Crystal Structure Theory and Applications, 2012, 1, 52-56
http://dx.doi.org/10.4236/csta.2012.13010 Published Online December 2012 (http://www.SciRP.org/journal/csta)
Conditions for Singularity of Twist Grain Boundaries
between Arbitrary 2-D Lattices
David Romeu1, Jose L. Aragón2, Gerardo Aragón-González3, Marco A. Rodríguez-Andrade4,5,
Alfredo Gómez1
1Departamento de Materia Condensada, Instituto de Física, Universidad Nacional Autónoma de México, México City, México
2Departamento de Nanotecnología, Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México,
México City, México
3Programa de Desarrollo Profesional en Automatización, Universidad Autónoma Metropolitana, México City, México
4Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional,
Unidad Profesional Adolfo López Mateos, México City, México
5Departamento de Matemática Educativa, Cinvestav-IPN, México City, México
Received September 29, 2012; revised October 31, 2012; accepted November 18, 2012
ABSTRACT
1
2tan N We have shown that the expression




2
derived by Ranganathan to calculate the angles at which there
exists a CSL for rotational interfaces in the cubic system can also be applied to general (oblique) two-dimensional lat-
tices provided that th e quantities

and


cos

 are rational numbers, with  ba and α is the angle between
the basis vectors a and b. In contrast with Ranganathan’s results, N; given by


2
tanN

 needs no longer be an in-
teger. Specifically, vectors a and b must have the form


1, 0a;


,tanr

b where r is an arbitrary rational number.
We have also shown that the interfacial classification of cubic twist interfaces based on the recurrence properties of the
O-lattice remains valid for arbitrary two-dimensional interfaces provided the above requirements on the lattice are met.
Keywords: Grain Boundaries; Crystallography of Interfaces; Coincidence Site Lattice
1. Introduction
In a now classic paper Ranganathan [1] showed that a
Coincidence Site Lattice exists between two rotated cu-
bic lattices when the rotation angle can be expressed as [1]
tan 2
yN
x




222
Nh kl


(1)
where x, y are coprime integers and is
an integer equal to the square of the magnitude of the
crystallographic rotation axis

hklc. He also provided
a procedure to find the index number (the quotient
between the areas of the unit cells of the CSL and the
cubic lattices) as a function of N, x and y. In this paper
we show that this equation is also valid for arbitrary
(oblique) two dimensional lattices provided their basis
vectors fulfill certain rationality conditions. Specifically,
we will show that a two-dimensional CSL exists when
the rotation angle θ given by

1
2tan N 


 (2)
where N is a real number that depends only on the lattice
and ξ is a rational number. Reducing the problem to two di-
mensions makes the generalization to arbitrary twist grain
boundaries tractable while it is not in itself major shortcom-
ing since twist interfaces are two dimensional systems.
Besides its mathematical novelty, this result makes it
possible to extend to arbitrary lattices a recent classifica-
tion scheme for cubic GBs [2] based on the recurrence
properties of the O-lattice [3] in combination with the
angular parameterization introduced by Equation (2). In
this scheme the angular space is partitioned into disjoint
intervals


12, 12xx   centered around every
integer x. This partitioning groups GBs into an effec-
tively finite number of equivalence classes [2], each
containing a special (singular) interface which is the
normal form of the class. Normal forms have the pro-
perty of having a particularly simple structure [4,5]
composed of structural elements (translational states) [4]
also presented in all the elements of each class (see Sec-
tion 5). Singular interfaces are located at the centre of
each interval at the angles
x

obtained by inserting in-
tegral values of ξ into Equation (2). The classification of
C
opyright © 2012 SciRes. CSTA

D. ROMEU ET AL. 53
interfaces is important since it allows, in principle, the
description of physical properties in terms of differences
and/or similarities with ideal defect-free structures akin
to the crystalline Bravais lattices. Other work in this di-
rection includes classification efforts based on symmetry
variants [6,7] and structural units [8,9]. Without a crys-
tallography of interfaces it is difficult to tell whether a
given property is specific of a particular interface or
shared by a whole collection of them.
In the following sections we review Ranganathan’s ap-
proach towards the derivation of Equation (1), we then
extend it to two dimensional interfaces to calculate the
explicit form that the basis vectors of 2D lattices must
have. Finally we show an example of a rhombic lattice
showing that an interface in the class x has common ele-
ments with the normal form of the class.
2. Ranganathan’s Method
In this section we extend Ranganathan’s approach to
two-dimensional lattices that are not planes of a cubic
lattice. We will see that Ranganathan’s approach still
gives coincidences but some rationality conditions must
be imposed on the lattices. Also, N is not any longer re-
lated to any crystallographic direction but depends ex-
clusively on the nature of the lattice.
In what follows we shall use L to indistinctly refer to a
lattice and its structure matrix. Let L a cubic lattice in
ordinary three-dimensional space and assume 2R

L
L

where R is a rotation through the angle θ around th e cr ys-
tallographic axis

hkl

20
. We want to know when such a
rotation leads to coincidences; in other words, when is
.
LL
The strategy followed by Ranganathan [1] involved
first asking when the (hkl) plane of a cubic crystal con-
tains a rectangular sublattice. He noticed that the answer
is always in the affirmative: the orthogonal vectors
0lk


a and 22 klhk hl



b are always
on the (hkl) plane.
But a rectangular cell is always symmetric under a re-
flection with respect to one side of the rectangle. Then,
for any vector
x
yPab with x and y integers (which
lies in L) the vector
x
y
Pab (reflection with re-
spect to the a side of the rectangle) is also in the lattice.
Since the effect of this reflection is the same of a ro tation
through taking P into P' (see Figure 1), we conclude that
such a rotation also brings the lattice into coincidence.
The angle θ is given by
tan 2




yyN
xx

b
a
with
22
ba
Figure 1. Ranganathan’s rectangle. For any lattice vector P =
xa + yb with x and y integers, the reflection with respect to
the a side of the rectangle: P' = xa − yb is also in the lattice.
The effect of this reflection is the same as that of a rotation
through  .
3. General Two-Dimensional Lattices
Following Ranganathan, we address here the problem of
finding a rectangular cell that spans a sublattice of a gen-
eral 2-D lattice. The lattice shall be referred to as L and
we give a spanning set


,ab
¢0
.
The problem is to find two vectors d1, d2 2L such
that 12

dd

. Then there must exist four intege rs u, v, x,
y such that
22
2
Nhkl
.


1
2
12
22
22
0
cos
uv
xy
uv xy
uxvyuy vx
uxvyuy vx




 


dab
dab
dda bab
ab ab
ab ab
0ab 1
where α is the angle between a and b.
There are several cases to consider:
1) In the trivial case , then

da
2
and

db.
2) The rhombic case

ab, in this case




0

ab ab so 1

dabdab and .
2
3) In the most general case where

ab
0 and

ab 0v0. Here there is a solution with (or u

)
provided that





2cos 0
cos 0
cos
ux uy
ux uy
x
y








aab
ba
ba
and this holds if

 
cos

ba is a rational number
mn with m and n integers. Solving for mn xy


0v
we have
x
m1u

,

,

yn
da.dn bma
and , and using
Equation (3) we have 1 and 2


12
, mnm
Sometimes it is convenient to use a rectangle that is not
the smallest (in area) possible. This is the choice

dadba (4)

D. ROMEU ET AL.
54
(see Figure 2).
Swapping the roles of a and b we see that


cos

ba must be also rational and, consequently,
22
ba
.
In what follows we will work in this general setting,
since the rectangular and rhombic cases are particular
instances.
3.1. Finding Coincidences
Once we have a rectangular cell spanned by


,dd
12
that generates a sublattice of L, then by Ranganathan’s
argument it follows that there is a rotation that produces
coincidences for every choice of integers x, y such that
2
1
2
y
x





d
d
tan (5)
so if we call yx

 and

2
21
Ndd we obtain
tan 2
N 




 (6)
which is identical to Equation (2) except now N depends
only on the lattice and does not have to be an integer,
from Equation (4) it follows that N must be rational. No-
tice that

12


tanN1 where

is the angle for
1

 so we can write
1
1tan
22
tan









m
(7)
For the second choice of rectangle 1

da,
, we have 122ndbman

bdd , hence 1 and
12
are parallel to a and b respectively so that the
angle between 12
and d is the same as the angle
between a and b. Using
d
dd
1
dd
12

  we obtain
1tan
22
tan








 (8)
Figure 2. Two-dimensional lattice generated by the vectors
(1,0) and (3/4, 6/4). Note the small rectangular cell with
dashed vertical line is spanned by d1 = (1,0) and d2 = (0,4).
The larger cell spanned by d3 = (3,0) and d4 = (0,4) has the
property that the angle between d3 + d4 and d3 is the same as
between a and b.
Notice that different choices of i will lead to the
same set of angles θ because Equation (5) implies that
given a rectangle 12 , if we choose a different one
31
d
,dd
h

dd42 k,

dd (for h, k integers or even rationals)
the set


4 1
3
tan 2













d
d
is equal to the set


2 1
1
tan 2
k
h













d
d
12
the angles obtained for a given ξ are, however, different.
So far we have provided coincidences only along a
line. Next we show that th ere is a genuine 2D CSL: con-
sider the vector
y
pxq
 Pdd
12
with x, y as before and p, q integers. This is a vector in
the lattice and it is orthogonal to
x
yPd d if and
only if p and q are chosen so as to satisfy
2
2
2
1
pN
q
d
d

(they will be assumed to be coprime).
This shows that we can always construct a lattice vec-
tor orthogonal to P and it gives rise to another line of
coincidences by exactly the same argument used to prove
that P gives rise to a line of coincidences. Then a full
two-dimensional CSL is ensured.

The area of the rectangle spanned by P and P is


222 2
12
xqy pdd
A
so

22 2
12
Aqx yN
dd

1q
 
 
an expression that leads to and agrees with the cor-
responding one in Ranganathan’s work since in his case
N is integer and, a fortiori, .
4. Lattice Basis Vectors
Using elementary trigonometric identities it can be
shown that the two-dimensional rotation matrix (with
respect to a given orthonormal basis) and Equation (2)
are given by
2
22
cos sin2
1
sin cos2
NN
RNNN
 
 

 








The lattice will be given by the basis

1,0a
and


cos ,sin

 b; note we loose no generality by

D. ROMEU ET AL. 55
taking the first vector to be unitary. The structure matrix
(the matrix having as columns the vectors of the lattice
basis) is given by


1cos
0sin







 L (9)
but, using Equations (6) and (8) and we get
1cos
0cosN







 L (10)
which equals
11
0
N
N







1N

 L (11)
or
1
0
L1
1
N
N
N










(12)
Consequently the rotation matrix with respect to the
lattice basis (also known as transition matrix) is
22
2
1
22
N
N







 

2
,,N
2
22
1
1
N
TNN




 






which is a rational matrix (a necessary and sufficient
condition for the existence of coincidences [10]) if

 and
cos
N1
r


 
r
 (14)
are rational numbers. Substituting in Equation (12),
the general form for the structure matrix L of an oblique
lattice that can give rise to a CSL through rotations ac-
quires the particularly simple form
1
0
r
rN
1
0tan
r
r






L


(15)
which depends only on the angle α and size ratio σ of a to b.
5. Singular vs Non-Singular Cases
Figure 3 illustrates that the parametrization of Equation
(2) works for non cubic lattices. Figure 3 shows an ex-
ample of the difference in structure between a normal
class interface and other interfaces in the class [2]. The
figure shows the 5

 and 513

 interfaces for a
lattice defined by the structure matrix which is of the
form 12.
Figure 3. Rotational interfaces of a non cubic lattice defined
by the structure matrix of equation 16. Top: singular inter-
face
 = 5 (= 3). Bottom: non singular interface  = 5 + 1/3
(= 301). Small open and filled circles (dichromatic pat-
tern) identify points of the rotated and unrotated lattices.
The length of the lines joining dichromatic pattern points
reveals the magnitude of the strain field. Large circles are
drawn in the middle of dichromatic points whose distance is
smaller than one atomic diameter and represent zones of
relatively low strain. Note the non-songular interface con-
tains geometrical features belonging to the singular inter-
face. The rectangle on the top and the rhomus at the bottom
are the corresponding CSL unit cells.
112
052





L (16)
The Figure shows that the singular interface has a
much simpler structure and that the non singular case has
structural units already presented in the singular one in
accordance with previous observations [4].
6. Conclusions


1
2tan
y
Nx 


2
We have shown that the expression
derived by Ranganathan to calculate the angles at which
there exists a CSL for rotational interfaces in the cubic
system can also be applied to general (oblique) two-di-
mensional lattices provided that the quantities

and


cos


are rational numbers, with
 ba
and α
is the angle between the basis vectors a and b. Note that
in contrast with Ranganathan’s results, N given by


2
tanN

 needs no longer be an integer. Addition-

D. ROMEU ET AL.
Copyright © 2 CSTA
56
[5] D. Romeu, “Interfaces and Quasicrystals as Competing
Crystal Lattices: Towards a Crystallographic Theory of
Interfaces,” Physical Review B, Vol. 67, No. 2, 2003, pp.
24202-24214. doi:10.1103/PhysRevB.67.024202
012 SciRes.

ally, we have shown that in order for two dimensional
lattices to give rise to a (twist) CSL and the basis vectors
a and b must have the form

,1, Nab1, 0, where
r is an arbitrary rational number. [6] R. C. Pond and W. Bollmann, “The Symmetry and Inter-
facial Structure of Bicrystals,” Philosophical Transac-
tions of the Royal Society, Vol. 292, No. 1395, 1979, pp.
449-472. doi:10.1098/rsta.1979.0069
We have also shown that the interfacial classification
based on the recurrence properties of the O-lattice re-
mains valid for arbitrary two-dimensional interfaces pro-
vided the above requirements on the lattice are met. [7] R. C. Pond and D. S. Vlachavas, “Bicrystallography,” Pro-
ceedings of the Royal Society, Vol. 386, No. 1790, 1983,
pp. 95-143. doi:10.1098/rspa.1983.0028
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