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# Background material

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.