506
F. Gentile et al.
14.2.2 Lattice Packings and SHSs
(Where it is introduced the mathematical representation of lattices. This is important
because SHSs may be artificially mimicked by a discrete subset of cylindrical micro
pillars, whose centers are placed in the plane as to recover a 2D lattice, as described
in the following sections of the text. In brief, here we translate the problem from
being expressed in terms of the sole φ, to a geometrical form.) A point lattice π, in a
bi-dimensional space, is a regularly spaced array of points. The mathematical form
of π is
π =
d
i=1
a i v i |a i ∝ Z
(14.4)
where d = 2 and {ν 1 , ν 2 } is a basis for R
2 . Every lattice in R 2 is completely determined given the fundamental parallelotope, that is the region of the plane that would
reproduce the original lattice by simple translations. The fundamental parallelotope
is thus the elementary cell there from the whole space can be reconstructed. Its
volume is uniquely determined by π, and the square of this volume is called determinant of the lattice [35]. Here, a representation of 2d lattices in a complex plane is
discussed. A lattice is defined by a pair of unit vector s, consider the first of these
with coordinates ν 1 = (1, 0) or, in a complex notation, simply ν 1 = 1. The position
of the second vector in the plane is thus sufficient for describing the whole lattice
geometry. The circumference in Fig. 14.2a describes all the configurations that are
isotropic, that is, the lengths in the ν 1 and ν 2 directions are the same. Thus, for
instance, a regular square pattern is the complex number ν 2 = i = e iπ/2 (where
i is the imaginary unit, and the Euler formula has been used), while an hexagonal
lattice finds adequate formalisation in the number ν 2 = e iπ/3 . In the realm of this
structure, the multiplication of a lattice by another holds the meaning of a rotation,
and thus, for instance, e iπ/2 × e iπ = e i3π/2 , and notice that e i3π/2 is still a square
lattice. Considering that, with [35], the area of the parallelotope may be written as
the determinant of the generator matrix M = [ν 1 , ν 2 ], for the present configuration
it follows that
A p
2
= det π = (det M)
2
=
v 12
v 21
v 12
v 22
2
=
1
cos ω
0
sin ω
2
= sin
2
ω (14.5)
and thus the determinant of the lattice is the square of the imaginary part of ν 2 . The
consequence of this is that the diagram of Fig. 14.2a finds a direct interpretation, that
is, all the points that are equally distant from the real axis describe the same pattern.
For instance, e iα/3 , e i2/3α , e i4/3α and e i5/3α are equivalent and all represent the
hexagonal lattice (Fig. 14.2b). This representation may be of some help in deriving
φ as a function of the architecture of the pillars. Consider an isotropic periodic array
of cylinders, the non dimensional diameter d of the cylinders, rescaled to the pitch,
is a number comprised between 0 and 1. The solid fraction φ may be accordingly
derived as α/2 2 d 2 /A p = α/2 2 d 2 / sin 2 ω.
F. Gentile et al.
14.2.2 Lattice Packings and SHSs
(Where it is introduced the mathematical representation of lattices. This is important
because SHSs may be artificially mimicked by a discrete subset of cylindrical micro
pillars, whose centers are placed in the plane as to recover a 2D lattice, as described
in the following sections of the text. In brief, here we translate the problem from
being expressed in terms of the sole φ, to a geometrical form.) A point lattice π, in a
bi-dimensional space, is a regularly spaced array of points. The mathematical form
of π is
π =
d
i=1
a i v i |a i ∝ Z
(14.4)
where d = 2 and {ν 1 , ν 2 } is a basis for R
2 . Every lattice in R 2 is completely determined given the fundamental parallelotope, that is the region of the plane that would
reproduce the original lattice by simple translations. The fundamental parallelotope
is thus the elementary cell there from the whole space can be reconstructed. Its
volume is uniquely determined by π, and the square of this volume is called determinant of the lattice [35]. Here, a representation of 2d lattices in a complex plane is
discussed. A lattice is defined by a pair of unit vector s, consider the first of these
with coordinates ν 1 = (1, 0) or, in a complex notation, simply ν 1 = 1. The position
of the second vector in the plane is thus sufficient for describing the whole lattice
geometry. The circumference in Fig. 14.2a describes all the configurations that are
isotropic, that is, the lengths in the ν 1 and ν 2 directions are the same. Thus, for
instance, a regular square pattern is the complex number ν 2 = i = e iπ/2 (where
i is the imaginary unit, and the Euler formula has been used), while an hexagonal
lattice finds adequate formalisation in the number ν 2 = e iπ/3 . In the realm of this
structure, the multiplication of a lattice by another holds the meaning of a rotation,
and thus, for instance, e iπ/2 × e iπ = e i3π/2 , and notice that e i3π/2 is still a square
lattice. Considering that, with [35], the area of the parallelotope may be written as
the determinant of the generator matrix M = [ν 1 , ν 2 ], for the present configuration
it follows that
A p
2
= det π = (det M)
2
=
v 12
v 21
v 12
v 22
2
=
1
cos ω
0
sin ω
2
= sin
2
ω (14.5)
and thus the determinant of the lattice is the square of the imaginary part of ν 2 . The
consequence of this is that the diagram of Fig. 14.2a finds a direct interpretation, that
is, all the points that are equally distant from the real axis describe the same pattern.
For instance, e iα/3 , e i2/3α , e i4/3α and e i5/3α are equivalent and all represent the
hexagonal lattice (Fig. 14.2b). This representation may be of some help in deriving
φ as a function of the architecture of the pillars. Consider an isotropic periodic array
of cylinders, the non dimensional diameter d of the cylinders, rescaled to the pitch,
is a number comprised between 0 and 1. The solid fraction φ may be accordingly
derived as α/2 2 d 2 /A p = α/2 2 d 2 / sin 2 ω.