Determine the repeat distance, linear density, and packing fraction for BCC lithium, which has a lattice parameter of 0.35089 nm, in the [100], [110], and [111] directions. Which of these directions is close packed?

The [111] direction is close packed; the linear packing factor is 1.

Explanation:

Lattice parameter = a₀ = 0.35089nm

Directions 100, 110 and 111

r =  3a₀  

        4

where r = repeat distance

r = 3(0.35089)

           4

r = 0.1519nm

For [100]; r = a₀ = 0.35089nm

Linear density = 1/a₀ = 2.85points/nm

linear density = 1/a₀ = 2.85 points/nm

linear packing fraction = (2)(0.1519)(2.85) = 0.866

For [110]; r = 2a₀ = 0.496 nm

linear density = 1/a₀ = 2.015 points/nm

linear packing fraction = (2)(0.1519)(2.015) = 0.612

For [111]: r = 3a₀/2 = 0.3039 nm

linear density= 2/3a₀ = 3.291 points/nm

linear packing fraction = (2)(0.1519)(3.291) = 1

Therefore, from the above, [111] direction is close packed; the linear packing factor is 1.

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