Input interpretation
silicon | Bravais lattice
Result
face-centered cubic
Common name
fcc
Description of lattice
basis | (-1 | -1 | 0) | (1 | -1 | 0) | (0 | 1 | -1) Gram matrix | (2 | 0 | -1 0 | 2 | -1 -1 | -1 | 2)
Lattice invariants
dimension | 3 determinant | 4 minimal squared norm | 2 kissing number | 12
Lattice-packing invariants
packing radius | 1/sqrt(2)≈0.707107 covering radius | 1 density | π/(3 sqrt(2))≈0.74048 center density | 1/(4 sqrt(2))≈0.176777 Hermite invariant | 2^(1/3)≈1.25992 thickness | (2 π)/3≈2.0944 volume | 2
Quadratic form and theta series
quadratic form | 2 x^2 - 2 x z + 2 y^2 - 2 y z + 2 z^2 theta series (closed series) | 1/2 (ϑ_3(0, e^(i π x))^3 + ϑ_4(0, e^(i π x))^3)
More properties
number of symmetries | 48
Common properties
even | integral | nonunimodular
Crystallographic properties
lattice system | cubic crystal system | cubic crystal family | cubic required point group symmetry | 4 3-fold rotation axes point groups | 5 space groups | 36
Point groups
crystal class | Schönflies | Hermann-Mauguin tetartoidal | T | 23 diploidal | T_h | m3^_ gyroidal | O | 432 tetrahedral | T_d | 4^_3m hexoctahedral | O_h | m3^_m
Space groups
crystal class | IUCr number | Hermann-Mauguin tetartoidal | 195 | F23 diploidal | 200 | 201 | Fd3^_ | Fm3^_ gyroidal | 207 | 208 | F4132 | F432 tetrahedral | 215 | 216 | F43c^_ | F43m^_ hexoctahedral | 221 | 222 | 223 | 224 | Fd3c^_ | Fd3m^_ | Fm3c^_ | Fm3m^_