NAME: GILAND T. CATOR
DATE:
SUBJECT: MEC 163
SCORE:
Introduction to Materials Science and Engineering (ENS 167/MEC136)
CHAPTER 3 QUIZ
B. total number of carbon atoms in the unit cell
1.) A. Coordination Number
8 atoms on each corner with
1/8 of its volume inside the
unit cell = 1 atom
3
1
2
6 atoms on each corner with
1/2 of its volume inside the
unit cell = 3 atom
4
4 atoms
TOTAL = 8 Carbon Atoms
2.) A.) (111)
B. (-1-1-1)
z
z
-x
(111)
y
y
x
(-1-1-1)
C.) (-12-1)
E.) (212)
z
z
(212)
y
y
(-12-1)
x
x
D.) (1-13)
z
(1-13)
y
x
3.) A. (1/2, 0, 1/2); (0, 0,1); (1, 1, 1)
Intercept: (∞, ∞, 1)
Reciprocal: (0, 0, 1)
Reduced: (0, 0, 1)
Enclosure: (0 0 1)
z
Intercept: (1/2, ∞, 1/2)
Reciprocal: (2, 0, 2)
Reduced: (1, 0, 1)
Enclosure: (1 0 1)
Intercept: (1, 1, 1)
Reciprocal: (1, 1, 1)
Reduced: (1, 1, 1)
Enclosure: (1 1 1)
y
x
B.) (0, 0, 1/2); (1, 0, 0); (1/2; 1/4; 0)
z
Intercept: (1, ∞, ∞)
Reciprocal: (1, 0, 0)
Reduced: (1, 0, 0)
Enclosure: (1 0 0)
Intercept: (1/2, 1/4, ∞)
Reciprocal: (2, 4, 0)
Reduced: (1, 2, 0)
Enclosure: (1 2 0)
Intercept: (∞, ∞, 1/2)
Reciprocal: (0, 0, 2)
Reduced: (0, 0, 1)
Enclosure: (0 0 1)
y
x
4.) (a) (1-1-1); (b) (10-2); (c) (1-21); (d) (21-3); (e) (421)
z
y
z
Intercept: (1, -1, -1)
Reciprocal: (1, -1, -1)
Reduced: (1, -1, -1)
Enclosure: (1-1-1)
x
y
Intercept: (1, ∞, -1/2)
Reciprocal: (1, 0, -2)
Reduced: (1, 0, -2)
Enclosure: (10-2)
x
b.) (10-2)
a.) (1-1-1)
z
y
x
c.) (1-21)
Intercept: (1, -1/2, 1)
Reciprocal: (1, -2, 1)
Reduced: (1, -2, 1)
Enclosure: (1-21)
z
d.) (21-3)
y
Intercept: (1/2, 1, -1/3)
Reciprocal: (2, 1, -3)
Reduced: (2, 1, -3)
Enclosure: (21-3)
x
z
x
Intercept: (1/4, 1/2, 1)
Reciprocal: (4, 2, 1)
Reduced: (4, 2, 1)
Enclosure: (421)
y
x
e.) (421)
5. Sketch the following planes in a hexagonal unit cell: (a) (0 1 -1 2); (b) (1 -1 0 1); (c) (2 -1 -1 2);
(d) (-1 -1 2 1).
b.) (1 -1 0 1)
𝑧
Intercept: ∞, 1, -1, 1/2)
Reciprocal: (0, 1, -1, 2)
Reduced: (0, 1, -1, 2)
Enclosure: (0 1 -1 2)
𝑎2
𝑧
𝑎3
𝑎1
Intercept:(1, -1, ∞, 1)
Reciprocal: (1, -1, 0, 1)
Reduced: (1, -1, 0, 1)
Enclosure: (1 -1 0 1)
𝑎2
𝑎3
𝑎1
a.) (0 1 -1 2)
Intercept:(1/2, -1, -1, 1/2)
Reciprocal: (2, -1, -1, 2)
Reduced: (2, -1, -1, 2)
Enclosure: (2-1-1 2)
𝑧
Intercept:(-1, -1, 1/2, 1)
Reciprocal: (-1, -1, 2, 1)
Reduced: (-1, -1, 2, 1)
Enclosure: (-1-1 2 1)
𝑧
𝑎2
𝑎2
𝑎3
𝑎3
𝑎1
𝑎1
c.)
(2-1-12)
d.) (-1-1 2 1)
6. Sketch the following directions in a hexagonal unit cell: (a) [1 1 -2 3]; (b) [-1 0 1 0]; (c) [1 -1 0 0];
(d) [-2 1 1 0].
𝑧
𝑧
0.
[1 1 -2 3]
𝑎2
𝑎2
[-1 0 1 0]
𝑎3
𝑎3
𝑎1
𝑎1
b.) [1 1 -2 3]
a.) [-1 0 1 0]
𝑧
𝑧
[-2 1 1 0]
𝑎2
𝑎3
𝑎2
𝑎3
[1 -1 0 0]
d.) [1 -1 0 0]
𝑎1
𝑎1
c.)
[-2 1 1 0]
7. The figure below shows the x-ray diffraction pattern (Intensity with diffraction angle 2 ) of an
unknown material showing several peaks as tabulated in table. (a) Using the information below,
determine the crystal structure (e.g., BCC, SC or FCC) of the unknown material. Show your
complete solution. (b) Index (i.e., give the h,k, and l indices of) each of these peaks. (c) Determine
the interplanar spacing for each of the peaks. (d) Determine the lattice constant of the unknown
material by using the monochromatic x-ray radiation having a wavelength 0.1542 nm. (e) Obtain the
atomic radius and determine the unknown material by using Table 3.1.
𝑟𝑎𝑡𝑖𝑜 1 =
𝑠𝑖𝑛2 𝜃𝑛
𝑑=
𝑠𝑖𝑛2 𝜃1
𝑛𝜆
2 sin 𝜃
2
𝑟𝑎𝑡𝑖𝑜 2 = 2 ×
𝑠𝑖𝑛 𝜃𝑛
2
𝑠𝑖𝑛 𝜃1
𝑎 = 𝑑√(ℎ2 + 𝑘 2 + 𝑙2 )
𝑚 = 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑜𝑓 𝑟𝑎𝑡𝑖𝑜 2
SC
𝑎 = 2𝑟 ⟹ 𝑟 =
𝑚 = (ℎ2 + 𝑘 2 + 𝑙2 )
PEAK
1
2
3
4
5
6
REMARKS
PEAK/2Θ
40.035
57.95
72.85
86.5
100.09
114.3
M
HKL
2
4
6
8
10
12
Θ(RAD)
0.349371433
0.505709368
0.635736453
0.754855225
0.873450399
0.997456095
SINΘ
0.342307281
0.484428139
0.593770251
0.685183226
0.766549291
0.840093786
MATERIAL
110 Molybdenum
200
211
220
310
222
𝑎
2
SIN^2 Θ
RATIO 1 RATIO 2 M
D
A
R
0.11717427
1
2
2 0.222607 0.314814 0.136318
0.23467062
2.0027
4.0055
4 0.157299 0.314598 0.136225
0.35256311
3.0088
6.0178
6 0.128332 0.314349 0.136117
0.46947605
4.0066
8.0133
8 0.111211 0.314553 0.136205
0.58759782
5.0147
10.029
10 0.099407 0.314351 0.136118
0.70575757
6.0231
12.046
12 0.090704 0.314208 0.136056
BCC
8. Describe the x-ray diffraction pattern that you can obtain from the following samples: (a) powder
samples; (b) single crystal sample; (c) amorphous sample (e.g., SiO2). You may show the expected
XRD pattern to support your answer. Do not forget to cite your sources (for copyright purposes), if
there is any.
A . powder samples
CP powdered sample
Calcium carbonate from ICSD powder
diffraction file database
The CP powdered sample and the calcium
carbonate powder have a clean pattern. There
are almost no humps and have a few pointy
peaks. The peaks can be identified easily. By this,
I can say that powdered samples are the best for
x ray diffraction testing.
B . single crystal sample
Single Crystal samples have a semi rough
pattern. It has few humps and have several
pointy peaks. It has
C . amorphous sample
Amorphous samples have a rough pattern.
Having many humps and have many peaks.
Which is difficult to determine the desired
peaks.
https://www.researchgate.net/figure/X-ray-diffraction-pattern-a-CP-powdered-sample-b-Calciumcarbonate-from-ICSD-powder_fig7_264627056
https://www.researchgate.net/figure/The-X-ray-power-diffraction-pattern-of-calculated-and-singlecrystal-samples-and_fig1_325374666
https://www.researchgate.net/figure/XRD-patterns-obtained-from-Mag-and-MagSiO2nanoparticles-XRD-peaks-with-corresponding_fig9_262804643