Example 21.6
Monochromatic X rays having a wavelength of 10.4 Å are preferentially dif-
fracted by a crystal at an angle of 25.5°.
a.Assuming that this is the first-order diffraction, what is the dspacing be-
tween the crystalline planes?
b.At what angle would the second-order diffraction be found?Solution
a.For n1 (that is, first-order diffraction), we can set up Bragg’s law as
10.4 Å 2 dsin (25.5°)
and the only unknown in this equation is d, the spacing between crystalline
planes. Solving for d:d
2si1
n0.
(
4
25
Å
.5°)
d12.1 Å
Notice how the units work out: the only unit is a unit of distance, Å.
b.Knowing the value for d, we can find the angle for the second-order dif-
fraction of these X rays. In this case,n2, and we can set up Bragg’s law as
2(10.4 Å) 2(12.1 Å) sin
Now the angle is the only unknown in the expression. Solving for sin :sin
^2
2(
(
1
1
0
2
.
.
4
1
Å
Å
)
)
sin
0.859
59.3°Notice that mathematically, only so many orders of diffraction may be pos-
sible for any given spacing of crystal planes and a given X-ray wavelength. In
the previous example, if you were trying to determine the angle of the third-
order diffraction, you would get to the expression
sin
3
2
(
(
1
1
0
2
.
.
4
1
Å
Å
)
)
where the 3 in the numerator represents the order n. Evaluating this fraction,
we get
sin
1.289Sine functions can’t get above a value of 1, so having a sine of 1.289 is physi-
cally impossible. This shows that the given plane of atoms can diffract 10.4-Å
X rays only to the first and second order.
Although Bragg’s law is the fundamental basis of experimental crystallogra-
phy, its simplicity is potentially misleading. For the simplest of cubic lattices
(like many of the solid noble gases), only one type of atom can make a plane
that refracts X rays. Consider a molecular crystalline solid like water, H 2 O: not
only is the crystal more complicated because it’s a molecular solid, but each
atom in the molecule can be used to define a regular matrix of atoms that can
act as a refracting plane. The diffraction of X rays by any random compound
21.5 Determination of Crystal Structures 743