ELECTRONS IN ATOMS - Chemical Fundamentals of Geology and Environmental Geoscience - page 86

Box 5.3 Diffraction

Diffraction is a phenomenon that occurs when electromag-
netic waves such as light interact with some regularly
repeated geometric pattern. A familiar example is provided
by the colour fringes seen in light reflected from the sur-
face of a compact disc or DVD. The key requirement for
diffraction to occur is that the repeat distance of the pat-
tern should be similar in magnitude to the wavelength(s) of
the light being diffracted. The digital signal on a CD is
etched in a spiral track with a constant spacing between
successive turns; in a manufactured diffraction grating the
same effect is achieved by engraving straight grooves on
a glass plate or mirror (with a typical spacing of 500-
1000 nm). In each case, the spacing of track or grooves
happens (or is designed) to lie within the wavelength range
of visible radiation (400-760 nm). For the same reason
X-rays ( λ = 10 -2 to 1 nm) are diffracted by the regularly
repeated atomic structure of a crystal (typical repeat dis-
tance 0.1 to 2 nm), which acts as a 3D diffraction
grating.
As a glance at any CD demonstrates, the effect of a
diffraction grating on visible radiation is very similar to
that of a prism: white light is spread out into a contin-
uum of different colours. The physical process involved,
however, is entirely different. Incoming waves are scat-
tered in all directions by individual grooves or tracks.
For rays scattered in certain specific directions, waves
of a particular wavelength emanating from neighbouring

grooves (or atoms in the case of X-ray diffraction) will
reinforce each other and give an enhanced intensity of
that wavelength or colour (see Figure 5.3.1), whereas in
other directions the neighbouring waves are 'out of
phase' and will eliminate each other, reducing the inten-
sity of that wavelength. When many wavelengths are
present in the incoming light beam, the effect is to dis-
perse one wavelength in one specific angular direction
but not in others, so that a spectrum of colours is
formed.
Diffraction finds many uses in spectrometry (Chapter 6)
for separating visible, ultraviolet or X-ray spectra (Box 6.3)
into spectral 'lines' to allow the emissions of individual
elements to be measured separately. The wavelength of
the line can be calculated from the angle of diffraction at
which the wavelength is detected and the spacing of the
grooves or atomic planes. This relationship is expressed
most simply for X-ray diffraction, through the well-known
Bragg equation:

n λ = 2sin d θ (5.3.1)

where n is a whole number (1, 2 …), d is the spacing
between adjacent planes of identical atoms in the crystal
and θ (Greek theta) is the diffraction angle at which the
X-ray wavelength λ will be diffracted with maximum
intensity. Using a crystal with known d -spacing, the Bragg

plucking, where the string will soon resume a station-
ary condition. A guitar string, however, has a restricted
length, and waves reaching the fixed ends of the string
are reflected back on themselves. The string is there-
fore deflected by waves travelling in opposite direc-
tions at the same time, and the overall effect is to set up
a stationary wave : the string vibrates rapidly up and
down within a stationary envelope that gradually con-
tracts with time. This envelope on the guitar string is
just visible to the naked eye. Its form is shown (much
exaggerated) at the top of Figure 5.2. The stationary
wave is the characteristic form adopted by any wave
that is trapped in a restricted region of space (like a
guitar string, or an organ pipe). Standing waves are
fundamental to the generation of musical notes, but
the phenomenon is not restricted to acoustic waves.

When an electron is captured by an atom, by the
electrostatic attraction of the nucleus, the attendant
wave becomes trapped within the volume of the atom.
It responds to confinement in the same way as the gui-
tar string: it becomes a stationary wave, an oscillating
disturbance inside a fixed envelope.
Before developing this analogy further, one must
acknowledge two obvious limitations:

(a) On the guitar we see a wave distributed along a
one-dimensional string, whereas the electron
must be treated as a wave in three-dimensional
space.
(b) The vibration of a plucked string decays away
quite rapidly because its energy is being dissipated
into the surrounding air, through which the sound