For an FCC single crystal, would you expect the surface energy for a (100) plane to be greater or less than that for a (111) plane?

The the surface energy for a (100) plane of FCC single crystal will be greater than that for a (111) plane.

Check Explanation for the reason why.

Explanation:

To explain, we would use the planar density to do this.

Planar density is the fraction of total crystallographic plane area that is occupied by atoms. The higher the planar density, the lower the surface energy.

Mathematically,

Planar density = Number of atoms centered on a given plane/Area of the plane

Planar density for the (100) plane

Number of atoms centred on the plane = 2

Area of the plane = (lattice parameter)² = (2R√2)² = 8 R²

Planar density = 2/8R² = 1/(4R²)

Planar density of the (111) plane

The (111) plane of FCC is attached to this solution

Number of atoms centered on the plane = (3×1/6) + (3×1/2) = 2 atoms

For the area, it is evident that it is an equilateral triangle with length a√2 where a = lattice parameter

So, area = bh/2

b = 2a

h will be obtained using Pythagoras theorem,

(a√2)² = h² + [(a/2)√2]²

2a² = h² + a²/2

h² = 2a² - a²/2 = 3a²/4

h = √(3a²/4) = (a/2)√3

Area of plane = (1/2)(2a)(a/2)(√3)

Area = (a²/2)√3

a = 2R √2

a² = 8 R²

Area = 4R²√3

Planar density = 2/(4R²√3) = 1/(2R²√3)

Planar density for (100) = 1/(4R²)

Planar density for (111) = 1/(2R²√3)

As planar density increases, atoms have lower surface energy because the atoms have more neighbouring atoms to bond with.

And from the calculation,

Planar density of (100) < Planar density of (111)

Surface energy of (100) > surface energy of (111)

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explanation: