## 6.2.3. MILLER INDEX OPTION, [M]

next, previous Section / Table of Contents / Index [M] This option allows you to choose Miller indices defining net plane normal vectors which determine the direction of net plane stacking used to build the lattice section. Here Miller indices can be given in different forms depending on the lattice type, see below. If BALSAC/LATTICE is started from scratch in an interactive session (without a structure input file) this option shows an explanatory text ========== MILLER INDEX OPTION ==================================== Define Miller indices for directions of net plane stacking to build the lattice section. Simple cubic, Bravais, and 4-index notation are available depending on the lattice type. Miller indices (H K L / 0 0 0 ) : =================================================================== asking for Miller indices given as integer triples or ========== MILLER INDEX OPTION ==================================== Define Miller indices for directions of net plane stacking to build the lattice section. Simple cubic, Bravais, and 4-index notation are available depending on the lattice type. Miller indices (L M N Q / 0 0 0 0 ) : =================================================================== for Miller indices given as integer quadruples in hexagonal lattices where 4-index notation is used depending on the type of lattice chosen in the previous lattice option, see below and Sec. 6.2.1. After correct input BALSAC moves to the section option, see Sec. 6.2.3. Selecting [M] from the main option menu of BALSAC/LATTICE shows the prompt ========== MILLER INDEX OPTION ==================================== Miller indices (H K L / hold kold lold ) : =================================================================== asking for three (integer) numbers h, k, l to define new Miller indices where hold, kold, lold are the present values. The indices are defined with respect to - simple cubic reciprocal lattice vectors for cubic lattices chosen by lattice type options [A], [B], [C], [E], [F], [G], [H], see Sec. 6.2.1. - Bravais reciprocal lattice vectors for cubic lattices chosen by lattice type options [L], [M], [O], [P], [Q], see Sec. 6.2.1. - Bravais reciprocal lattice vectors for all other lattice types. Note that Miller index input will be different for lattice type options [N], [R], [S], see below and Sec. 6.2.1. For hexagonal lattices using 4-index notation of Miller indices (lattice type options [N], [R], [S]) the above prompt is replaced by ========== MILLER INDEX OPTION ==================================== Miller indices (L M N Q / lold mold nold qold ) : =================================================================== asking for four (integer) numbers to define new Miller indices where lold, mold, nold, qold are the present values. The indices l, m, n, q are connected with the standard Bravais lattice Miller indices h, k, l by ( l, m, n, q ) = ( h, k-h, -k, l ) . The 4-index input is checked for consistency ( l+m+n = 0 ). In case of error a warning WARNING: indices inconsistent ( l m' n q ) used instead ** Press [C], L-click to continue ** is issued and revised indices where m' = -(l+n) (shown in the warning) will be used for input. Note that Miller indices used in internal computations are always determined from indices l, n, q only. After Miller indices have been given BALSAC returns to the BALSAC/LATTICE main option menu. For dummy input h,k,l = 0,0,0 (l,m,n,q = 0,0,0,0) the index prompt is cancelled keeping the previous Miller index values and returning to the BALSAC/LATTICE main option menu. Note that the above definition of the 4-index notation refers to a strict definition of reciprocal lattice vectors by vector products of R1, R2, R3 where G1, G2 form an angle of 120 degrees (R1, R2 form an angle of 60 degrees), see Sec. 5.3. An alternative description of the hexagonal lattice starts from lattice vectors R1', R2', R3' where R1', R2' form an angle of 120 degrees and as a consequence G1' and G2' form an angle of 60 degrees. This corresponds in effect to a transformation G1' = G1 + G2 G2' = G2 G3' = G3 and lattice planes (h k l) would be expressed by (h' k' l') with h'= h k'= k - h l' = l As a consequence, the 4-index notation ( L', M', N', Q' ) would read ( L', M', N', Q' ) = (h', k', -h'-k', l') which is identical to definitions used in a number of surface science papers and books (cp. "Low-Energy Electron Diffraction" by Van Hove et al. as referenced below). However, in this program we use the former definition for internal consistency of BALSAC. For cubic lattices an internal transformation is used to connect between Bravais (fcc or bcc) and simple cubic (sc) Miller indices where a) (h,k,l) <--> (h',k',l') fcc sc h' -1.0 1.0 1.0 h h 0.0 0.5 0.5 h' k' = 1.0 -1.0 1.0 x k k = 0.5 0.0 0.5 x k' l' 1.0 1.0 -1.0 l l 0.5 0.5 0.0 l' b) (h,k,l) <--> (h',k',l') bcc sc h' 0.0 1.0 1.0 h h -0.5 0.5 0.5 h' k' = 1.0 0.0 1.0 x k k = 0.5 -0.5 0.5 x k' l' 1.0 1.0 0.0 l l 0.5 0.5 -0.5 l' If for given simple cubic (h',k',l') indices the fcc or bcc indices (h,k,l) are non-integer they will be multiplied by 2 to yield integer values. next, previous Section / Table of Contents / Index