FCC , BCC and SC Lattices Derived from the Coxeter-Weyl groups and Quaternions

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine extended Coxeter groups 3 3 3 ( ) and W( ) ( ) W A B Aut A  . It is naturally expected that these rank-3 Coxeter-Weyl groups define the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices which have extensive applications in materials science. The imaginary quaternionic units are used to represent the root systems of the rank-3 Coxeter-Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The group elements are written explicitly in terms of pairs of quaternions which constitute the binary octahedral group. The constructions of the vertices of the Wigner-Seitz cells have been presented in terms of quaternionic imaginary units. This is a new approach which may link the lattice dynamics with quaternion physics. Orthogonal projections of the lattices onto the Coxeter plane represent the square and honeycomb lattices.


( ) and W( ) ( ) W A
B Aut A  define the point tetrahedral and octahedral symmetries of the cubic lattices which have an enormous number of applications in materials science.In this paper we explicitly show that the root lattice and the weight lattice of the affine Coxeter group 3 ( ) a WA describe the fcc and bcc lattices respectively.We point out that there is a natural correspondence between the octahedral symmetry of the fcc and bcc lattices and the binary octahedral group of quaternions.The paper is organized as follows.

T
In Section 2 we explore how the honeycomb lattice and the square lattice in two dimensions are respectively described by the affine extensions of the Coxeter-Weyl groups WA and prove that the root lattice and the weight lattice describe the fcc and the bcc lattices respectively.In Section 4 we point out that the regular polyhedra can be obtained as the orbits of the Coxeter-Weyl group 3 () WB and furthermore we demonstrate that its affine extension 3 () a WB describes the sc and the bcc lattices as the root and the weight lattices of 3 B expressed in terms of quaternions.In Section 5 we study the orthogonal projections of 33 and AB lattices onto the Coxeter plane, displaying respectively the square and hexagonal lattices and emphasize the importance of the dihedral subgroups of the Coxeter-Weyl groups in projection techniques.Some conclusive remarks will be presented in Section 6 regarding the applications of the quaternionic constructions of the 3D lattices.

Construction of the honeycomb and square lattices as the affine Coxeter-Weyl groups W a (A 2 ) and W a (B 2 )
The Coxeter-Weyl groups are generated by reflections with respect to some hyperplanes represented by vectors (also called roots in the literature of Lie algebras).If 12 , ,..., n r r r represent the reflection generators, then the presentation of the Coxeter-Weyl group ( ) WG is given by 12 ( ) , ,..., ( ) where ij m is an integer label with The vectors 12 and  (hereafter called "roots") orthogonal to the planes with respect to which the reflection generators reflect an arbitrary vector  as 2( , ) , 1, 2. ( , ) The Cartan matrix C with the matrix elements correspond to the generating vectors of the reciprocal lattice.Now we discuss the lattices associated with each Coxeter-Weyl group.

The lattice determined by the affine group W a (A 2 )
The Cartan matrix C and the metric G are given as follows The plot of the polygons (triangle or hexagon) described by the orbits of the group 2 () WA will be given in orthonormal basis which can be obtained by using the eigenvectors of the Cartan matrix [ The group generators act on the roots as: ,,, r r r r D .The matrix representations of the generators in the root space can be written as The affine Coxeter group 2 () a WA includes another generator 0 r which reflects the vectors with respect to a plane bisecting the vector 12   .This corresponds to a translation on an arbitrary vector  , as     determines the primitive cell of the direct lattice which is an hexagon.The whole lattice is the infinite set of hexagons as shown in Figure 2. It is interesting that the honeycomb lattice does in fact exist as graphene made of carbon atoms [19].The weight vectors can be determined from the relation where the reciprocal lattice 2 A  (weight lattice) vectors are given by aa  .We will define an orbit in the dual space (reciprocal space) as which represents an isogonal hexagon with two edge lengths in general.When we have 12 aa  , the orbit is a regular hexagon

(1,1)
A determined by the root system which is invariant under the group We assume that the reader is familiar with the concept of nearest neighbor region which is called the Wigner-Seitz cell or Brillouin zone by the crystallographers, but is also known as the Voronoi cell or Drichlet region by mathematicians.The Wigner-Seitz cell of the root lattice 2 A is the cell which can be determined from its primitive cell (see Figure 3).The Wigner-Seitz cell

The lattice determined by the affine group W a (B 2 )
The roots of 2 B consist of long and short roots; in a particular orthonormal system with ( , )  , the simple roots can be written as 1 1 2 2 2 , l l l     .The Cartan matrix, its inverse and the matrix G are given by 1 The group generators 12 and rr act in the root space as which can be represented by the matrices They generate the dihedral group 4 D of order 8, which is the symmetry of the square.The root system consists of two orbits which can also be written in the orthonormal basis as   , corresponding to the centers of the edges and the vertices of a square respectively.It is clear that the root lattice is generated by the short roots of 2 B .The primitive cell is the square represented by the vectors 12 { , } ll .The translation generator is obtained as the addition of the highest short root

Construction of the fcc and the bcc lattices as the affine Coxeter-Weyl group W a (A 3 )
A brief introduction is necessary for those readers not familiar with the quaternions.The quaternions are the extension of the complex number system discovered by Hamilton in 1853 [20] with three imaginary units ,, i j k satisfying the well-known relations In what follows we will use a different notation for the quaternionic units ,, i e j e k e    .Let 0 ii q q q e  , ( 1, 2,3) i  be a real unit quaternion with its conjugate defined by 0 ii q q q e  and its norm 1 qq qq  .The quaternionic imaginary units now satisfy the relations where ij  and ijk  are the Kronecker and Levi-Civita symbols, and summation over the repeated indices is understood.
The unit quaternions form a group isomorphic to the special unitary group (2) SU .With the definition of the scalar quaternions generate the four-dimensional Euclidean space and the unit quaternions 1, , and e e e form an orthonormal basis.From now on it is understood that a quaternion is a 4D-Euclidean vector.The Coxeter-Dynkin diagram of 3 A with the quaternionic simple roots is given in Figure 5.A with quaternionic simple roots.
An arbitrary quaternion  when reflected by the operator r with respect to the hyperplane orthogonal to quaternion  is given in terms of quaternion multiplication [21] as The bracket on the right of the equation ( 12) is self explanatory and should not be confused with the commutator notation.The Cartan matrix of the Coxeter-Dynkin diagram 3 A and its inverse matrix are given respectively by the matrices 2 1 0 The generators of the Coxeter group 3 () WA are then given in the notation of ( 12) by The reflection generators operate on the quaternionic imaginary units as follows: Here and TT  are the sets of quaternions where T represents the binary tetrahedral group of order 24, and the union of the set O T T   represents the binary octahedral group [22] of order 48 .When the simple roots are chosen 1 1 2 2 ,, e e e e e e          as in the Figure 5 then the weight vectors are determined from (13) as The orbits generated by these vectors would lead to the set of vectors They respectively represent the vertices of a tetrahedron, an octahedron and another tetrahedron.The orbits 33 (1,0,0) and (0,0,1) AA together represent the vertices of a cube.Therefore the symmetry of the union of the orbits 33 ( Note that from now on we are using the group notation instead of ( 15), but with the same meaning.It can be proved that the Coxeter-Weyl group 3 () WA given by ( 15) is isomorphic to tetrahedral group S represents the symmetric group of four objects which permutes the vertices of a tetrahedron.To understand this better, let us denote the vertices of the tetrahedron by letters; ,, In this notation they represent the generators of the group 4 r r r permutes all four letters in the order ACDB.The group 3 () Aut A has three maximal subgroups, each is of order 24.
a) The usual one is the

( ) {[ , ] [ , ] } W A T T T T   
tetrahedral group derived from the Coxeter-Dynkin diagram and it is the symmetry of a tetrahedron as we have seen.b) The group

( ) {[ , ] [ , ]} W A C T T T T  
is the proper rotational subgroup of the octahedral group 3 () Aut A .In another paper [23] we have proved that it is the symmetry of the snub cube, a chiral polyhedron, and one can construct the vertices of the snub cube by the action of the group on a vector.The group

{[ , ] [ , ] }
is the pyritohedral group representing symmetry of a pyritohedron, an irregular dodecahedron with irregular pentagonal faces which occurs in pyrites.For the interested reader we give here the matrix representations of the generators of the group

The fcc lattice as the root lattice of A 3
The group generators of the affine 3 () WAare the three reflection generators 1 2 3 0 , , and r r r r , where 0 r acts like a translation by the highest root . A general vector of the root lattice is then given by The root system of 3 A is the orbit A which corresponds to the origin, the root system represents the "non-conventional" face centered cubic cell [24].It is easy to prove that the Wigner-Seitz cell of the fcc lattice is the Catalan solid, the rhombic dodecahedron, dual to the Archimedean solid cuboctahedron determined by the root system of 3 A .The vertices of the rhombic dodecahedron which constitutes the Wigner-Seitz cell can then be written as the union of the orbits 333 (1, 0, 0) (0,1, 0) (0, 0,1) whose quaternionic vertices are given in (18).The rhombic dodecahedron represented by the set of vertices is depicted in Figure 6.
The centers of the faces of the rhombic dodecahedron are the vertices given by
A general vector of the lattice is given by , , , q a a a a a a In terms of quaternions, a weight lattice vector consists of the linear combinations of the quaternionic units; the coefficients of the unit vectors are either all integers or half integers.Recall that the root lattice consists of the quaternions obtained as the linear coefficients of the quaternionic imaginary units with integer coefficients only.We note that the orbits 3 3 3 (1,0,0) (0,0,0) (0,0,1) form the body centered cubic cell where the origin represents the central lattice point.One can easily prove that the nearest lattice points to the origin are the orbits 333 (1,0,0) (0,1,0) (0,0,1) AAA .From (18) we deduce that the distance to the lattice points represented by the vertices of the square 33 (1,0,0) 0,0,1) AA  is 3 0.866 2  , but the next nearest points are on the orbit 3 (0,1,0) A at a distance 1.As usual, the walls of the Wigner-Seitz cell bisect the lines joining these nearest points to the origin.It is easy to show that the intersections of these planes determine the vertices of the Wigner-Seitz cell as the orbit 3 1 (1,1,1) 4 A , which is a truncated octahedron as shown in Figure 7, and the vertices of which are given by 3 The generators, is given in (19).The weight vectors It is clear from (28) that they constitute two different orbits; however the second set is merely a linear combination of the first set of vectors.
By the affine extension of 3 B we note that the quaternionic highest root is

The sc lattice as the root lattice of B 3
The root lattice is generated by the short roots 1 2 3 ,, e e e of 3 B and a general root lattice vector is given in terms of quaternionic units as The weight lattice is exactly the same lattice as determined by the weight lattice 3 A  , since the weight vectors will be the linear combinations of the quaternionic imaginary units with either integer coefficients or half integer coefficients.Here the nearest neighbors of the origin are the vectors

Projections of the A 3 and B 3 lattices onto their Coxeter planes
Projection of higher dimensional lattices to lower dimensional spaces is an interesting technique in the description of quasicrystallography.In reference [18] we proposed that in the projection technique the Coxeter number and the Coxeter integers play important roles.We here describe the technique in applying it to 3 A and 3 B lattices although the result is not a quasi crystal.Every Coxeter group has a maximal dihedral subgroup of order 2h where h is the Coxeter number.It goes as follows: partition the simple roots into two sets of roots so that each set consists of orthogonal roots.In the case of 3 A and 3 B these sets are 1 3 2 { , } and    .We now consider the case of 3 A where we define the Coxeter plane determined by the vectors given by which is orthogonal to the vector 3 . We now define the generators A of order 8.The orthogonal set of vectors in the Coxeter plane can be taken as the quaternionic unit vectors 23 and ee .Let us recall that the root lattice vectors are given by For the orthogonal projection of the lattice 3 B , we follow the same technique; however the group generated by the generators 12 and RR is the dihedral group 6 D of order 12 since

Conclusion
The crystallographic Coxeter groups, in other words, the Coxeter-Weyl groups, are the skeleton of the Lie groups which may have many applications in quasicrystallography.We have shown in this paper that the Coxeter-Weyl groups acting in 2D and 3D directly describe the square, hexagonal, sc, fcc and bcc lattices.The advantage here is that they can be associated with some finite subgroups of quaternions and the vertices can be represented by quaternionic imaginary units.The symmetries of all these lattices have been explicitly demonstrated.This is a novel aspect of the quaternionic lattices in which not only the lattice vectors but also the group elements are represented by the quaternions.Since the quaternions can be used to describe the spin 2 1 states of electrons or some atoms we anticipate that the lattice structures in 2D and/or 3D with explicit spin dependence can be described by the quaternionic lattice structures.Another interesting aspect of this work is that we have developed an orthogonal projection technique of the 33 and AB lattices leading to the square and the honeycomb lattices respectively.It is an interesting observation that, if one views the 3D lattices from the Coxeter plane, one may observe the square or honeycomb lattice behaviors in the 33 and AB lattices respectively.

22 (
Dynkin diagram symmetry in terms of quaternions.We explicitly display how those polyhedra possessing tetrahedral symmetry can be constructed as the orbits of the group 3 ( )

, 2 ( 1 , 1 )
each representing an equilateral triangle, constitutes another hexagon dual to the hexagon A and describes the primitive cell of the weight lattice.Since both the root lattice and the weight lattice are made of hexagons they can be transformed to each other by a change of scale and the lattice point symmetry.Note that the hexagon described by two triangles is invariant under the group 2 () Aut A which involves the Dynkin diagram symmetry.

22 ( 1 2 ( 1 , 1 ) 2 1 3 A
,0) (0,1) AA  is a scaled copy of the dual of the primitive cell A after a 0 30 rotation.It is obvious that the primitive cell of the weight lattice is the hexagon22 (1,0) (0,1) AA  .It can be shown that the Wigner-Seitz cell of the weight lattice is the orbit , which is a scaled copy of the root polyhedron (hexagon in this case) by the Coxeter number 3 of the group 2 () WA .We will see that this is a general property of the n A series of the Coxeter-Weyl groups.

Figure 3 . 22 (
Figure 3.The Wigner-Seitz cell 1 2 1 + l  to any vector.Therefore the root lattice is represented either by the vectors 1 l m m  .Then the Wigner-Seitz cell is a square with the vertices

Figure 4 .
Figure 4.The square lattice as the root lattice of 2 B .

T
is used by crystallographers and 4 e e e        consisting of 12 quaternions representing the centers of the edges of a cube.Including the orbit 3 (0, 0, 0) rhombic dodecahedron tiles the 3D Euclidean space, however, it is not vertex transitive under the group 3 () Aut A , but rather face transitive.

Figure 7 . 4 .
Figure 7. Truncated octahedron.4. Construction of the sc and the bcc lattices as the affine Coxeter-Weyl groups W a (B 3 )

Figure 8 .
Figure 8.The Coxeter-Dynkin diagram of 3 B with quaternionic simple roots.

B represents the truncated octahedron which is obtained as the orbit 3 ( 3 B
consists of long roots and short roots of norms 2 and 1 respectively.The octahedral group

2  at the point 2 2 
12 H ee  .Therefore the generator 0 r corresponding to the highest root 12 H ee  leads to a translation 0 1 2 r e e      .This represents a reflection of the origin with respect to the plane bisecting the vector .A general root lattice vector is then given by

4 . 2
has the same generator vectors 1 2 3 , , e e e .The Wigner-Seitz cell can be determined around the origin by bisecting the lines joining the origin to the vertices 1 2 3,, e e e    .Intersecting planes determine the vertices of the Wigner-Seitz cell which represents the cube given by the orbit The bcc lattice as the weight lattice of B 3

3 1
have seen in the case of3 A  , the Wigner-Seitz cell is the truncated octahedron whose vertices are determined as the orbit  .The orthogonal projection of the 3A lattice means one takes only the set of vectors 2 a square lattice, or in other words, the root lattice of the Coxeter-Weyl group 2 B as shown in Figure4.Projection of the weight lattice onto the Coxeter plane also includes the set of quaternions23 and ee with half integer coefficients as shown in Figure4.

generate the dihedral group 6 D
of order 12.This is the symmetry of a hexagon.The third vector is orthogonal to the Coxeter plane.The projection of the 3 B lattice onto the Coxeter plane will lead to a honeycomb lattice as shown in Figure2.
WA of order 24 isomorphic to the tetrahedral group, the elements of which can be written compactly by the set 3 3 () S , permuting the four letters.To see a few examples, note that 12 rr fixes B but permutes ACD.Similarly 23 rr leaves A invariant but permutes BDC.The Coxeter element