Quaternionic Root Systems and Subgroups of the SQU Journal for Science, 2014, 19(1), 95-104 ©2014 Sultan Qaboos University 95 FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and Quaternions Nazife Özdeş Koca*, Mehmet Koca and Muna Al-Sawafi Department of Physics, College of Science, Sultan Qaboos University, P.O. Box: 36, PC 123, Al-Khod, Muscat, Sultanate of Oman. *E-mail: nazife@squ.edu.om. ABSTRACT: 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. Keywords: Coxeter groups; Lattices; Quaternions. ( المشحقة مهFCC( والمكعب مركزي الىجه )BCC( والمكعب مركزي الجسم )SCشبكات المكعب البسيط ) لكىاجيرويىنويل وا -مجمىعات كىسيحر وظيفة كىجا، مهمث كىجا و مىى الصىافية يٍ انشبكت (FCC( وانًكعب يشكضي انىخه )BCC( وانًكعب يشكضي اندسى )SC: نقذ حى بُاء شبكاث انًكعب انبسٍط ) ملخص (affine extended Coxeterانًًخذة )انًخشابطت واألساسٍت وانشبكت انخثاقهٍت نًدًىعاث كىسٍخش ( راث انًشحبت انثانثت حًخهك حُاظش )حًاثم( سباعً Coxeter-Weylوٌم ) -.إٌ يدًىعاث كىسٍخش و ( quaternionsوثًاًَ نشبكت انًكعب وانخً نذٌها حطبٍقاث واسعت فً يدال عهى انًىاد. إٌ وحذاث انعذد انًشكب انخخٍهً كىاحٍشٍَىٌ ) انًشحبت انثانثت إلٌداد يخدهاث انشبكت. عُاصش و( رCoxeter-Dynkinدٌُكٍ )-و األساسً نًخططاث كىسٍخشُظااناسخخذيج نخًثم صٌخض -انًدًىعت كخبج بذالنت أصواج يٍ كىاحٍشٍَىٌ حشكم انًدًىعت انثًاٍَت انثُائٍت. كًا حى حًثٍم سؤوط )قًى( خالٌا وٌدُش (Wigner-Seitzٍبذالنت وحذاث انعذد انًشكب انخخ )( ٌهً كىاحٍشٍَىquaternions وهزا انُهح اندذٌذ قذ ٌشبط بٍٍ دٌُايٍكٍت انشبكت .) ( ًٌثم شبكت انًشبع وانشبكت انسذاسٍت )خهٍت انُحم(.Coxeterيع فٍضٌاء انكىاحٍشٍَىٌ. اإلسقاط انعًىدي نهشبكت عهى سطح كىسٍخش ) انًشكبت. يدًىعاث كىكسخش، انشبٍكت انفشاغٍت و انشباعٍاث مفحاح الكلمات: 1. Introduction he Lie groups based on the Coxeter-Weyl groups [1] have had a great impact in high energy physics, describing its Standard Model [2-4] and its extension to the Grand Unified theories [5-7]. The Coxeter-Weyl groups acting as discrete groups in the 3D Euclidean space generate certain orbits [8-9] which describe the molecular structures [10] and viral symmetries [11-12]. Higher dimensional lattices described by the affine extension of the Coxeter-Weyl groups can be used to describe the quasicrystal structures when projected into lower dimensions [13-16]. The rank-3 Coxeter-Weyl groups 3 3 3 ( ) 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 W A 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 mailto:nazife@squ.edu.om NAZIFE OZDES KOCA ET AL 69 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 2 2 ( ) and ( )W A W B respectively. Section 3 is devoted to the construction of the Coxeter-Weyl group 3 ( ) W A and its extension 3 ( )Aut A by 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 ( ) W A 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 ( )W B and furthermore we demonstrate that its affine extension 3 ( ) a W B 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 3 3 and A B 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. 2. Construction of the honeycomb and square lattices as the affine Coxeter-Weyl groups Wa (A2) and Wa (B2) 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 1 2 , ,..., n r r r represent the reflection generators, then the presentation of the Coxeter-Weyl group ( ) W G is given by 1 2 ( ) , ,..., ( ) 1ij m n i j W G r r r r r    (1) where ij m is an integer label with 1, 2, 3, 4 ii ij m m  and 6 for i j representing respectively no line, one line, two lines (or label 4) and three lines (or label 6) between the nodes of the Coxeter-Dynkin diagrams which determine the crystallographic groups. In case ij m takes other positive integer values they correspond to non-crystallographic Coxeter groups which are out of the scope of this work. The Coxeter-Dynkin diagrams representing the groups 2 2 ( ) and ( )W A W B that we will study in this section are given in Figure 1. (a) (b) Figure 1. The Coxeter-Dynkin diagrams (a) for 2 2 and (b) for A B . The vectors 1 2 and   (hereafter called „roots‟) orthogonal to the planes with respect to which the reflection generators reflect an arbitrary vector  as 2( , ) , 1, 2. ( , ) i i i i i r i           (2) The Cartan matrix C with the matrix elements 2( , ) ( , ) i j ij j j C      and the metric G defined by matrix elements 1 ( , ) ( ) 2 j j ij ij G C     are important for the description of the Coxeter-Weyl groups and the corresponding lattices. The Cartan matrix and the matrix G represent the Gram matrices of the direct lattice and the reciprocal lattice respectively [17]. We take the roots i  as the generating vectors of the direct lattice. The weights i  spanning the dual space and satisfying the scalar product 2 ( , ) and ( , ) with ( , ) j i j ij i j ij j j j G             correspond to the generating vectors of the reciprocal lattice. Now we discuss the lattices associated with each Coxeter-Weyl group. 2.1 The lattice determined by the affine group Wa(A2) The Cartan matrix C and the metric G are given as follows 2 1 2 11 , 1 2 1 23 C G              . (3) FCC, BCC AND SC LATTICES 69 The plot of the polygons (triangle or hexagon) described by the orbits of the group 2 ( )W A will be given in orthonormal basis which can be obtained by using the eigenvectors of the Cartan matrix [18] 1 1 2 2 1 2 1 1 ˆ ˆ( ), ( ) 2 6 x x       (4) and in the orthonormal basis the roots read as 1 2 1 3 1 3 ( , ), ( , ) 2 22 2     . (5) The group generators act on the roots as: 1 1 1 1 2 2 1 2 1 1 2 2 2 2 , , ,r r r r                 which generate the group 2 ( )W A of order 6, which is isomorphic to the dihedral group 3 D . Including the Dynkin diagram symmetry which implies 1 2 :   , the generators generate a group of order 12, which is isomorphic to the dihedral group 6 D . The matrix representations of the generators in the root space can be written as 1 2 1 0 1 1 0 1 , , 1 1 0 1 1 0 r r                         . (6) The affine Coxeter group 2 ( ) a W A includes another generator 0 r which reflects the vectors with respect to a plane bisecting the vector 1 2   . This corresponds to a translation on an arbitrary vector  , as 1 2 ( )     . The repeated applications of the generators will generate the root lattice 2 A , where an arbitrary lattice vector is given by 1 1 2 2 , p b b   with 1 2 , b b  (the set of integers). The root system consisting of the vectors 1 2 1 2 , , ( )       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]. Figure 2. The honeycomb lattice. The weight vectors can be determined from the relation 2 1 i ij j j G     where the reciprocal lattice 2A  (weight lattice) vectors are given by 1 1 2 2 1 2 ( , ) q a a a a    with 1 2 , a a  . We will define an orbit in the dual space (reciprocal space) as 22 1 2 1 2 ( )( , ) ( , ) A W A a a a a which represents an isogonal hexagon with two edge lengths in general. When we have 1 2 a a , the orbit is a regular hexagon 2 (1,1) A determined by the root system which is invariant under the group 2 2 2 ( ) ( ) :Aut A W A C generated by the matrices in (6). The orbits 2 21 2 ( , 0) and (0, ) A A a a represent triangles. The union of the orbits 2 2 (1, 0) (0,1) A A  , each representing an equilateral triangle, constitutes another hexagon dual to the hexagon 2 (1,1) 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. 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 2 2 (1, 0) (0,1) A A  which can be determined from its primitive cell (see Figure 3). The Wigner-Seitz cell 2 2 (1, 0) (0,1) A A  is a scaled copy of the dual of the primitive cell 2 (1,1) A after a 0 30 rotation. It is obvious that the primitive cell of the weight lattice is the hexagon 2 2 (1, 0) (0,1) A A  . NAZIFE OZDES KOCA ET AL 69 It can be shown that the Wigner-Seitz cell of the weight lattice is the orbit 2 1 (1,1) 3 A , which is a scaled copy of the root polyhedron (hexagon in this case) by the Coxeter number 3 of the group 2 ( )W A . We will see that this is a general property of the n A series of the Coxeter-Weyl groups. Figure 3. The Wigner-Seitz cell 2 2 (1, 0) (0,1) A A  inscribed in the primitive cell of the root lattice. 2.2 The lattice determined by the affine group Wa (B2) The roots of 2 B consist of long and short roots; in a particular orthonormal system with ( , ) i j ij l l  , 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 1 1 1 1 2 2 2 , ,1 1 2 1 11 2 2 2 C C G                           . (7) The group generators 1 2 and r r act in the root space as 1 1 1 1 2 1 2 2 1 1 2 2 2 2 , , 2 ,r r r r                 (8) which can be represented by the matrices 1 2 1 0 1 2 , 1 1 0 1 r r              . (9) They generate the dihedral group 4 D of order 8, which is the symmetry of the square. The root system consists of two orbits 2 1 2 1 1 2 { , ( + )} and { , ( +2 )}         which can also be written in the orthonormal basis as 1 2 1 2 { , } and { }l l l l    , 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 1 2 { , }l l  . The translation generator is obtained as the addition of the highest short root 1 2 1 + l   to any vector. Therefore the root lattice is represented either by the vectors 1 1 2 2 1 2 + ; ,b b b b   or, equivalently, by 1 2 2 1 2 + ; , l m l m l m m  . Then the Wigner–Seitz cell is a square with the vertices 2 1 2 1 1 (0,1) { } 2 2 B l l   as shown in Figure 4. Figure 4. The square lattice as the root lattice of 2 B . The weight lattice is represented by the vectors 1 1 2 2 1 2 +a , , a a a   . Then the lattice vectors in the orthonormal basis will be 1 2 2 1 2 + ; 2 , 2 l m l m l m m  so that it involves the union of the sets with 1 2 and m m are either both FCC, BCC AND SC LATTICES 66 integers or both half integers. This indicates also that the root lattice is a sublattice of the weight lattice. Since the weight lattice and the root lattice are equivalent to each other up to a scale transformation and a rotation by 0 45 , it suffices to talk about the root lattice only. 3. Construction of the fcc and the bcc lattices as the affine Coxeter-Weyl group Wa (A3) 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 2 2 2 1, 1i j k ijk      . In what follows we will use a different notation for the quaternionic units 1 2 3 , ,i e j e k e   . Let 0 i i q q q e  , ( 1, 2, 3)i  be a real unit quaternion with its conjugate defined by 0 i i q q q e  and its norm 1qq qq  . The quaternionic imaginary units now satisfy the relations i j ij ijk k e e e    , )3,2,1,,( kji (10) 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 product 1 1 ( , ) ( ) ( ) 2 2 p q pq qp pq qp    (11) quaternions generate the four-dimensional Euclidean space and the unit quaternions 1 2 3 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. 21 ee  23 ee  12 ee  Figure 5. The Coxeter diagram 3 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 [ , ] 2 2 2 2 r             . (12) 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 1 2 1 0 1 2 C             , 1 3 2 1 1 2 4 2 4 1 2 3 C            . (13) The generators of the Coxeter group 3 ( )W A are then given in the notation of (12) by 1 1 2 1 2 1 1 [ ( ), ( )] 2 2 r e e e e      , 2 3 2 3 2 3 2 1 2 1 1 1 1 1 [ ( ), ( )] , [ ( ), ( )] . 2 2 2 2 r e e e e r e e e e           (14) The reflection generators operate on the quaternionic imaginary units as follows: 1 1 2 3 3 2 1 1 2 3 3 1 2 3 3 : , ; : , ; : , .r e e e e r e e e e r e e e e       They generate the Coxeter group 3 ( )W A of order 24 isomorphic to the tetrahedral group, the elements of which can be written compactly by the set 3 ( ) {[ , ] [ , ] }W A p p t t    , , p T t T   . (15) NAZIFE OZDES KOCA ET AL 011 Here and T T  are the sets of quaternions {T 1 2 3 1 2 3 1 1, , , , ( 1 )} 2 e e e e e e        , 1 1 1 1 1 1 1 2 3 2 3 1 3 1 22 2 2 2 2 2 { ( 1 ), ( ), ( 1 ), ( ), ( 1 ), ( )}e e e e e e e e eT              (16) 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 3 2 3 2 1 , ,e e e e e e        as in the Figure 5 then the weight vectors are determined from (13) as 1 1 2 3 2 3 3 1 2 3 1 1 (1, 0, 0) ( ), (0,1, 0) , (0, 0,1) ( ) 2 2 e e e e e e e             . (17) The orbits generated by these vectors would lead to the set of vectors 3 3 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 1 (1, 0, 0) { ( ), ( ), ( ), ( )}, 2 2 2 2 (0,1, 0) { , , }, 1 1 1 1 (0, 0,1) { ( ), ( ), ( ), ( )}. 2 2 2 2 A A A e e e e e e e e e e e e e e e e e e e e e e e e e e e                           (18) They respectively represent the vertices of a tetrahedron, an octahedron and another tetrahedron. The orbits 3 3 (1, 0, 0) and (0, 0,1) A A together represent the vertices of a cube. Therefore the symmetry of the union of the orbits 3 3 (1, 0, 0) (0, 0,1) A A  requires the Dynkin diagram symmetry 1 1 [ , ]e e    which extends the Coxeter group 3 ( )W A to the octahedral group 3 3 2 ( ) ( ) :Aut A W A C whose quaternionic representation is given by 3 ( ) {[ , ] [ , ] [ , ] [ , ]}Aut A T T T T T T T T         . (19) 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 ( )W A given by (15) is isomorphic to tetrahedral group 3 4 ( ) d W A T S  where the notation d T is used by crystallographers and 4 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; 1 2 3 1 2 3 1 2 3 1 2 3 1 1 1 1 ( ), ( ), ( ), ( ). 2 2 2 2 A e e e B e e e C e e e D e e e              It is clear now that the reflection generators can be written in the permutation notations 1 2 3 , , A B C D A B C D A B C D r r r C B A D A B D C A D C B                      . (20) In this notation they represent the generators of the group 4 S , permuting the four letters. To see a few examples, note that 1 2 r r fixes B but permutes ACD. Similarly 2 3 r r leaves A invariant but permutes BDC. The Coxeter element 1 2 3 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 3 ( ) {[ , ] [ , ] }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 3 2 ( ) {[ , ] [ , ]}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 4 2 {[ , ] [ , ] } h T A C T T T T      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 3 ( )Aut A acting on the quaternionic imaginary units. FCC, BCC AND SC LATTICES 010 1 2 3 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 , 0 0 1 , 1 0 0 , 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 r r r                                                (21) 3.1 The fcc lattice as the root lattice of A3 The group generators of the affine 3 ( ) a W A are the three reflection generators 1 2 3 0 , , and r r r r , where 0 r acts like a translation by the highest root 1 2 3 2 3H e e        , that is, 0 H r    . A general vector of the root lattice is then given by 1 1 2 2 3 3 1 2 3 ; , ,p b b b b b b      which can be written in terms of quaternions as 1 1 2 2 3 3 1 2 3 ; , ,p m e m e m e m m m    . (22) The root system of 3 A is the orbit 3 1 2 2 3 3 1 (1, 0,1) { , , } A e e e e e e       consisting of 12 quaternions representing the centers of the edges of a cube. Including the orbit 3 (0, 0, 0) 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 3 3 3 (1, 0, 0) (0,1, 0) (0, 0,1) A A A   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 3 1 2 2 3 3 1 1 1 (1, 0,1) { , , } 2 2 A e e e e e e       . Note that the rhombic dodecahedron tiles the 3D Euclidean space, however, it is not vertex transitive under the group 3 ( )Aut A , but rather face transitive. Figure 6. The rhombic dodecahedron, Wigner –Seitz cell of the fcc lattice. 3.2 The bcc lattice as the weight lattice of A3 The weight lattice 3 A  is generated by three weight vectors 1 2 3 (1, 0, 0), (0,1, 0), (0, 0,1).     A general vector of the lattice is given by 1 1 2 2 3 3 1 2 3 , , ,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) A A A   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 3 3 3 (1, 0, 0) (0,1, 0) (0, 0,1) A A A   . From (18) we deduce that the distance to the lattice points represented by the vertices of the square 3 3 (1, 0, 0) 0, 0,1) A A  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 1 2 2 3 3 1 1 2 2 3 3 1 1 1 (1,1,1) { 2 , 2 , 2 , 2 , 2 , 2 } 4 4 A e e e e e e e e e e e e             . (23) NAZIFE OZDES KOCA ET AL 011 Figure 7. Truncated octahedron. 4. Construction of the sc and the bcc lattices as the affine Coxeter-Weyl groups Wa (B3) The Coxeter-Dynkin diagram of 3 B leading to the octahedral group 3 ( ) h W B O is shown in Figure 8. Figure 8. The Coxeter-Dynkin diagram of 3 B with quaternionic simple roots. The Cartan matrix of the Coxeter-Dynkin diagram of 3 B and its inverse matrix are given by 2 1 0 1 2 2 0 1 2 C             , 1 1 1 1 1 1 1 2 1 2 2 , 1 2 1 1 3 1 3 1 1 2 2 2 4 C G                          . (24) The generators, 1 1 2 1 2 1 1 [ ( ), ( )] 2 2 r e e e e      ; 2 2 3 2 3 1 1 [ ( ), ( )] 2 2 r e e e e      ; 3 3 3 [ , ]r e e    (25) generate the octahedral group which can be written as 3 3 4 2 ( ) ( ) {[ , ] [ , ] [ , ] [ , ] }W B Aut A S C p p p p t t t t          , ,p T t T  . (26) where the shorthand notation 3 ( ) {[ , ] [ , ] [ , ] [ , ] }W B T T T T T T T T         is given in (19). The weight vectors 1 2 3 , and    are determined from the simple roots 1 1 2 2 2 3 3 3 , ,e e e e e       as 1 3 2 1 2 3 1 2 3 1 , , ( ) 2 e e e e e e         . (27) The orbits 3 3 (1, 0, 0) , (0,1, 0) , B B and 3 (0, 0,1) B represent an octahedron, a cuboctahedron and a cube respectively. Here the orbit 3 (1,1, 0) B represents the truncated octahedron which is obtained as the orbit 3 (1,1,1) A under the group 3 ( )W A . Note that 3 B consists of long roots and short roots of norms 2 and 1 respectively. The octahedral group 3 ( ) h W B O in (26) generates a root system consisting of the roots 3 3 1 2 3 1 2 2 3 3 1 (1, 0, 0) (0,1, 0) { , , } { , , } B B e e e e e e e e e            . (28) 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 1 2H e e   . Therefore the generator 0 r corresponding to the highest root 1 2H e e   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 2  at the point 2 2  . A general root lattice vector is then given by 1 1 2 2 3 3 , p b b b     with 1 2 3 , , b b b  . FCC, BCC AND SC LATTICES 012 4.1 The sc lattice as the root lattice of B3 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 1 1 2 2 3 3 1 2 3 ; , ,p m e m e m e m m m    . Including the origin the simple cube has the vertices 1 2 3 1 2 2 3 3 1 1 2 3 0, , , , , , , e e e e e e e e e e e e     . (29) The reciprocal lattice 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 3 1 2 3 1 (0, 0,1) ( ) 2 B e e e    . (30) 4.2 The bcc lattice as the weight lattice of B3 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 1 2 3 1 ( ) 2 e e e   and 1 2 3 { , , }e e e   . As we have seen in the case of 3 A  , the Wigner-Seitz cell is the truncated octahedron whose vertices are determined as the orbit 3 1 (1,1, 0) 4 B . 5. Projections of the A3 and B3 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 1 1 3 2 2 2 2 3 1 ( } 2e , 2 e e          (31) which is orthogonal to the vector 3 1 3 1 1 ( } 2e 2      . We now define the generators 1 1 3 2 2 and R r r R r  which satisfy the relation 4 1 2 ( ) 1R R  .Orthogonal projection will be made onto the Coxeter plane determined by the vectors 1 2 and   in which 1 2 and R R act as reflection generators. 1 2 and R R do in fact generate the dihedral subgroup 4 D of 3 A of order 8. The orthogonal set of vectors in the Coxeter plane can be taken as the quaternionic unit vectors 2 3 and e e . Let us recall that the root lattice vectors are given by 1 1 2 2 3 3 ;p m e m e m e   1 2 3 , ,m m m  . The orthogonal projection of the 3 A lattice means one takes only the set of vectors 2 2 3 3 2 3 ; ,m e m e m m  which constitutes a square lattice, or in other words, the root lattice of the Coxeter-Weyl group 2 B as shown in Figure 4. Projection of the weight lattice onto the Coxeter plane also includes the set of quaternions 2 3 and e e with half integer coefficients as shown in Figure 4. For the orthogonal projection of the lattice 3 B , we follow the same technique; however the group generated by the generators 1 2 and R R is the dihedral group 6 D of order 12 since 6 1 2 ( ) 1R R  . Here the Coxeter plane is determined by the vectors 1 1 2 3 1 ( 2 ) 3 e e e    and 2 2 3 e e   . Since the angle between them is 0 150 , the reflection generators generate the dihedral group 6 D of order 12. This is the symmetry of a hexagon. The third vector 3 1 2 3 1 ( ) 3 e e e     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 Figure 2. NAZIFE OZDES KOCA ET AL 013 6. 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 3 3 and A B lattices leading to the square and the honeycomb lattices respectively. 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