Page 130 - Geotechnical Engineering Soil and Foundation Principles and Practice

Soil Minerals
Soil Minerals 125

Figure 6.3
Derivation of the
Bragg Law
relating diffraction
angle to d-spacing.

Bragg, father and son, suggested that even though X-rays actually come from
point sources, namely electrons in crystalline atoms, the rays may be thought of as
reflecting from atomic planes. As a wave front is reflected from adjacent planes,
the interlayer distance imposes an extra path length that is shown in Fig. 6.3.
In order for rays to reinforce, that extra path length must equal an exact number
of wavelengths. This enables writing an equation:
nl ¼ 2d sin ð6:1Þ

where n is a whole number, l is the X-ray wavelength, d is the distance between
identical crystal planes, and is the diffraction angle. This is the Bragg Law.
When a reflection occurs, the interlayer distance d is obtained by measuring the
angle and knowing the X-ray wavelength.

If a reflection is obtained with n ¼ 1, according to eq. (6.1) another reflection
should be obtained with n ¼ 2, or at exactly one-half of the d-spacing. This is
called a second-order reflection. In general as the order increases the reflection
becomes weaker, but that is not always the case. Reflection intensities are modeled
with Fourier analyses for determination of crystal structures.

˚
˚
The unit of measure is the Angstom, or A, which equals 10 8 cm or 10 nm
(nanometers). As a reference, the diameter of an oxygen ion in crystal structures is
˚
2.64 A.
Example 6.1
Mica, the shiny, flakey mineral in mica schist, is closely related to clay minerals and gives
˚
an X-ray diffraction angle 2 ¼ 8.88 with an X-ray wavelength of 1.54 A. At what angle
may one expect a second-order reflection?

Answer: Equation (6.1) solved for the first-order reflection is

ð1Þð1:54Þ¼ 2ðdÞ sin ð4:4 Þ
˚
from which d ¼ 10.0 A. Substituting this value in eq. (6.1) and solving for n ¼ 2 gives
ð2Þð1:54Þ¼ 2ð10:0Þ sin
2 ¼ 17:7

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