# Body Centered Cubic (bcc)

Besides the simple cubic (sc) and the face centered cubic (fcc) lattices there is another cubic Bravais lattice called **b**ody **c**entered **c**ubic (bcc) lattice. Unlike the simple cubic lattice it has an additional lattice point located in the center of the cube.^{[1]}

## 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

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.^{[3]}^{[4]}

## Coordination Number

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

[1] | Oxford Solid State Basics Oxford 2013 (ch. 12.2.1) |

[2] | Festkörperphysik De Gruyter 2014 (p. 68) |

[3] | Festkörperphysik De Gruyter 2014 (ch. 1.2.3) |

[4] | Festkörperphysik De Gruyter 2014 (p. 68) |

[5] | Festkörperphysik De Gruyter 2014 (ch. 1.2.3) |

[6] | Festkörperphysik De Gruyter 2014 (p. 68) |