Example

Let consider phase transition in compounds with general formula M₃B₇O₁₃X The high symmetry phase is with group of symmetry $G = T_d^5 = F\bar 4 3c$. Let also consider phase transition with 16-component order parameter characterized by wave vector $\vec k = \left( \frac{1}{2} \frac{1}{2} \frac{1}{2} \right)$. The irrep $D^{\vec k v} = L_3$ is 8-dimensional complex representation. The kernel of this representation $\mbox{ \rm Ker }D_G^f$ is an invariant subgroup of index 192, so the image of D_G^j, $\mbox{ \rm Image }(L_3 \oplus (L_3)^* ) = A$ is matrix group formed by 192 matrix, each with dimensions $(16 \times 16)$. From the tables (Appendix A5 in the thesis) we can see that this is group H192a. This is the group $A \cong G / \mbox{ \rm Ker }D_G^f$.

In order to find the groups of symmetry of all possible phases for PT with this order parameter we have to check the set of group-theoretical criteria for the homomorphic image H192a, instead of space group T_d^h.

i) All subgroups A' of the image A=H192a are tabulated already. There are 356 such subgroups. All they falls into 70 classes of conjugated subgroups. From all the 70 classes for only 15 condition $\mbox{ \rm Core }A' = C_1$ is satisfied. The group-subgroup relations of these groups are shown in Table 2

 

 

1: 1

2: - 1

3: - - 1

4: - - - 1

5: - 2 - - 1

6: 2 - - - - 1

7: 2 - - - - - 1

8: 2 - - - - - - 1

9: - - - - - - - - 1

10: - 2 - - - - - - - 1

11: - - 2 - - - - - - - 1

12: - - - 4 - - - - - - - 1

13: - - 6 4 - - - - - - - - 1

14: - - - 4 - - - - - - - - - 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

ii) In the next step we have to check subduction multilicities $D_A^f \downarrow A'$, where $D_A^f = \mbox{ \rm Image }( L_3 \oplus L_3^* ) \cong H192a$ and to find the maximal subgroups (i.e. the isotropy ones). The isotropy subgroups are listed in Table 3

 

 

iii) In the last step we have to find the space groups which corresponds to founded already abstract subgroups A'. The results are show in Table 4

 

A'_l H'_l V'/V a'₁, a'₂, a'₃ Origin shift

A'₀ C₁¹ 8 (0,1,1),(1,0,1),(1,1,0) (0,0,0)

$$(\bar 2,0,0),(0,0,2),(1,1,0) (\frac{3}{4},\frac{1}{4},0)$$ A'₁ C₂³ 8

$$(\bar 2,0,0),(0,0,2),(1,1,0)$$ A'₂ C₂³ 8 (0,0,0)

$$(\frac{1}{2},\frac{1}{2},1),(\frac{1}{2},\bar \frac{1}{2},0),(1,1,0)$$ A'₃ C₁¹ 4 (0,0,0)

$$(1,1,0),(\bar 1,0,1),(2,\bar 2,2) (0,0,\frac{1}{2})$$ A'₄ C₃⁴ 8

$$(\frac{3}{4},\frac{1}{4},\frac{3}{4})$$ A'₅ D₂⁷ 8 (2,0,0),(0,2,0),(0,0,2)

$$(1,1,0),(0,0,2),(\frac{1}{2},\bar \frac{1}{2},0)$$ A'₆ C₂³ 4 (0,0,0)

$$(0,0,2),(1,1,0),(\bar \frac{1}{2},\frac{1}{2},0) (\frac{1}{8},\frac{1}{8},0)$$ A'₇ C_s⁴ 4

$$(1,1,0),(\bar 1,1,0),(0,0,2) (\frac{1}{4},\frac{3}{4},\frac{1}{2})$$ A'₈ S₄² 8

$$(1,1,0),(\bar 1,1,0),(0,0,2) (0,0,\frac{1}{4})$$ A'₉ S₄² 8

A'₁₀ D₂⁷ 8 (2,0,0),(0,2,0),(0,0,2) (0,0,0)

$$(1,\frac{1}{2},\frac{1}{2}),(\frac{1}{2},1,\frac{1}{2}),(\frac{1}{2},\frac{1}{2},1)$$ A'₁₁ C₁¹ 2 (0,0,0)