New solid phase of dipolar systems
The systems of molecules with a permanent dipole moment have solid phases with various crystal symmetries. In particular, the solid phases of the simplest of these systems, the dipolar hard sphere model, have been extensively studied in the literature. The article presents Monte Carlo simulation re...
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irk-123456789-1570062019-06-20T01:28:05Z New solid phase of dipolar systems Levesque, D. The systems of molecules with a permanent dipole moment have solid phases with various crystal symmetries. In particular, the solid phases of the simplest of these systems, the dipolar hard sphere model, have been extensively studied in the literature. The article presents Monte Carlo simulation results which, at low temperature, point to the stability of a polarized solid phase of dipolar hard spheres with the unusual number of eleven nearest neighbors, the so-called primitive tetragonal packing or tetragonal close packing. Системи молекул з постiйним дипольним моментом мають твердi фази з кристалами рiзної симетрiї. Зокрема, твердi фази найпростiшої з таких систем, моделi дипольних твердих сфер, широко вивченi у лiтературi. В статтi представленi результати моделювання методом Монте-Карло, якi при низькiй температурi вказують на стабiльнiсть поляризованої твердої фази дипольних твердих сфер з незвичайним числом одинадцяти найближчих сусiдiв, так званої примiтивної тетрагональної упаковки або тетрагональної закритої упаковки. 2017 Article New solid phase of dipolar systems / D. Levesque // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33601: 1–8 . — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 61.50.Ah, 64.70.K-, 81.30.Dz DOI:10.5488/CMP.20.33601 arXiv:1710.01100 http://dspace.nbuv.gov.ua/handle/123456789/157006 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The systems of molecules with a permanent dipole moment have solid phases with various crystal symmetries.
In particular, the solid phases of the simplest of these systems, the dipolar hard sphere model, have been extensively studied in the literature. The article presents Monte Carlo simulation results which, at low temperature,
point to the stability of a polarized solid phase of dipolar hard spheres with the unusual number of eleven
nearest neighbors, the so-called primitive tetragonal packing or tetragonal close packing. |
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Levesque, D. New solid phase of dipolar systems Condensed Matter Physics |
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Levesque, D. |
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Levesque, D. |
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New solid phase of dipolar systems |
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New solid phase of dipolar systems |
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New solid phase of dipolar systems |
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New solid phase of dipolar systems |
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New solid phase of dipolar systems |
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new solid phase of dipolar systems |
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Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/157006 |
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New solid phase of dipolar systems / D. Levesque // Condensed Matter Physics. — 2017. — Т. 20, № 3. — С. 33601: 1–8
. — Бібліогр.: 22 назв. — англ. |
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Condensed Matter Physics |
| work_keys_str_mv |
AT levesqued newsolidphaseofdipolarsystems |
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2025-07-14T09:21:14Z |
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Condensed Matter Physics, 2017, Vol. 20, No 3, 33601: 1–8
DOI: 10.5488/CMP.20.33601
http://www.icmp.lviv.ua/journal
New solid phase of dipolar systems
D. Levesque
Laboratoire de Physique Théorique, CNRS, Université de Paris-Sud, Université de Paris-Saclay,
Bâtiment 210, 91405 Orsay Cedex, France
Received April 12, 2017, in final form May 17, 2017
The systems of molecules with a permanent dipole moment have solid phases with various crystal symmetries.
In particular, the solid phases of the simplest of these systems, the dipolar hard sphere model, have been exten-
sively studied in the literature. The article presents Monte Carlo simulation results which, at low temperature,
point to the stability of a polarized solid phase of dipolar hard spheres with the unusual number of eleven
nearest neighbors, the so-called primitive tetragonal packing or tetragonal close packing.
Key words: simulation, solid phases, dipolar hard sphere
PACS: 61.50.Ah, 64.70.K-, 81.30.Dz
1. Introduction
This work is dedicated to Jean-Pierre Badiali and our friendly collaboration.
The solid phases of dipolar spherical molecules have been studied both by numerical simulations
[1–4] and theoretical approaches [5, 6]. At low temperatures, these phases are polarized and ferroelectric.
They have crystal symmetries seemingly induced by the strong dipolar head-to-tail interaction which,
in polarized phases at these temperatures, modifies the unit cells of the body centered cubic (bcc), face
centered cubic (fcc) or hexagonal close packed (hcp) lattices which are expected to be the stable solid
phases of particles with a spherical symmetry. Hence, on the basis of the density functional theory,
following the density, temperature and dipolar moment values, the body centered tetragonal (bct), body
centered orthorhombic (bco), face centered tetragonal or orthorhombic lattices have been estimated to
be the stable solid phases of the Stockmayer system as well as the fundamental states of the dipolar soft
sphere system [7]. Furthermore, the ground state of electrorheological fluids has been investigated; in
[8, 9], supposing that the ground state lattice is one among simple cubic, fcc, bcc, hcp or bct, it was
established that the bct lattice has a lower energy.
The solid phases of the dipolar hard sphere (DHS) system have been studied in references [10–12] by
numerical simulations. However, at the solid phase density, in Monte Carlo (MC) simulations realized
in the canonical (NVT) ensemble, the constant value and shape of the volume with periodic boundary
conditions and the hard core pair potential strongly preclude in the MC sampling the initial configuration,
of a given crystal symmetry, to evolve towards the configurations of a lattice with a different symmetry
and lower free energy. The MC simulations in the isothermal constant pressure (NpT) ensemble where
the size and shape of the volume with periodic boundary conditions change, seem to make easier the
transition between the initial configuration and the configurations of a thermodynamically more stable
lattice. In practice, a very tight packing of dipolar hard spheres in solid polarized phases slows or thwarts
such an evolution. Furthermore, a relative stability of phases estimated from their free energies is not
obtained from the NVT or NpT simulations.
In order to circumvent this last shortcoming of the MC sampling in the NVT and NpT ensembles and
then to remove the uncertainty on the relative thermodynamic stability of solid polarized phases, recent
works [11–13] have estimated the free energy differences between solid phases of specified symmetry for
the DHS system. These studies rely on the thermodynamic integration scheme [14] and on the Jarzynski
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
33601-1
https://doi.org/10.5488/CMP.20.33601
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
D. Levesque
relation [15, 16]. In the two approaches, the free energy difference ∆FBA = FB − FA between two
equilibrium states A and B of a system at density ρ and temperature T is computed. ∆FBA is given in the
thermodynamic integration scheme by:
∆FBA =
λB∫
λA
dλ
〈 ∂Uλ
∂λ
〉
λ
. (1.1)
Here, 〈. . .〉λ is the canonical average of the partial derivative with respect to λ of the internal energy Uλ
of the system evolving through a set of equilibrium states from the state A to state B when the parameter
λ varies from λA to λB. Such a computation of ∆FBA supposes that the transition from UλA to UλB can
be made by using a parametrisation with a unique parameter and that the number of degrees of freedom
stays identical in the two states A and B. For the computation of solid phase free energies in [11, 12], the
internal energyUλA is that of a solid harmonic model whileUλB is that of a system of particles interacting
by a pair potential, the crystal symmetries of the A, intermediate and B states being supposed identical.
However, since in the harmonic solid model (state A), the particles do not have rotational degrees of
freedom, also, in state B, the particles do not have such degrees of freedom. Such a constraint, if the
state B corresponds to a system of polar particles, implies that the particle dipole moments remain fixed
(cf. [12]).
The Jarzynski relation writes:
〈exp(−W/kBT)〉 = exp(−∆FBA/kBT), (1.2)
in which the average 〈exp(−W/kBT)〉, where kB is the Boltzmann constant, is estimated from the works
W effectuated along a set of paths inducing a transition between the states A and B, paths which can
involve stable and metastable states. However, clearly, the absolute free energy of state B will be obtained
only when that of the state A is known. In [13], ∆FBA is estimated from equation (1.2) by computing the
work W needed to progressively modify the volume shape of systems with periodic boundary conditions
from an initial shape compatible with the lattice symmetry of state A to a shape compatible with the
lattice symmetry of state B.
In both computation methods of ∆FBA, defects can appear in the particle arrangements when λ or
the volume shape vary. In particular, in the ∆FBA evaluation scheme based on the Jarzynski relation, it is
needed to check if the state B has the expected symmetry. It is one of the aims of this work to discuss to
what extent this identification is unambiguous.
Section 2 is devoted to presenting the details of the tests which can be used to proceed to this
identification. In section 3, it is shown on the basis of these tests that at high density and high dipole
moment values, a DHS stable solid phase can be of a tetragonal close packing (tcp) type.
2. Symmetry tests
The MC simulations for estimating 〈exp(−W/kBT)〉 were realized for systems of parallelepipedic
volume V with periodic boundary conditions and N ∼ 1000 dipolar hard spheres interacting by the
potential
v(ri j, si, sj) = vhs(ri j) +
µ2
r3
i j
[
si · sj −
3si · (ri − rj)sj · (ri − rj)
r2
i j
]
. (2.1)
vhs(r) is a hard sphere potential of diameterσ and the second term on the r.h.s. of equation (2.1) the dipolar
interaction. ri and si are, respectively, the vector position of the sphere i and a unit vector associated
with the orientation of its dipole moment µi = µ si , µ = |µi | and ri j = |rj − ri |. The reduced density
ρ∗ = ρσ3 and the reduced dipole moment µ∗ = µ/
√
kBTσ3 characterize the states of the DHS system.
Furthermore, the value of N and the periodic boundary conditions must be chosen such that they are
compatible with the crystal symmetry of the initial state A and that expected for the final state B.
From the pressure tensor associated with the interparticle interactions v(ri j, si, sj) which includes
both the contributions of vhs(ri j) and dipolar interaction, it is straightforward, as described in [17–19],
33601-2
New solid phase of dipolar systems
to compute the work needed to change the shape of a DHS system by a series of small homogeneous
deformations. These deformations generate the transition between states A and B of the same V , N and
T values but with different periodic boundary conditions. Obviously, there are many types of such paths
allowing one to generate a transition between the volumes of the same size but with different shapes.
In [13], to maintain the system in a solid phase and to minimize the formation of defects, the transition
paths between the states A and B at constant T prevented the system from melting by keeping the density
variation in the limit of a few percent with respect to the identical density of the initial states A and final
states B.
The identification of the lattice symmetry for a configuration relies on the pair distribution func-
tion g(r), the structure factor, the positions and numbers of particles present in the neighboring shells,
as well as on the types of Voronoi polyhedra associated with the nearest neighbors of a particle. Clearly,
these properties are not independent, but their simultaneous use allows one to reduce the ambiguities
resulting from the fluctuations in the local arrangement of the particles.
For instance, to determine the crystal symmetry of DHS configurations at ρ∗ = 1.06−1.26 and
µ∗ = 1.0−3.0, a simple way is to compare their two-body distribution functions g(r) with those of
solid HS systems, computed by MC simulations in the canonical ensemble for solid states of the known
symmetry. Indeed, as mentioned above, at these densities, the constant volume and hard core packing
effects seem a priori to preclude any symmetry change. Figure 1 shows, at these densities, such results
for µ∗ = 0.0, i.e., a HS system, obtained after 40 · 106 MC trial moves in the NVT ensemble, when, in the
initial configurations, the HS are located on a bcc, bct, fcc or hcp lattice. The functions g(r) of equilibrium
bcc configurations at both densities clearly differ from those computed for the other crystal symmetries,
while the positions of the two first peaks in g(r) at r < 1.6 for the bct, fcc and hcp configurations are
almost identical, the third peaks at r = 1.8−1.9 differing only by their amplitudes. It is only for r > 1.9
that, in particular, the g(r) functions of the fcc solid are clearly distinct from those of the bct and hcp
solids, whereas the bct and hcp g(r) functions stay more similar.
Additional information to interpret these g(r) data can be obtained by computing the number of
neighbors nk involved in the g(r)’s k-th peak and, also, the average locations of the neighbors around
one HS in these equilibrium solid configurations. nk is defined by
nk = 4πρ
r
sup
k∫
r inf
k
g(r)r2dr, (2.2)
Figure 1. (Color online) Left-hand panel. At ρ∗ = 1.06, pair distribution functions g(r) of a HS
system obtained from MC NVT simulations in volumes with periodic boundary conditions and initial
configurations corresponding to bcc lattice: black dot and solid line, bct lattice: red square and solid line,
fcc lattice: green diamond and solid line, hcp lattice: blue up triangle and solid line. Right-hand panel.
Same at ρ∗ = 1.26.
33601-3
D. Levesque
where r inf
k
and rsup
k
are the positions of the g(r) local minima surrounding its k-th peak. For a configuration
of N HS, the average position r̄i of the i-th neighbor can be defined as:
r̄i =
∑
j=1,N |rj − rji |
N
, (2.3)
where rj is the position of the HS particle j, while rji is that of its i-th neighbor sorted into the increasing
values of |rj − rji |, so i = 1 is the nearest neighbor, i = 2 the second nearest neighbor, . . . . These
average positions indicate whether the numbers of particles in the g(r) peaks are in agreement with those
expected in the neighboring shells for the symmetry of the considered solid phase. Indeed, due to the
density fluctuations, the neighbor average positions in a shell present a dispersion reflecting the g(r) peak
width and, in the r̄i graphs, the neighboring shells are indicated by a gap or a variation of the curves slope.
Results of r̄i computations are presented in figures 2 and 3 for the equilibrium HS configurations
at ρ∗ = 1.06 and 1.26 obtained from HS initial configurations located on a bcc, bct, fcc or hcp lattice.
At ρ∗ = 1.06, figure 2 clearly shows the important dispersion of the first, second and third neighbor
shells for the noncompact initial bcc and bct configurations. From the r̄i plot for the bcc equilibrium
configurations, up to 2.5, it appears that the two first shells overlap and the 4 to 6 th shells are largely
spread. Furthermore, the nk values indicate that the g(r)’s first peak involves 14 neighbors and the second
peak involves 46 neighbors. Similarly, in the bct configuration, the two first shells overlap, the g(r) first
peak involving 12 neighbors and the second, third and fourth peak 6, 20 and 18 neighbors, respectively.
In the compact fcc and hcp configurations, the two first peaks in g(r) correspond to 12 and 6 neighbors
in agreement with the values expected for a perfect crystal. This agreement holds for the third, fourth and
fifth shells in the fcc configuration in spite of the shell broadening. In the hcp configuration, the third
to sixth shells overlap to form the g(r) third peak. For this solid low density, figures 1 and 2 give clear
evidence that in the bct configuration, the local structure of the three first neighboring shells rearranges
towards that found for the hcp configuration.
Figure 2. (Color online) For a HS system at ρ∗ = 1.06, comparison of the r̄i values for bcc, bct, fcc and
hcp configurations: black dots perfect lattice, red dots: equilibrium configuration.
33601-4
New solid phase of dipolar systems
Figure 3. (Color online) For a HS system at ρ∗ = 1.26, comparison of the r̄i values for bcc, bct, fcc and
hcp configurations: black dots perfect lattice, red dots: equilibrium configuration.
At ρ∗ = 1.26 compared to ρ∗ = 1.06, the shell spreading is reduced. In the fcc g(r) plot, the peak
positions and widths correspond to the positions and expected numbers of neighbors of a perfect fcc solid
up to r > 3.0. For the hcp configuration, the g(r) third and fourth peaks result from the overlaps of the
third and fourth shells, and of the fifth and sixth shells, respectively. Clearly, the HS local arrangement in
the equilibrium bct configuration is very similar, almost identical to that obtained in the equilibrium hcp
configurations. In the equilibrium bcc configurations, the g(r) first and second peaks involve together
∼ 14 neighbors, about 10−11 in the first and 3−4 in the second peak, these neighbor numbers and the
position of the second peak being similar to those expected in a bct crystal.
The Voronoi tessellation of the HS considered solid configurations give little additional information
on their lattice symmetry. In the fcc and hcp configurations, at both densities, the face, edge and vertice
numbers of the Voronoi polyhedra significantly differ for most HS from the values characterising the
polyhedra in the fcc and hcp perfect lattices. A similar result is obtained for the bct configurations. It is
only for the bcc configuration that the 14 HS neighbors, present in the g(r) first peak at ρ∗ = 1.06 and the
first and second peaks at ρ∗ = 1.26, delimit, for more than 50% of HS, a Voronoi polyhedra with face,
edge and vertice numbers in agreement with the polyhedra of a bcc perfect lattice.
The analysis of HS solid configurations with regard to the identification of their crystal symmetry,
shows that in solid noncompact configurations, the HS local arrangement, compared to that of perfect
lattices, can be strongly modified by a large overlap of the neighboring shells. These modifications
can induce an evolution of the local arrangement towards that of more stable configurations, as seems
to happen for the bct configurations, where the local arrangement becomes of hcp type at the two
considered densities. Noticeably, in spite of the nearest neighbor shell spreading, in the tessellation of the
bcc configurations, most Voronoi polyhedra have 6 faces with 4 edges and 8 faces with 6 edges disposed
as in the polyhedra of the bcc and bct perfect lattices.
In conclusion, the association of g(r) data with computed values of nk and r̄i and, possibly, the
Voronoi tessellation allows one to obtain, at the equilibrium in a solid phase, a definite characterization
of the HS local arrangement, but, of the crystal symmetry only for HS compact configurations.
33601-5
D. Levesque
3. Symmetry of DHS polarized solids
The previous analysis made on the HS solid configurations is used now to study the symmetries of
the DHS configurations at ρ∗ = 1.26 with µ∗ , 0, the aim being to achieve a better characterization of
these symmetries than in [13]. In the latter work, as mentioned above, the symmetries were supposed to
be those associated with the periodic boundary conditions of the volume V enclosing the DHS system in
the final states B of the transition paths used to compute ∆FBA by equation (1.2). In these computations,
N , the numbers of DHS in V and periodic boundary conditions were chosen in such a way that the B
states can be bct, bco, fcc or hcp configurations without defects. By comparing the ∆FBA values obtained
with respect to an identical initial A state, namely a bcc configuration at equilibrium, it was possible to
estimate the relative stability of the B states of different symmetries.
Hence, it was obtained that the more stable configuration at ρ∗ = 1.26 at µ∗ = 1.0 was bcc and at
µ∗ = 2.0 and 3.0 bco with a ratio of the volume edges Ly/Lx along the y and x directions ∼ 1.10. The
g(r) functions of such configurations are given in figure 4. In figure 5, similarly to figures 2 and 3, there
are given the numbers of neighbors present in the peaks of these functions. At µ∗ = 1.0, as in the HS
system at ρ∗ = 1.26, the first shells are modified. The first peak involves 11 neighbors and the second,
third and fourth peaks 3, 6 and 12 neighbors. This important modification of the local arrangement with
respect to that of the bcc crystal remains, as for the HS configuration, compatible with a tessellation made
of Voronoi polyhedra with 14 faces (8 with 6 edges and 6 with 4 edges) typical of the tessellation of bcc
and bct crystals. These polyhedra around a specific DHS in the configuration are strongly distorted, but
when the N polyhedra are averaged, the average polyhedra are close to that of a bcc perfect crystal. On the
basis of this latter result, it seems still reasonable to label these unpolarized DHS configurations as bcc.
For µ∗ = 2.0, Ly/Lx = 1.10 and ρ∗ = 1.26, the bco local arrangement is formed, in the perfect crystal,
by shells of 2, 8, 2, 2, 4, 4, 8, 4, . . . neighbors, which are located at 1.019, 1.021, 1.190, 1.309, 1.567,
1.659, 1.766, 1.769, . . . . It is characterized, possibly very close, by shells of only a few neighbors which
are expected largely to overlap due to the local fluctuation densities. Figure 4 shows how the noncompact
bco structure is modified by the DHS dipolar interaction. The values of the number of neighbors in the
first g(r) peaks are equal to 12, 6, 20, 18, 14, 16, . . . values close to those obtained up to the fifth peak
for the HS g(r) hcp function at ρ∗ = 1.26: 12, 6, 20, 18, 12, 20, . . . . A similar agreement is found for the
peak positions. These data show that in the DHS bco configuration at µ∗ = 2.0 and ρ∗ = 1.26, similar
to the HS bct configurations at this density (cf. figure 3), the DHS local arrangement evolves towards an
arrangement typical of an hcp configuration.
Figure 4. (Color online) At ρ∗ = 1.26 and µ∗ = 1.0, 2.0 and 3.0, pair distribution functions g(r) of DHS
configurations with the lowest value of ∆FBA. µ∗ = 1.0 — black line, µ∗ = 2.0 — red line, µ∗ = 3.0 —
green line. Blue line: g(r) of tcp equilibrium configuration obtained by MC NVT simulation at µ∗ = 3.0.
33601-6
New solid phase of dipolar systems
Figure 5. (Color online) For a DHS system at ρ∗ = 1.26 and µ∗ = 1.0, 2.0 and 3.0, comparison of the r̄i
values for the configurations with minimal ∆FBA values. Left-hand panel, black dots: bcc configuration
(minimal ∆FBA at µ∗ = 1.0), red dots: configuration (state B) obtained at the end of the transition path
bcc-bco Ly/Lx = 1.10 (minimal ∆FBA at µ∗ = 2.0). Right-hand panel, green dots: configuration (state
B) obtained at the end of the transition path bcc-bco Ly/Lx = 1.10 (minimal ∆FBA at µ∗ = 3.0). Blue
dots: r̄i values of the tcp perfect lattice.
Clearly, figure 4 and figure 5 establish how the characteristics of the equilibrium DHS configurations
at µ∗ = 3.0 and ρ∗ = 1.26 differ from those of bcc, fcc or hcp configurations. The g(r) function has 8
well defined peaks between r = 1.0 and r = 2.2 corresponding to the nk values of 11, 2, 3, 4, 14, 8, 10,
8 neighbors. The maximum positions of these peaks are, respectively, 1.0, 1.26, 1.46, 1.62, 1.78, 1.92,
2.04 and 2.16. These nk values and positions are in agreement with those expected at low temperature for
a configuration, referred to as primitive tetragonal close packing (tcp) in the literature [20–22]. In this type
of configuration, the nine first shells correspond to 11, 2, 3, 4, 4, 10, 8, 8, 2, 8 neighbors, and are located,
for the considered density, at 1.029, 1.260, 1.455, 1.627, 1.757, 1782, 1.901, 2.036, 2.058, and 2.161.
The fifth and sixth shells and eighth and ninth shells, considering their respective very close positions,
are expected, at finite temperature, to overlap and, hence, leading to the 14 and 10 neighbors in the g(r)
sixth and eighth peaks. The g(r) computed by MC simulation for DHS system in tcp configurations and
plotted in figure 5, unambiguously confirms a full agreement between the g(r) functions of a tcp solid
and those of the states B with bco periodic boundary condition obtained at the end of the transition paths.
In [13], it was concluded from ∆FBA computations that these configurations with local tcp symmetry
were the most stable among the equilibrium configurations with bct, bco, fcc and hcp periodic boundary
conditions obtained from transition paths starting from bcc configurations. In addition, in this reference,
this result was supported by the fact that in NpT simulations, at µ∗ = 3.0 and ρ∗ = 1.26, the bct initial
configurations also evolved to configurations with g(r) functions similar to those of a tcp solid (cf. figure 4
of [13]).
From this analysis of the local arrangements made on the configurations estimated to be the most
stable at µ∗ = 3.0 and ρ∗ = 1.26, it is possible to conclude that in this domain of density and µ∗ value,
the DHS solid phase has a tcp symmetry.
4. Conclusion
In section 2, it was shown that the identification of the lattice symmetry of noncompact HS solid
phases made only on the basis of the g(r) functions is not obvious. The local arrangement in such phases
around a given HS, presents large overlaps of the nearest neighbor shells, so that the nk values of the g(r)
peaks for r < 2.0 do not correspond to those expected in bcc or bct solid phases, but rather those of fcc
33601-7
D. Levesque
or hcp phases. However, to determine if these HS noncompact configurations eventually would evolve
towards compact configurations without defects of fcc or hcp type, prohibitively long simulations seem
required. A similar uncertainty is encountered to characterise the lattice symmetry of DHS bct or bco
configurations, as shown above for the DHS systems at µ∗ = 1.0 and 2.0 and ρ∗ = 1.26. Thus, as already
mentioned, in [13], these configurations with the lower values of ∆FBA were labelled according to the
periodic boundary conditions of the states terminating the transition paths. Then, it is remarkable that,
for µ∗ = 3.0 and ρ∗ = 1.26, on the basis of the excellent agreement between the g(r), nk and r̄i values
with those of tcp solid phase, it can be concluded almost unambiguously that this phase is a stable phase
of the DHS system.
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Нова тверда фаза дипольних систем
Д. Левек
Лабораторiя теоретичної фiзики, Нацiональний центр наукових дослiджень Францiї, Унiверситет
Парi-Сюд, Унiверситет Парi-Саклє, Орсе, Францiя
Системи молекул з постiйним дипольним моментом мають твердi фази з кристалами рiзної симетрiї.
Зокрема, твердi фази найпростiшої з таких систем, моделi дипольних твердих сфер, широко вивченi у
лiтературi. В статтi представленi результати моделювання методом Монте-Карло, якi при низькiй тем-
пературi вказують на стабiльнiсть поляризованої твердої фази дипольних твердих сфер з незвичайним
числом одинадцяти найближчих сусiдiв, так званої примiтивної тетрагональної упаковки або тетраго-
нальної закритої упаковки.
Ключовi слова: комп’ютерне моделювання, твердi фази, дипольна тверда сфера
33601-8
https://doi.org/10.1103/PhysRevLett.68.2043
https://doi.org/10.1103/PhysRevA.46.7783
https://doi.org/10.1103/PhysRevE.48.3728
https://doi.org/10.1103/PhysRevE.49.5131
https://doi.org/10.1063/1.448764
https://doi.org/10.1063/1.480531
https://doi.org/10.1103/PhysRevE.63.021203
https://doi.org/10.1103/PhysRevLett.67.398
https://doi.org/10.1103/PhysRevA.46.R719
https://doi.org/10.1103/PhysRevE.61.R2188
https://doi.org/10.1103/PhysRevLett.94.138303
https://doi.org/10.1103/PhysRevE.72.051402
https://doi.org/10.1080/00268976.2011.610368
https://doi.org/10.1063/1.448024
https://doi.org/10.1103/PhysRevLett.78.2690
https://doi.org/10.1103/PhysRevE.56.5018
https://doi.org/10.1063/1.1605941
https://doi.org/10.1063/1.2202352
https://doi.org/10.1063/1.2214719
https://doi.org/10.1016/0025-5408(81)90051-9
https://doi.org/10.1107/S0567740882007547
https://doi.org/10.1016/0022-4596(83)90192-5
Introduction
Symmetry tests
Symmetry of DHS polarized solids
Conclusion
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