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A quasicrystalline structure for the cholesteric fog phase
Volume 115, number 9
PHYSICS LETTERS A
12 May 1986
A QUASICRYSTALLINE STRUCTURE FOR THE CHOLESTERIC FOG PHASE R.M. H O R N R E I C H
and S. S H T R I K M A N
Department of Electronics, The Weizrnann Institute of Science, 76100 Rehovot, Israel Received 30 January 1986; accepted for publication 10 March 1986
Using Landau theory, the free energy of a cholesteric system with broken (non-symmorphic) icosahedral symmetry is calculated to third order and compared with that of a bcc structure. It is argued that this new quasicrystalline structure is compatible with the experimentally observed anomalous fog phase (blue phase 11I).
The Landau theory of phase transitions has proved to be an excellent basis for the theoretical description of cholesteric blue phases (BP) [1]. A universal phase diagram, including the two experimentally observed cubic BP as well as the disordered and helicoidal phases, has been obtained and shown to be consistent with the main experimental results [1,2]. This theoretical framework has also been used to study the effect of external fields on this phase diagram. A hexagonal structure was predicted [3] and found experimentally [4] above a threshold field. The main outstanding problem in the theory of BP is a description for the apparently amorphous phase (known variously as gray [2], fog [5], and BP III [6] ) which exists just below the disordered phase at sufficiently short pitch. We here argue that this anomalous phase can also be understood within the Landau theory. Our model has features which also appear in models [7,8] put forward recently to explain the existence of quasicrystalline metallic structures [9] but, in addition, has several novel elements. We show that the suggested structure is consistent with experimental results and indicate lines of research which could provide further verification. The Landau average free energy density for cholesterics, in suitable units, can be written as [1]
f = v - l f dr [~(W/] x 2 + /a2 i],l ---~/aij/al'l/ali + (/a2,)2 ],
2geijl/ainlajn,l)
Here #i/(r), the order parameter, is proportional to the anisotropic part of the dielectric tensor, t is a reduced temperature variable, (whose unit is approximately 0.5 K), the chirality parameter K = qC~R is the product of the helicoidal (C) phase wave vector qc and the racemic mixture correlation length ~R at t = t C = I, and/ai/,l =- 3/aij/O(Xl/~R)" (An additional term, proportional to/ai],i/al[,l, does not contribute to f for any of the C or BP model structures.) Clearly, the (K, t) plane phase diagram obtained by minimizing (1) is universal as f contains no additional parameters. For cubic BP [1],/aij(r) is taken to be a linear combination of plane waves [/am (o, n)] exp(iqa n - r). Here +-qon are the allowed wave vectors for a given (simple or body-centered cubic) Bravais lattice, o is the sum of the squares of the Miller indices, and n (= 1,2, ...) labels the wave vectors for given e. The tensor amplitudes [~am(O, n)] are taken exclusively from the low-lying branch of the excitation spectrum, i.e., proportional to an m = 2 basis tensor. We thus have [/a2(o,n)] exp(iqa n. r) = ½/a2(o) i - 1 0 0 X exp [i ff2(on)] exp(iqo n • r) = PZ(O)[M2(a, n)] exp[itb2(on)] exp(iqo n • r).
(1)
0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
Both the basis tensor [M2(tl , n)] and the phase ~b2(on ) in eq. (2) are def'med in a local axis system 451
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associated with each qon" There are also selection rules [10,11] which result in/a2(o ) vanishing for particular values of o. For both scalar [7,8,12] and tensor order parameters [1 ], the initial approach to selecting possible structures has been to minimize the quadratic portion of the free energy by taking all the wave vectors to have the same magnitude and then to maximize the (negative) cubic term. This leads to structures composed, in reciprocal space, of wave vector sets which form equilateral triangles. In three dimensions, the simplest such structure has six wave vectors forming the edges of a regular tetrahedron and is bcc. One can, however, envision an alternate structure in which the wave vectors are the thirty edges of a regular icosahedron. While lacking translational symmetry, this structure would exhibit Bragg scattering. It has, for this reason, been of interest in connection with the observations of Shechtman et al. [9] on A1-Mn alloys. For the scalar order parameter case, a comparison of icosahedral and bcc free energies was made by Alexander and McTague [12] (The bcc structure had been considered earlier by Baym et al. [13] ). They normalized the order parameter magnitude so that the quadratic free energy contributions of both structures were identical and then compared the cubic: terms. They found that the bcc phase was always energetically preferred. More recently, Kalugin et al. [8] considered a modified icosahedral model having two wave vector sets. They found that when these are chosen to be the vertex and edge vectors of the icosahedron which are, to within 5%, of equal magnitude, the icosahedral phase is energetically preferred to be bcc one. As in earlier work, only quadratic and cubic contributions to fwere considered. Turning to cholesteric systems, which are characterized by a tensor rather than scalar order parameter, Kleinert and Maki [14] claimed that the simple icosahedral structure (i.e., with thirty edge vectors +q2n only) has a higher energy than bcc 05 . In analogy with the scalar case, one might expect that by extending this model to include the additional set of twelve +q In vertex vectors, a stable phase could be obtained. The calculation is as follows: The ratio r =-qln/q2n = sin(27r/5) = 0.951 is very close to unity. The quadratic part of f is minimal when q = qc, so we set qln = (1 - t$)qc, q2n = (1 + 6)qc, with ~ = (1 - r)/ (1 + r) = 0.025. The icosahedral (IC) order parameter is 452
12 May 1986
lai/(r) = , ~
1
6
#2(1) ~
n=l
{IM2(1, n)]
X exp [i ~2(ln)] exp(iq In" r) + c.c.} +~
1
15 #2(2/~
n=l
{1M2(2, n)l
X exp [i ff 2(2n)] exp(iq 2n" r) + c.c.),
(3)
where c.c. denotes complex conjugate and the quadratic part f2 of f i n eq. (1)becomes (f2)IC ~<¼[t- (1 - 6 2 ) K 2] [#2(1) + #2(2)],
(4)
with 6 2 = 6.3 X 10 -4. Since 6 2 < 10 -3, the quadratic portion o f f is essentially at its absolute minimum value in the region of interest [1 ], n ~ 1. Consider now the third-order contribution to f. For the simple ~2(1) = 6 = 0) icosahedron, the thirty components of/ai/form twenty equilateral triangles, each of which, in isolation, hating the maximum (negative) contribution, -(27x/if/80)#32(2). However, since each edge is shared by two, non-coplanar triangles, the cubic term is necessarily reduced in magnitude [1 ]. For the phases chosen in ref. [14] (all •2(2n) = 7r, point symmetry 235), the reduction factor is 7x/~/27. (Note, however, that these phases do not minimize f for the simple icosahedral structure [15,16]). Including the vertex vectors in the order parameter gives an additional thirty triangles. However, for 235 point symmetry, #2(1)= 0, and these thirty triangles cannot contribute to the cubic term. Thus, unlike the scalar order-parameter case, adding the second set of Fourier components does not stabilize the icosahedral structure in a cholesteric system. There is, however, another possibility; to break the 235 rotational symmetry of the order parameter such that each of the thirty triangles formed by vertex and edge vectors contributes maximally to the third-order term. This can be done as the q2n leg in each of these triangles is not shared. This leg's associated phase, ff2(2n), is fixed to maximize this triangle's contribution and the amplitudes #2(1), #2(2) associated with each wave vector are chosen to be n-independent. The structure defined by this procedure (broken icosahedral or BIC) has the following properties: Its free energy Js independent of ff2(ln), as any
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shft in those phases can be compensated by suitably changing the ff2(2n). This reflects the degeneracy of the system with respect to three phonon and three phason modes at zero wave vector. By a suitable choice of ~b2(ln), a point symmetry 32 about the origin can be obtained [16]. However, this is not the BIC symmetry group which includes, in addition, combinations of rotations with zero wave vector phasons. In particular, the structure is invariant under the operation 5¢, a rotation of 27r/5 about q In followed by an appropriate shift of the phases. The overall symmetry, 235¢, is thus similar to that of a non-symmorphic space group. (In a six-dimensional description [7], it would be non-symmorphic.) It follows that free energy contributions from equivalent (under 235 point symmetry) sections of the BIC structure (i.e., triangles, loops) are equal. This does not hold for cubic (32) point symmetry, where all phases are not symmetry prescribed and the/12(2 ) are not all equal. In analogy with the scalar model [8] and using methods given elsewhere [ 1], the BIC average free energy to third order is fBIC = ¼(t - g2)/a2 -- 2.825c2s/a 3 + 0.450s3tt 3 ,
(5)
where 0 2 =/a2(1) + U2(2), O2(1)=pe,
U2(2)=gs,
c 2 +s 2 = 1.
(6)
The magnitude of the cubic term in eq. (5) is maximized by setting s 2 = 0.288, giving fBIC = ¼(t - gZ)/a2 -- 1.010//3 .
(6a)
The equivalent result for 05 is [1] f0 s = ~(t - K2)U2 - 1.016ta 3.
(6b)
Thus, to this order, the BIC and 05 structures are essentially degenerate. Further, the lowest lying allowed harmonic for 05 occurs at q/qc = ~ and does not contribute significantly [1]. For BIC, on the other hand, there are seven harmonics whose wave vectors, together with edge and vertex vectors, form triangles and which lie at or below q/qc = x/3-. The one at q/qc = 1.176, for example, forms sixty triangles which contribute to the cubic term in f. There is thus good reason to expect that BIC, not 05 , is thermodynamically stable immediately below the disordered
12 May 1986
phase for [1] r ~> 1.2. (For K ~ oo, however, 05 is stable [1,17] .) However, free energy comparisons have their limitations. The energy differences between the C, simple cubic 02, and body-centered 08 structures are known to be small (less than 2%) over substantial regions of the K - t plane and the phase boundaries are very sensitive to small energy shifts [1 ]. Thus higher order terms in lai/and its spatial derivatives can be significant. In particular, including the fourth-order term in the one and two harmonic calculations favors a bcc structure in both the scalar [18] and tensor [15,16] models. (Indeed, in the former, it results in the bcc rather than icosahedral structure being energetically favored [18].) We believe, however, that the results (6a) of our calculation indicates that the BIC structure can describe BP III. The BIC structure is also consistent with experimental results for BP III. In particular, (1) BIC and BP III [2,5,6] are both non-birefringent. (2) Both BIC and BP III [19] will exhibit strong optical activity. (3) The I-BIC latent heat will be approximately equal to that of [1] 1-05 and considerably larger than those of 0 2 - 0 8 and 0 8 - C , in qualitative agreement with experimental results [20]. (4) BIC will have phason (Goldstone) modes, indicating that its elasticity will be markedly lower than that of the cubic BP, in agreement with experiment [20]. (5) In principal, BIC should exhibit characteristic Bragg scattering, similar to that reported by Shechtman et al. [9] for A1-Mn alloys. In their BP III study, Meiboom and Sammon [6] measured as a function of wavelength X the transmitted (i.e., nonscattered) intensity which decreases monotonically as 3, is reduced as a consequence of additional Bragg scattering and an intrinsic X-dependence [11 ]. More recently, Demikhov et al. [21 ] measured scattered intensity directly as a function of X and found a broad peak in BP III with a half-width approximately 3.8 times greater than in BP I or BP II. The peak was centered at q/qc = 0.96. In either measurement, the nature of the experimental spectrum depends critically upon the ratio of the Bragg peak width to the spachag between the two low-lying components of the order parameter. Qualitatively, the scattered intensity 453
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PHYSICS LETTERS A
will be greatest near X = XC, where the two principal amplitudes/a2(1),/a2(2 ) o f the order parameter are found. Theoretically, two distinct "steps" (peaks) should appear in the transmitted (scattered) intensity, but these, we believe, were "washed o u t " in refs. [6,21] as a consequence o f the experimental resolution, which is evidently insufficient to resolve the small spacing between the two Fourier components. Also, small quasicrystallite size (see below) and the large number o f allowed harmonics (as compared with the cubic BP [10] will further smear the spectrum. For these reasons, the observed spectra [6,21 ] are consistent with a BIC structure for BP III. Further experimental verification is, however, clearly desirable. In the optical analog o f classical D e b y e - S c h e r r e r X-ray diffraction [21 ], a crucial factor is quasicrystallite size, which must be sufficiently large if closely spaced Bragg peaks are to be resolved. This problem does not arise for the cubic BP where the peaks are well spaced. Theoretical calculations o f harmonic peak intensities [1 ] could be helpful here. One could also attempt to grow relatively large single "quasicrystals" of BP III, as has been done successfully for the cubic BP [22]. Experimental diffraction patterns could then be compared with those of a BIC phase. It may also be possible to detect, b y light scattering, the phason modes which, in principle, exist in the incommensurate BIC structure [23]. To summarize, we have argued using Landau theory that a structure with broken icosahedral symmetry describes the anomalous BP III observed experimentally in cholesteric systems. Experimental confirmation could provide an unambiguous example of a thermodynamically stable phase with long-range quasicrystaUine ordering. Discussions with D. Mukamel are gratefully acknowledged. Models built b y D. Leibovitz contributed substantially to our understanding o f the BIC structure. This work was supported in part b y the Commission for Basic Research o f the Israel Academy of Arts and Sciences.
References [ 1 ] H. Grebel, R.M. Hornreich and S. Shtrikman, Phys. Rev. A 28 (1983) 1114, 3669(E) (in eqs. (A1) and (A3), replace K ~ by K~R); 30 (1984) 3264. 454
12 May 1986
[2] H. Stegemeyer and K. Bergmann, in: Liquid crystals of one-and-two-dimensional order, eds. W. Helfrich and G. Heppke (Springer, Berlin, 1980) p. 161. [3] R.M. Hornreich, M. Kugler and S. Shtrikman, Phys. Rev. Lett. 54 (1985) 2099; J. Phys. (Paris) 46 (1985) C3-47. [4] P. Pieranskii, P.E. Cladis and R. Barbet-Massin, J. Phys. (Paris) Lett. 46 (1985) L-973. [5] M. Marcus, J. Phys. (Paris) 42 (1981) 61. [6] S. Meiboom and S. Sammon, Phys. Rev. A 24 (1981) 468. [7] D. Levine and P.J. Steinhardt, Phys. Rev. Lett. 53 (1984) 2477; P. Bak, Phys. Rev. Lett. 54 (1985) 1517; N.D. Mermin and S.D. Troian, Phys. Rev. Lett. 54 (1985) 1524; M.V. Jaric, Phys. Rev. Lett. 55 (1985) 607; D.R. Nelson and S. Sachdev, Phys. Rev. B 32 (1985) 4592. [8] P.A. Kaluglin, A.Yu. Kitaev and L.S. Letitov, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 119 [JETP Lett. 41 (1985) 145 ]. [9] D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [10] R.M. Hornreich and S. Shtrikman, Phys. Lett. A 82 (1981) 20; Phys. Rev. A 28 (1983) 1791, 3669(E). [ 11 ] V.A. Belyakov, V.E. Dmitrenko and S.M. Osadchii, Zh. Eksp. Theor. Fiz. 83 (1982) 585; 84 (1982) 1920(E) [Sov. Phys. JETP 56 (1982) 322; 57 (1982) ll17(E)]. [12] S. Alexander and J. McTague, Phys. Rev. Lett. 41 (1978) 702. [13] G. Baym, H.A. Bethe and C.J. Pethick, Nucl. Phys. A 175 (1971) 225. [14] J. Kleinert and K. Maki, Fortschr. Phys. 29 (1981) 219. [15] R.M. Hornreich and S. Shtrikman, submitted to Phys. Rev. Lett. [ 16] D. Rokshar, private communication. [ 17 ] D.C. Wright and N.D. Mermin, Phys. Rev. A 31 (1985) 3498. [18] O. Biham, D. Mukamel and S. Shtrikman, submitted to Phys. Rev. Lett. [19] P.J. Collings, Phys. Rev. A 30 (1984) 1990. [20] R.N. Kleinman, D.J. Bishop, R. Pindak and P. Taborek, Phys. Rev. Lett. 53 (1984) 2137. [21 ] E.I. Demikhov, U.K. Dolganov and S.P. Krylova, Pis'ma Zh. Eksp. Theor. Fiz. 42 (1985) 15 [JETP Lett. 42 (1985) 161. [22] Th. Btumel and H. Stegemeyer, J. Cryst. Growth 66 (1984) 163; R. Barbet-Massin, P.E. Cladis and P. Pieranskii, Phys. Rev. A 30 (1984) 1161. [23] V.A. Golovko and A.P. Levanyuk, Light scattering near phase transitions, eds. H.Z. Cummins and A.P. Levanyuk (North-Holland, Amsterdam, 1983) p. 169.
PHYSICS LETTERS A
12 May 1986
A QUASICRYSTALLINE STRUCTURE FOR THE CHOLESTERIC FOG PHASE R.M. H O R N R E I C H
and S. S H T R I K M A N
Department of Electronics, The Weizrnann Institute of Science, 76100 Rehovot, Israel Received 30 January 1986; accepted for publication 10 March 1986
Using Landau theory, the free energy of a cholesteric system with broken (non-symmorphic) icosahedral symmetry is calculated to third order and compared with that of a bcc structure. It is argued that this new quasicrystalline structure is compatible with the experimentally observed anomalous fog phase (blue phase 11I).
The Landau theory of phase transitions has proved to be an excellent basis for the theoretical description of cholesteric blue phases (BP) [1]. A universal phase diagram, including the two experimentally observed cubic BP as well as the disordered and helicoidal phases, has been obtained and shown to be consistent with the main experimental results [1,2]. This theoretical framework has also been used to study the effect of external fields on this phase diagram. A hexagonal structure was predicted [3] and found experimentally [4] above a threshold field. The main outstanding problem in the theory of BP is a description for the apparently amorphous phase (known variously as gray [2], fog [5], and BP III [6] ) which exists just below the disordered phase at sufficiently short pitch. We here argue that this anomalous phase can also be understood within the Landau theory. Our model has features which also appear in models [7,8] put forward recently to explain the existence of quasicrystalline metallic structures [9] but, in addition, has several novel elements. We show that the suggested structure is consistent with experimental results and indicate lines of research which could provide further verification. The Landau average free energy density for cholesterics, in suitable units, can be written as [1]
f = v - l f dr [~(W/] x 2 + /a2 i],l ---~/aij/al'l/ali + (/a2,)2 ],
2geijl/ainlajn,l)
Here #i/(r), the order parameter, is proportional to the anisotropic part of the dielectric tensor, t is a reduced temperature variable, (whose unit is approximately 0.5 K), the chirality parameter K = qC~R is the product of the helicoidal (C) phase wave vector qc and the racemic mixture correlation length ~R at t = t C = I, and/ai/,l =- 3/aij/O(Xl/~R)" (An additional term, proportional to/ai],i/al[,l, does not contribute to f for any of the C or BP model structures.) Clearly, the (K, t) plane phase diagram obtained by minimizing (1) is universal as f contains no additional parameters. For cubic BP [1],/aij(r) is taken to be a linear combination of plane waves [/am (o, n)] exp(iqa n - r). Here +-qon are the allowed wave vectors for a given (simple or body-centered cubic) Bravais lattice, o is the sum of the squares of the Miller indices, and n (= 1,2, ...) labels the wave vectors for given e. The tensor amplitudes [~am(O, n)] are taken exclusively from the low-lying branch of the excitation spectrum, i.e., proportional to an m = 2 basis tensor. We thus have [/a2(o,n)] exp(iqa n. r) = ½/a2(o) i - 1 0 0 X exp [i ff2(on)] exp(iqo n • r) = PZ(O)[M2(a, n)] exp[itb2(on)] exp(iqo n • r).
(1)
0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(2)
Both the basis tensor [M2(tl , n)] and the phase ~b2(on ) in eq. (2) are def'med in a local axis system 451
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associated with each qon" There are also selection rules [10,11] which result in/a2(o ) vanishing for particular values of o. For both scalar [7,8,12] and tensor order parameters [1 ], the initial approach to selecting possible structures has been to minimize the quadratic portion of the free energy by taking all the wave vectors to have the same magnitude and then to maximize the (negative) cubic term. This leads to structures composed, in reciprocal space, of wave vector sets which form equilateral triangles. In three dimensions, the simplest such structure has six wave vectors forming the edges of a regular tetrahedron and is bcc. One can, however, envision an alternate structure in which the wave vectors are the thirty edges of a regular icosahedron. While lacking translational symmetry, this structure would exhibit Bragg scattering. It has, for this reason, been of interest in connection with the observations of Shechtman et al. [9] on A1-Mn alloys. For the scalar order parameter case, a comparison of icosahedral and bcc free energies was made by Alexander and McTague [12] (The bcc structure had been considered earlier by Baym et al. [13] ). They normalized the order parameter magnitude so that the quadratic free energy contributions of both structures were identical and then compared the cubic: terms. They found that the bcc phase was always energetically preferred. More recently, Kalugin et al. [8] considered a modified icosahedral model having two wave vector sets. They found that when these are chosen to be the vertex and edge vectors of the icosahedron which are, to within 5%, of equal magnitude, the icosahedral phase is energetically preferred to be bcc one. As in earlier work, only quadratic and cubic contributions to fwere considered. Turning to cholesteric systems, which are characterized by a tensor rather than scalar order parameter, Kleinert and Maki [14] claimed that the simple icosahedral structure (i.e., with thirty edge vectors +q2n only) has a higher energy than bcc 05 . In analogy with the scalar case, one might expect that by extending this model to include the additional set of twelve +q In vertex vectors, a stable phase could be obtained. The calculation is as follows: The ratio r =-qln/q2n = sin(27r/5) = 0.951 is very close to unity. The quadratic part of f is minimal when q = qc, so we set qln = (1 - t$)qc, q2n = (1 + 6)qc, with ~ = (1 - r)/ (1 + r) = 0.025. The icosahedral (IC) order parameter is 452
12 May 1986
lai/(r) = , ~
1
6
#2(1) ~
n=l
{IM2(1, n)]
X exp [i ~2(ln)] exp(iq In" r) + c.c.} +~
1
15 #2(2/~
n=l
{1M2(2, n)l
X exp [i ff 2(2n)] exp(iq 2n" r) + c.c.),
(3)
where c.c. denotes complex conjugate and the quadratic part f2 of f i n eq. (1)becomes (f2)IC ~<¼[t- (1 - 6 2 ) K 2] [#2(1) + #2(2)],
(4)
with 6 2 = 6.3 X 10 -4. Since 6 2 < 10 -3, the quadratic portion o f f is essentially at its absolute minimum value in the region of interest [1 ], n ~ 1. Consider now the third-order contribution to f. For the simple ~2(1) = 6 = 0) icosahedron, the thirty components of/ai/form twenty equilateral triangles, each of which, in isolation, hating the maximum (negative) contribution, -(27x/if/80)#32(2). However, since each edge is shared by two, non-coplanar triangles, the cubic term is necessarily reduced in magnitude [1 ]. For the phases chosen in ref. [14] (all •2(2n) = 7r, point symmetry 235), the reduction factor is 7x/~/27. (Note, however, that these phases do not minimize f for the simple icosahedral structure [15,16]). Including the vertex vectors in the order parameter gives an additional thirty triangles. However, for 235 point symmetry, #2(1)= 0, and these thirty triangles cannot contribute to the cubic term. Thus, unlike the scalar order-parameter case, adding the second set of Fourier components does not stabilize the icosahedral structure in a cholesteric system. There is, however, another possibility; to break the 235 rotational symmetry of the order parameter such that each of the thirty triangles formed by vertex and edge vectors contributes maximally to the third-order term. This can be done as the q2n leg in each of these triangles is not shared. This leg's associated phase, ff2(2n), is fixed to maximize this triangle's contribution and the amplitudes #2(1), #2(2) associated with each wave vector are chosen to be n-independent. The structure defined by this procedure (broken icosahedral or BIC) has the following properties: Its free energy Js independent of ff2(ln), as any
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PHYSICS LETTERS A
shft in those phases can be compensated by suitably changing the ff2(2n). This reflects the degeneracy of the system with respect to three phonon and three phason modes at zero wave vector. By a suitable choice of ~b2(ln), a point symmetry 32 about the origin can be obtained [16]. However, this is not the BIC symmetry group which includes, in addition, combinations of rotations with zero wave vector phasons. In particular, the structure is invariant under the operation 5¢, a rotation of 27r/5 about q In followed by an appropriate shift of the phases. The overall symmetry, 235¢, is thus similar to that of a non-symmorphic space group. (In a six-dimensional description [7], it would be non-symmorphic.) It follows that free energy contributions from equivalent (under 235 point symmetry) sections of the BIC structure (i.e., triangles, loops) are equal. This does not hold for cubic (32) point symmetry, where all phases are not symmetry prescribed and the/12(2 ) are not all equal. In analogy with the scalar model [8] and using methods given elsewhere [ 1], the BIC average free energy to third order is fBIC = ¼(t - g2)/a2 -- 2.825c2s/a 3 + 0.450s3tt 3 ,
(5)
where 0 2 =/a2(1) + U2(2), O2(1)=pe,
U2(2)=gs,
c 2 +s 2 = 1.
(6)
The magnitude of the cubic term in eq. (5) is maximized by setting s 2 = 0.288, giving fBIC = ¼(t - gZ)/a2 -- 1.010//3 .
(6a)
The equivalent result for 05 is [1] f0 s = ~(t - K2)U2 - 1.016ta 3.
(6b)
Thus, to this order, the BIC and 05 structures are essentially degenerate. Further, the lowest lying allowed harmonic for 05 occurs at q/qc = ~ and does not contribute significantly [1]. For BIC, on the other hand, there are seven harmonics whose wave vectors, together with edge and vertex vectors, form triangles and which lie at or below q/qc = x/3-. The one at q/qc = 1.176, for example, forms sixty triangles which contribute to the cubic term in f. There is thus good reason to expect that BIC, not 05 , is thermodynamically stable immediately below the disordered
12 May 1986
phase for [1] r ~> 1.2. (For K ~ oo, however, 05 is stable [1,17] .) However, free energy comparisons have their limitations. The energy differences between the C, simple cubic 02, and body-centered 08 structures are known to be small (less than 2%) over substantial regions of the K - t plane and the phase boundaries are very sensitive to small energy shifts [1 ]. Thus higher order terms in lai/and its spatial derivatives can be significant. In particular, including the fourth-order term in the one and two harmonic calculations favors a bcc structure in both the scalar [18] and tensor [15,16] models. (Indeed, in the former, it results in the bcc rather than icosahedral structure being energetically favored [18].) We believe, however, that the results (6a) of our calculation indicates that the BIC structure can describe BP III. The BIC structure is also consistent with experimental results for BP III. In particular, (1) BIC and BP III [2,5,6] are both non-birefringent. (2) Both BIC and BP III [19] will exhibit strong optical activity. (3) The I-BIC latent heat will be approximately equal to that of [1] 1-05 and considerably larger than those of 0 2 - 0 8 and 0 8 - C , in qualitative agreement with experimental results [20]. (4) BIC will have phason (Goldstone) modes, indicating that its elasticity will be markedly lower than that of the cubic BP, in agreement with experiment [20]. (5) In principal, BIC should exhibit characteristic Bragg scattering, similar to that reported by Shechtman et al. [9] for A1-Mn alloys. In their BP III study, Meiboom and Sammon [6] measured as a function of wavelength X the transmitted (i.e., nonscattered) intensity which decreases monotonically as 3, is reduced as a consequence of additional Bragg scattering and an intrinsic X-dependence [11 ]. More recently, Demikhov et al. [21 ] measured scattered intensity directly as a function of X and found a broad peak in BP III with a half-width approximately 3.8 times greater than in BP I or BP II. The peak was centered at q/qc = 0.96. In either measurement, the nature of the experimental spectrum depends critically upon the ratio of the Bragg peak width to the spachag between the two low-lying components of the order parameter. Qualitatively, the scattered intensity 453
Volume 115, number 9
PHYSICS LETTERS A
will be greatest near X = XC, where the two principal amplitudes/a2(1),/a2(2 ) o f the order parameter are found. Theoretically, two distinct "steps" (peaks) should appear in the transmitted (scattered) intensity, but these, we believe, were "washed o u t " in refs. [6,21] as a consequence o f the experimental resolution, which is evidently insufficient to resolve the small spacing between the two Fourier components. Also, small quasicrystallite size (see below) and the large number o f allowed harmonics (as compared with the cubic BP [10] will further smear the spectrum. For these reasons, the observed spectra [6,21 ] are consistent with a BIC structure for BP III. Further experimental verification is, however, clearly desirable. In the optical analog o f classical D e b y e - S c h e r r e r X-ray diffraction [21 ], a crucial factor is quasicrystallite size, which must be sufficiently large if closely spaced Bragg peaks are to be resolved. This problem does not arise for the cubic BP where the peaks are well spaced. Theoretical calculations o f harmonic peak intensities [1 ] could be helpful here. One could also attempt to grow relatively large single "quasicrystals" of BP III, as has been done successfully for the cubic BP [22]. Experimental diffraction patterns could then be compared with those of a BIC phase. It may also be possible to detect, b y light scattering, the phason modes which, in principle, exist in the incommensurate BIC structure [23]. To summarize, we have argued using Landau theory that a structure with broken icosahedral symmetry describes the anomalous BP III observed experimentally in cholesteric systems. Experimental confirmation could provide an unambiguous example of a thermodynamically stable phase with long-range quasicrystaUine ordering. Discussions with D. Mukamel are gratefully acknowledged. Models built b y D. Leibovitz contributed substantially to our understanding o f the BIC structure. This work was supported in part b y the Commission for Basic Research o f the Israel Academy of Arts and Sciences.
References [ 1 ] H. Grebel, R.M. Hornreich and S. Shtrikman, Phys. Rev. A 28 (1983) 1114, 3669(E) (in eqs. (A1) and (A3), replace K ~ by K~R); 30 (1984) 3264. 454
12 May 1986
[2] H. Stegemeyer and K. Bergmann, in: Liquid crystals of one-and-two-dimensional order, eds. W. Helfrich and G. Heppke (Springer, Berlin, 1980) p. 161. [3] R.M. Hornreich, M. Kugler and S. Shtrikman, Phys. Rev. Lett. 54 (1985) 2099; J. Phys. (Paris) 46 (1985) C3-47. [4] P. Pieranskii, P.E. Cladis and R. Barbet-Massin, J. Phys. (Paris) Lett. 46 (1985) L-973. [5] M. Marcus, J. Phys. (Paris) 42 (1981) 61. [6] S. Meiboom and S. Sammon, Phys. Rev. A 24 (1981) 468. [7] D. Levine and P.J. Steinhardt, Phys. Rev. Lett. 53 (1984) 2477; P. Bak, Phys. Rev. Lett. 54 (1985) 1517; N.D. Mermin and S.D. Troian, Phys. Rev. Lett. 54 (1985) 1524; M.V. Jaric, Phys. Rev. Lett. 55 (1985) 607; D.R. Nelson and S. Sachdev, Phys. Rev. B 32 (1985) 4592. [8] P.A. Kaluglin, A.Yu. Kitaev and L.S. Letitov, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 119 [JETP Lett. 41 (1985) 145 ]. [9] D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [10] R.M. Hornreich and S. Shtrikman, Phys. Lett. A 82 (1981) 20; Phys. Rev. A 28 (1983) 1791, 3669(E). [ 11 ] V.A. Belyakov, V.E. Dmitrenko and S.M. Osadchii, Zh. Eksp. Theor. Fiz. 83 (1982) 585; 84 (1982) 1920(E) [Sov. Phys. JETP 56 (1982) 322; 57 (1982) ll17(E)]. [12] S. Alexander and J. McTague, Phys. Rev. Lett. 41 (1978) 702. [13] G. Baym, H.A. Bethe and C.J. Pethick, Nucl. Phys. A 175 (1971) 225. [14] J. Kleinert and K. Maki, Fortschr. Phys. 29 (1981) 219. [15] R.M. Hornreich and S. Shtrikman, submitted to Phys. Rev. Lett. [ 16] D. Rokshar, private communication. [ 17 ] D.C. Wright and N.D. Mermin, Phys. Rev. A 31 (1985) 3498. [18] O. Biham, D. Mukamel and S. Shtrikman, submitted to Phys. Rev. Lett. [19] P.J. Collings, Phys. Rev. A 30 (1984) 1990. [20] R.N. Kleinman, D.J. Bishop, R. Pindak and P. Taborek, Phys. Rev. Lett. 53 (1984) 2137. [21 ] E.I. Demikhov, U.K. Dolganov and S.P. Krylova, Pis'ma Zh. Eksp. Theor. Fiz. 42 (1985) 15 [JETP Lett. 42 (1985) 161. [22] Th. Btumel and H. Stegemeyer, J. Cryst. Growth 66 (1984) 163; R. Barbet-Massin, P.E. Cladis and P. Pieranskii, Phys. Rev. A 30 (1984) 1161. [23] V.A. Golovko and A.P. Levanyuk, Light scattering near phase transitions, eds. H.Z. Cummins and A.P. Levanyuk (North-Holland, Amsterdam, 1983) p. 169.