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In a body-centred cubic (BCC) unit cell, if the number of nearest neighbours, next-nearest neighbours, and next-to-next-nearest neighbours of an atom are x, y, and z respectively, find the value of x*y/z.

1. 16
2. 12
3. 24
4. 8

Correct answer: 16

Solution

In a BCC lattice, considering an atom at a corner: 1st nearest neighbours (x): the 8 body-centre atoms of the 8 surrounding unit cells, at distance a*sqrt(3)/2. So x = 8. 2nd nearest neighbours (y): 6 atoms at face centres of adjacent cells... actually the 6 nearest corner atoms along face diagonals at distance a. So y = 6. 3rd nearest neighbours (z): 12 atoms at distance a*sqrt(2) (edge midpoints of surrounding cells, or atoms two edges away). So z = 12. Therefore x*y/z = 8*6/12 = 4... Hmm that gives 4 not 16. Let me recount. In BCC: corner atom sees body-centre of each unit cell. 1st nearest: body centres, 8 neighbours at a*sqrt(3)/2. x=8. 2nd nearest: 6 corner atoms at distance a. y=6. 3rd nearest: 12 at distance a*sqrt(2). z=12. x*y/z = 48/12 = 4. But for x*y/z = 16, we need different values. Alternatively: x=8, y=6, z=3: 48/3=16. Or from body-centre atom: 1st nearest = 8 corners (x=8); 2nd nearest = 6 body-centre atoms of adjacent cells at distance a (y=6); 3rd nearest = 12 atoms at a*sqrt(2)... still z=12. The answer 16 may correspond to a different set: perhaps z=3 (not standard). Standard x*y/z = 8*6/12 = 4.

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