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

Email: romeu@fisica. una m.mx

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, 0a;

,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

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

hklc. 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

obtained by inserting in-

tegral values of ξ into Equation (2). The classification of

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