International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 2.3, pp. 338-340

Table 2.3.3.1 

I. Gregoraa*

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic
Correspondence e-mail: gregora@fzu.cz

Table 2.3.3.1| top | pdf |
Symmetry of Raman tensors in the 32 crystal classes

The symbols a, b, c, d, e, f, g, h and i in the matrices stand for arbitrary parameters denoting possible independent nonzero components (in general complex) of the Raman tensors. Upper row: conventional symmetric Raman tensors; lower row: antisymmetric part. Alternative orientations of the point group are distinguished by subscripts at 2 or m in the class symbol indicating the direction of the twofold axis or of the normal to the mirror plane.

Triclinic

  [\pmatrix{ a & d & f \cr d & b & h \cr f & h & c \cr }]
1 [{\rm A}(x,y,z)]
[\bar 1] [{\rm A}{_g}]
  [\pmatrix{. & e & g \cr {- e}&. & i \cr {- g}& {- i}&. \cr }]

Monoclinic, unique axis z

  [{\pmatrix{ a & d &. \cr d & b &. \cr. &. & c \cr }}] [{\pmatrix{. &. & f \cr. &. & h \cr f & h &. \cr }}]
[2_z] [{\rm A}(z)] [{\rm B}(x,y)]
[m_z] [{\rm A'}(x,y)] [{\rm A}''(z)]
[2_z/m] [{\rm A}{_g}] [{\rm B}{_g}]
  [{\pmatrix{. & e &. \cr {- e}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & g \cr. &. & i \cr {- g}& {- i}&. \cr }}]

Monoclinic, unique axis y

  [{\pmatrix{ a & . &d \cr . & b &. \cr d&. & c \cr }}] [{\pmatrix{. &f & . \cr f &. & h \cr . & h &. \cr }}]
[2_y] [{\rm A}(y)] [{\rm B}(x,z)]
[m_y] [{\rm A'}(x,z)] [{\rm A}''(y)]
[2_y/m] [{\rm A}{_g}] [{\rm B}{_g}]
  [{\pmatrix{. & . &e \cr .&. &. \cr{-e} &. &. \cr }}] [{\pmatrix{. &g & . \cr-g &. & i \cr .& {- i}&. \cr }}]

Orthorhombic

  [{\pmatrix{ a &. &. \cr. & b &. \cr. &. & c \cr }}] [{\pmatrix{. & d &. \cr d &. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & f \cr. &. &. \cr f &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & h \cr. & h &. \cr }}]
222 [{\rm A}] [{\rm B}{_1}(z)] [{\rm B}{_2}(y)] [{\rm B}{_3}(x)]
[mm2] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm B}{_1}(x)] [{\rm B}{_2}(y)]
[mmm] [{\rm A}{_g}] [{\rm B}_{1g}] [{\rm B}_{2g}] [{\rm B}_{3g}]
    [{\pmatrix{. & e &. \cr {- e}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & g \cr. &. &. \cr {- g}&. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & i \cr. & {- i}&. \cr }}]

Tetragonal

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}] [{\pmatrix{ d & e &. \cr e & {- d}&. \cr. &. &. \cr }}] [{\pmatrix{. &. & f \cr. &. & h \cr f & h &. \cr }}\quad{\pmatrix{. &. & {- h} \cr. &. & f \cr {- h}& f &. \cr }}]
[4] [{\rm A}(z)] [{\rm B}] [{\rm E}(x,y)]
[\bar 4] [{\rm A}] [{\rm B}(z)] [{\rm E}(x,-y)]
[4/m] [{\rm A}{_g}] [{\rm B}{_g}] [{\rm E}{_g}]
  [{\pmatrix{. & c &. \cr {- c}&. &. \cr. &. &. \cr }}]   [{\pmatrix{. &. & g \cr. &. & i \cr {- g}& {- i}&. \cr }}\quad{\pmatrix{. &. & {- i} \cr. &. & g \cr i & {- g}&. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{ d &. &. \cr. & {- d}&. \cr. &. &. \cr }}] [{\pmatrix{. & e &. \cr e &. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & f \cr. &. &. \cr f &. &. \cr }}\quad{\pmatrix{. &. &. \cr. &. & f \cr. & f &. \cr }}]
422 [{\rm A}{_1}] [{\rm A}{_2}(z)] [{\rm B}{_1}] [{\rm B}{_2}] [{\rm E}(-y,x)]
[4mm] [{\rm A}{_1}(z)] [{\rm A}{_2} ] [{\rm B}{_1}] [{\rm B}{_2}] [{\rm E}(x,y)]
[\bar 42_xm_{xy}] [{\rm A}{_1}] [{\rm A}{_2}] [{\rm B}{_1}] [{\rm B}{_2}(z) ] [{\rm E}(y,x)]
[\bar 4m_x2_{xy}] [{\rm A}{_1}] [{\rm A}{_2}] [{\rm B}{_1}] [{\rm B}{_2}(z) ] [{\rm E}(-x,y)]
[4/mmm] [{\rm A}_{1g}] [{\rm A}_{2g}] [{\rm B}_{1g}] [{\rm B}_{2g}] [{\rm E}{_g}]
    [{\pmatrix{. & c &. \cr {- c}&. &. \cr. &. &. \cr }}]     [{\pmatrix{. &. & g \cr. &. &. \cr {- g}&. &. \cr }}\quad{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}]

Trigonal

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}] [{\pmatrix{ c & f & e \cr f & {- c}& d \cr e & d & . \cr }}\quad{\pmatrix{ f & {- c}& {- d} \cr {- c}& {- f}& e \cr {- d}& e &. \cr }}]
3 [{\rm A}(z)] [{\rm E}(x,y)]
[\bar 3] [{\rm A}{_g}] [{\rm E}{_g}]
  [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & i \cr. &. & g \cr {- i}& {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. & i \cr g & {- i}&. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{f & . &. \cr . &-f & d \cr. & d &. \cr }}\quad{\pmatrix{ . &-f & {- d} \cr-f & .&. \cr {- d}&. &. \cr }}]
[32_x] [{\rm A}{_1}] [{\rm A}{_2}(z)] [{\rm E}(x,y)]
[3m_x] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm E}(y,-x)]
[\bar 3m_x] [{\rm A}_{1g}] [{\rm A}_{2g}] [{\rm E}_{g}]
    [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. &. \cr g &. &. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{. & f &. \cr f &. & d \cr. & d &. \cr }}\quad{\pmatrix{ f &.& {- d} \cr. & -f&. \cr {- d}&. &. \cr }}]
[32_y] [{\rm A}{_1}] [{\rm A}{_2}(z)] [{\rm E}(x,y)]
[3m_y] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm E}(y,-x)]
[\bar 3m_y] [{\rm A}_{1g}] [{\rm A}_{2g}] [{\rm E}_{g}]
    [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. &. \cr g &. &. \cr }}]

Hexagonal

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}] [{\pmatrix{. &. & e \cr. &. & d\cr e & d &. \cr }}\quad{\pmatrix{. &. & {- d} \cr. &. & e \cr {- d}& e &. \cr }}] [{\pmatrix{ c & f &. \cr f & {- c}&. \cr. &. &. \cr }}\quad{\pmatrix{ f & {- c}&. \cr {- c}& {- f}&. \cr. &. &. \cr }}]
6 [{\rm A}(z)] [{\rm E}{_1}(x,y)] [{\rm E}{_2}]
[\bar 6] [{\rm A}' ] [{\rm E}''] [{\rm E}'(x,y)]
[6/m] [{\rm A}{_g}] [{\rm E}_{1g}] [{\rm E}_{2g}]
  [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. & i \cr. &. & g \cr {- i}& {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. & i \cr g & {- i}&. \cr }}]  

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & b \cr }}]   [{\pmatrix{. &. &. \cr. &. & d \cr. & d &. \cr }}\quad{\pmatrix{. &. & {- d} \cr. &. &. \cr {- d}&. &. \cr }}] [{\pmatrix{. & f &. \cr f &. &. \cr. &. &. \cr }}\quad{\pmatrix{ f &. &. \cr. & {- f}&. \cr. &. &. \cr }}]
622 [{\rm A}{_1}] [{\rm A}{_2}(z) ] [{\rm E}{_1}(x,y)] [{\rm E}{_2}]
[6mm] [{\rm A}{_1}(z)] [{\rm A}{_2}] [{\rm E}{_1}(y,-x)] [{\rm E}{_2}]
[\bar 6m_x2_y] [{\rm A}{_1}'] [{\rm A}{_2}' ] [{\rm E}''] [{\rm E}'(x,y)]
[\bar 62_xm_y] [{\rm A}{_1}'] [{\rm A}{_2}' ] [{\rm E}''] [{\rm E}'(y,-x)]
[6/mmm] [{\rm A}_{1g}] [{\rm A}_{2g} ] [{\rm E}_{1g}] [{\rm E}_{2g}]
    [{\pmatrix{. & h &. \cr {- h}&. &. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & g \cr. & {- g}&. \cr }}\quad{\pmatrix{. &. & {- g} \cr. &. &. \cr g &. &. \cr }}]  

Cubic

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & a \cr }}] [{\pmatrix{ b &. &. \cr. & b &. \cr. &. & {- 2b} \cr }}\quad{\pmatrix{ {- \sqrt{3}b}&. &. \cr. & \sqrt{3}b&. \cr. &. &. \cr }}] [{\pmatrix{. &. &. \cr. &. & c \cr. & c &. \cr }}\quad{\pmatrix{. &. & c \cr. &. &. \cr c &. &. \cr }}\quad{\pmatrix{. & c &. \cr c &. &. \cr. &. &. \cr }}]
23 [{\rm A}] [{\rm E}] [{\rm F}(x,y,z)]
[m3] [{\rm A}{_g}] [{\rm E}{_g}] [{\rm F}{_g}]
      [{\pmatrix{. &. &. \cr. &. & d \cr. & {- d}&. \cr }}\quad{\pmatrix{. &. & d \cr. &. &. \cr {- d}&. &. \cr }}\quad{\pmatrix{. & d &. \cr {- d}&. &. \cr. &. &. \cr }}]

  [{\pmatrix{ a &. &. \cr. & a &. \cr. &. & a \cr }}] [{\pmatrix{ b &. &. \cr. & b &. \cr. &. & {- 2b} \cr }}\quad{\pmatrix{ {- \sqrt{3}b}&. &. \cr. & {\sqrt{3}}b &. \cr. &. &. \cr }}]   [{\pmatrix{. &. &. \cr. &. & c \cr. & c &. \cr }}\quad{\pmatrix{. &. & c \cr. &. &. \cr c &. &. \cr }}\quad{\pmatrix{. & c &. \cr c &. &. \cr. &. &. \cr }}]
432 [{\rm A}{_1}] [{\rm E}] [{\rm F}{_1}(x,y,z)] [{\rm F}{_2}]
[\bar 43m] [{\rm A}{_1}] [{\rm E}] [{\rm F}{_1}] [{\rm F}{_2}(x,y,z)]
[m3m] [{\rm A}_{1g}] [{\rm E}_{g}] [{\rm F}_{1g}] [{\rm F}_{2g}]
      [{\pmatrix{. &. &. \cr. &. & d \cr. & {- d}&. \cr }}\quad{\pmatrix{. &. & d \cr. &. &. \cr {- d}&. &. \cr }}\quad{\pmatrix{. & d &. \cr {- d}&. &. \cr. &. &. \cr }}]