3.70
The diffraction peaks labeled in Figure 3.33 correspond to the reflection rules for an FCC
metal (
h
,
k
,
l
unmixed, as shown in Table 3.4). What would be the
hkl
indices for the three
lowest diffractionangle peaks for a BCC metal?
SOLUTION
First note an error in the problem statement in the text, which is corrected above.
Diffraction in
dices are indicated by
hkl
with NO adornments, that is, no enclosing parentheses or brackets, by
the convention of the International Union of Crystallography, the professional society charged
with such standards (see
http://it.iucr.org/Ab/ch1o1v0001/ch1o1.pdf
).
A typical "diffraction pattern" is shown in Figure 333, which shows diffraction "peaks" or "re
flections" occurring at different angles (
θ
) from a crystal, in this case, aluminum.
The positions
of the peaks, denoted by an angle measurement on the
x
axis, are established by the symmetry of
the crystalline lattice associated with the sample.
Crystals with different lattices (say, FCC
vs
BCC) produce different diffraction patterns, and diffraction is the experimental method by which
such structural distinctions are made.
The governing mathematical expression relating diffraction angles to the families of (
hkl
) planes
responsible for diffraction is known as Bragg’s Law.
It appears as equation (3.5) in the text on p.
93,
where
λ
is the wavelength of the radiation used for the diffraction experiment.
Furthermore, the
d
spacings are related to the lattice constant (
a
) of a cubic lattice (FCC or BCC) by equation
(3.6) of the text, found on p. 94,
Combining these two equations generates an expression relating the diffraction angle
θ
to the
hkl
indices of the diffraction peaks,
which indicates that the peaks occurring at the smallest values of
θ
are those having the smallest
values of
h
2
+
k
2
+
l
2
.
n
λ
= 2
d
hkl
sin
θ
d
hkl
=
a
√
h
2
+
k
2
+
l
2
sin
θ
=
n
λ
2
a
h
2
+
k
2
+
l
2
Problem 3.70
J. F. Shackelford,
Introduction to Materials Science for Engineers
, 7
th
Edition, Prentice Hall, New Jersey (2009)
Problem 3.70 Solution
Professor R. Gronsky
page 1 of 2
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But there is one more important consideration in predicting diffraction.
Sometimes diffraction
results in
destructive
interference, rather than the
constructive
interference.
Under conditions of
destructive interference, no peaks are seen.
This phenomenon is described by the socalled “re
flection rules” listed in Table 3.4 (also known as the “structure factor rules”) at the bottom of p.
94.
The rules predict the values of
h
,
k
, and
l
for forbidden reflections (diffraction not observed)
and allowed reflections (diffraction is observed).
Different crystal structures produce different
diffraction patterns, and peaks are indexed according to these rules.
So, because the sample is a
BCC crystal, it will show diffraction peaks from families of (
hkl
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 Fall '08
 GRONSKY
 Materials Science, Diffraction, Professor R. Gronsky, J. F. Shackelford

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