PHY.F20 Molecular and Solid State Physics

NaCl

NaCl is a crystal structure with a face centered cubic Bravais lattice and two atoms in the basis.

Crystal structure: NaCl
Bravais lattice: face centered cubic
Space group: 225 (F m -3 m),   Strukturbericht: B1,   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 = 0.564 $ nm
Atomic density $n_{\text{atoms}}=8/a^3$


The lattice paramters 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,

$\vec{B}_1=(0,0,0)$,   $\vec{B}_2=(0.5,0,0)$.

Space group: 225 (F m -3 m)
192 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
49	z,y+1/2,-x+1/2	4-fold screw axis|translation: 0 1/2 0
50	y,x+1/2,-z+1/2	2-fold screw axis|translation: 1/4 1/4 0
51	x,z+1/2,-y+1/2	C4 axis
52	z,x+1/2,-y+1/2	3-bar axis|inversion center at 0 1/2 0
53	y,z+1/2,-x+1/2	3-bar axis|inversion center at 1/2 1/2 0
54	x,y+1/2,-z+1/2	b-glide plane|translation: 0 1/2 0
55	z,-y+1/2,x+1/2	2-fold screw axis|translation: 1/4 0 1/4
56	y,-x+1/2,z+1/2	4-fold screw axis|translation: 0 0 1/2
57	x,-z+1/2,y+1/2	C4 axis
58	z,-x+1/2,y+1/2	3-bar axis|inversion center at 1/2 0 1/2
59	y,-z+1/2,x+1/2	3-bar axis|inversion center at 0 0 1/2
60	x,-y+1/2,z+1/2	c-glide plane|translation: 0 0 1/2
61	-z,y+1/2,x+1/2	4-fold screw axis|translation: 0 1/2 0
62	-y,x+1/2,z+1/2	4-fold screw axis|translation: 0 0 1/2
63	-x,z+1/2,y+1/2	2-fold screw axis|translation: 0 1/2 1/2
64	-z,x+1/2,y+1/2	3-bar axis|inversion center at -1/2 0 1/2
65	-y,z+1/2,x+1/2	3-bar axis|inversion center at -1/2 1/2 0
66	-x,y+1/2,z+1/2	n-glide plane|translation: 0 1/2 1/2
67	-z,-y+1/2,-x+1/2	2-fold screw axis|translation: -1/4 0 1/4
68	-y,-x+1/2,-z+1/2	2-fold screw axis|translation: -1/4 1/4 0
69	-x,-z+1/2,-y+1/2	C2 axis
70	-z,-x+1/2,-y+1/2	3-bar axis|inversion center at 0 1/2 0
71	-y,-z+1/2,-x+1/2	3-bar axis|inversion center at 0 0 1/2
72	-x,-y+1/2,-z+1/2	Ci: 0 1/4 1/4
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+1/2,y,-x+1/2	C4 axis
98	y+1/2,x,-z+1/2	2-fold screw axis|translation: 1/4 1/4 0
99	x+1/2,z,-y+1/2	4-fold screw axis|translation: 1/2 0 0
100	z+1/2,x,-y+1/2	3-bar axis|inversion center at 1/2 1/2 0
101	y+1/2,z,-x+1/2	3-bar axis|inversion center at 1/2 0 0
102	x+1/2,y,-z+1/2	a-glide plane|translation: 1/2 0 0
103	z+1/2,-y,x+1/2	2-fold screw axis|translation: 1/2 0 1/2
104	y+1/2,-x,z+1/2	4-fold screw axis|translation: 0 0 1/2
105	x+1/2,-z,y+1/2	4-fold screw axis|translation: 1/2 0 0
106	z+1/2,-x,y+1/2	3-bar axis|inversion center at 1/2 -1/2 0
107	y+1/2,-z,x+1/2	3-bar axis|inversion center at 0 -1/2 1/2
108	x+1/2,-y,z+1/2	n-glide plane|translation: 1/2 0 1/2
109	-z+1/2,y,x+1/2	C4 axis
110	-y+1/2,x,z+1/2	4-fold screw axis|translation: 0 0 1/2
111	-x+1/2,z,y+1/2	2-fold screw axis|translation: 0 1/4 1/4
112	-z+1/2,x,y+1/2	3-bar axis|inversion center at 0 0 1/2
113	-y+1/2,z,x+1/2	3-bar axis|inversion center at 0 1/2 1/2
114	-x+1/2,y,z+1/2	c-glide plane|translation: 0 0 1/2
115	-z+1/2,-y,-x+1/2	C2 axis
116	-y+1/2,-x,-z+1/2	2-fold screw axis|translation: 1/4 -1/4 0
117	-x+1/2,-z,-y+1/2	2-fold screw axis|translation: 0 -1/4 1/4
118	-z+1/2,-x,-y+1/2	3-bar axis|inversion center at 0 0 1/2
119	-y+1/2,-z,-x+1/2	3-bar axis|inversion center at 1/2 0 0
120	-x+1/2,-y,-z+1/2	Ci: 1/4 0 1/4
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+1/2,y+1/2,-x	4-fold screw axis|translation: 0 1/2 0
146	y+1/2,x+1/2,-z	2-fold screw axis|translation: 1/2 1/2 0
147	x+1/2,z+1/2,-y	4-fold screw axis|translation: 1/2 0 0
148	z+1/2,x+1/2,-y	3-bar axis|inversion center at 0 1/2 -1/2
149	y+1/2,z+1/2,-x	3-bar axis|inversion center at 1/2 0 -1/2
150	x+1/2,y+1/2,-z	n-glide plane|translation: 1/2 1/2 0
151	z+1/2,-y+1/2,x	2-fold screw axis|translation: 1/4 0 1/4
152	y+1/2,-x+1/2,z	C4 axis
153	x+1/2,-z+1/2,y	4-fold screw axis|translation: 1/2 0 0
154	z+1/2,-x+1/2,y	3-bar axis|inversion center at 1/2 0 0
155	y+1/2,-z+1/2,x	3-bar axis|inversion center at 1/2 0 1/2
156	x+1/2,-y+1/2,z	a-glide plane|translation: 1/2 0 0
157	-z+1/2,y+1/2,x	4-fold screw axis|translation: 0 1/2 0
158	-y+1/2,x+1/2,z	C4 axis
159	-x+1/2,z+1/2,y	2-fold screw axis|translation: 0 1/4 1/4
160	-z+1/2,x+1/2,y	3-bar axis|inversion center at 0 1/2 1/2
161	-y+1/2,z+1/2,x	3-bar axis|inversion center at 0 1/2 0
162	-x+1/2,y+1/2,z	b-glide plane|translation: 0 1/2 0
163	-z+1/2,-y+1/2,-x	2-fold screw axis|translation: 1/4 0 -1/4
164	-y+1/2,-x+1/2,-z	C2 axis
165	-x+1/2,-z+1/2,-y	2-fold screw axis|translation: 0 1/4 -1/4
166	-z+1/2,-x+1/2,-y	3-bar axis|inversion center at 1/2 0 0
167	-y+1/2,-z+1/2,-x	3-bar axis|inversion center at 0 1/2 0
168	-x+1/2,-y+1/2,-z	Ci: 1/4 1/4 0
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