1.10 please Given a cubic lattice, indicate how many equivalent planes [parts (a

Question

1.10 please

Given a cubic lattice, indicate how many equivalent planes [parts (a) to (c)] or equivalent directions [parts (d) to (f)] are associated with each of the following designations: {100}, {110}, {111}, (100), (110), (111). Treating atoms as rigid spheres with radii equal to one-half the distance between nearest neighbors, show that the ratio of the volume occupied by the atoms to the total available volume in the various crystal structures is: Pi/6 or 52% for the simple cubic lattice, Square root 3/8 or 68% for the body centered cubic lattice, Square root 2 Pi/6 or 74% for the face centered cubic lattice, Square root 3 Pi/16 or 34% for the diamond lattice.

Explanation / Answer

Treating atom as sphere , V = 4/3 pi r^3

distance between nearest neighbours for a body centred cubic lattice= sqroot ( 3)a/2

half of distance between nearest neighbours for a body centred cubic lattice = sqroot(3) a/4

Vof sphere = 4/3 pi ( sqroot(3) a/4)^3= sqroot(3) pi / 16 or 34 %

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