MSE 421/521 Structural Characterization
MSE 421/521
Spring 2016
Dr. Rick Ubic
Homework assignment 2
Due 22 February 2016
1. Briefly explain how characteristic x-rays are generated in a diffractometer. Your
answer should explain the difference between K, L and M series x-rays and the
difference between α and β x-rays (e.g., Kα and Kβ x-rays).
5
2. Even the weather can affect the long-term stability of the measured intensity of xrays from a well-stabilized tube because a change in humidity or barometric
pressure changes the absorption of x-rays. What is the percentage change in the
measured intensity of Cr Kα resulting from a 3% drop in pressure over a 12-hour
period (a not uncommon event)? Assume a path length in air of 27 cm and take µ of
air for Cr Kα = 3.48x10-2 cm-1.
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MSE 421/521 Structural Characterization
3. Calculate the possible structure factors (F2) for the following reflection types from
ZnS with the zincblende structure (fcc, space group F 43m ) in terms of fZn, fS, and
hkl. Your answer should have no imaginary parts.
a) h + k + l = 2n+1
b) h + k + l = 2n, n = 2m+1 (i.e., h + k + l = odd multiple of 2)
c) h + k + l = 2n, n = 2m (i.e., h + k + l = even multiple of 2)
In which case would the structure factor, even if not the diffracted intensity, likely be
greatest?
Atomic Positions:
S:
000
½½0
Zn:
¼¼¼ ¾¾¼
½0½
¾¼¾
Note: eix = cos(x) + isin(x) and
0½½
¼¾¾
(A + Beix)2 = (A + Beix)(A + Be-ix)
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MSE 421/521 Structural Characterization
4. Considering an XRD peak at 2θ = 60° with FWHM of B = ∆2θ = 0.5° obtained using
Cu Kα radiation (λ = 1.540562 Å), calculate the magnitude of the strain in the
specimen assuming the case of non-uniform strain. Take the instrumental
broadening component to be 0.15° and neglect crystallite size effects (i.e., assume
crystallites are large).
5
5. Consider the same peak as in Q.4 but assume a strain-free specimen and that the
extra peak broadening is due to the effect of crystallite size, then calculate this size.
5
6. How might the effects of both strain and crystallite size be determined independently
when they are both present in a specimen?
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MSE 421/521 Structural Characterization
7. Use the Nelson-Riley function and the three strongest peaks in the x-ray diffraction
(XRD) pattern for ferrite (below) to calculate the value of the cubic lattice constant.
Show ALL of your work, but you may use a software package to generate the
required graph and curve fit. Submit both your data and graph with your answer.
bcc iron
80
λ = 1.540562 Å
44.68°
70
Intensity
60
50
40
30
82.34°
20
116.38°
10
0
40
60
80
Diffraction Angle (2θ)
According to PDF 6-696:
PeakList
h
1
2
2
2
3
2
k
1
0
1
2
1
2
l
0
0
1
0
0
2
d
2.0268
1.4332
1.1702
1.0134
0.9064
0.8275
I
100
20
30
10
12
6
100
120
44