Last Name:
First Name:
Student #:
MSE 231H1F Characterization of Materials
2004 / 2005
Quiz # 1
Time: 50 minutes
Date: November 12, 2004
______________________________________________________________________________
  2d sin 
d
N
Fhkl   f n e 2 i ( hun  kvn lwn )
a
h2  k 2  l 2
[UVW ] (hkl )  hU  kV  lW  0
1
e i  e3 i  e5 i  1
e n i  (1) n , where n is any in teger
e2 i  e4 i  e6 i  1
e n i  e  n i , where n is any in teger
b1  2
b 2  2
b3  2
a 2  a3

a1 a 2  a 3
a 3  a1

a1 a 2  a 3
a1  a 2

a1 a 2  a 3


bi a j  2 ij
 ij  0,
for i  j;
 ij  1,
for i  j

c  d  c2 d3  c3d 2 , c3d1  c1d3 , c1d 2  c2 d1
c d  c1d1  c2d2  c3d3
______________________________________________________________________________
Answer all questions. The marks for each part are shown in the left margin. There is a total of
30 marks. You have 50 minutes to complete this quiz.
Page 1 of 4
Last Name:
First Name:
Student #:
The Ni-base superalloys have been designed for high temperature applications, e.g. turbine
blades in jet engines. Ni3Al is an important strengthening phase in these materials. Its structure
is shown below:
a = 0.36 nm
10
1)
Ni3Al has the L12 structure, with 1 Al atom located at 0 0 0 and 3 Ni atoms located at
½ ½ 0, ½ 0 ½, and 0 ½ ½. Derive the diffraction rules for this crystal using the
structure factor.
Fhkl  f Al e2 i (0)  f Ni (e2 i ( h / 2 k / 2)  e2 i ( h / 2l / 2)  e2 i ( k / 2l / 2) )
 f Al  f Ni (e i ( h k )  e i ( h l )  e i ( k l ) )
Case1: (mixed)
Fhkl  f Al  f Ni (1  1  1)  f Al  f Ni
Fhkl 2   f Al  f Ni 
2
Case2: (Unmixed)
Fhkl  f Al  f Ni (1  1  1)  f Al  3 f Ni
Fhkl 2   f Al  3 f Ni 
2
Page 2 of 4
Last Name:
2)
Student #:
Draw a schematic diagram of the powder X-ray diffraction pattern for the Ni3Al phase
over the 2 range of 30 to 55. Label and give the diffraction angle for each peak in this
range (use an X-ray wavelength of 0.154 nm and a lattice parameter of 0.36 nm).
Use the following equations to find values of “d” and “2”
a
d
h2  k 2  l 2
  2d sin 
h2+k2+l2
1
2
3
4
5
Plane
100
110
111
200
210

12.35
17.6
21.7
25.3
28.57
d
0.360
0.255
0.208
0.180
0.161
2
24.70
35.21
43.48
50.65
57.14
1
0.8
Intensity
10
First Name:
0.6
0.4
0.2
0
30
35
40
45
50
55
2 theta (degree)
Page 3 of 4
Last Name:
First Name:
Student #:
5
3)
Draw a schematic diagram of the 9 central spots in the electron diffraction pattern of a
single crystal of Ni3Al oriented with the [010] direction parallel to the incident electron
beam. Label each diffraction spot.
5
4)
TiO2 is polymorphic. One form of TiO2 is the mineral rutile, which is used extensively as
a pigment because of its high refractive index (2.70). Rutile has the tP6 crystal structure.
Derive the reciprocal lattice vectors for rutile using the following direct lattice primitive
vectors:
a1  axˆ; a 2  ayˆ ; a3  czˆ
a2  a3  acxˆ
a1 (a2  a3 )  a 2 c
2 acxˆ 2 xˆ
 b1  2 
ac
a
a3  a1  acyˆ
2 acyˆ 2 yˆ

a 2c
a
2
a1  a2  a zˆ
 b2 
 b3 
2 a 2 zˆ 2 zˆ

a 2c
c
Page 4 of 4