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PHY.K02UF Molecular and Solid State Physics | ||||
Simple cubic is Bravais lattice. When there is just one atom in the basis, it is also a crystal structure. Of the elements, only α-polonium forms this crystal structure.
Crystal structure: simple cubic (sc)
Bravais lattice: simple cubic (cP)
Space group: 221 (P m 3 m),
Point group: m3m ($O_h$) six 2-fold rotations, four 3-fold rotations, three 4-fold rotations, nine mirror planes, inversion
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=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\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 also,
\begin{equation} \vec{a}_1=a\hat{x},\quad \vec{a}_2=a\hat{y},\quad\vec{a}_3=a\hat{z}. \end{equation}Space group: 221 (P m 3 m)
48 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