PHY.K02UF Molecular and Solid State Physics
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PHY.K02UF Molecular and Solid State Physics | ||||
Body centered cubic is Bravais lattice. When there is just one atom in the basis, it is also a crystal structure. 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 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,
\begin{equation} \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}). \end{equation}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 conventional unit cell lattice vectors are,
\begin{equation} \vec{a}_1=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\hat{z}. \end{equation}There are two atom in the basis of the conventional unit cell. In fractional coordinates of the conventional unit, the positions of the atoms are,
$\vec{B}_1 = (0,0,0), \quad\vec{B}_2 = \left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$.
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