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Theory of crystal ordering: Hexagonal symmetry
Volume 98A, number 1,2
PHYSICS LETTERS
3 October 1983
THEORY OF CRYSTAL ORDERING: HEXAGONAL SYMMETRY A.C. MITUS1 and A.Z. PATASHINSKII
Institute of Nuclear Physics, 630090, Noyosibirsk 90, USSR Received 14 July 1983
Using the statistical-mechanics theory of crystal ordering, the case of the hexagonal symmetry of the local order parameter (the sixth-rank tensor) is studied. The crystals metastability interval is small: E = 0.04. The phase diagram is constructed and the explanation of non-existence of mesophase in hcp crystals is proposed.
1. The statistical theory of crystal ordering, developed in refs. [1,2], describes the liquid—crystal (melting) phase transition in terms of the local order parameter field A (x). The simplest case of cubic symmetry ofthe parameter A (x)was studied in refs. [1,2]. In case oflower point-symmetry of the parameter A (x) a set of phase transitions (PT) in the crystalline phase may exist [3]. The general case of the lowest point-symmetry, the triclinic one, was studied in ref. [4]. Briefly, the interaction ofvarious point-symmetry local order parameters results in the existence ofthe effective “external” field acting on one of the local order parameters. The “external” field is due to the global ordering of the other local order parameter. Simple symmetry considerations allow one to determine the symmetry of the low-temperature phase once the symmetry of the higher-temperature phase and the hamiltonian are known. In order to discuss such a problem, the cases of the highest point symmetries of the field A (x), namely the cubic (m3m) and the hexagonal (6/mmm) ones, should be examined. Recently, the phase diagram of the system with local hexagonal symmetry was qualitatively studied via the Landau free energy expansion by Toner [5]. The stability of the various phases depends on the values of two free parameters which describe the weight with which the nematic and hexagonal second-order invariants enter the expression for the free energy. 1
The aim of this paper is to present the statisticalmechanics theory of the PT from the disordered to the ordered phase in the case oflocal hexagonal symmetry of the field A (x). 2. The local order parameter is the smallest-rank irreducible tensor which reflects the characteristic anisotropy of the atoms’ arrangement. Unlike the other point-symmetries, in the case of 6/mmm symmetry it is necessary to introduce the sixth-rank tensors (the hexagonal tensors of rank I = 4,2 are uniaxial). In order to construct the local order parameter consider the simple geometric S figure of fig. 1. Note that the S figure does not give rise to a simple hexagonal Bravais lattice.Nevertheless, the S figure reflects properly the characteristic arrangement of atoms in a 6/mmm-symmetiy crystal. One can hope that an aria-
1/
.
X
On leave of absence from Physics Dept. of Technical University, Wroclaw, Wybrze~e,Wyspianskiego 27, Poland.
0.031-9163/83/0000—0000/s 03.00 © 1983 North-Holland
Fig. 1. The 6/mmm symmetry S figure.
31
Volume 98A, number 1,2
PHYSICS LETTERS
logous treatment using a more refmed S figure will resultin the existence ofsmall (numerical) corrections to our present results. Our choice of the S figure is for computational simplifications. In this paper it is supposed that the geometric characteristics of the S figure are those of the elementary cell of the hcp crystal and the values a = 16/3 andB = 3.16 are used. Let us construct the tensor ~ ~(0) n(37~ !.LP
=
(1)
~
where n~is the radius vector of the ith atom of the figure 5, and the summation extends over all such atoms. In the rigid form approximation [1,2] one has H(x) = ~(x)~(0),
(2)
where the tilde denotes the irreducible part of a given tensor and R(x) describes a rotation in point x. The tensor has two independent components. For example, forHç°~ 0~ 1111 andH~ 3333 one has, in the cartesian coordinate system of fig. 1. =
(1/231)(281
—
5B);
= (16/231)(—10 ÷ B). (3) The mean-field approximation (MFA) equations are
~O~333
3 October 1983
components is for computational simplifications only. Numerical analysis shows that the system of 10 equations, eq. (4), is equivalent to a system of two equations for a(T) and b(T). These equations were solved numerically. The integration was carried out using the Gauss method. 3. At high temperatures the only solution of the MFA equations (4) is a(T) = b(fl = 0. At T = 0 one finds the obvious solution (8) and a non-trivial one a(0)O,
b(0)=B—10.
Its physical meaning will be discussed below. In fig. 2 a plot of the functions a(fl, b(T) is given. The stability of the various phases appearing in the model can be studied via the thermodynamicpotential ~(a, b, 1): ~(a, b, 1)
=
2 ÷562a2
(8/231)(b
—
2Oab)
—
Tln Z.
(10) The system of two equations, a 4/aa = 0 and a ~/ab = 0 is equivalent to the MFA equations (4). One finds that curve 1 of fig. 2 describes asolutionwhichis stable for temperatures T < T~,and metastable for T~
P~lha~Mp = z1f D~H,~P&)exp[_TlHMFA(j)], (4) where the MFA hamiltonian is —HMFA&) =Ha~pj)ha~çy~p,
(9)
1
(5)
and the self-consistence condition takes the form: ha~.y~p = v ~
(6)
v is the coordination number, Z = fD~’exp(—T’HMFA).
-
At T = 0 the global ordering in the system is described by formula (3). For T* 0 the following parametrisation for the mean-field tensor hMp is used:
b(T) -
B
h111111(T) = (~/231)[281a(T) 5b(fl] (7a) 3l)[—l0a(fl + b(~], (7b) h333333(fl = (l6v/2 with a(fl, b(T) satisfying the obvious conditions —
a(0)1,
,
b(0)=~B.
(8) Fig. 2. The temperature dependence of scalar amplitudes
The choice ofh 111111 and h333333 as independent 32
a(7), b(T).
Volume 98A, number 1,2
PHYSICS LETTERS
respectively. The solution given by curve 2 of fig. 2 is metastable, for all temperatures. For both curve 1 and 2, the “negative” and “positive” (i.e. below or above the T axis) solutions corresponding to the same temperature are characterized by the same value of~,and are physically equivalent. Consider, for example, the positive and negative solutions, given by curve 1, at T = 0. The former describes the S figure of fig. 1, the latter the same S figure in the coordinate system rotated round theZ axis by ir/2. The temperature interval of the metastable (overheated) crystal, found with the help of ~ is E
(T~— T~)/T~ = 0.04.
(11)
Note that result (11) is valid for ideal hcp crystals only. In general, E is a function of parameters describing the size and shape of hexagonal clusters (e.g. C/A, see below), The value ofE is twice as much as it is in the cubicsymmetry case [1,2]. The effect of the broadening of the crystal’s metastabiity interval is due to the growing complexity of the order parameter and reflects the well-known properties of the q-state Potts model. Namely, in this model the metastabiity interval ofthe ordered phase increases with increasing q. The case of cubic symmetry of the field A (x) can be described in terms of the q = 4 Potts model [1]. In order to repro. duce the above results, the Potts model with q > 4 should be used. 4. Consider the problem of the existence of stable mesophase-like solutions of eqs. (4). The mesophase is characterized by the orientation in space of its order parameter, the director. Such a phase is described by a solution of the following type: a = 0, b ‘ 0. As seen from fig. 2, and formula (9), such a solution exists. It is, nevertheless, metastable. It explains the well-known experimental fact that for systems which crystallize into a hcp lattice the liquid—crystal mesophase does not exist. This result is of purely geometric origin, Namely, the real hexagonal crystals have a hcp lattice built from two simple hexagonal Bravais lattices. The simple hexagonal Bravais lattice is characterized by the ratio C/A (C, A are standard parameters of the hexagonal-lattice elementary cell). In the case of ideal crystals C/A = 1.63. For real crystal this number varies from 1.56 (Be) to 1.89 (Cd) [6]. In fig. 3 the phase diagram of the system is shown.
~
3 October 1983
A
~ G/’nmTn
Cuc~uLd
~ ULLILIILIIIIILIL 1711171L171L1 T “S’”’.’
diagram of the system with
Fig. 3. The phase
local hexagonal
symmetry.
The shadowed area corresponds to real hexagonal crystals; the marked part of the T axis represents the “plate-like” liquid crystal [7]. For C/A ~ ~ the thermodynamically stable mesophase appears. The reasons for this behaviour are as follows. The effective hamiltonian of the space arrangement can be split into two parts, the self-crystalline and the liquid—crystal one [4]. The interaction constants depend on geometrical characteristics of the cluster only. In the case ofhexagonal crystals these coefficients depend on the ratio C/A. When C/A > ~ the liquid—crystal term in the effective hamiltonian plays the central role at temperatures not too low. For real crystals, with C/A having nearly the ideal value, the self-crystalline part in the hamiltonian is dominant. Some further details can be found in preprint [8]. For substances with a more complex structure the effective hexagonal local order parameter, with C/A ~ 2~/~7i, can occur. In such a case the stable liquid— crystal mesophase may appear in the system (see the phase diagram, fig. 3). The local ordering is described in terms of the basic functions ofthe irreducible representations ofthe 3-D rotation group SO(3). As stated in refs. [1,2], the two equivalent realizations, the irreducible cartesian tensor and the spherical harmonic (spherical multipole) ones can be used. The latter one was used in a recent paper of Haymet [9] where the icosahedral local symmetry case was studied and a value E = 0.052 was found. The tensorial properties ofhisparameter are that of a rank-6 irreducible tensor. The small difference between this and our results [see (11)] is probably due to the fact that the icosahedron is not characterized by an ideal hcp lattice ratio C/A = ,
33
Volume 98A, number 1,2
PHYSICS LETTERS
References [11 A.C. Mitu~and A.Z. Patashinskii, Soy. Phys. JETP ~ (1981) 789. [2] A.C. Mitu~and A.Z. Patashinskii, Phys. Lett. 87A (1982) 179. [3] A.C. Mitu~and A.Z. Patashinskii, Fiz. Tverd. Tela 24 (1982) 3344.
34
3 October 1983
[4] A.C. Mitu~and A.Z. Patashinskii, Fiz. Tverd. Tela 24 (1982) 2881. [5] J. Toner, Phys. Rev. A27 (1983) 1157. [6] N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Reinhait and Wilson, New York, 1976). [7] R. Alben, Phys. Rev. Lett. 30 (1973) 778. [8] A.C. Mitui and A.Z. Patashinskii, preprint INP 82-111, Novosibirsk (1982). [9] A.D.J. Haymet, Phys. Rev. B., to be published.
PHYSICS LETTERS
3 October 1983
THEORY OF CRYSTAL ORDERING: HEXAGONAL SYMMETRY A.C. MITUS1 and A.Z. PATASHINSKII
Institute of Nuclear Physics, 630090, Noyosibirsk 90, USSR Received 14 July 1983
Using the statistical-mechanics theory of crystal ordering, the case of the hexagonal symmetry of the local order parameter (the sixth-rank tensor) is studied. The crystals metastability interval is small: E = 0.04. The phase diagram is constructed and the explanation of non-existence of mesophase in hcp crystals is proposed.
1. The statistical theory of crystal ordering, developed in refs. [1,2], describes the liquid—crystal (melting) phase transition in terms of the local order parameter field A (x). The simplest case of cubic symmetry ofthe parameter A (x)was studied in refs. [1,2]. In case oflower point-symmetry of the parameter A (x) a set of phase transitions (PT) in the crystalline phase may exist [3]. The general case of the lowest point-symmetry, the triclinic one, was studied in ref. [4]. Briefly, the interaction ofvarious point-symmetry local order parameters results in the existence ofthe effective “external” field acting on one of the local order parameters. The “external” field is due to the global ordering of the other local order parameter. Simple symmetry considerations allow one to determine the symmetry of the low-temperature phase once the symmetry of the higher-temperature phase and the hamiltonian are known. In order to discuss such a problem, the cases of the highest point symmetries of the field A (x), namely the cubic (m3m) and the hexagonal (6/mmm) ones, should be examined. Recently, the phase diagram of the system with local hexagonal symmetry was qualitatively studied via the Landau free energy expansion by Toner [5]. The stability of the various phases depends on the values of two free parameters which describe the weight with which the nematic and hexagonal second-order invariants enter the expression for the free energy. 1
The aim of this paper is to present the statisticalmechanics theory of the PT from the disordered to the ordered phase in the case oflocal hexagonal symmetry of the field A (x). 2. The local order parameter is the smallest-rank irreducible tensor which reflects the characteristic anisotropy of the atoms’ arrangement. Unlike the other point-symmetries, in the case of 6/mmm symmetry it is necessary to introduce the sixth-rank tensors (the hexagonal tensors of rank I = 4,2 are uniaxial). In order to construct the local order parameter consider the simple geometric S figure of fig. 1. Note that the S figure does not give rise to a simple hexagonal Bravais lattice.Nevertheless, the S figure reflects properly the characteristic arrangement of atoms in a 6/mmm-symmetiy crystal. One can hope that an aria-
1/
.
X
On leave of absence from Physics Dept. of Technical University, Wroclaw, Wybrze~e,Wyspianskiego 27, Poland.
0.031-9163/83/0000—0000/s 03.00 © 1983 North-Holland
Fig. 1. The 6/mmm symmetry S figure.
31
Volume 98A, number 1,2
PHYSICS LETTERS
logous treatment using a more refmed S figure will resultin the existence ofsmall (numerical) corrections to our present results. Our choice of the S figure is for computational simplifications. In this paper it is supposed that the geometric characteristics of the S figure are those of the elementary cell of the hcp crystal and the values a = 16/3 andB = 3.16 are used. Let us construct the tensor ~ ~(0) n(37~ !.LP
=
(1)
~
where n~is the radius vector of the ith atom of the figure 5, and the summation extends over all such atoms. In the rigid form approximation [1,2] one has H(x) = ~(x)~(0),
(2)
where the tilde denotes the irreducible part of a given tensor and R(x) describes a rotation in point x. The tensor has two independent components. For example, forHç°~ 0~ 1111 andH~ 3333 one has, in the cartesian coordinate system of fig. 1. =
(1/231)(281
—
5B);
= (16/231)(—10 ÷ B). (3) The mean-field approximation (MFA) equations are
~O~333
3 October 1983
components is for computational simplifications only. Numerical analysis shows that the system of 10 equations, eq. (4), is equivalent to a system of two equations for a(T) and b(T). These equations were solved numerically. The integration was carried out using the Gauss method. 3. At high temperatures the only solution of the MFA equations (4) is a(T) = b(fl = 0. At T = 0 one finds the obvious solution (8) and a non-trivial one a(0)O,
b(0)=B—10.
Its physical meaning will be discussed below. In fig. 2 a plot of the functions a(fl, b(T) is given. The stability of the various phases appearing in the model can be studied via the thermodynamicpotential ~(a, b, 1): ~(a, b, 1)
=
2 ÷562a2
(8/231)(b
—
2Oab)
—
Tln Z.
(10) The system of two equations, a 4/aa = 0 and a ~/ab = 0 is equivalent to the MFA equations (4). One finds that curve 1 of fig. 2 describes asolutionwhichis stable for temperatures T < T~,and metastable for T~
P~lha~Mp = z1f D~H,~P&)exp[_TlHMFA(j)], (4) where the MFA hamiltonian is —HMFA&) =Ha~pj)ha~çy~p,
(9)
1
(5)
and the self-consistence condition takes the form: ha~.y~p = v ~
(6)
v is the coordination number, Z = fD~’exp(—T’HMFA).
-
At T = 0 the global ordering in the system is described by formula (3). For T* 0 the following parametrisation for the mean-field tensor hMp is used:
b(T) -
B
h111111(T) = (~/231)[281a(T) 5b(fl] (7a) 3l)[—l0a(fl + b(~], (7b) h333333(fl = (l6v/2 with a(fl, b(T) satisfying the obvious conditions —
a(0)1,
,
b(0)=~B.
(8) Fig. 2. The temperature dependence of scalar amplitudes
The choice ofh 111111 and h333333 as independent 32
a(7), b(T).
Volume 98A, number 1,2
PHYSICS LETTERS
respectively. The solution given by curve 2 of fig. 2 is metastable, for all temperatures. For both curve 1 and 2, the “negative” and “positive” (i.e. below or above the T axis) solutions corresponding to the same temperature are characterized by the same value of~,and are physically equivalent. Consider, for example, the positive and negative solutions, given by curve 1, at T = 0. The former describes the S figure of fig. 1, the latter the same S figure in the coordinate system rotated round theZ axis by ir/2. The temperature interval of the metastable (overheated) crystal, found with the help of ~ is E
(T~— T~)/T~ = 0.04.
(11)
Note that result (11) is valid for ideal hcp crystals only. In general, E is a function of parameters describing the size and shape of hexagonal clusters (e.g. C/A, see below), The value ofE is twice as much as it is in the cubicsymmetry case [1,2]. The effect of the broadening of the crystal’s metastabiity interval is due to the growing complexity of the order parameter and reflects the well-known properties of the q-state Potts model. Namely, in this model the metastabiity interval ofthe ordered phase increases with increasing q. The case of cubic symmetry of the field A (x) can be described in terms of the q = 4 Potts model [1]. In order to repro. duce the above results, the Potts model with q > 4 should be used. 4. Consider the problem of the existence of stable mesophase-like solutions of eqs. (4). The mesophase is characterized by the orientation in space of its order parameter, the director. Such a phase is described by a solution of the following type: a = 0, b ‘ 0. As seen from fig. 2, and formula (9), such a solution exists. It is, nevertheless, metastable. It explains the well-known experimental fact that for systems which crystallize into a hcp lattice the liquid—crystal mesophase does not exist. This result is of purely geometric origin, Namely, the real hexagonal crystals have a hcp lattice built from two simple hexagonal Bravais lattices. The simple hexagonal Bravais lattice is characterized by the ratio C/A (C, A are standard parameters of the hexagonal-lattice elementary cell). In the case of ideal crystals C/A = 1.63. For real crystal this number varies from 1.56 (Be) to 1.89 (Cd) [6]. In fig. 3 the phase diagram of the system is shown.
~
3 October 1983
A
~ G/’nmTn
Cuc~uLd
~ ULLILIILIIIIILIL 1711171L171L1 T “S’”’.’
diagram of the system with
Fig. 3. The phase
local hexagonal
symmetry.
The shadowed area corresponds to real hexagonal crystals; the marked part of the T axis represents the “plate-like” liquid crystal [7]. For C/A ~ ~ the thermodynamically stable mesophase appears. The reasons for this behaviour are as follows. The effective hamiltonian of the space arrangement can be split into two parts, the self-crystalline and the liquid—crystal one [4]. The interaction constants depend on geometrical characteristics of the cluster only. In the case ofhexagonal crystals these coefficients depend on the ratio C/A. When C/A > ~ the liquid—crystal term in the effective hamiltonian plays the central role at temperatures not too low. For real crystals, with C/A having nearly the ideal value, the self-crystalline part in the hamiltonian is dominant. Some further details can be found in preprint [8]. For substances with a more complex structure the effective hexagonal local order parameter, with C/A ~ 2~/~7i, can occur. In such a case the stable liquid— crystal mesophase may appear in the system (see the phase diagram, fig. 3). The local ordering is described in terms of the basic functions ofthe irreducible representations ofthe 3-D rotation group SO(3). As stated in refs. [1,2], the two equivalent realizations, the irreducible cartesian tensor and the spherical harmonic (spherical multipole) ones can be used. The latter one was used in a recent paper of Haymet [9] where the icosahedral local symmetry case was studied and a value E = 0.052 was found. The tensorial properties ofhisparameter are that of a rank-6 irreducible tensor. The small difference between this and our results [see (11)] is probably due to the fact that the icosahedron is not characterized by an ideal hcp lattice ratio C/A = ,
33
Volume 98A, number 1,2
PHYSICS LETTERS
References [11 A.C. Mitu~and A.Z. Patashinskii, Soy. Phys. JETP ~ (1981) 789. [2] A.C. Mitu~and A.Z. Patashinskii, Phys. Lett. 87A (1982) 179. [3] A.C. Mitu~and A.Z. Patashinskii, Fiz. Tverd. Tela 24 (1982) 3344.
34
3 October 1983
[4] A.C. Mitu~and A.Z. Patashinskii, Fiz. Tverd. Tela 24 (1982) 2881. [5] J. Toner, Phys. Rev. A27 (1983) 1157. [6] N.W. Ashcroft and N.D. Mermin, Solid State Physics (Holt, Reinhait and Wilson, New York, 1976). [7] R. Alben, Phys. Rev. Lett. 30 (1973) 778. [8] A.C. Mitui and A.Z. Patashinskii, preprint INP 82-111, Novosibirsk (1982). [9] A.D.J. Haymet, Phys. Rev. B., to be published.