PHY.K02UF Molecular and Solid State Physics

Simple cubic

Simple cubic is Bravais lattice. When there is just one atom in the basis, it is also a crystal structure. Of the elements, only α-polonium forms this crystal structure.

Crystal structure: simple cubic (sc)
Bravais lattice: simple cubic (cP)
Space group: 221 (P m 3 m),
Point group: m3m ($O_h$)   six 2-fold rotations, four 3-fold rotations, three 4-fold rotations, nine mirror planes, inversion

The lattice parameters are:

\[ \begin{equation} a = b = c, \hspace{0.5cm} \alpha = 90^{\circ} , \,\beta = 90^{\circ} ,\,\gamma = 90^{\circ} . \end{equation} \]

The primitive lattice vectors are,

\begin{equation} \vec{a}_1=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\hat{z}. \end{equation}

There is one atom in the basis. In fractional coordinates of the conventional unit, the position of this atom is,

$\vec{B}_1 = (0,0,0)$.

The conventional unit cell lattice vectors are also,

\begin{equation} \vec{a}_1=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\hat{z}. \end{equation}

Space group: 221 (P m 3 m)
48 symmetry operations:

1	x,y,z	identity
2	z,y,-x	C4 axis
3	y,x,-z	C2 axis
4	x,z,-y	C4 axis
5	z,x,-y	3-bar axis|inversion center at 0 0 0
6	y,z,-x	3-bar axis|inversion center at 0 0 0
7	x,y,-z	mirror plane
8	z,-y,x	C2 axis
9	y,-x,z	C4 axis
10	x,-z,y	C4 axis
11	z,-x,y	3-bar axis|inversion center at 0 0 0
12	y,-z,x	3-bar axis|inversion center at 0 0 0
13	x,-y,z	mirror plane
14	-z,y,x	C4 axis
15	-y,x,z	C4 axis
16	-x,z,y	C2 axis
17	-z,x,y	3-bar axis|inversion center at 0 0 0
18	-y,z,x	3-bar axis|inversion center at 0 0 0
19	-x,y,z	mirror plane
20	-z,-y,-x	C2 axis
21	-y,-x,-z	C2 axis
22	-x,-z,-y	C2 axis
23	-z,-x,-y	3-bar axis|inversion center at 0 0 0
24	-y,-z,-x	3-bar axis|inversion center at 0 0 0
25	-x,-y,-z	Ci: 0 0 0
26	-z,-y,x	4-bar axis|inversion center at 0 0 0
27	-y,-x,z	mirror plane
28	-x,-z,y	4-bar axis|inversion center at 0 0 0
29	-z,-x,y	C3 axis
30	-y,-z,x	C3 axis
31	-x,-y,z	C2 axis
32	-z,y,-x	mirror plane
33	-y,x,-z	4-bar axis|inversion center at 0 0 0
34	-x,z,-y	4-bar axis|inversion center at 0 0 0
35	-z,x,-y	C3 axis
36	-y,z,-x	C3 axis
37	-x,y,-z	C2 axis
38	z,-y,-x	4-bar axis|inversion center at 0 0 0
39	y,-x,-z	4-bar axis|inversion center at 0 0 0
40	x,-z,-y	mirror plane
41	z,-x,-y	C3 axis
42	y,-z,-x	C3 axis
43	x,-y,-z	C2 axis
44	z,y,x	mirror plane
45	y,x,z	mirror plane
46	x,z,y	mirror plane
47	z,x,y	C3 axis
48	y,z,x	C3 axis