in rare-gas crystals
with the hexagonal close-packed (hcp) structure by using classical simulation techniques. Our results
are discussed in the light of recent experimental and theoretical studies on hcp 4He, an archetypal
quantum crystal. According to our simulations classical hcp rare-gas crystals present a strong tendency
towards dislocation dissociation into Shockley partials in the basal plane, similarly to what is observed
in solid helium. This is due to the presence of a low-energy metastable stacking fault, of the order
of 0.1 mJ/m2, that can get further reduced by quantum nuclear effects. We compute the minimum
shear stress that induces glide of dislocations within the hcp basal plane at zero temperature, namely,
the Peierls stress, and find a characteristic value of the order of 1 MPa. This threshold value is similar
to the Peierls stress reported for metallic hcp solids (Zr and Cd) but orders of magnitude larger
than the one estimated for solid helium. We find, however, that in contrast to classical hcp metals
but in analogy to solid helium, glide of edge dislocations can be thermally activated at very low
temperatures, T�10 K, in the absence of any applied shear stress.

Full text

crystals
Article
Dislocation Structure and Mobility in Hcp Rare-Gas
Solids: Quantum versus Classical
Santiago Sempere 1, Anna Serra 2ID , Jordi Boronat 1and Claudio Cazorla 3,*ID
1
Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain;
ssemllag@gmail.com (S.S.); jordi.boronat@upc.edu (J.B.)
2Departament d’Enginyeria Civil i Ambiental, Universitat Politècnica de Catalunya, Campus Nord C2,
E-08034 Barcelona, Spain; a.serra@upc.edu
3School of Materials Science and Engineering, The University of New South Wales Australia,
Sydney 2052, Australia
*Correspondence: c.cazorla@unsw.edu.au
Received: 13 December 2017; Accepted: 20 January 2018; Published: 29 January 2018
Abstract:
We study the structural and mobility properties of edge dislocations in rare-gas crystals
with the hexagonal close-packed (hcp) structure by using classical simulation techniques. Our results
are discussed in the light of recent experimental and theoretical studies on hcp
4
He, an archetypal
quantum crystal. According to our simulations classical hcp rare-gas crystals present a strong tendency
towards dislocation dissociation into Shockley partials in the basal plane, similarly to what is observed
in solid helium. This is due to the presence of a low-energy metastable stacking fault, of the order
of 0.1 mJ/m
2
, that can get further reduced by quantum nuclear effects.
We compute
the minimum
shear stress that induces glide of dislocations within the hcp basal plane at zero temperature, namely,
the Peierls stress, and find a characteristic value of the order of 1 MPa.
This threshold
value is similar
to the Peierls stress reported for metallic hcp solids (Zr and Cd) but orders of magnitude larger
than the one estimated for solid helium. We find, however, that in contrast to classical hcp metals
but in analogy to solid helium, glide of edge dislocations can be thermally activated at very low
temperatures, T∼10 K, in the absence of any applied shear stress.
Keywords: dislocations; rare-gas solids; molecular dynamics; quantum nuclear effects
1. Introduction
Dislocations are line defects related to the accommodation of plastic deformation in crystals. They are
characterized by the Burgers vector that represents the magnitude and direction of the lattice distortion
along the dislocation line. Dislocations are ubiquitous in materials and can alter significantly their
physical properties. Due to their fundamental and technological interests,
the structure
and mobility of
dislocations in classical metals with the three elemental crystal structures, namely, face-centered cubic (fcc),
body-centered cubic (bcc), and hexagonal close-packed (hcp) (e.g., Al, Fe, and Zr), have been extensively
investigated both with theory and experiments (see [1–4] and references therein).
Quantum and classical solids are fundamentally different. In quantum crystals, typically
4
He
and H
2
, the kinetic energy per particle in the
T→
0 limit is much larger than
kBT
(where
kB
is the
Boltzmann constant) and the fluctuations of the atoms around the equilibrium lattice sites are up
to
∼
10% of the distance to the neighboring lattice sites [
5
–
9
]. In classical crystals, on the contrary,
those quantities are practically negligible at low temperatures. Because of these important differences,
one might expect finding disparate dislocation phenomena in the two types of crystals. In fact,
Haziot et al.
analysed the plastic properties of hcp
4
He by means of direct stress-strain measurements
and found that the resistance to shear along directions contained in the basal plane nearly vanishes at
T≈
0.1 K due to the free glide of dislocations [
10
]. This intriguing effect, which has been termed as
Crystals 2018,8, 64; doi:10.3390/cryst8020064 www.mdpi.com/journal/crystals

Crystals 2018,8, 64 2 of 19
“giant plasticity”, disappears in the presence of
3
He impurities or when the temperature is raised [
10
].
Meanwhile, in a recent quantum Monte Carlo simulation study on hcp
4
He [
11
],
Landinez-Borda et al.
have shown that the Peierls stress for the flow of dislocations in the basal plane is nominally zero,
essentially due to the zero-point motion of the atoms. The “giant plasticity” observed in solid helium,
therefore, appears to be a manifestation of its quantum character.
Quantum and classical solids, however, also present some similarities as regards dislocation
behavior. For instance, Landinez-Borda et al. have shown that in solid helium either screw or edge
dislocations with Burgers vectors contained in the hcp basal plane tend to dissociate into Shockley
partial dislocations separated by ribbons of fcc-like stacking fault [
11
]. The same behavior is observed
in classical metals with the hcp structure like, for instance, Zr [
2
,
12
,
13
]. Nevertheless, since the nature
of the atomic interactions in rare gases and metallic systems are so different, the physical origins of
such similarities (or the differences explained above) are not totally understood. Actually, studies
on the structure and mobility of dislocations in classical hcp rare-gas solids are to the best of our
knowledge absent in the literature (probably due to the lack of related applied interests). Consequently,
straightforward and physically insightful comparison between classical and quantum hcp rare-gas
crystals in terms of dislocation behavior is not possible.
In this article, we analyze the structure and mobility of edge dislocations in a model hcp rare-gas
crystal with classical simulation methods, and compare our results to those obtained in other hcp
crystals and solid helium. We focus on the atomic structure and glide of edge dislocations with
Burgers vector contained in the basal plane, as this type of line defect and dislocation motion are most
likely to occur in solid helium [10,11,14,15]. Our results reveal a strong tendency towards dislocation
dissociation into Shockley partials separated by wide regions of fcc-like stacking fault, in analogy to
what occurs in solid helium. We find that the Peierls stress for the glide of edge dislocations in the
hcp basal plane amounts to
∼
1 MPa, which is very similar in magnitude to the values reported for
classical metals with the hcp structure (e.g., Zr and Cd) [
12
,
13
]. However, in contrast to other classical
solids but in analogy to solid helium, edge dislocations in hcp rare gases turn out to be extremely
mobile: they can diffuse with an approximate velocity of 50 m/s in the absence of any applied stress at
temperatures as low as 25 K (that is, well below the corresponding Debye temperature
ΘD∼
65 K [
16
]).
We rationalize the origins of this effect in terms of the exceptionally weak interatomic interactions in
rare gases.
The organization of this article is as follows. In the next section, we provide the technical details
of our classical simulations and explain the methods that we have employed to analyze the structure
and mobility of dislocations. Then, we present our results and discuss them in the light of previous
classical and quantum simulation studies. Special consideration is put on the technical aspects of the
calculations as regards the impact of finite-size effects, relaxation of the simulation cell, and sampling
of different thermodynamic ensembles with molecular dynamics. Finally, we summarize our main
findings and conclusions in Section 4.
2. Methods Outline
2.1. Classical Simulations
All the geometry relaxations and molecular dynamics (MD) simulations were performed with
the LAMMPS code [
17
]. Our model hcp crystal consists of xenon (Xe) atoms interacting through a
pairwise Lennard–Jones (L–J, 6–12) potential with parameters
e=
0.01881 eV and
σ=
4.06 Å [
18
–
20
].
We note that the employed L–J pairwise potential, in spite of being analytically simple, was originally
devised to reproduce a considerable amount of experimental data measured in solid Xe, including the
elastic constants, sound velocities, and equation of state, among others. (The ground-state structure
of solid Xe is known to be cubic fcc; however, since our focus here is on classical hcp rare gases,
we chose the species with the largest possible atomic weight and most intense atomic forces as an
upper-bound). A particle–particle particle-mesh
k
-space solver was used to compute the long-range

Crystals 2018,8, 64 3 of 19
van der Waals interactions and forces beyond a cut-off distance of 20 Å at each relaxation and
MD step
.
The initial dislocation configuration was generated by removing a
(
11
2
0
)
semi-plane of Xe atoms
from an orthorhombic simulation box containing a perfect hcp lattice; the generated Burgers vector
then was equal to
b= (a
, 0, 0
)
as expressed in Cartesian coordinates, where
a=
4.26 Å represents
the equilibrium in-plane lattice parameter. (It was checked that, upon full geometry optimization,
the relaxed system was identical to that obtained when starting from an initial configuration generated
with the Osetsky and Bacon’s method [21].)
The geometry relaxations were performed with a conjugate gradient algorithm and convergence
was reached after the forces on the atoms and mechanical stresses were smaller than 10
−10
eV/Å and
10
−8
eV/Å
3
, respectively. Regarding the MD simulations, the pressure and/or temperature of the
system were kept fluctuating around a set-point value by using thermostatting and/or barostatting
techniques in which some dynamic variables are coupled to the particle velocities and/or simulation
box dimensions. Large simulation boxes containing several thousands of atoms were employed
in the dynamical simulations, and periodic boundary conditions normally were applied along the
three Cartesian directions. Examples of the simulation-cell dimensions considered in this study
are
Lx=
614.64,
Ly=
61.27, and
Lz=
577.59 Å (344,544 atoms),
Lx=
104.38,
Ly=
107.20,
and Lz=101.13 Å (18,242 atoms)
, and
Lx=
43.00,
Ly=
45.75, and
Lz=
43.26 Å (1368 atoms).
Newton’s equations of motion were integrated by using the customary Verlet’s algorithm with a time
step of 10−3ps.
2.2. Analysis Methods
In order to identify with precision the structure and position of the edge dislocation in our model
crystal, we employed three different analysis methods that are briefly explained next.
2.2.1. Differential Displacement Analysis (DD)
The presence of line defects makes the atoms contained in the slip plane of the dislocation to
displace. We can quantify the spread of such a disregistry through the distribution of partial Burgers
vector components,
[bx
,
by]
, within the glide plane [
12
,
22
]. These partial components can be expressed as:
bi(x) = d(∆ui)
dx , (1)
where
x
represents the direction perpendicular to the dislocation line contained in the hcp basal plane,
i=x,y, and ∆uiis the atomic disregistry. The latter quantity can be defined as:
∆ui=uabove,i−ubelow,i, (2)
where
up,i=rdisl oc
p,i−rper f
p,i
,
rdisl oc
p,i
are the positions of the atoms above/below the glide plane in the
system containing the dislocation, and
rper f
p,i
the positions of those same atoms in the perfect-lattice
system. In a general case, the partial
x
components of the Burger vectors describe the edge part of the
dislocation, whereas the
y
components the screw. The components of the total Burger vector contained
in the simulation cell,
[bT
x
,
bT
y]
, then can be computed by integrating the respective partial components
along the glide direction as:
bT
i=ZLx
0bi(x)dx , (3)
where
Lx
represents the size of the simulation box along the
x
-direction. In our particular case,
bT
y
should be always equal to zero, as we are dealing exclusively with edge dislocations. The presented
differential displacement (DD) analysis is especially useful for detecting the presence of stacking fault
ribbons bounded by two partial dislocations, and for estimating the width of dislocation cores.

Crystals 2018,8, 64 4 of 19
2.2.2. Nearest Neighbor Analysis (NN)
In this method, the number of atoms within a certain radial distance from a selected atom,
nc
,
is computed
. The cut-off distance defining such an interval normally is chosen to be a value between
the distances to the first and second shells of atomic nearest neighbors. In the particular case of hcp
systems, a possible definition of the cut-off distance is [23]:
rhcp
c=1
2 1+r4+2x2
3!a, (4)
where
x= (c/a)/
1.633 and
c
and
a
represent the hcp lattice parameters. In the perfect hcp lattice,
the number of nearest neighbors is 12 for every atom; in the system containing the dislocation,
the atoms displaying
nc6=
12 values then can be identified with a highly distorted region of the crystal
like, for instance, the core of the dislocation. The nearest neighbor (NN) analysis turns out to be very
useful for locating dislocation cores and hence monitoring the motion of line defects.
2.2.3. Common Neighbor Analysis (CNA)
This method, which originally was introduced by Honeycutt and Andersen [
24
], consists of
creating a 4-index sequence for each pair of atoms,
α
and
β
. The first index is equal to “1” if
α
and
β
are nearest neighbors, “2” otherwise (two atoms are nearest neighbors if the distance between them is
smaller than a certain cut-off value, e.g., see Section 2.2.2 ). The second index adopts a value that is
equal to the number of common nearest neighbors shared by
α
and
β
(e.g., “4” in a perfect hcp or fcc
lattice when the first sequence index is equal to “1”). The third index indicates the number of bonds
between common neighbors. The fourth index is introduced to differentiate diagrams with same first,
second and third indexes but with different types of bonds between common neighbors. For instance,
in a perfect fcc system, all the 4-index sequences ascribed to nearest neighbors are equal to “1421”; in a
perfect hcp system, half of the sequences ascribed to nearest neighbors are equal to “1421” while the
other half are equal to “1422”; in a perfect bcc system, we find 4-index sequences describing nearest
neighbors that are equal to “1441” and “1661”. The common neighbor analysis (CNA) method turns
out to be very useful for locating dislocation cores and also stacking faults.
3. Results and Discussion
3.1. Edge Dislocation Structure
The shortest perfect Burgers vector in an hcp lattice is
b=1
3h
11
2
0
i
, and the most common
dislocation slip planes are the basal,
(
0001
)
, and prism,
{
10
1
0
}
, planes [
2
,
12
,
25
]. The preference of the
glide plane is determined by the energy and stability of a stacking fault. If a low-energy metastable
stacking fault with vector
1
3h
1
1
00
i
exists,
I2
, then the dislocation normally dissociates in the basal plane
into two Shockley partial dislocations bounding a ribbon of fcc-like stacking fault [
2
,
12
,
25
]. Such a
dissociation process is described in crystallographic notation as:
1
3[1120]→1
3[1010] + 1
3[0110]. (5)
The resulting system geometry generally consists of two partial dislocations lying on the basal plane
at
±
30
◦
of the initial Burgers vector
b
(or
±
60
◦
, if referred to the dislocation line) and with partial
Burgers vectors |bp|=|b|/√3.
In Figure 1a,b, we represent the final relaxed configuration of our model hcp Xe solid in which we
initially created an edge dislocation with its line oriented along the
y
-direction. The full relaxation was
performed via minimization of all the atomic forces,
Fi
, and mechanical stresses,
σij
(see Section 2.1).
By using the CNA analysis method (see Section 2.2.3), we are able to distinguish the atoms that belong
to the dislocation core (green) or to the fcc-like stacking fault (blue), and those that render the usual hcp

Crystals 2018,8, 64 5 of 19
ordering (yellow). The relaxed structure clearly shows two Shockley partial dislocations oriented as
+
30
◦
and
−
30
◦
with respect to the initial Burgers vector
b= (a
, 0, 0
)
, and a ribbon of fcc-like stacking
fault between them; we note that the same structural behavior is observed also in classical metallic
(e.g., Zr [2,12]) and quantum rare-gas (e.g., 4He [11]) hcp crystals.
Figure 1.
Sketch of a fully relaxed system containing an edge dislocation from different views. Green
spheres represent atoms belonging to the dislocation core, blue spheres atoms belonging to the
fcc-like stacking fault, and yellow spheres atoms with common hcp atomic coordination features.
Red arrows represent the Burgers vectors of the partial dislocations. (
a
,
b
) represent different views of
the simulation cell.
In order to provide a quantitative description of the relaxed dislocation configuration,
we employed
the differential displacement (DD) analysis method (Section 2.2.1). In Figure 2a,b, we plot
the relative displacement of the atoms delimiting the glide plane, and, in Figure 2c,d,
the corresponding
partial Burgers vector components
[bx
,
by]
. It is shown that, as expected, integration of
bx
and
by
along

Crystals 2018,8, 64 6 of 19
the
x
-direction leads to non-zero edge and null screw total dislocation components, respectively.
The width of the resulting fcc-like stacking fault,
ωs f
, as deduced from the distance between the two
maxima in Figure 2c, is approximately equal to 50
a
. The width of the dislocation core, which can be
defined as the region in which the atomic disregistry is greater than the half of its maximum, is found
to be
∼
12.5
a
. This latter quantity has an unusually large value, which indicates the presence of very
mobile dislocations (we will comment again in this point in Section 3.3).
Figure 2.
Relative displacement (
a
,
b
) and differential displacement (
c
,
d
) of the atoms above and
below the glide plane of the edge dislocation in the
x
- and
y
-directions. The simulation cell contains
n= 344,544 atoms and is fully relaxed.
Concerning the technical aspects involved in the simulation of dislocations, we have analyzed the
effects of reducing the size of the simulation cell on the determination of the final
equilibrium state
.
This type of analysis is especially useful for interpreting the results obtained in quantum and
first-principles simulations where, due to the high computational expense involved, one only can
handle systems made up of few hundreds or thousands of atoms [
11
,
26
]. Figures 3and 4show the
DD analysis performed in two simulation cells containing 18,424 and 1368 atoms, respectively. In the
n
= 18,424 case, we have also analyzed the effects of constraining the shape of the simulation cell to
orthorhombic, that is, of not relaxing it (hence
σij 6=
0). In Figure 3a–d (blue lines), it is appreciated that
ωs f
now is equal to 12
a
and the width of the dislocation core is
∼
5
a
. These values are significantly
smaller than the results obtained in the simulation cell containing 344,544 atoms, which in principle
are not affected by finite-size errors. Nevertheless, integration of the corresponding
bx
and
by
partial
Burgers vector components along the
x
direction still provides non-zero edge and null screw total
dislocation components, and the two Shockley partial dislocations can be clearly differentiated in the
DD plots shown in Figure 3. Meanwhile, it is found that when the shear stresses on the simulation cell
are not minimized the separation between the two partial dislocations reduces to
approximately 10a
.
In addition, the orientation of the two partial dislocations changes from
+
30
◦
and
−
30
◦
to
−
30
◦
and
+
30
◦
,
as compared
to the minimum-energy case
σij =
0. Nevertheless, it may be reasonably concluded
that, in the particular case of simulating edge dislocations, the inaccuracies deriving from the use of
relatively small orthorhombic boxes containing up to
∼
10
4
atoms are not critical. In the
n=
1368 case
(see Figure 4), by contrast, it is found that the edge dislocation hardly can get dissociated owing to
the limited size of the simulation cell, which artificially prevents the appearance of any stacking fault
(that is, only one diffuse maximum is appreciated in Figure 4c). Moreover, integration of the
by
partial
Burgers vector component along the
x
-direction neither provides an exact null value for the total screw

Crystals 2018,8, 64 7 of 19
dislocation component (see Figure 4d). In view of the results enclosed in Figures 2–4,
we may
conclude
that the use of small simulation cells containing just up to
∼
1000 atoms is likely to produce unrealistic
dislocation configurations (see, for instance, Ref. [26]).
Figure 3.
Relative displacement (
a
,
b
) and differential displacement (
c
,
d
) of the atoms above and
below the glide plane of the edge dislocation in the
x
and
y
directions. The simulation cell contains
n= 18,424 atoms
. Green and blue lines represent the results obtained in a non-relaxed (
σij 6=
0) and a
fully relaxed (σij =0) simulation cell, respectively.
Figure 4.
Relative displacement (
a
,
b
) and differential displacement (
c
,
d
) of the atoms above and
below the glide plane of the edge dislocation in the
x
- and
y
-directions. The simulation cell contains
n=1368 atoms and is fully relaxed.
In order to get quantitative insight into the metastable stacking fault that induces the dissociation
of the edge dislocation into Shockley partials within the basal plane, we have computed the stacking

Crystals 2018,8, 64 8 of 19
fault energy in our model hcp crystal as a function of the fault plane displacement,
γ(f)
[
12
].
For this
calculation, first we rigidly displace one half of the crystal with respect to the other over a grid of 10
4f
points spanning all possible faults within the
x
–
y
plane. Subsequently, at each
f
point,
the atoms
are
allowed to relax perpendicular to the fault plane, which is along the
z
-direction, by potential energy
minimization (i.e., zero-temperature conditions are assumed). Our simulation cell contains a total
of 8100 Xe atoms, and we apply periodic boundary conditions over the
x
–
y
fault plane and rigid
boundary conditions along
z
. Our stacking fault energy results are represented in Figure 5, for which a
spline-based interpolation has been used in order to provide smooth iso-
γ
contours. We actually find
a metastable stacking fault at
f=1
3h
1
1
00
i
, which is
I2
, similarly to what has been reported by other
authors for classical hcp metals [
12
]. According to our calculations, the energy of the
I2
stacking fault,
γs f
, is equal to 0.094 mJ/m
2
. (It is worth noting that the numerical accuracy in our
γs f
estimations is
below 0.001 mJ/m
2
.) We also calculated the energy of the metastable stacking fault associated to an
edge dislocation with its line laying on the hcp prism plane (see, for instance, Figure 2b in Ref. [
12
]).
In that case, we obtained a
γs f
value of 15 mJ/m
2
, which is about three orders of magnitude larger
than the value calculated for the basal plane. This result shows a major tendency towards dislocation
dissociation into Shockley partials in the basal plane.
0.0 0.2 0.4 0.6 0.8 1.0
x/a
0.0
0.2
0.4
0.6
0.8
1.0
y/a
0
5
10
15
20
25
30
35
40
45
γ (mJ/m2)
Figure 5.
The
γ
-surface of the analyzed classical hcp rare-gas crystal. Perfect hcp stacking positions
correspond to the four corners of the plot while large white spheres indicate metastable fault positions.
Iso-γcurves are represented with solid black lines at 5 and 10 mJ/m2intervals.
As expected, the
γs f
values calculated in Xe turn out to be extremely small as compared
to those obtained in other hcp crystals where the interactions between atoms are much stronger
(
e.g., γs f ∼100 mJ/m2
in Zr [
12
,
27
,
28
]). We note that Keyse and Venables already measured more
than 30 years ago the stacking fault energy in fcc Xe at low temperatures by means of transmission
electron microscopy techniques [
29
]. In particular, they found a
γs f
value of 1.96
±
0.65 mJ/m
2
at
a temperature of 25
±
5 K, which is about two (one) orders of magnitude larger (smaller) than the
stacking fault energy that we have determined for the basal (prism) plane in the hcp phase. The reasons
behind these discrepancies may be possibly understood in terms of the different crystal structure
considered in our calculations and also of likely inaccuracies present in the employed interaction
pairwise potential.

Crystals 2018,8, 64 9 of 19
Once the metastable stacking fault energy is known, we can estimate from elastic theory the
expected equilibrium distance between the Shockley partial dislocations, which is the width of the
resulting fcc-like stacking fault, with the formula [1,2]:
ωelas
s f =Gb2
p
4πγs f
, (6)
where likely elastic anisotropic effects have been disregarded,
G
represents the shear modulus of the
system (which we estimate here to be 200 MPa), and
bp
the modulus of the corresponding partial
Burgers vector. By performing the necessary numerical substitutions, we find that in the basal plane
ωelas
s f
is equal to 24
a
. We recall that, in the larger simulation cell considered in this study, we have
found that
ωs f
approximately amounts to 50
a
(see Figure 2c,d), which turns out to be of the same order
of magnitude and larger than
ωelas
s f
. Consequently, the results obtained in the 344,544-atoms system
may be considered to be virtually free of finite-size bias.
Recently, Landinez-Borda et al. have estimated an almost vanishing
γs f
value of 0.002 mJ/m
2
in solid
4
He at ultralow temperatures by using quantum Monte Carlo simulation techniques [
11
].
In an attempt to quantify the importance of quantum nuclear effects on the stacking fault energy of
solid helium, we have performed analogous classical
γ(f)
calculations to those described for Xe but
considering the same volume conditions, interatomic interactions, and atomic mass than in Ref. [
11
].
Our classical calculations in solid 4He render a stacking fault energy of 0.003 mJ/m2, which is orders
of magnitude smaller than the value estimated in solid Xe. By comparing this result to the stacking
fault energy calculated by Landinez-Borda et al., we may conclude that quantum nuclear effects are
responsible for a
γs f
reduction of the
∼
30% . Consequently, quantum nuclear fluctuations further
contribute to the dissociation of edge dislocations into partials in solid
4
He and probably also in any
other quantum crystal (e.g., H2, Ne and LiH [5,30,31]).
3.2. The Peierls Stress
The Peierls stress,
τP
, is key to quantifying the resistance of a crystal to the motion of dislocations.
τPnormally
is referred to the critical stress that induces glide of dislocations in the absence of thermal
excitations. Here, we use two different methods to evaluate the Peierls stress in our model crystal as
concerns the motion of edge dislocations in the basal plane. We note that, since the glide of dislocations
involves the breaking and formation of atomic bonds, the value of
τP
in principle is expected to depend
strongly on the crystal structure and strength of the interatomic forces.
3.2.1. Method A: Fixed Boundary Conditions
We first employ the usual method found in classical simulation studies based on force fields
(see, for instance, Refs. [
12
,
21
]), which is briefly described next. The simulation cell is divided into three
main parts: “U”, the upper region containing frozen atoms, “L”, the lower region containing frozen
atoms, and “M”, the rest of the simulation cell containing mobile atoms (see Figure 6). Regions U and
L consist of several layers of atoms that are displaced together as a block. Essentially, a shear strain
deformation is applied on the simulation cell perpendicular to the dislocation line and the resulting
stresses are monitored. In our particular case, the dislocation line is parallel to the
y
-axis; hence,
we first displace the U slab a small distance along the
x
-direction,
∆u
, and then proceed to minimize
the potential energy of the atoms in M while keeping the L slab fixed. Periodic boundary conditions
are applied just along the
x
- and
y
-directions. The applied mechanical strain is straightforwardly
calculated as
∆η=∆u/Lz
, where
Lz
is the length of the simulation along the
z
-direction, and the
accompanying shear stress is
σxz =Fx/LxLy
, where
Fx
is the sum of all the forces along the
x
-direction
exerted on the atoms in region U. By iteratively repeating this procedure, we can reproduce with detail
the dependence of the shear stress on
η
. For small cell distortions,
σxz
is expected to increase almost
linearly, as it follows from elastic theory; however, when
η
is large enough so that it induces the glide

Crystals 2018,8, 64 10 of 19
of dislocations, the shear stress should decrease sharply. The maximum value of
σxz
just before that
sudden drop can be identified with the Peierls stress.
Figure 6.
Sketch of the system used to estimate the Peierls stress with Method
A
. Three main parts
are differentiated: the upper part “U”, the lower part “L”, and the region with mobile atoms “M”.
“P” indicates application of periodic boundary conditions and the dashed line the orientation of the
edge dislocation.
In Figure 7, we show the
σxz(η)
results obtained in a large simulation cell containing 344,544 atoms
by adopting two different
∆η
increments; the thickness of the upper region U,
dU
, was safely fixed
to 5
c
in both cases [
21
]. As can be appreciated in Figure 7a (case
∆η
= 8
×
10
−4
), a regular pattern
emerges that follows elastic theory at small cell deformations (that is,
σxz ∝η
) and which allows for an
estimation of the Peierls stress (as identified with the
σxz
maximum) of
τP=
6.0
±
0.1 MPa.
When the
employed
∆η
is significantly reduced (case
∆η=
1
×
10
−4
, see Figure 7b), the obtained
σxz(η)
curve
shows more irregularities and the expected elastic behavior is reproduced at conditions
η>
0.001.
This outcome reflects the intricacies found in the relaxation of such a large simulation cell, which at
small
η
values may easily end up on metastable configurations. In this latter case, we estimate a Peierls
stress (as identified with the σxz maximum) of τP=7.4 ±0.1 MPa.
Figure 7.
Evolution of the shear stress expressed as a function of strain for a simulation cell containing
n
= 344,544 atoms in which the thickness of the “U” region is taken to be
dU=
5
c
. Results obtained
with ∆η=8×10−4and 1 ×10−4are shown in (a,b), respectively (see text).

Crystals 2018,8, 64 11 of 19
In order to assess the effects of finite-size bias on the estimation of
τP
, we repeated the same
calculations in a smaller simulation cell containing
n
= 18,424 atoms. The stress profiles that we
obtained in this case (see Figure 8a) are intricate and do not allow for a clear estimation of the Peierls
stress (that is, an unambiguous maximum appearing periodically is missing). Such a finite-size effect is
related to the fact that the width of the fcc-like stacking fault is already of the same order of magnitude
than the characteristic size of the simulation cell (that is,
∼
100 Å). In order to somehow determine
τP
from the results shown in Figure 8a, we monitored the position of the two partial dislocations
with the differential displacement (DD) and nearest neighbor (NN) analysis methods (see Section 2.2
and Figure 8b), and averaged the value of the shear stress over the set of strain points at which the
partial dislocations change their position. (In the NN case, for the sake of simplicity, we have averaged
the position of all the atoms exhibiting a nearest neighbor number different from 12; consequently,
we obtain the center position of the stacking fault.) By proceeding like this, we obtained a Peierls stress
of
τP=
1.3
±
0.2 MPa, which is about 5 times smaller than the one estimated in the
n
= 344,544 atoms
simulation cell. Our results, therefore, show that finite-size errors affect critically the estimation of
τP
when using Method A(in agreement with previous conclusions by Osetsky and Bacon [21]).
Figure 8.
(
a
) evolution of the shear stress expressed as a function of strain for a simulation cell
containing
n
= 18,424 atoms in which the thickness of the “U” part is equal to
dU=
2.5
c
. Several
mechanical strain steps,
∆e
, are considered; (
b
) position of the corresponding dissociated edge
dislocation expressed as a function of strain. “DD” and “NN” stand for the analysis methods of
differential displacement and nearest neighbors, respectively.
3.2.2. Method B: Periodic Boundary Conditions
As we have just shown in the previous section, Method
A
requires of very large simulation
cells (i.e.,
N∼
10
5
–10
6
atoms) in order to remove all possible finite-size bias affecting the estimation
of
τP
. This technical aspect suggests that accurate calculation of the Peierls stress with Method
A
and quantum atomistic or electronic first-principles simulation techniques (in which typically

Crystals 2018,8, 64 12 of 19
N∼102–103
atoms [
11
,
32
]) is hardly achievable in practice. Moreover, by construction customary
electronic first-principles techniques (e.g., plane-wave density functional theory [
5
,
33
]) generally
demand the application of periodic boundary conditions in all directions in order to ensure the
periodicity and continuity of the electrostatic potential in space. Therefore, it is desirable to work out
reliable τPcomputational methods in which all boundaries of the simulation cell are treated equally.
Here, we present a method in which a particular tilt is introduced in the simulation cell containing
the edge dislocation and the accompanying change in the total energy is monitored upon constrained
relaxation of the system; the relaxation is performed by applying periodic boundary conditions
along the three Cartesian directions and optimizing all degrees of freedom of the system except the
initial tilt
. The resulting stresses then can be calculated numerically with the well-known expressions
from elastic theory. (A similar approach has been employed by Wang et al. to investigate the
dynamics of screw dislocations in bcc tantalum [
34
]; in our case, however, the simulations are
strictly performed at zero-temperature conditions.) Specifically, the lattice vectors describing our
simulation cell are
a1= (Lx
, 0, 0
)
,
a2= (x2
,
Ly
, 0
)
, and
a3= (x3
,
y3
,
Lz)
, where
x3
represents the
introduced tilt. The corresponding
shear strain is
ηxz =x3/Lz
, and the resulting shear stress can be
estimated as [35–37]:
σxz =1
V
∂E
∂ηxz , (7)
where Vrepresents the volume of the system and Ethe corresponding total energy.
We enclose the total energy and shear stress results obtained in a simulation cell containing
18,424 atoms in Figure 9a,b. In both cases, the profiles that we obtain as a function of applied shear
strain are periodic, in contrast to what we found with Method
A
when using a simulation cell of
the same dimensions (see Figure 8), and elastic behavior is observed for small system deformations.
By identifying the global maximum in the
σxz
curve with the Peierls stress, we obtain a value of
3.40
±
0.01 MPa (see Figure 9b). This value is roughly two times smaller than the free-of-bias result
obtained with Method
A
in the simulation cell containing
n
= 344,544 atoms. Therefore, we may
conclude that, as compared to Method
A
, finite-size bias appear to affect less critically the calculation
of
τP
with Method
B
(we recall that, with Method
A
, we obtained a Peierls stress of 1.3
±
0.2 MPa in a
same simulation cell containing 18,424 atoms (see previous Section 3.2.1)).
In spite of this favorable outcome, we should acknowledge that the use of periodic boundary
conditions in systems containing dislocations is not exempt of important limitations. For instance,
periodic boundary conditions are inconsistent with the existence of a net Burgers vector in the
simulation cell; consequently, a dipole or quadrupole of dislocations needs to be introduced in
the system [
38
,
39
]. In order to further incorporate this technical aspect on the calculation of
τP
with
Method B
, we constructed a large simulation cell of
n
= 247,680 atoms containing a dislocation
quadrupole. Upon introduction of a moderate tilt and by proceeding to relax the system, however,
we found that all the dislocations merged into a big stacking fault and were annihilated (we note that
a similar behavior has been observed also by Wang et al. in bcc tantalum [
34
]). This outcome suggests
that, unfortunately, even larger simulation cells are required to correct for the inaccuracies associated
to Method B.
The main conclusions emerging from Sections 3.2.1 and 3.2.2 is that the Peierls stress in our model
hcp crystal is of the order of 1 MPa. This result is orders of magnitude smaller than the
τP
values
reported for archetypal crystals with cubic symmetry (e.g., Fe and Mo) [
13
]; however, to our surprise,
it is very similar in magnitude to the Peierls stresses found in classical metals with the hcp structure
(e.g., Zr, Cd, and Mg) [
12
,
13
]. Very recently, Landinez-Borda et al. have shown in solid helium that
τP
nominally amounts to zero, that is, dislocations can move freely throughout the crystal in the absence
of thermal excitations and shear stresses [
11
]. The authors of that study have argued that such an
effect is quantum in nature as is essentially originated by zero-point fluctuations. Our Peierls stress
results obtained in classical rare-gas hcp solids come to corroborate Landinez-Borda et al.’s conclusion,

Crystals 2018,8, 64 13 of 19
as we have demonstrated that weak interparticle interactions alone cannot render practically vanishing
τPvalues.
Figure 9.
Energy (
a
) and shear stress (
b
) expressed as a function of shear strain for a simulation
cell containing
n
= 18,424 atoms in which periodic boundary conditions are applied along the three
Cartesian directions.
3.3. Dislocation Mobility: Finite-T Simulations
We have estimated the basal mobility of an edge dislocation in our model hcp rare-gas solid at
T6=
0 conditions by performing molecular dynamics (MD) simulations in a simulation cell containing
n
= 18,424 atoms. Fully periodic boundary conditions are employed and a
z
-vacuum slab is introduced
in order to avoid the presence of additional dislocations in the upper and lower edges of the simulation
cell (which otherwise would interact with the principal dislocation). No external stresses are considered
in our MD simulations, which are performed in the canonical,
(N
,
V
,
T)
, microcanonical,
(N
,
V
,
E)
,
and isothermal-isobaric,
(N
,
P
,
T)
ensembles. For the cases in which the volume is fixed, we use the
simulation cell obtained through full geometry relaxation of the system. Likewise, the pressure is set to
zero in the
(N
,
P
,
T)
calculations [
17
]. All simulations are performed with a time step of 10
−3
ps and last
for a total of 800 ps. The position of the (dissociated) edge dislocation is monitored with three different
methods: the differential displacement (DD, Section 2.2.1), the nearest neighbor (NN, Section 2.2.2),
and the common neighbor (CNA, Section 2.2.3). In the DD case, due to the high sensitivity of this
method to thermal fluctuations, we have averaged the positions of the atoms over five consecutive
time steps. In the NN case, for the sake of simplicity, we have averaged the position of the atoms
with nearest neighbor number different from 12, hence we have determined the center position of the
stacking fault; we note that the NN method provides inaccurate results in the
(N
,
P
,
T)
simulations
owing to the fluctuations of the simulation cell, thus that particular case must be disregarded in what
follows (shown here just for completitude).
In Figures 10 and 11, we represent the position of the (dissociated) edge dislocation expressed as
a function of time at a temperature of 25 and 50 K, respectively. It is appreciated that, in spite of the
absence of shear stresses, the dislocation moves at temperatures as low as 25 K, which are well below
the corresponding Debye temperature (
ΘD∼
65 K [
16
]). (Cautiously, we have monitored the size of
the fluctuating stresses in our MD simulations, which are null in average, and checked that in fact

Crystals 2018,8, 64 14 of 19
they are not responsible for the observed dislocation glide (e.g.,
σ
fluctuations
τP
in the
(N
,
P
,
T)
runs). The partial dislocations move either to the left or to the right along the
x
-direction with equal
probability, which is fully consistent with the absence of applied stresses. In our MD simulations,
the partial
dislocations practically remain rigid along the
y
-direction as no kinks or jogs are observed
along their dislocation lines (although for a more detailed analysis of the structural properties of the
mobile dislocations the dimensions of our simulation cell should probably need to be increased). It is
worth noting that, in the
(N
,
V
,
E)
simulations, we have selected
E=
0 and assigned initial velocities
to the atoms reproducing the temperatures chosen in the
(N
,
V
,
T)
and
(N
,
P
,
T)
runs; consequently,
after initializing the
(N
,
V
,
E)
simulations, half of the total kinetic energy is transformed into potential
and the effective temperature of the system is halved (i.e.,
T=
12.5 and 25 K, respectively). In spite of
such a reduction in temperature, one still can observe in Figure 9b that the dislocation remains mobile
in the
(N
,
V
,
E)
simulations. These results clearly make evident a very low resistance of the rare-gas
lattice to
dislocation glide
. From Figures 10 and 11 we can also estimate the width of the fcc-like
stacking fault in our MD simulations,
ωs f
, which corresponds to the position difference between
the two
dislocation cores
.
We enclose
our averaged
ωs f
results in Table 1, expressed as a function of
temperature and simulation ensemble. It is appreciated that all three ensembles provide consistent
results and that, as expected, thermal excitations tend to increase the ωs f fluctuations.
Figure 10.
Position of the dissociated edge dislocation expressed as a function of time for a simulation
cell containing
n
= 18,424 atoms. Molecular dynamics simulations have been performed in the three
thermodynamic ensembles (
a
)
NVT
, (
b
)
NV E
, and (
c
)
NPT
; the temperature has been fixed to
25 K, the pressure to zero, and the volume to the equilibrium one. In the
NV E
case,
an equilibrium
temperature of 12.5 K is reached. “DD”, “NN”, and “CNA” stand for the analysis methods of differential
displacement, nearest neighbors, and common neighbor, respectively.

Crystals 2018,8, 64 15 of 19
Figure 11.
Position of the edge dissociated dislocation expressed as a function of time for a simulation
cell containing
N=
18424 atoms. Molecular dynamics simulations have been performed in the
three thermodynamic ensembles (
a
)
NVT
, (
b
)
NV E
, and (
c
)
NPT
; the temperature has been fixed to
50 K, the pressure to zero, and the volume to the equilibrium one. In the
NV E
case, an equilibrium
temperature of 25 K is reached. “DD”, “NN”, and “CNA” stand for the analysis methods of differential
displacement, nearest neighbors, and common neighbor, respectively.
In Figure 12, we show the time-accumulated average displacement of the dissociated edge
dislocations,
∆x
, expressed as a function of time, temperature, and simulation ensemble. In particular,
we calculate this quantity with the formula:
∆x(t) = |x(t)−x(t−δt)|+∆x(t−δt), (8)
where
x(t)
corresponds to the average position of the atoms belonging to the dislocations along the
x
direction at time
t
(that is, as shown in Figures 10 and 11) and
δt
is equal to a time step in our molecular
dynamic simulations. A series of kinks appear in the figure that are a consequence of the dislocation
cores passing through the boundaries of the simulation cell (that is, due to use of periodic boundary
conditions during the crossing of an edge, some of the atoms belonging to the same dislocation core are
located in one extreme of the box, whereas the rest remain in the opposite boundary).
From the
slope of
the linear fits to the
∆x
data points, which are not affected by such a periodic boundary artifact, we can
deduce the module of the average diffusion velocity of the dislocations,
vd
, as a function of temperature
and simulated thermodynamic ensemble. The results enclosed in Table 1show that simulations
performed both in the
(N
,
V
,
T)
and
(N
,
P
,
T)
ensembles render very similar
vd
values;
by contrast
,

Crystals 2018,8, 64 16 of 19
simulations performed in the
(N
,
V
,
E)
ensemble systematically provide smaller dislocation diffusion
velocities due to the effective reduction in the temperature of the system (
see preceding paragraph
).
Interestingly, our MD results suggest a square root-like dependence of
vd
on temperature,
vd∝√T
.
For instance
, in the
(N
,
V
,
T)
and
(N
,
P
,
T)
cases, we realize that
vd(2T)'√2vd(T)
,
and, when
comparing those results with the values obtained in the
(N
,
V
,
E)
ensemble, we consistently find
that
vNVT
d(T)'√2vNV E
d(T)
. This behavior appears to depart significantly from the Arrhenius-like
relation that is expected for thermally activated dislocations, namely
vd∝exp [−∆G/kBT]
, and which
has been observed in other materials at high temperatures [2,40,41].
0 100 200 300 400 500 600 700
Time
(
ps
)
0
100
200
300
400
500
600
700
800
∆
x
(
Å
)
NVT (25K)
NPT (25K)
NVE (25K)
NVT (50K)
NPT (50K)
NVE (50K)
Figure 12.
Time-accumulated average displacement of the dissociated edge dislocations expressed as
a function of time, temperature, and simulated thermodynamic ensemble (
n= 18,424 atoms
). Solid
lines represent the actual dislocation positions and dashed lines are linear fits performed on regions in
which the dislocation motion is not disturbed by the simulation cell boundaries. Dislocation diffusion
velocities are deduced directly from the slope of the linear fits.
Table 1.
Average width of the fcc-like stacking fault,
ωs f
, and edge dislocation diffusion velocity,
vd
,
expressed as a function of temperature, and simulated thermodynamic ensemble (
n
= 18,424 atoms).
ωs f
results are expressed in units of lattice parameter
a
and the figures within parentheses indicate the
corresponding statistical uncertainty.
T=25K
vd(m/s)ωs f (a)
NVT 57.2 (0.5)12.0 (0.3)
NVE 37.0 (0.5)12.1 (0.6)
NPT 53.1 (0.5)12.1 (0.6)
T=50K
vd(m/s)ωs f (a)
NVT 79.6 (0.5)11.9 (3.3)
NVE 56.2 (0.5)11.9 (3.3)
NPT 76.1. (0.5)11.9 (3.3)
It is physically insightful to compare the
vd
values obtained in our model rare-gas solid with
those reported for other materials with the hcp structure. In the case of Zr, Khater and Bacon have
estimated edge dislocation velocities within the basal plane of about 100 ms
−1
at room temperature
and practically vanishing applied shear stresses (see Figure 7a in Ref. [
12
]). In the present case, similar
vd
values are obtained already at a much lower temperature of 50 K and nominally zero mechanical
stress. This comparison comes to show that edge dislocations in classical hcp rare-gas solids are much
more mobile than in structurally analogous metals. The origins of such differences may reside on the

Crystals 2018,8, 64 17 of 19
interatomic interactions (mind that the atomic masses of Zr and Xe atoms are roughly comparable),
which in the case of rare gases are extremely weak [33].
4. Conclusions
We have presented a comprehensive computational study on the structural and mobility
properties of edge dislocations in classical rare-gas solid with the hcp structure. We have shown
that dissociation of edge dislocations into Shockley partials, as induced by the presence of a low-energy
metastable stacking fault, is a common process in hcp rare-gas crystals. On the other hand, we have
inferred that quantum nuclear effects further enhance the dissociation of edge dislocations into
partials as they tend to decrease the energy of the actual stacking fault. A dislocation-related quantity
that indirectly appears to be drastically affected by quantum nuclear effects is the Peierls stress,
τP
.
While we have calculated a Peierls stress value of the order of 1 MPa in our model classical rare-gas
crystal, other researchers have estimated a practically vanishing
τP
in archetypal quantum crystal
hcp
4
He. Meanwhile, the mobility of edge dislocations in rare-gas solids in general is very large,
owing to the characteristic weak interactions between atoms. In the present case, we have found that
glide of dislocations can be activated at temperatures as low as
∼
10 K in the absence of any applied
shear stress, achieving large dislocation diffusion velocities of the order of 10 ms
−1
. Furthermore,
our molecular dynamics results suggest that the diffusion velocity of edge dislocations depends on
temperature as the square root, namely
vd∝√T
, in contrast to what has been observed in other
crystals.
The conclusions
presented in this study provide valuable new insights into the structure
and mobility of edge dislocations in rare-gas solids, and allow for a quantitative assessment of the
importance of quantum nuclear effects in solid 4He dislocation behavior.
Acknowledgments:
This research was supported by the Australian Research Council under the Future Fellowship
funding scheme (Grant No. FT140100135). J.B. and A.S. acknowledge financial support from MINECO (Spain)
Grants No. FIS2014-56257-C2-1-P and No. FIS2015-69017-P. Computational resources and technical assistance
were provided by the Australian Government and the Government of Western Australia through Magnus under
the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre.
Author Contributions:
S.S., C.C., and J.B. conceived and designed the study; S.S. and C.C. performed the
calculations; S.S., C.C., A.S. and J.B. analyzed the results and wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
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