--- title: "BCC chromium: packing density and coordination number" subject: "Electrical Engineering Material" chapter: 1 key: "semester_3/electrical_engineering_material:bcc-chromium-packing" source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/bcc-chromium-packing site: BCT Engineering --- # BCC chromium: packing density and coordination number Electrical Engineering Material · Chapter 1: Theory of Metals - Asked in: 2071 Chaitra, 2+4 marks - Video: [Structure of metals | lattice types | body-centered cubic, face-centered, hexagonal | bcc, fcc, hcp](https://www.youtube.com/watch?v=zuu_tcdcje0) — tec-science - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/bcc-chromium-packing - Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-1 ## Question Define lattice and basis of a crystal and draw a neat diagram of body centered cubic structure of chromium and determine its packing density and state its co-ordination number. ## Short answer A lattice is a periodic array of points; a basis is the atom(s) at each point. Chromium’s BCC structure has 2 atoms per cell, coordination number 8, and packing density √3π/8 ≈ 0.68 (68%). ## Answer #### Lattice and basis - **Lattice:** an infinite, regular, periodic arrangement of points in space, each point having identical surroundings. It is a purely geometrical framework. - **Basis:** the atom or group of atoms placed at each lattice point. $$ \text{Crystal} = \text{Lattice} + \text{Basis} $$ For chromium, the lattice is BCC and the basis is a single Cr atom. #### BCC structure of chromium One Cr atom at each of the 8 corners and one at the body centre. Atoms touch along the **body diagonal**. ```text o-----------o /| /| / | / | o = Cr atom at corner o-----------o | @ = Cr atom at centre | | @ | | | o--------|--o atoms touch along | / | / body diagonal |/ |/ o-----------o ``` #### Packing density Atoms per cell: $N = 8\times\frac{1}{8} + 1 = 2$. Along the body diagonal: $\sqrt{3}\,a = 4R \Rightarrow a = \dfrac{4R}{\sqrt{3}}$. $$ \begin{aligned} \text{APF} &= \frac{N\cdot\frac{4}{3}\pi R^3}{a^3} = \frac{2\times\frac{4}{3}\pi R^3}{\left(\frac{4R}{\sqrt{3}}\right)^3} \\ &= \frac{\frac{8}{3}\pi R^3\times3\sqrt{3}}{64R^3} = \frac{\sqrt{3}\,\pi}{8} = 0.68 \end{aligned} $$ So 68 % of the volume of the cell is filled by atoms. (For Cr, $R \approx 0.125$ nm gives $a = 4R/\sqrt{3} \approx 0.289$ nm.) #### Coordination number The body-centre atom touches all 8 corner atoms, and each corner atom touches the centre atoms of the 8 cells sharing that corner. So the **coordination number is 8**. **Answer:** packing density $= 0.68$, coordination number $= 8$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the description Crystal = Lattice + Basis, what is the basis? A. The infinite array of identical points in space B. The atom or group of atoms placed at each lattice point C. The unit cell edge length D. The number of nearest neighbours Answer: B. The lattice is the geometrical array of points; the basis is the atom or group of atoms attached to every point. For Cr it is a single atom. (Key point: Lattice vs basis) 2. How many atoms belong to one BCC unit cell? A. 1 B. 2 C. 4 D. 9 Answer: B. 8 corners x 1/8 + 1 body centre = 2 atoms. (Key point: Atoms per BCC cell) 3. Along which direction do atoms touch in BCC, and what relation does this give between a and R? A. Cube edge: a = 2R B. Face diagonal: sqrt(2)a = 4R C. Body diagonal: sqrt(3)a = 4R D. Body diagonal: sqrt(3)a = 2R Answer: C. The body diagonal carries two corner atoms and the centre atom, a length of 4R, so sqrt(3)a = 4R. (Key point: Touching direction) 4. What are the packing density and coordination number of BCC chromium? A. 0.74 and 12 B. 0.68 and 8 C. 0.52 and 6 D. 0.68 and 12 Answer: B. APF = sqrt(3)pi/8 = 0.68 and the centre atom touches 8 corner atoms, so the coordination number is 8. 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