Characterization of coincidence site lattices of oblique planar lattices Marco Antonio Rodríguez -Andrade1 Gerardo Aragón -González2 José Luis Aragón Vera3

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Characterization of coincidence site lattices of oblique

planar lattices

Marco Antonio Rodríguez-Andrade(1), Gerardo Aragón-González(2*), José Luis Aragón Vera(3)

(1)Departamento de Matemáticas, ESFM, IPN

(2)PDPA. UAM-Azcapotzalco.

(3)Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México.

Apartado Postal 1-1010, 76000 Querétaro, México.

*E-mail: gag@correo.azc.uam.mx.

Abstract

Coincidence site lattices of oblique planar lattices are algebraically characterized

using as basic tool the Cartan-Dieudonné theorem, that is, the decomposition of

an orthogonal transformation as a product of reflections. The case of rectangular

lattices is worked out in detail. We use the rectangular lattices for obtain the

characterization of the corresponding obliques.

1. Preliminaries

A lattice



, consists of all integer linear combinations of a basis

 

1 2 n

a ,a ,…,a

, that is,

Z Z Z Z = =   a a a

. In terms of group structure, leaving aside its geometric meaning, a

lattice



is a finitely-generated free Abelian group. In this sense, a subset



  

is called sublattice

if it is a subgroup of finite index, where the index is the number of distinct cosets of





, denoted

 

:



. The following two general definitions are central for the coincidence problem [1]:

Definition 1. Two lattices



and



are called commensurate, denoted by

  

, if and only if

  

is a sublattice of both



and



Definition 2. Let



be a lattice in

. An orthogonal transformation

()T O n

is called a

coincidence isometry of



if and only if

T  

. The integer

 

( ): :TT =    

is called the

coincidence index of

with respect to



. If

is not a coincidence isometry then

( ):T = 

. Two

useful sets are also defined:

( )   ( ) ( )

 

: ( )| ( ) , : |det( ) 1 .OC T O n T SOC T OC T =      =   =

Since we will be dealing with quotient groups, some words about the order of these groups should be

said. The elements of the quotient group

( ) ( )

( )

/TT   

will be denoted by

 

, where

( )

aT

, that is,

 

( ) ( ) ( )

 

.a b T a b T=   −   

In this case we have a finite Abelian group with

identity

 

( )

0T=   

. Since all the elements of this group have finite order, given

x

, the order

( )





, denoted as

( )

o T x





, is finite and defined as

( )

( ) ( )

 

mino T x k N kT x T



=    



(1)

In Ref. [2], the general problem of coincidence lattices in several planar lattices was worked out using

Clifford algebras. In what follows we summarize the main results (without requiring Clifford

algebras) that will be used in this work.

Suppose that

aR

is a non-zero vector. By



we denote a simple reflection (a reflection by a

hyperplane orthogonal to

) . Thus, if

denotes the orthogonal complement to

, then:

( ) ( )

( )

; if H





= − = 



a a a

a a w w w

x x - 2 a

||a||

(2)

By the Cartan Theorem[3], any given orthogonal transformation in

can be written as a product of

simple reflections [4]. In our particular case,

aR

and the hyperplane is a line perpendicular to

Also, we can always find

x,y R

such that



=xy

In [2, Proposition 7] it was proved that for any lattice



( )





is a reflection then

there exists a scalar





such that



c

. Since





, this condition can be rephrased as

follows:

Lemma 1. A simple reflection

( )





if and only if there exists a vector

c

such that



Now, for a rotation



(henceforth, rotation with angle



) it was shown in the same reference

(Proposition 11) that for simple lattices generated by the basis

 

a ,a

, the coincidence site lattice

(CSL) problem for this case is solved by the following Lemma:

Lemma 2.

( )

T SOC





if and only if there exists a vector

c

such that



where

b = e

(square)

b = a

(rectangular) and

2 1 2

b = a -a

(rhombic [5]).

This result arose from the fact that in the rhombic and hexagonal lattice the vectors

1 1 2

d = a +a

and

2 1 2

d = a -a

are orthogonal and, in fact,

 

d ,d

defines a rectangular sublattice of the rhombic

lattice. Notice the resemblance with the approach followed by Ranganathan [6].

2. The CSL problem for the oblique lattice

Let

( )



= 

Ze Z a

be an oblique lattice with



R

and

( ) ( )

cos sin



a = e + e

. Define

1 cos

0 sin





=



and let

1 cos

cos



  



=



be the structure matrix with respect to the ordered

basis

 



=B e a

. Clearly, the square case corresponds to



and





; the rectangular case to





; the hexagonal case to



and





; and the rhombic case to



The next Theorem characterizes the coincidence orthogonal transformations:

Theorem 1.

( )

,T OC

( )

= 12

Za + Za

if and only if

 

( )



Q

with respect to the basis

 

B = a ,a .

Proof. The condition is necessary. We now use the fact that there is exists an isomorphism between

( )

/T   

and

( ) ( )

( )

/TT   

(see [1]). Let

( )

T OC

, the quotient group

( ) ( )

( )

 

/T T T



    = 



is finite, where the equivalence class is given by:

( ) ( ) ( ) ( )

 

;T b T T T



=  −    



a b a

and

 

( )

0T=   

. Thus, the elements of this group have

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摘要：

CharacterizationofcoincidencesitelatticesofobliqueplanarlatticesMarcoAntonioRodríguez-Andrade(1),GerardoAragón-González(2*),JoséLuisAragónVera(3)(1)DepartamentodeMatemáticas,ESFM,IPN(2)PDPA.UAM-Azcapotzalco.(3)CentrodeFísicaAplicadayTecnologíaAvanzada,UniversidadNacionalAutónomadeMéxico.ApartadoPost...

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Characterization of coincidence site lattices of oblique planar lattices Marco Antonio Rodríguez -Andrade1 Gerardo Aragón -González2 José Luis Aragón Vera3

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