1

MSE 230 Assignment 3 Solutions Fall 2013 Problem 1.

[1 2 1]z

x

y

(1 1 1)

(2 0 2)

[2 0 1] (1 2 1)

Origin

[1 0 1]

2

2.* a) Applying the cross to the two directions gives you a new direction that is perpendicular to both directions: [3 1 2] × [11 1 ] → [1 5 4] For cubic structures, 1 5 4[ ] is perpendicular to 1 5 4( ) . I used the drawing below to illustrate this point in class, just change the directions to those used in this problem and the same thing applies.

You can check your answer using the dot product. If the direction lies with a given plane the dot product is equal to zero.

[3 1 2] • [1 5 4] = - 3 - 5 + 8 = 0

[1 1 1 ] • [1 5 4] = -1+ 5 - 4 = 0

b) In this case we use the definition of the dot product: u•v = u v cosθ where theta is the angle between vectors u and v. [3 1 2] • [11 1 ] = 0 = 14 3 cosθ cosθ = 0 θ = 90°

[111]111( )

3

3. The (011) is shown in the cube below. Note that the plane contains cube edges <100>, cube face diagonals <110> and cube body diagonals <111>. We can determine the specific directions within each family of directions from the drawing. We have chose the correct directions if the dot product of the direction with [011] equals zero (e.g., the directions lies within the (011) plane.

4. To get started, recall that the interplanar spacing (d) gets larger as the diffraction angle gets smaller. So asking for reflections with a d-spacing larger then 0.15 nm down the number of peaks we are looking for. Further, specifying a FCC structure means that the Miller index values for h, k and l are either all odd or all even. With this in mind, the solution is below.

dhkl =a

h2 + k2 + l2 0.15nm <

0.45 nmh2 + k2 + l2

→ h2 + k2 + l2 ≤ 3

Plane h2 + k2 + l2 FCC/h,k,l = d(nm) 2θ from nλ = 2dsin θ all odd, all even? 100 1 No 110 2 No 111 3 Yes 0.26 34.5 200 2 Yes 0.225 40.0 210 5 No 211 6 No 220 8 Yes 0.16 57.9 300 9 No 221 too large, >3

x

y

z [1 0 0][1 0 0]

[0 1 1][0 1 1]

[1 1 1][1 1 1][1 1 1][1 1 1]

4

5. a) Thermal expansion during heating increases the d-spacing between planes of atoms.

Referring to Bragg’s law, λ=2dsinθ, increasing d decreases sinθ. This will cause the x-ray peaks to shift to slightly lower values of 2θ.

b) When T>Tm iron melts to form a liquid (Tm(iron)=1538°C). Liquids are amorphous

(no long range order), and long range ordering is required to form sharp peaks.

I

2θ10 20 30 40 50 60

(200)(111) (220)