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