textbooks.) Matter is usually relatively transparent to X rays to different degrees;
this characteristic is what makes X rays useful for medical purposes. But in or
around 1912, the German physicist Max von Laue (Figure 21.14) reasoned that
in crystals, the rows of atoms (or ions or molecules) were spaced at distances
that corresponded to the wavelength of X rays, so crystals should diffractXrays.
In 1912, experiments with crystals of copper sulfate and, later, zinc sulfide con-
firmed this idea.
This result was furthered by von Laue and also by the father-and-son team
of William Henry Bragg (Figure 21.15a) and William Lawrence Bragg (Figure
21.15b). In particular, in 1915 the Braggs worked out a simple relationship be-
tween the wavelength, , of monochromatic X rays that are preferentially dif-
fracted by a crystal lattice, the distance,d(sometimes called the d spacing), be-
tween the planes that the crystal lattice makes, and the angle, , between the
crystal planes and the incoming monochromatic X rays. The relationship is
n 2 dsin (21.5)
where nis an integer (0, 1, 2, 3, and so on). Equation 21.5 is known as Bragg’s
law of diffraction.For their work on X rays and crystals, von Laue won the 1914
Nobel Prize in physics and the two Braggs won the next year’s prize. (The
younger Bragg was only 25 at the time and still holds the record for the
youngest Nobel laureate. The Braggs’ award-winning work had just been pub-
lished earlier that year.)
Ifnequals 0, the process is equivalent to reflection of light at a surface,
which is not a new physical phenomenon; it will not be considered further. If
nequals 1, then X rays will be preferentially diffracted only if their wavelength
is equal to 2dtimes the sine of the angle the X rays make with the plane of the
crystalline species:
2 dsin (21.6)
The physical reason for this preferential diffraction is shown in Figure 21.16.
If monochromatic X rays were approaching a crystal (which again is normally
very transparent to X-ray radiation) at some random angle (Figure 21.16a),
then radiation reflected from multiple planes of crystalline species would
combine destructivelyand no coherent, measurable diffracted X rays would be
produced.
On the other hand, if the angle is just right, then the X rays reflected from
multiple planes would constructivelyinterfere, and the X ray would be dif-
fracted, as shown in Figure 21.16b. A detector at just the right angle would
measure X rays coming from the crystal as the crystalline lattice diffracts the
radiation. The 2 is present in equations 21.5 and 21.6 because the radiation
must travel twice the distance between the diffracting layers. The mathemati-
cal relationship between and can actually be derived from geometric prin-
ciples (which is what the Braggs did). In Figure 21.16, because there is only one
wavelength of light between adjacent layers of crystal,nequals 1 and this dif-
fraction is called first-order diffraction.(Unless otherwise stated, we will assume
that a diffraction of X rays from crystal is a first-order diffraction.)*
The more general form of Bragg’s law, equation 21.5, allows for different,
but integral,numbers of wavelengths of X rays to satisfy this constructive21.5 Determination of Crystal Structures 741*Destructive interference can also occur between reflections off different planes that are
a distance of /2 apart. For example, in face-centered cubic crystals, some diffractions that
satisfy Bragg’s law are not detected because of these destructive interferences.Figure 21.15 William Henry Bragg (1862–
1942) and William Lawrence Bragg (1890–1971)
were the father-and-son team that took von Laue’s
idea and expressed it in a simple mathematical
form so it could be applied to any crystalline solid.
Their Nobel Prize in physics followed the year
after von Laue’s. At an age of 25 years, William
Lawrence Bragg is the youngest person to be
named a Nobel laureate.
(b)
(a)