LOCAL STABILITY OF QUASICRYSTALS

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LOCAL STABILITY OF QUASICRYSTALS

O. Biham, D. Mukamel, S. Shtrikman

To cite this version:

O. Biham, D. Mukamel, S. Shtrikman. LOCAL STABILITY OF QUASICRYSTALS. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-245-C3-249. �10.1051/jphyscol:1986326�. �jpa-00225738�

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LOCAL S T A B I L I T Y OF QUASICRYSTALS

0. BIHAM*, D. MUKAMEL* and S. SHTRIKMAN*'

'~epartment of Physics, Weizmann Institute of Science, Rehovot, Israel

'"~epartment of Electronics, W e i z m a m Institute of Science, Rehovot , f srael

Abstract

-

The local stability of the icosahedral quasicrystals to small de- formations w h i c m e them commensurate in one or more directions is conside- red. It is shown that icosahedral structures are generically stable to such deformations.

The experimental observation of quasicrystalline solids with icosahedral', decago- na12 and other symmetries, has stimulated extensive theoretical studies. In particular the global stability of the icosahedral phase has recently been considered by several au- t h o r ~ ~ - ~ . These studies, which are based on the Landau theory, show that the icosahedral phase may be energetically competitive with the bcc and the other crystalline solids. The analysis was carried out by considering the local density p(?) as an order parameter and constructing a Landau free energy density. The free energy of the various crystalline ( bcc, fcc etc.) and quasicrystalline ( icosahedral, decagonal etc.) phases has been calculated and the globally stable structure has been found. However, the question of stability of quasicrystals has not been analyzed before. In particular, these structures may exhibit an arbitrarily small distortion and become periodic (i.e. commensurate) in one or more directions. In this paper we analyze the energetics of such distortions. We find that quasicrystals, such as icosahedral or decagonal phases are generically stable to these deformations. They may therefore exist as thermodynamically stable phases

'.

Consider a phase transition between a liquid and a crystalline (or quasicrystalline) phase. Let p ( 3 be the corresponding order parameter, with p ( 3 ^{= 0}in the liquid phase, and p ( 3 # 0 in the low temperature phase. Let F{p(F)) = f{p(F))dFbe the Landau free energy functional associated with the transition, where f { p ( q ) is the free energy density.

This density is invariant under the symmetry group of the liquid (namely the centrosym- metric Euclidean group), and it may be expanded in powers of p ( 3 and its derivatives. It takes the form:

1 1 ⁰⁰

f ( p ( 3 ) = --k;(Vp)' ₂

+

a ( V 2 ~ ) 2

+

^anpn

n=O

where we have included only the lowest order terms in the gradient operator V , which are needed to describe the transition in a consistent way. In this expansion a1 = 0 since the order parameter corresponds to the symmetry breaking component of the density. We con-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986326

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C3-246 JOURNAL DE PHYSIQUE

sider the transition into the icosahedral structure and analyze the stability of this structure using the free energy density (1). In the icosahedral phase we take P(?) of the form:

where the six reciprocal vectors connect the center of the icosahedron to its vertices in the upper half space ( z > 0) with ~ & l ⁼ko (see Fig. la). The amplitudes pe and the phases pe are real quantities. In general one should include in the expansion (2) higher harmonics of the six basic vectors &. These harmonics are not expected to change the re- sults of the present local stability analysis and they have therefore been omitted. For a structure with icosahedral symmetry one has to take pe = po, l = 1,.

.

. ,6. To study the stability of this structure one assumes that the phases pe are slowly varying functions of r', and expands the free energy density in pe and their gradients, keeping the amplitudes pe constant. These six degrees of freedom provide six gapless modes 5 , namely three phonons,

ii, and three phasons, 5, which are linear combinations of pe. To second order in V p e , the elastic free energy density takes the form :

where a and b are.constants. One may then express pe in terms of the phonon and phason variables ii and 17. Such analysis has been carried out in previous studies for both decag*

nal and icosahedral systems

'.

However, by inserting the density function (2) into the Lan- dau free energy (I), one finds that in addition to the elastic free energy density fel, the free energy exhibits other terms which are functions of pe themselves (and not their gradi- ents!). For example, the term pn in (1) will generate terms of the form:

cos

[(c

^{n t h )}ⁱ⁺

c

n e ~ e ]

where ne are integers satisfying

x

⁽^ne ⁽⁼n. For the set of

&

considered here, the sum

x

^neked is non zero for any set of integers ne (except nl = .

. .

⁼^no= 0). Therefore, for pe = const. such a term yields a zero contribution to the free energy, when integrated over P: However, the structure p ( 3 may deform and the phases pe may become r' dependent.

In particular, by considering pe = Zt

-

Fwhere &, e = 1 , . . . , 6 are constant vectors one replaces the basic wavevectors

&

by the vectors:

For certain choices of the vectors Ge one may find non zero integers ne such that

...

x n e k i = 0. In this case the argument of the cosine function in (4) is a constant, yield- ing a gain of order -& in the free energy. On the other hand, the free energy cost of this distortion is of the order of

C

/&I2.

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tors &;,

. . .

, of this structure when it is distorted along the five fold axis in the z direction.

In the following we estimate the energy balance associated with such distortions. As an example we consider a distortion of the icosahedral structure which makes it commensurate along the five fold axis in the z direction. Such distortion may be induced by a term of the form (4) with nl = q and n2 =

. .

. ⁼^ne⁼^p. The term in the free energy

associated with a distortion which makes kl, commensurate with k&, L = 2,.

. .

^,6^is:

where n = q

+

^5p.This term yields non-zero contribution to the free energy if:

For simplicity we consider a distortion for which GI # ⁰^{and Zt =}^0, ^L^{= 2,.}

. .

^,6.The elastic free energy cost associated with this distortion is

In order t o find the free energy cost of this distortion we have t o estimate the difference )&/5 - p l q [ between the irrational number &/5 and the closest rational with denomina- tor q. Note that @/5 is a quadratic irrational lo. For any quadratic irrational T there is a

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C3-248 JOURNAL DE PHYSIQUE

constant C ( T ) such that"

17 - p/ql > c(4/q2

for all the rationals plq. Therefore, the parameter Zl satisfies:

and the elastic free energy cost of this distortion is of the order of l / n 4 . Combining this with a gain of -anp$, associated with the commensurability energy, we find that the free energy difference between the icosahedral phase and the distorted phase is

For small order parameter po < 1 , to which this analysis applies, the gain decays exponentially with n, while the loss decays only algebraically. Therefore A f is expected to be positive for large n , and the icosahedral structure is stable to deformations which make it periodic along the z axis with very long periods. If the coefficients a , are such that A f > ⁰also for small n, the icosahedral structure is stable! On the other hand, if for some small n = no Af becomes negative, the icosahedral phase becomes unstable, exhibiting a commensurate distortion along the z axis. Both possibilities are generic and therefore we conclude that icosahedral structures 9 exist as stable phases. To complete this analysis,one has to consider other possible distortions not necessarily along the five fold axis. The elastic energy associated with such distortion takes the form:

Since the incommensurate ratios of the icosahedron in the x, y and z directions are quadratic irrationals, the former analysis can be applied in each axis direction separately.

Therefore, the icosahedral structure may be stable to deformations in all directions. Sim- ilar analysis for planar quasicrystals with n >6 fold rotation axis shows that these struc- tures are also generically locally stable.

It has recently been obsenred that in addition to icosahedral structures, AI-Mn alloys exhibit other quasicrystalline structures. In particular, electron diffraction studies indi- cate that there exist a phase (the T-phase) which is periodic along one axis (say z) and quasiperiodic in the perpendicular plane . The symmetry along the z axis was found to be 10-fold. It can be shown that this structure may be obtained when an icosahedral structure is deformed along one of its five fold axes, and becomes periodic along that direction ',I2.

Acknowledgements: This work was supported in part by the Israel Commission for Basic Research and by Minerva Foundation, Munich, Germany.

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10. Quadratic irrational is an irrational number which satisfies a n algebraic equation of the second degree with rational coefficients.

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