On the role of distortion in the hcp vs fcc competition in rare-gas solids
As a prototype of initial or intermediate structure somewhere in between the hcp and fcc lattices we consider a distorted bcc crystal. We calculate the temperature and pressure dependences of the lattice parameters for heavier rare gas solids Ar, Kr, Xe in the quasiharmonic approximation applying th...
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irk-123456789-1184802017-05-31T03:03:28Z On the role of distortion in the hcp vs fcc competition in rare-gas solids Krainyukova, N.V. 8th International Conference on Cryocrystals and Quantum Crystals As a prototype of initial or intermediate structure somewhere in between the hcp and fcc lattices we consider a distorted bcc crystal. We calculate the temperature and pressure dependences of the lattice parameters for heavier rare gas solids Ar, Kr, Xe in the quasiharmonic approximation applying the Aziz potentials and confirm that in line with the previously found prevalence of hcp over fcc the hcp structure is still dominant in the bulk over the wide P–T ranges analyzed. The situation is different for confined clusters up to 105 atoms where owing to the specific surface energetics and terminations the structures with five-fold symmetry comprising fcc fragments are dominant. As a next step we consider a free relaxation of differently distorted bcc clusters, and show that two types (monoclinic and orthorhombic) of the initial distortion and its degree is a driving force for the hcp vs fcc final realizations. Possible energetic links between the initial and final structures are shown and analyzed. 2011 Article On the role of distortion in the hcp vs fcc competition in rare-gas solids / N.V. Krainyukova // Физика низких температур. — 2011. — Т. 37, № 5. — С. 547–550. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 61.50.Ah, 61.50.Ks, 64.70.K–, 71.15.Nc http://dspace.nbuv.gov.ua/handle/123456789/118480 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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8th International Conference on Cryocrystals and Quantum Crystals 8th International Conference on Cryocrystals and Quantum Crystals |
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8th International Conference on Cryocrystals and Quantum Crystals 8th International Conference on Cryocrystals and Quantum Crystals Krainyukova, N.V. On the role of distortion in the hcp vs fcc competition in rare-gas solids Физика низких температур |
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As a prototype of initial or intermediate structure somewhere in between the hcp and fcc lattices we consider a distorted bcc crystal. We calculate the temperature and pressure dependences of the lattice parameters for heavier rare gas solids Ar, Kr, Xe in the quasiharmonic approximation applying the Aziz potentials and confirm that in line with the previously found prevalence of hcp over fcc the hcp structure is still dominant in the bulk over the wide P–T ranges analyzed. The situation is different for confined clusters up to 105 atoms where owing to the specific surface energetics and terminations the structures with five-fold symmetry comprising fcc fragments are dominant. As a next step we consider a free relaxation of differently distorted bcc clusters, and show that two types (monoclinic and orthorhombic) of the initial distortion and its degree is a driving force for the hcp vs fcc final realizations. Possible energetic links between the initial and final structures are shown and analyzed. |
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Article |
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Krainyukova, N.V. |
| author_facet |
Krainyukova, N.V. |
| author_sort |
Krainyukova, N.V. |
| title |
On the role of distortion in the hcp vs fcc competition in rare-gas solids |
| title_short |
On the role of distortion in the hcp vs fcc competition in rare-gas solids |
| title_full |
On the role of distortion in the hcp vs fcc competition in rare-gas solids |
| title_fullStr |
On the role of distortion in the hcp vs fcc competition in rare-gas solids |
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On the role of distortion in the hcp vs fcc competition in rare-gas solids |
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on the role of distortion in the hcp vs fcc competition in rare-gas solids |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2011 |
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8th International Conference on Cryocrystals and Quantum Crystals |
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| citation_txt |
On the role of distortion in the hcp vs fcc competition in rare-gas solids / N.V. Krainyukova // Физика низких температур. — 2011. — Т. 37, № 5. — С. 547–550. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT krainyukovanv ontheroleofdistortioninthehcpvsfcccompetitioninraregassolids |
| first_indexed |
2025-07-08T14:04:27Z |
| last_indexed |
2025-07-08T14:04:27Z |
| _version_ |
1837087827091783680 |
| fulltext |
© N.V. Krainyukova, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 5, p. 547–550
On the role of distortion in the hcp vs fcc competition
in rare-gas solids
N.V. Krainyukova
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: ninakrai@yahoo.com
Received December 10, 2010
As a prototype of initial or intermediate structure somewhere in between the hcp and fcc lattices we consider
a distorted bcc crystal. We calculate the temperature and pressure dependences of the lattice parameters for
heavier rare gas solids Ar, Kr, Xe in the quasiharmonic approximation applying the Aziz potentials and confirm
that in line with the previously found prevalence of hcp over fcc the hcp structure is still dominant in the bulk
over the wide P–T ranges analyzed. The situation is different for confined clusters up to 105 atoms where owing
to the specific surface energetics and terminations the structures with five-fold symmetry comprising fcc frag-
ments are dominant. As a next step we consider a free relaxation of differently distorted bcc clusters, and show
that two types (monoclinic and orthorhombic) of the initial distortion and its degree is a driving force for the hcp
vs fcc final realizations. Possible energetic links between the initial and final structures are shown and analyzed.
PACS: 61.50.Ah Theory of crystal structure, crystal symmetry; calculations and modeling; crystal growth;
61.50.Ks Crystallographic aspects of phase transformations; pressure effects;
64.70.K– Solid-solid transitions;
71.15.Nc Total energy and cohesive energy calculations.
Keywords: rare-gas solids, phase transformations, lattice distortion.
1. Introduction
The problem of the hcp vs fcc competition in rare gas
solids (RGS) has a long history [1]. Only some specific
theories based on a first-principle local density approxima-
tion [2,3] or density-functional perturbation theory [4] con-
firm that the fcc structure may be expected at ambient
pressures (in line with experiment) but it is still unclear
why the theories based on the pair potentials like Lennard-
Jones (LJ) predict that the hcp structure is the only possi-
bility while experimental observations evidence in favor of
the absolutely dominant fcc structure. Some certain pro-
gress is attained accounting not only two- but also many-
body interactions [1,5,6]. Alternatively the explanation of
such a discrepancy based on the specific surface energetics
and terminations elaborated in particular in our works
[7,8], which show that the structures with five-fold sym-
metry comprising fcc fragments are dominant up to
105 atoms, is also valid. The transformations of the fcc
structure towards the hcp meet large barriers, therefore for
larger sizes they are impeded. Within last two and half
decades several experimental groups observed the transi-
tion from the fcc to the hcp structure in RGS under high
pressures [9–12]. These observations have in particular
revealed some features, which are not typical for usual
structural phase transitions. One is the sluggishness of such
transformations, which stretch within several dozens GPa,
the other is a persistence of such transitions until their
completion. The purpose of our work is to make the bulk-
related calculations within a relatively wide P–T range (in
order to frame some important trends) applying more rea-
listic Aziz potentials, to analyze the mechanisms of the
fcc–hcp transformations elaborating some new ideas and to
discuss the possibilities of their experimental confirmation.
2. The energetics of rare-gas solids
It is well known that for the LJ potential the hcp is more
favorable than the fcc by ~0.01% [1]. We try to analyze the
same with the Aziz pair potential:
2
6,8,10
2
( ) [exp (α β ) ( ) ] ,
exp [ ( * / 1) , *( ) ,
1, *,
n
c n
c
U R R R f R C R
r R R rf R
R r
−
∞
= − − γ −
⎧ − − <⎪= ⎨
>⎪⎩
∑ ∑
N.V. Krainyukova
548 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5
here R is an interatomic distance, α, β, γ, Cn and r* are
constants, which can be found for Ar, Kr in [13] and for
Xe in [14]. We consider the free energy (per atom):
kin( )F U r F PV= + + , (1)
where the potential energy U(r) is calculated applying the
Aziz potential and a summation runs over the infinite crys-
tal; P, V and T are pressure, volume (per atom) and tem-
perature respectively. We are searching for a minimum of
F. The kinetic energy Fkin is calculated within the quasi-
harmonic approach by analogy with [15] applying the
similar method, i.e.,
1 1
kin
0 0
3 ωωρ(ω)dω 3 ln 1 exp ρ(ω)dω.
2
F T
T
Θ ⎡ Θ ⎤⎛ ⎞= + − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∫ ∫
In [15] it was shown that the Debye formula can be suc-
cessfully replaced for numerous applications by the Ein-
stein approach:
kin
3 3 ln 1 exp .
2
F T
T
Θ ⎡ Θ ⎤⎛ ⎞= + − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
As Θ we used 2 2 2~ / ( / ),R d dR R dU dR−Θ i.e., we calcu-
late Θ in a quasiharmonic approach suggested by LJ and
Devonshire [16]. For the Θ calculations we used the expe-
rimental Θ values [17] as references, which were found at
ambient pressures and T approaching zero. The free energy
(1) is a function of two variables P, T and one parameter V;
the latter can be easily expressed via the nearest neighbor
distance RNN for both fcc and hcp lattices. The validity of
our method is confirmed by a good agreement of the calcu-
lated and experimental nearest neighbor distances as func-
tions of temperature (Fig. 1). These calculations in particu-
lar showed that the LJ potential with the improved
Fig. 1. (Color online) Temperature dependences of calculated
(this work) and experimental [17] nearest neighbor distances for
Ar at 1 bar. The inset shows the structure identifications. For
comparison we show the calculated values for 1 kbar. The dotted
line presents the calculation performed for the LJ potential with
improved parameters [15].
0 20 40 60 80 100 120
7.0
7.1
7.2
7.3
7.4
7.5
hcp calc
fcc calc
exp 1 bar
LJ mod
Ar
1 kbar
1 bar
exp
T, K
R
N
N
,
a.
u
.
Fig. 2. (Color online) The temperature dependences of the free
energy F and the difference F between the hcp and the fcc (in
favor of the hcp) for Ar. The insets show the structure and pres-
sure identifications.
0 200 400
–1.6
–1.4
–1.2
–1.0
–0.8
–0.6
–0.4
–0.2
0 20 40 60 80
–1060
–1040
–1020
–1000
–980
–960
–940
1 bar
1 kbar
15 kbar
Ar
hcp
fcc
Ar
1 bar
T, K T, K
F
,
K
F
F
h
cp
fc
c
–
,
K
0
0
0
2
2
2
4
4
4
6
6
6
8
8
8
10
10
10
12
12
12
14
14
14
16
16
16
–0.4
–0.2
0
–0.4
–0.2
0
–0.2
0
Ar
Kr
= 1 KT
Xe
P, kbar
P, kbar
P, kbar
F
F
h
cp
fc
c
–
,
K
F
F
h
cp
fc
c
–
,
K
F
F
h
cp
fc
c
–
,
K
Fig. 3. (Color online) The difference F between the hcp and the
fcc (in favor of the hcp) as a function of pressure for Ar, Kr and
Xe.
On the role of distortion in the hcp vs fcc competition in rare-gas solids
Fizika Nizkikh Temperatur, 2011, v. 37, No. 5 549
parameters (adjusted to the experimental data under study
[15]) produces also a rather good coincidence with expe-
riment.
The next figure (Fig. 2) demonstrates the temperature
dependences of the free energy and the difference F be-
tween hcp and fcc lattices at various pressures. Each de-
pendence terminates at the point where the second deriva-
tive of the free energy with respect to interatomic distances
attains zero, i.e., the crystal becomes unstable at such a
point. We should mention here that the Θ values at relevant
temperatures are still positive and sufficiently large. We
can see from these dependences that the difference be-
tween the hcp and the fcc (in favor of the hcp) is not re-
duced with temperature but grows up. The similar effect is
observed for the increasing pressure (Fig. 3) for all solid
rare gases. These results confirm a growing gain of the hcp
at elevated pressures in line with the experimental observa-
tions [9–12] but do not explain the observation of the fcc at
ambient pressures. We assume that it can be due to a size-
dependent effect mentioned above [7,8] that was confirm-
ed in particular in our previous studies of the size depen-
dence of the formed structures in impurity-helium solids
[18] but apparently other reasons may not be excluded.
3. Structural transformations
The next and important question is: how to observe the
fcc–hcp transformation and is it possible at all because
barriers are large? The relevant mechanisms were studied
within a first-principle local density approximation [2]
where two paths were suggested for an explanation of un-
usual features typical for the fcc–to–hcp phase transition
(see Introduction). At lower pressures authors suggest the
sliding-like scheme but at growing P they assume that the
intermediate face-centered orthorhombic (fco) structure
may be responsible for the transformation mechanism be-
cause according to their calculations this phase becomes
more favorable than the fcc at some pressure. As an argu-
ment in favor of this concept they calculated the barriers
for the sliding mechanism and showed that such barriers
crucially increase with P. Unfortunately they have not cal-
culated the barriers for the fcc-fco transition, and as it will
be shown below these barriers also significantly increase
with P.
We have found previously that the bcc crystallites may
freely relax towards the hcp structure under the applied LJ
and Aziz potentials [19,20]. In some particular distortion
cases an increasing contribution of the fcc phase was found
[20]. More specifically the fraction of the fcc component in
a final (after relaxation) crystal gradually increases if we
expand the crystal along c (Fig. 4, orthorhombic distor-
tion). A similar result can be attained if we gradually re-
duce the angle between a and b from 90° to 70.5° (Fig. 4,
monoclinic distortion). As we can see in the lower panels
of Fig. 4 in both cases the crystal after allowed relaxation
is enriched with stacking faults (SF). This finding implies
that in a real experiment both orthorhombic and mono-
clinic distortions result after relaxation in abundances of
SF that make them undistinguishable from each other.
We calculated also barriers, which prevent the fcc–hcp
transformation for both monoclinic and orthorhombic dis-
tortions (the top panels in Fig. 5). We see that for the or-
thorhombic distortion barriers are not lower than for the
monoclinic one and grow up with pressure that does not
support the main argument in favor of two paths between
the fcc and hcp structures at increasing pressures suggested
in [2] and discussed at the beginning of this section. But
apparently even more important is the other result shown
in the lower panels of Fig. 5. We analyzed the volume re-
laxation under both distortions and have found that the
volume grows up for the monoclinic distortion. It implies
in particular that if the transformation fcc–hcp is initiated
at growing pressures it can result in the ‘negative’ com-
pressibility that is impossible. Therefore the monoclinic
distortion is realistic only at decompression (if the fcc–hcp
transformation is invertible). In contrast with the monoc-
linic distortion the orthorhombic one (in a path from the
fcc) results in the lattice compression that is consistent
with an increasing P (which is necessary for the transition
observation according to experiment). Moreover this find-
ing may explain the sluggishness of the fcc–hcp transition,
because the additional pressure is needed to compensate
the volume reduction. The estimations show that in the
Fig. 4. (Color online) A general scheme, this describes mutual
transformations fcc–bcc–hcp. The orthorhombic cell is cut from
4 bcc cubes. For the hcp lattice the cell is rectangular, c coincides
with ahcp, b coincides with chcp and every 2nd (010) plane is dis-
placed by a/6; for the fcc lattice a = b = c = afcc or the same fcc
lattice can be obtained if we reduce the angle between a and b
from 90° to 70.5° and transform all parameters in such a way that
the plane (010) will be a basal plane (a). The results of a free
relaxation of the undistorted bcc lattice (b, the final hcp structure)
and of the bcc distorted in monoclinic (c) and orthorhombic (d)
ways. As dark atoms we show atoms with the hcp-like environ-
ment, light gray atoms have the fcc-like environment.
N.V. Krainyukova
550 Fizika Nizkikh Temperatur, 2011, v. 37, No. 5
point of the minimum of V/Vhcp (Fig. 5) at increasing pres-
sures, PΔV (ΔV is negative) can be close to the relevant
barrier that facilitates the transformation.
4. Summary
Applying the Aziz potential we show that the competi-
tion between hcp and fcc is still in favor of hcp (as for the
LJ potential), the difference increases with T and growing
pressures. A size-dependent energetic prevalence of fcc is
likely an important reason for the mentioned dominant
observation of fcc in RGS although other reasons may not
be excluded. As concerns the mechanism of the possible
fcc–hcp transformation we show that the known scheme of
sliding along the basal planes is plausible only for decom-
pression while at growing pressures we suggest as much
more probable an alternative possibility when the bcc dis-
torted in a orthorhombic way participates as a third coun-
terparty in such a transformation.
The author is grateful to D. Lee, M.A. Strzhemechny,
Yu.A. Freiman, V. Kiryukhin, and V. Khmelenko for fruit-
ful discussion of this work.
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Fig. 5. (Color online) Calculated barriers for the monoclinic and
orthorhombic distortions (see the figure for identifications). The
calculation for the pure bcc crystal at P = 0 is shown as a di-
amond (a). The volume variations are shown in the panels (b).
Orthorhombic distortion ( ) and ( ), monoclinic distortion ( ).
1.00
1.02
1.04
0
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1.0
1.0
0
50
100
Distortion degree, relat. units
Distortion degree, relat. units
P = 0
P =15 kbar
bcc
Ar
a
b
fc
c
h
cp
V
/V
h
cp
U
–
U
h
cp
,
K
|