Chemistry · Ch 6 — Solid State
Body Centered Cubic Arrangement
Body Centered Cubic Arrangement
In the body-centred cubic (bcc) arrangement, the spheres of the first layer (A-type) are slightly separated from one another, and the second layer is formed by placing its spheres into the depressions between the spheres of layer A. The third layer repeats the first, so the pattern ABABAB... continues throughout the crystal. Each sphere in this arrangement has a coordination number of 8 -- four neighbours in the layer above and four in the layer below.
Packing efficiency. Here the spheres touch along the leading body diagonal of the cube. Using the cube's vertices A-H (edge length a): in triangle ABC (a face of the cube), Pythagoras gives the face diagonal AC^2 = AB^2 + BC^2 = a^2 + a^2 = 2a^2, so AC = root-2 * a. Then, in triangle ACG (running through the cube's interior), the body diagonal AG^2 = AC^2 + CG^2 = 2a^2 + a^2 = 3a^2, so AG = root-3 * a.
Since the spheres touch along this body diagonal, AG spans 4 sphere radii: root-3 * a = 4r, giving r = (root-3 / 4) a. The volume of one sphere is then:
A bcc unit cell contains exactly 2 spheres (Section 6.5.2), so the total sphere volume in the cell is 2 times the above, i.e. (root-3 * pi * a^3)/8. Dividing by the cell volume a^3 and multiplying by 100: …
What this figure shows. A cluster of pink spheres (Layer a) forming the top and bottom of a stack, with a row of smaller green spheres (Layer b) nested in the depressions between them at the middle -- a legend at the side keys 'Layer a' (pink, top and bottom) and 'Layer b' (green, middle) -- captioned 'Body Centered Cubic (BCC)'. Unlike the direct, gap-free stacking of the simple-cubic figure, the spheres of layer a here are drawn slightly separated from one another, with the green layer-b spheres sitting distinctly lower, nestled into the depressions between them -- visually showing why this ABABAB-pattern stack packs more efficiently (68%) …
What this figure shows. A cube with vertices labelled A, B, C, D, E, F, G, H, a green sphere of radius r at every corner and a pink sphere of radius 2r occupying the body centre, touching all four body-diagonal corner spheres; the front face's diagonal is marked root-2 a and the three visible edges are each marked a, setting up triangle ABC (giving face diagonal AC = root-2 a) and triangle ACG (giving body diagonal AG = root-3 a), from which root-3 a = 4r and the bcc packing efficiency of 68% both follow. …