Allowed Space Groups for Superlattices 1009

The following table gives allowed space groups for all combinations of superlattices on a substrate lattice of a given space group. The orientation of substrate lattices (basis vectors a, b) is assumed to be such that a points along the (horizontal) x direction, except for centered rectangular groups cm, cmm where (a + b) points along x. The angles 'any', 0, 45, 90 indicate possible azimuths of mirror (m) and glide (g) planes in the superlattices, relative to the substrate lattice. Note that symmetry degenerate orientations, such as 120 degree rotations for hexagonal lattices, are not included in the list.

Table A:  Superlattice space groups [1] to [9] (oblique, rectangular)

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substrate lattice         combined superlattice and substrate lattice
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[1]p1       [2]p2       [3]pm      [4]pg      [5]cm      [6]pmm      [7]pmg       [8]pgg       [9]cmm

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[1]p1                             any          -              -              -              -               -                 -                  -                  -

[2]p2                             any          any         -              -              -               -                 -                  -                  -

[3]pm m=0                    any          -              0              0              0              -                 -                  -                  -

[4]pg g=0                     any          -              -              0              -               -                 -                  -                  -

[5]cm m=0                    any          -              0              0              0              -                 -                  -                  -

[6]pmm m=0,90            any          any         0,90         0,90         0,90         0,90            0,90            0,90            0,90

[7]pmg m=0, g=90       any          any         0              0,90         0              -                 m=0,g=90   0,90            -

[8]pgg g=0,90              any          any         -              0,90         -               -                 -                  0,90            -

[9]cmm m=0,90            any          any         0,90         0,90         0,90         0,90            0,90            0,90            0,90

[10]p4                           any          any         -              -              -               -                 -                  -                  -

[11]p4m m=0,45           any          any         0,45         0,45         0,45         0,45            0,45            0,45            0,45

[12]p4g g=0                 any          any         45            0,45         45            45               45               0,45            45

[13]p3                           any          -              -              -              -               -                 -                  -                  -

[14]p3m1 m=90            any          -              90            90            90            -                 -                  -                  -

[15]p31m m=0              any          -              0              0              0              -                 -                  -                  -

[16]p6                           any          any         -              -              -               -                 -                  -                  -

[17]p6m m=0,90           any          any         0,90         0,90         0,90         0,90            0,90            0,90            0,90

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Table B:  Superlattice space groups [10] to [17] (square, hexagonal)

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substrate lattice         combined superlattice and substrate lattice
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[10]p4     [11]p4m  [12]p4g      [13]p3     [14]p3m1   [15]p31m   [16]p6     [17]p6m

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[1]p1                            -               -              -                  -              -                  -                 -               -

[2]p2                             -               -              -                  -              -                  -                 -               -

[3]pm m=0                    -               -              -                  -              -                  -                 -               -

[4]pg g=0                     -               -              -                  -              -                  -                 -               -

[5]cm m=0                    -               -              -                  -              -                  -                 -               -

[6]pmm m=0,90            -               -              -                  -              -                  -                 -               -

[7]pmg m=0, g=90       -               -              -                  -              -                  -                 -               -

[8]pgg g=0,90              -               -              -                  -              -                  -                 -               -

[9]cmm m=0,90            -               -              -                  -              -                  -                 -               -

[10]p4                           any          -              -                  -              -                  -                 -               -

[11]p4m m=0,45           any          0,45         0,45            -              -                  -                 -               -

[12]p4g g=0                 any          -              0                 -              -                  -                 -               -

[13]p3                           -               -              -                  any         -                  -                 -               -

[14]p3m1 m=90            -               -              -                  any         0                 90               -               -

[15]p31m m=0              -               -              -                  any         90               0                 -               -

[16]p6                           -               -              -                  any         -                  -                 any          -

[17]p6m m=0,90           -               -              -                  any         0,90            0,90            any          0,90

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