ELECTRONS IN ATOMS - Chemical Fundamentals of Geology and Environmental Geoscience - page 87

equation can be used to investigate the various
wavelengths present in a complex X-ray spectrum from a
geological specimen (Box 6.4); conversely, if the X-ray
spectrum consists of a single known wavelength from an
X-ray tube, the equation provides a tool for investigating
the atomic structure of an unknown crystalline material

by measuring the d -spacings of various sets of atomic
planes. Since the work of the Braggs in the 1930s, X-ray
diffraction has been an essential technique for
determining the atomic structure of minerals, and for the
identification of fine-grained crystalline materials such
as clay minerals (Box 8.2).

(a) (b) (c)

Incident light

Scattered ray

Incident ray

Figure 5.3.1 The principle of the diffraction grating illustrated by a row of atoms. (a) Incident ray, and ray
scattered by atom at an arbitrary angle. (b) Waves scattered from neighbouring atoms such that they are in phase 1
(constructive interference - rays pull together - enhanced ray diffracted in this angular direction). (c) Waves
scattered from neighbouring atoms such that they are exactly out of phase 2 (destructive interference - rays cancel
each other out - zero intensity observed in this direction). Broad bars illustrate relative positions of equivalent
wave-fronts.

1

Meaning the wavefronts of the two rays coincide in space and reinforce each other (Figure 5.3.1b).

2

Meaning the wavefronts of one ray are displaced relative to those of the other (Figure 5.3.1c) and tend to eliminate each other.

reaches our ears. The electron standing wave does
not experience this 'damping' effect, and continues
vibrating.

notes from the same guitar string one can shorten its
length by pressing it down at some intermediate point,
but it is more relevant to consider what other notes are
obtainable on the open (full-length) string. It is possi-
ble to generate a number of higher notes from the open
guitar string by making it adopt alternative forms of
stationary waves called harmonics , as described in the
caption to Figure 5.2.
Because the ends of the string are immobilized, only
a restricted number of harmonics is possible. These all
have wavelengths ( λ 2 , λ 3 , etc.) and frequencies ( v 2 , v 3 ,
etc.) that are related in simple ways to the fundamen-
tal, as shown in Figure 5.2. Whereas in the fundamen-
tal mode the string is stationary only at its ends, the
harmonics share the property of having intermediate
points where the string remains stationary too (that is,

The characteristics of these two types of standing
waves differ in detail, but the underlying principles
are identical.

Harmonics

The stationary wave shown at the top of Figure 5.2 is
the simplest possible mode of vibration, obtained by
plucking the string at its centre. This waveform, the
fundamental , produces the lowest musical note obtain-
able from the string (frequency v 0 ). To produce higher