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PHY.K02UF Molecular and Solid State Physics

## Body centered cubic

Body centered cubic is both a crystal structure and a Bravais lattice and has one atom in the basis. Na, K and Cr form this crystal structure.

Crystal structure: body centered cubic (bcc)
Bravais lattice: body centered cubic
Space group: 229 (I m -3 m),   Strukturbericht: A2,   Pearson symbol: cI2
Point group: m3m ($O_h$)   six 2-fold rotations, four 3-fold rotations, three 4-fold rotations, nine mirror planes, inversion
Lattice constant: $a_{Na} = 0.429$ nm   $a_{K} =0.533$ nm   $a_{Cr} =0.291$ nm
Atomic density $n_{atoms} = 2/a^3$

The lattice parameters of the conventional unit cell are:

$$$a = b = c, \hspace{0.5cm} \alpha = 90^{\circ} , \,\beta = 90^{\circ} ,\,\gamma = 90^{\circ} .$$$

The primitive lattice vectors are,

$$\vec{a}_1=\frac{a}{2}(\hat{x}+\hat{y}-\hat{z}),\quad \vec{a}_2=\frac{a}{2}(-\hat{x}+\hat{y}+\hat{z}),\quad\vec{a}_3=\frac{a}{2}(\hat{x}-\hat{y}+\hat{z}).$$

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 asymmetric unit contains one atom at fractional coordinates (0,0,0).

The positions of the other atoms can be determined by applying the symmetries of the space group.

Space group: 229 (I m -3 m)
96 symmetry operations:

1	x,y,z		identity
2	-x,-y,z		C2 axis
3	-x,y,-z		C2 axis
4	x,-y,-z		C2 axis
5	z,x,y		C3 axis
6	z,-x,-y		C3 axis
7	-z,-x,y		C3 axis
8	-z,x,-y		C3 axis
9	y,z,x		C3 axis
10	-y,z,-x		C3 axis
11	y,-z,-x		C3 axis
12	-y,-z,x		C3 axis
13	y,x,-z		C2 axis
14	-y,-x,-z	C2 axis
15	y,-x,z		C4 axis
16	-y,x,z		C4 axis
17	x,z,-y		C4 axis
18	-x,z,y		C2 axis
19	-x,-z,-y	C2 axis
20	x,-z,y		C4 axis
21	z,y,-x		C4 axis
22	z,-y,x		C2 axis
23	-z,y,x		C4 axis
24	-z,-y,-x	C2 axis
25	-x,-y,-z	Ci: 0 0 0
26	x,y,-z		mirror plane
27	x,-y,z		mirror plane
28	-x,y,z		mirror plane
29	-z,-x,-y	3-bar axis|inversion center at 0 0 0
30	-z,x,y		3-bar axis|inversion center at 0 0 0
31	z,x,-y		3-bar axis|inversion center at 0 0 0
32	z,-x,y		3-bar axis|inversion center at 0 0 0
33	-y,-z,-x	3-bar axis|inversion center at 0 0 0
34	y,-z,x		3-bar axis|inversion center at 0 0 0
35	-y,z,x		3-bar axis|inversion center at 0 0 0
36	y,z,-x		3-bar axis|inversion center at 0 0 0
37	-y,-x,z		mirror plane
38	y,x,z		mirror plane
39	-y,x,-z		4-bar axis|inversion center at 0 0 0
40	y,-x,-z		4-bar axis|inversion center at 0 0 0
41	-x,-z,y		4-bar axis|inversion center at 0 0 0
42	x,-z,-y		mirror plane
43	x,z,y		mirror plane
44	-x,z,-y		4-bar axis|inversion center at 0 0 0
45	-z,-y,x		4-bar axis|inversion center at 0 0 0
46	-z,y,-x		mirror plane
47	z,-y,-x		4-bar axis|inversion center at 0 0 0
48	z,y,x		mirror plane
49	x+1/2,y+1/2,z+1/2	translation: 1/2 1/2 1/2
50	-x+1/2,-y+1/2,z+1/2	2-fold screw axis|translation: 0 0 1/2
51	-x+1/2,y+1/2,-z+1/2	2-fold screw axis|translation: 0 1/2 0
52	x+1/2,-y+1/2,-z+1/2	2-fold screw axis|translation: 1/2 0 0
53	z+1/2,x+1/2,y+1/2	3-fold screw axis|translation: 1/2 1/2 1/2
54	z+1/2,-x+1/2,-y+1/2	3-fold screw axis|translation: 1/6 -1/6 1/6
55	-z+1/2,-x+1/2,y+1/2	3-fold screw axis|translation: -1/6 1/6 1/6
56	-z+1/2,x+1/2,-y+1/2	3-fold screw axis|translation: 1/6 1/6 -1/6
57	y+1/2,z+1/2,x+1/2	3-fold screw axis|translation: 1/2 1/2 1/2
58	-y+1/2,z+1/2,-x+1/2	3-fold screw axis|translation: -1/6 1/6 1/6
59	y+1/2,-z+1/2,-x+1/2	3-fold screw axis|translation: 1/6 1/6 -1/6
60	-y+1/2,-z+1/2,x+1/2	3-fold screw axis|translation: 1/6 -1/6 1/6
61	y+1/2,x+1/2,-z+1/2	2-fold screw axis|translation: 1/2 1/2 0
62	-y+1/2,-x+1/2,-z+1/2	C2 axis
63	y+1/2,-x+1/2,z+1/2	4-fold screw axis|translation: 0 0 1/2
64	-y+1/2,x+1/2,z+1/2	4-fold screw axis|translation: 0 0 1/2
65	x+1/2,z+1/2,-y+1/2	4-fold screw axis|translation: 1/2 0 0
66	-x+1/2,z+1/2,y+1/2	2-fold screw axis|translation: 0 1/2 1/2
67	-x+1/2,-z+1/2,-y+1/2	C2 axis
68	x+1/2,-z+1/2,y+1/2	4-fold screw axis|translation: 1/2 0 0
69	z+1/2,y+1/2,-x+1/2	4-fold screw axis|translation: 0 1/2 0
70	z+1/2,-y+1/2,x+1/2	2-fold screw axis|translation: 1/2 0 1/2
71	-z+1/2,y+1/2,x+1/2	4-fold screw axis|translation: 0 1/2 0
72	-z+1/2,-y+1/2,-x+1/2	C2 axis
73	-x+1/2,-y+1/2,-z+1/2	Ci: 1/4 1/4 1/4
74	x+1/2,y+1/2,-z+1/2	n-glide plane|translation: 1/2 1/2 0
75	x+1/2,-y+1/2,z+1/2	n-glide plane|translation: 1/2 0 1/2
76	-x+1/2,y+1/2,z+1/2	n-glide plane|translation: 0 1/2 1/2
77	-z+1/2,-x+1/2,-y+1/2	3-bar axis|inversion center at 1/4 1/4 1/4
78	-z+1/2,x+1/2,y+1/2	3-bar axis|inversion center at -1/4 1/4 3/4
79	z+1/2,x+1/2,-y+1/2	3-bar axis|inversion center at 1/4 3/4 -1/4
80	z+1/2,-x+1/2,y+1/2	3-bar axis|inversion center at 3/4 -1/4 1/4
81	-y+1/2,-z+1/2,-x+1/2	3-bar axis|inversion center at 1/4 1/4 1/4
82	y+1/2,-z+1/2,x+1/2	3-bar axis|inversion center at 1/4 -1/4 3/4
83	-y+1/2,z+1/2,x+1/2	3-bar axis|inversion center at -1/4 3/4 1/4
84	y+1/2,z+1/2,-x+1/2	3-bar axis|inversion center at 3/4 1/4 -1/4
85	-y+1/2,-x+1/2,z+1/2	c-glide plane|translation: 0 0 1/2
86	y+1/2,x+1/2,z+1/2	d-glide plane|translation: 1/2 1/2 1/2
87	-y+1/2,x+1/2,-z+1/2	4-bar axis|inversion center at 0 1/2 1/4
88	y+1/2,-x+1/2,-z+1/2	4-bar axis|inversion center at 1/2 0 1/4
89	-x+1/2,-z+1/2,y+1/2	4-bar axis|inversion center at 1/4 0 1/2
90	x+1/2,-z+1/2,-y+1/2	a-glide plane|translation: 1/2 0 0
91	x+1/2,z+1/2,y+1/2	d-glide plane|translation: 1/2 1/2 1/2
92	-x+1/2,z+1/2,-y+1/2	4-bar axis|inversion center at 1/4 1/2 0
93	-z+1/2,-y+1/2,x+1/2	4-bar axis|inversion center at 0 1/4 1/2
94	-z+1/2,y+1/2,-x+1/2	b-glide plane|translation: 0 1/2 0
95	z+1/2,-y+1/2,-x+1/2	4-bar axis|inversion center at 1/2 1/4 0
96	z+1/2,y+1/2,x+1/2	d-glide plane|translation: 1/2 1/2 1/2