Chapter 3

1. Determine the linear density (atoms/nm) for BCC [100], [110], and [111] directions in terms of atomic radius R.

In[4]:= LD[Natoms_, length_] := Natoms / length;

In[5]:= aBCC = 4 * r / Sqrt[3];

[100] (1 pt)

Natoms = 1;length = aBCC;LD[Natoms, length]

3

4 r

Sqrt[3] / (4. * r)

0.433013

r

[110] (1 pt)

Natoms = 1;length = Sqrt[2] * aBCC;LD[Natoms, length]

32

4 r

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Sqrt[3 / 2] / (4. * r)

0.306186

r

[111] (1 pt)

In[6]:= Natoms = 2;length = 4 * r;LD[Natoms, length]

Out[8]=1

2 r

2. For the BCC (100) and (110) planes, (a) Make a sketch of the plane. (b) Determine the planar density (atoms/nm2) in terms of atomic radius R. (c) Out of the {100}, {110}, and {111} direction families, which direction families fall within the plane?

PD[Natoms_, Aplane_] := Natoms / Aplane;aBCC = 4 * r / Sqrt[3];

(100)

(a) See sol’n 1 (1 pt)

(b) (1 pt)

Natoms = 1;Aplane = aBCC^2;PD[Natoms, Aplane]

3

16 r2

3 / (16. r^2)

0.1875

r2

2 hw3solutions.nb

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(c) {100} and {110} (see sol’n 1) (1 pt)

(110)

(a) See sol’n 1 (1 pt)

(b) (1 pt)

Natoms = 2;Aplane = aBCC * Sqrt[2] * aBCC;PD[Natoms, Aplane]

3

8 2 r2

3 / (8 * Sqrt[2.] r^2)

0.265165

r2

(c) {100}, {110}, and {111} (see sol’n 1) (1 pt)

3. Aluminum is FCC with an atomic radius of 0.184 nm. (a) Compute the interplanar spacing d111 for the (111) set of planes. (b) When monochromatic x-radiation with a wavelength of 0.1542 nm is used, what is the diffraction angle 2θ for the first-order reflection of the (111) plane?

(a) (1 pt)

dhkl[a_, h_, k_, l_] := a / Sqrt[h^2 + k^2 + l^2];r = 0.184;aFCC = 2 Sqrt[2] r;dAl111 = dhkl[aFCC, 1, 1, 1]

0.300471

0.300 nm

hw3solutions.nb 3

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(b) (2 pt)

theta[n_, λ_, dhkl_] := ArcSin[n * λ / (2 * dhkl)] / Degree;n = 1; (*first order reflection*)λ = 0.1542;theta111 = theta[n, λ, dAl111]

14.8683

diffractionangle = 2 * theta111

29.7365

29.7°

4 hw3solutions.nb

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