# Background material

Comments on stability of quasicrystals

From Jaric^{[1]}

Steinhardt found in direct simulations that a twodimensional,

two-component Lennard-Jones gas has

an at least quasistable equilibrium state corresponding

to a Penrose lattice. s A more phenomenological approach

was taken by Bak [6], Mermin and Troian [7] and

Kalugin, Kitaev, and Levitov [8], who based their investigations

on the Landau theory of solidification as formulated

by Alexander and McTague. [9] In order to

bypass the original conclusion [9] that a body-centeredcubic

crystalline structure should generally be favored,

they either included higher-order terms in the Landau

expansion, [6] or they introduced an additional component

to the density.[7,8]

also

In order to stabilize the icosahedral structure, Bak6

extended previous assumptions by adding a fifth-degree

term to the expansion. He then considered

only a contribution arising from those q; which form a

regular pentagon. There are no such q; for the bcc set,

but they exist in the icosahedral set. Therefore, this

fifth-order coupling can be chosen to make [QC energy lower thatn bcc]Mermin and Troian7 stabilized the icosahedral translational

ordering in another way. They assumed that C

is constant over its entire domain and they introduced

a second component (order parameter) p(k) which

selects another wave-vector magnitude such that

0 & k & 2q. They also assumed that the ordering

of p(k) is induced by the ordering of p(q) and they

effectively integrated out the p(k) component. In this

way they arrived at a theory which is equivalent to a

theory with a single-component q and with the quartic

coupling sharply peaked around C (k2/q2 —1,—1)

= C ( + 1/ J5, —1) which favors icosahedral ordering.In a related approach, Kalugin, Kitaev, and Levitov8

argued that the k components which are second harmonics

of the q components must be included into the

analysis since, in contrast to the ordinary crystals, for

the icosahedral vertex model $k^2/q^2 \approx 1$, and the

minimum of A (q) need not differentiate k and q.

They also concluded that an icosahedral structure is

stabilized.

Jaric's own work builds upon bond orintational order analysis

Several years ago Nelson and Toner[11] studied cubic

bond-orientational order, while more recently

Steinhardt, Nelson, and Ronchetti investigated the

short-range icosahedral bond-orientational order. 10

Penrose lattices4 and the experimentally observed

quasicrystals' both exhibit long-range icosahedral

bond-orientational order which coexists with longrange

icosahedral translational order.

His rigorous group theoretical approach allows to find large ranges of *globally* stable icosahedral QC phase.

From Katz-Lebedev-Muratovic^{[2]}

The icosahedral quasicrystal Y is still metastable for the simplest anisotropy of 2 of the form of

(2.54). At the same time it is not difficult to imagine the anisotropy of 2, rendering the icosahedral

phase absolutely stable. For this purpose it is necessary that the vertex A should have sufficiently

deep minima for the angles between the basis vectors of 36°and 72°.

They also have more detained discussion in sections 3.2.3 and 3.2.4.

**There seems to be a discrepancy in the papers: some claim that observed icosahedral QC have 15 q vectors (correspond to edges of icosahedron) [Bak, Kalugin-Kirtaev-Lewvitov], while others only 12 (vectros from the center of icosahedron to vertices) [Mermin-Troian]. The former arrangement is good for 3rd and 5th order GL invariants. The latter only for 4th order one.**