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HOMEWORK ASSIGNMENT #5
MEMS1054 AY2101
Return by: Next WEDNESDAY!
NAME:
1) A diffraction pattern has been obtained from a powder sample of a material with a cubic
crystal structure using an Xray powder diffractometer instrument with CuK
α
Xradiation,
wavelength
λ
=0.1542nm=1.542Å.
The experimental Xray diffraction pattern exhibits seven maxima for the following angles
θ
:
14.89˚
21.31˚
26.43˚
30.93˚
35.08˚
39.01˚
42.84˚
Determine the Bravais lattice and the corresponding lattice parameters of this material with a
cubic crystal structure.
You can assume that it has a monoatomic (elemental) motif. The
interplanar spacing of a plane (hkl), d
hkl
, for a cubic crystal relates to the lattice parameters as
d
hkl
2
= a
0
2
/(h
2
+ k
2
+ l
2
).
Suggested Solution on Next Page!
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View Full Document Answer to Q1)
(There are different ways to arrive at the correct answers to this question, namely cI lattice with
a
o
=4.243Å=0.4243nm. Suggested solutions are outlined below)
There are three possible Bravais lattices in the cubic crystal system: cP, cI and cF.
With a monoatomic motif this implies that atoms are distributed only on the lattice points of the
respective Bravais lattice, namely, Plattice unit cell with one atom at 000, Ilattice unit cell with
two atoms at 000 and at
½
½
½
, and Flattice unit cell with four atoms at 000 and at
½
½
0 and at
0
½
½
and at
½
0
½
, respectively. The corresponding structure factors for a P, I and Flattice
can then be calculated and are
cP: F
hkl
= fa
, where fa is the respective atomic scattering factor for Xrays;
cI:
F
hkl
= fa(1 + (1)
(h+k+l)
)= 0, if h+k+l=2n+1, here n is an integer,
and
F
hkl
= fa(1 + (1)
(h+k+l)
)= 2fa, if h+k+l=2n, here n is an integer;
cF: F
hkl
= fa(1 + (1)
(h+k)
+ (1)
(k+l)
+ (1)
(h+l)
)= 0, if h, k, l are a mix of even and odd integers,
and
F
hkl
= fa(1 + (1)
(h+k)
+ (1)
(k+l)
+ (1)
(h+l)
)= 4fa, if h, k, l are either all even or all odd integers;
Thus, the seven diffraction maxima for the three possible cubic Bravais lattices, cP, cI and cF,
would have different indices, hkl, as summarized below in the table, providing different
“fingerprints” in the ratio of the angles of the diffraction maxima. The table also includes the d
spacings determined from the experimental Xray diffraction pattern,
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This note was uploaded on 01/07/2011 for the course MEMS Mems1054 taught by Professor Wiezorek during the Fall '10 term at Pittsburgh.
 Fall '10
 Wiezorek

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