PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
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