Physics in a nutshell

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# Body Centered Cubic (bcc)

Besides the simple cubic (sc) and the face centered cubic (fcc) lattices there is another cubic Bravais lattice called body centered cubic (bcc) lattice. Unlike the simple cubic lattice it has an additional lattice point located in the center of the cube.

## Conventional Unit Cell

For the conventional unit cell a cubic one is chosen because it represents the symmetry of the underlying structure best. As before we denote the length of its edges by the letter $a$. The conventional unit cell contains 8 lattice points at the vertices, each being shared by 8 cells and another lattice point that is completely inside the conventional unit cell. So the number $N$ of poitns per unit cell adds up to \begin{align} N = 8 \cdot \frac{1}{8} + 1 = 2. \end{align}

## Packing Density When the lattice points are inflated gradually, at some point they start to touch each other along the diagonals of the cube. One can now interpret the atoms as close packed spheres with a radius defined geometrically by ${4r = \sqrt{3}a}$ ${\Leftrightarrow \; r = \frac{\sqrt{3}}{4}a}$. 

The packing density $\varrho$ is the ratio of the volume filled by the spherical atoms within a unit cell to the total volume $V_\text{uc}$ of the unit cell. When considering a one-atomic basis there are $n=2$ points per unit cell with a volume of $V_\text{sph} = \frac{4}{3} \pi r^3$ each. Thus for the packing density one obtains \begin{align} \varrho &= \frac{n \cdot V_\text{sph}}{V_\text{uc}} = \frac{ 2 \cdot \frac{4}{3} \pi \cdot \left( \frac{\sqrt{3}}{4} \right)^3 a^3}{a^3} \nonumber \\[1ex] &= \frac{\sqrt{3} \pi}{8} \approx 68\% \end{align} which is slightly less than the highest possible value of 74% which we obtained for the close-packed structures.

## Coordination Number Each lattice points has 8 nearest neighbours (in the centers of the neighboured cubes) and 6 next-nearest neighbours (located at the neighboured vertices of the lattice).

In the bcc structure each atom has $c_1 = 8$ nearest neighbours (coordination number) at a distance of \begin{align} d_{c_1} = 2r = \frac{\sqrt{3}}{2}a \approx 0.866a \end{align} and $c_2 = 6$ next-nearest neighbours at a distance of \begin{align} d_{c_2} = a \approx 2.3r \approx 1.15 \, d_{c_1} . \end{align} It is remarkable that there is a smaller number of nearest neighbours compared to the close-packed structures but for the bcc structure the next-nearest neighbours are only slightly further away which makes it possible for those to participate in bonds as well.

## References

  S. H. Simon Oxford Solid State Basics Oxford 2013 (ch. 12.2.1)  S. Hunklinger Festkörperphysik De Gruyter 2014 (p. 68)  R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.2.3)  S. Hunklinger Festkörperphysik De Gruyter 2014 (p. 68)  R. Gross, A. Marx Festkörperphysik De Gruyter 2014 (ch. 1.2.3)  S. Hunklinger Festkörperphysik De Gruyter 2014 (p. 68)

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