PHY.F20 Molecular and Solid State Physics

Diamond crystal structure

Diamond is a crystal structure with a face centered cubic Bravais lattice and two atoms in the basis. Carbon, silicon germanium, and α-tin form this crystal structure.

Crystal structure: Diamond
Bravais lattice: face centered cubic
Space group: 227 (F d -3 m),  Strukturbericht: A4,   Pearson symbol: cF8
Point group: m3m (Oh)   six 2-fold rotations, four 3-fold rotations, three 4-fold rotations, nine mirror planes, inversion
Lattice constant: $a$   $a_{\text{C}}=0.3567$ nm,  $a_{\text{Si}}=0.5431$ nm, $a_{\text{Ge}}=0.5658$ nm,  $a_{\alpha\text{-Sn}}=0.646$ nm
Atomic density $n_{\text{atoms}}=8/a^3$
Density of surface atoms
 (100): $\large \frac{2}{a^2}$    (110): $\large \frac{4}{\sqrt{2}a^2}$    (111): $\large \frac{4}{\sqrt{3}a^2}$

The lattice parameters of the conventional unit cell 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,

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

There are two atoms in the basis. In fractional coordinates of the conventional unit, the positions of the two atoms are,

$\vec{B}_1=(0,0,0)$,   $\vec{B}_2=(0.25,0.25,0.25)$.

The asymmetric unit contains one atom at fractional coordinate (0,0,0). The positions of the other atoms can be determined by applying the symmetries of the space group.

A CIF file contains the lattice parameters, the asymmetric unit, and all of the space group symmetries.

Space group: 227 (F d -3 m)
192 symmetry operations:

1	x,y,z			identity
2	z+1/4,y+1/4,-x+1/4	4-fold screw axis|translation: 0 1/4 0
3	y+1/4,x+1/4,-z+1/4	2-fold screw axis|translation: 1/4 1/4 0
4	x+1/4,z+1/4,-y+1/4	4-fold screw axis|translation: 1/4 0 0
5	z+1/4,x+1/4,-y+1/4	3-bar axis|inversion center at 1/12 1/3 -1/12
6	y+1/4,z+1/4,-x+1/4	3-bar axis|inversion center at 1/3 1/12 -1/12
7	x+1/4,y+1/4,-z+1/4	n-glide plane|translation: 1/4 1/4 0
8	z+1/4,-y+1/4,x+1/4	2-fold screw axis|translation: 1/4 0 1/4
9	y+1/4,-x+1/4,z+1/4	4-fold screw axis|translation: 0 0 1/4
10	x+1/4,-z+1/4,y+1/4	4-fold screw axis|translation: 1/4 0 0
11	z+1/4,-x+1/4,y+1/4	3-bar axis|inversion center at 1/3 -1/12 1/12
12	y+1/4,-z+1/4,x+1/4	3-bar axis|inversion center at 1/12 -1/12 1/3
13	x+1/4,-y+1/4,z+1/4	n-glide plane|translation: 1/4 0 1/4
14	-z+1/4,y+1/4,x+1/4	4-fold screw axis|translation: 0 1/4 0
15	-y+1/4,x+1/4,z+1/4	4-fold screw axis|translation: 0 0 1/4
16	-x+1/4,z+1/4,y+1/4	2-fold screw axis|translation: 0 1/4 1/4
17	-z+1/4,x+1/4,y+1/4	3-bar axis|inversion center at -1/12 1/12 1/3
18	-y+1/4,z+1/4,x+1/4	3-bar axis|inversion center at -1/12 1/3 1/12
19	-x+1/4,y+1/4,z+1/4	n-glide plane|translation: 0 1/4 1/4
20	-z+1/4,-y+1/4,-x+1/4	C2 axis
21	-y+1/4,-x+1/4,-z+1/4	C2 axis
22	-x+1/4,-z+1/4,-y+1/4	C2 axis
23	-z+1/4,-x+1/4,-y+1/4	3-bar axis|inversion center at 1/12 1/12 1/12
24	-y+1/4,-z+1/4,-x+1/4	3-bar axis|inversion center at 1/12 1/12 1/12
25	-x+1/4,-y+1/4,-z+1/4	Ci: 1/12 1/12 1/12
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
49	z+1/4,y-1/4,-x-1/4	4-fold screw axis|translation: 0 -1/4 0
50	y+1/4,x-1/4,-z-1/4	C2 axis
51	x+1/4,z-1/4,-y-1/4	4-fold screw axis|translation: 1/4 0 0
52	z+1/4,x-1/4,-y-1/4	3-bar axis|inversion center at 1/12 -1/12 -1/12
53	y+1/4,z-1/4,-x-1/4	3-bar axis|inversion center at -1/12 -1/3 -1/12
54	x+1/4,y-1/4,-z-1/4	n-glide plane|translation: 1/4 -1/4 0
55	z+1/4,-y-1/4,x-1/4	C2 axis
56	y+1/4,-x-1/4,z-1/4	4-fold screw axis|translation: 0 0 -1/4
57	x+1/4,-z-1/4,y-1/4	4-fold screw axis|translation: 1/4 0 0
58	z+1/4,-x-1/4,y-1/4	3-bar axis|inversion center at -1/12 -1/12 -1/3
59	y+1/4,-z-1/4,x-1/4	3-bar axis|inversion center at 1/12 -1/12 -1/12
60	x+1/4,-y-1/4,z-1/4	n-glide plane|translation: 1/4 0 -1/4
61	-z+1/4,y-1/4,x-1/4	4-fold screw axis|translation: 0 -1/4 0
62	-y+1/4,x-1/4,z-1/4	4-fold screw axis|translation: 0 0 -1/4
63	-x+1/4,z-1/4,y-1/4	2-fold screw axis|translation: 0 -1/4 -1/4
64	-z+1/4,x-1/4,y-1/4	3-bar axis|inversion center at 1/3 1/12 -1/12
65	-y+1/4,z-1/4,x-1/4	3-bar axis|inversion center at 1/3 -1/12 1/12
66	-x+1/4,y-1/4,z-1/4	n-glide plane|translation: 0 -1/4 -1/4
67	-z+1/4,-y-1/4,-x-1/4	2-fold screw axis|translation: 1/4 0 -1/4
68	-y+1/4,-x-1/4,-z-1/4	2-fold screw axis|translation: 1/4 -1/4 0
69	-x+1/4,-z-1/4,-y-1/4	C2 axis
70	-z+1/4,-x-1/4,-y-1/4	3-bar axis|inversion center at 1/12 -1/3 1/12
71	-y+1/4,-z-1/4,-x-1/4	3-bar axis|inversion center at 1/12 1/12 -1/3
72	-x+1/4,-y-1/4,-z-1/4	Ci: 1/12 -1/12 -1/12
73	-z,-y+1/2,x+1/2		4-bar axis|inversion center at -1/4 1/4 1/4
74	-y,-x+1/2,z+1/2		g-glide plane|translation: -1/4 1/4 1/2
75	-x,-z+1/2,y+1/2		4-bar axis|inversion center at 0 0 1/2
76	-z,-x+1/2,y+1/2		3-fold screw axis|translation: -1/3 1/3 1/3
77	-y,-z+1/2,x+1/2		C3 axis
78	-x,-y+1/2,z+1/2		2-fold screw axis|translation: 0 0 1/2
79	-z,y+1/2,-x+1/2		g-glide plane|translation: -1/4 1/2 1/4
80	-y,x+1/2,-z+1/2		4-bar axis|inversion center at -1/4 1/4 1/4
81	-x,z+1/2,-y+1/2		4-bar axis|inversion center at 0 1/2 0
82	-z,x+1/2,-y+1/2		C3 axis
83	-y,z+1/2,-x+1/2		3-fold screw axis|translation: -1/3 1/3 1/3
84	-x,y+1/2,-z+1/2		2-fold screw axis|translation: 0 1/2 0
85	z,-y+1/2,-x+1/2		4-bar axis|inversion center at 1/4 1/4 1/4
86	y,-x+1/2,-z+1/2		4-bar axis|inversion center at 1/4 1/4 1/4
87	x,-z+1/2,-y+1/2		mirror plane
88	z,-x+1/2,-y+1/2		C3 axis
89	y,-z+1/2,-x+1/2		C3 axis
90	x,-y+1/2,-z+1/2		C2 axis
91	z,y+1/2,x+1/2		g-glide plane|translation: 1/4 1/2 1/4
92	y,x+1/2,z+1/2		g-glide plane|translation: 1/4 1/4 1/2
93	x,z+1/2,y+1/2		n-glide plane|translation: 0 1/2 1/2
94	z,x+1/2,y+1/2		3-fold screw axis|translation: 1/3 1/3 1/3
95	y,z+1/2,x+1/2		3-fold screw axis|translation: 1/3 1/3 1/3
96	x,y+1/2,z+1/2		translation: 0 1/2 1/2
97	z-1/4,y+1/4,-x-1/4	4-fold screw axis|translation: 0 1/4 0
98	y-1/4,x+1/4,-z-1/4	C2 axis
99	x-1/4,z+1/4,-y-1/4	4-fold screw axis|translation: -1/4 0 0
100	z-1/4,x+1/4,-y-1/4	3-bar axis|inversion center at -1/3 -1/12 -1/12
101	y-1/4,z+1/4,-x-1/4	3-bar axis|inversion center at -1/12 1/12 -1/12
102	x-1/4,y+1/4,-z-1/4	n-glide plane|translation: -1/4 1/4 0
103	z-1/4,-y+1/4,x-1/4	2-fold screw axis|translation: -1/4 0 -1/4
104	y-1/4,-x+1/4,z-1/4	4-fold screw axis|translation: 0 0 -1/4
105	x-1/4,-z+1/4,y-1/4	4-fold screw axis|translation: -1/4 0 0
106	z-1/4,-x+1/4,y-1/4	3-bar axis|inversion center at -1/12 1/3 1/12
107	y-1/4,-z+1/4,x-1/4	3-bar axis|inversion center at 1/12 1/3 -1/12
108	x-1/4,-y+1/4,z-1/4	n-glide plane|translation: -1/4 0 -1/4
109	-z-1/4,y+1/4,x-1/4	4-fold screw axis|translation: 0 1/4 0
110	-y-1/4,x+1/4,z-1/4	4-fold screw axis|translation: 0 0 -1/4
111	-x-1/4,z+1/4,y-1/4	C2 axis
112	-z-1/4,x+1/4,y-1/4	3-bar axis|inversion center at -1/12 1/12 -1/12
113	-y-1/4,z+1/4,x-1/4	3-bar axis|inversion center at -1/12 -1/12 -1/3
114	-x-1/4,y+1/4,z-1/4	n-glide plane|translation: 0 1/4 -1/4
115	-z-1/4,-y+1/4,-x-1/4	C2 axis
116	-y-1/4,-x+1/4,-z-1/4	2-fold screw axis|translation: -1/4 1/4 0
117	-x-1/4,-z+1/4,-y-1/4	2-fold screw axis|translation: 0 1/4 -1/4
118	-z-1/4,-x+1/4,-y-1/4	3-bar axis|inversion center at 1/12 1/12 -1/3
119	-y-1/4,-z+1/4,-x-1/4	3-bar axis|inversion center at -1/3 1/12 1/12
120	-x-1/4,-y+1/4,-z-1/4	Ci: -1/12 1/12 -1/12
121	-z+1/2,-y,x+1/2		4-bar axis|inversion center at 0 0 1/2
122	-y+1/2,-x,z+1/2		g-glide plane|translation: 1/4 -1/4 1/2
123	-x+1/2,-z,y+1/2		4-bar axis|inversion center at 1/4 -1/4 1/4
124	-z+1/2,-x,y+1/2		C3 axis
125	-y+1/2,-z,x+1/2		3-fold screw axis|translation: 1/3 -1/3 1/3
126	-x+1/2,-y,z+1/2		2-fold screw axis|translation: 0 0 1/2
127	-z+1/2,y,-x+1/2		mirror plane
128	-y+1/2,x,-z+1/2		4-bar axis|inversion center at 1/4 1/4 1/4
129	-x+1/2,z,-y+1/2		4-bar axis|inversion center at 1/4 1/4 1/4
130	-z+1/2,x,-y+1/2		C3 axis
131	-y+1/2,z,-x+1/2		C3 axis
132	-x+1/2,y,-z+1/2		C2 axis
133	z+1/2,-y,-x+1/2		4-bar axis|inversion center at 1/2 0 0
134	y+1/2,-x,-z+1/2		4-bar axis|inversion center at 1/4 -1/4 1/4
135	x+1/2,-z,-y+1/2		g-glide plane|translation: 1/2 -1/4 1/4
136	z+1/2,-x,-y+1/2		3-fold screw axis|translation: 1/3 -1/3 1/3
137	y+1/2,-z,-x+1/2		C3 axis
138	x+1/2,-y,-z+1/2		2-fold screw axis|translation: 1/2 0 0
139	z+1/2,y,x+1/2		n-glide plane|translation: 1/2 0 1/2
140	y+1/2,x,z+1/2		g-glide plane|translation: 1/4 1/4 1/2
141	x+1/2,z,y+1/2		g-glide plane|translation: 1/2 1/4 1/4
142	z+1/2,x,y+1/2		3-fold screw axis|translation: 1/3 1/3 1/3
143	y+1/2,z,x+1/2		3-fold screw axis|translation: 1/3 1/3 1/3
144	x+1/2,y,z+1/2		translation: 1/2 0 1/2
145	z-1/4,y-1/4,-x+1/4	4-fold screw axis|translation: 0 -1/4 0
146	y-1/4,x-1/4,-z+1/4	2-fold screw axis|translation: -1/4 -1/4 0
147	x-1/4,z-1/4,-y+1/4	4-fold screw axis|translation: -1/4 0 0
148	z-1/4,x-1/4,-y+1/4	3-bar axis|inversion center at 1/12 -1/12 1/3
149	y-1/4,z-1/4,-x+1/4	3-bar axis|inversion center at -1/12 1/12 1/3
150	x-1/4,y-1/4,-z+1/4	n-glide plane|translation: -1/4 -1/4 0
151	z-1/4,-y-1/4,x+1/4	C2 axis
152	y-1/4,-x-1/4,z+1/4	4-fold screw axis|translation: 0 0 1/4
153	x-1/4,-z-1/4,y+1/4	4-fold screw axis|translation: -1/4 0 0
154	z-1/4,-x-1/4,y+1/4	3-bar axis|inversion center at -1/12 -1/12 1/12
155	y-1/4,-z-1/4,x+1/4	3-bar axis|inversion center at -1/3 -1/12 -1/12
156	x-1/4,-y-1/4,z+1/4	n-glide plane|translation: -1/4 0 1/4
157	-z-1/4,y-1/4,x+1/4	4-fold screw axis|translation: 0 -1/4 0
158	-y-1/4,x-1/4,z+1/4	4-fold screw axis|translation: 0 0 1/4
159	-x-1/4,z-1/4,y+1/4	C2 axis
160	-z-1/4,x-1/4,y+1/4	3-bar axis|inversion center at -1/12 -1/3 -1/12
161	-y-1/4,z-1/4,x+1/4	3-bar axis|inversion center at -1/12 -1/12 1/12
162	-x-1/4,y-1/4,z+1/4	n-glide plane|translation: 0 -1/4 1/4
163	-z-1/4,-y-1/4,-x+1/4	2-fold screw axis|translation: -1/4 0 1/4
164	-y-1/4,-x-1/4,-z+1/4	C2 axis
165	-x-1/4,-z-1/4,-y+1/4	2-fold screw axis|translation: 0 -1/4 1/4
166	-z-1/4,-x-1/4,-y+1/4	3-bar axis|inversion center at -1/3 1/12 1/12
167	-y-1/4,-z-1/4,-x+1/4	3-bar axis|inversion center at 1/12 -1/3 1/12
168	-x-1/4,-y-1/4,-z+1/4	Ci: -1/12 -1/12 1/12
169	-z+1/2,-y+1/2,x		4-bar axis|inversion center at 1/4 1/4 1/4
170	-y+1/2,-x+1/2,z		mirror plane
171	-x+1/2,-z+1/2,y		4-bar axis|inversion center at 1/4 1/4 1/4
172	-z+1/2,-x+1/2,y		C3 axis
173	-y+1/2,-z+1/2,x		C3 axis
174	-x+1/2,-y+1/2,z		C2 axis
175	-z+1/2,y+1/2,-x		g-glide plane|translation: 1/4 1/2 -1/4
176	-y+1/2,x+1/2,-z		4-bar axis|inversion center at 0 1/2 0
177	-x+1/2,z+1/2,-y		4-bar axis|inversion center at 1/4 1/4 -1/4
178	-z+1/2,x+1/2,-y		3-fold screw axis|translation: 1/3 1/3 -1/3
179	-y+1/2,z+1/2,-x		C3 axis
180	-x+1/2,y+1/2,-z		2-fold screw axis|translation: 0 1/2 0
181	z+1/2,-y+1/2,-x		4-bar axis|inversion center at 1/4 1/4 -1/4
182	y+1/2,-x+1/2,-z		4-bar axis|inversion center at 1/2 0 0
183	x+1/2,-z+1/2,-y		g-glide plane|translation: 1/2 1/4 -1/4
184	z+1/2,-x+1/2,-y		C3 axis
185	y+1/2,-z+1/2,-x		3-fold screw axis|translation: 1/3 1/3 -1/3
186	x+1/2,-y+1/2,-z		2-fold screw axis|translation: 1/2 0 0
187	z+1/2,y+1/2,x		g-glide plane|translation: 1/4 1/2 1/4
188	y+1/2,x+1/2,z		n-glide plane|translation: 1/2 1/2 0
189	x+1/2,z+1/2,y		g-glide plane|translation: 1/2 1/4 1/4
190	z+1/2,x+1/2,y		3-fold screw axis|translation: 1/3 1/3 1/3
191	y+1/2,z+1/2,x		3-fold screw axis|translation: 1/3 1/3 1/3
192	x+1/2,y+1/2,z		translation: 1/2 1/2 0