soon reinforced by the experimental discovery that a
beam of electrons can be diffract ed by a crystal lattice.
Diffraction is a phenomenon peculiar to waves,
which arises when any kind of wave encounters
a periodic (regularly repeated) structure whose
repeat-distance is similar in magnitude to the wave-
length of the wave concerned (Box 5.3). In the case of a
crystal, this periodic structure is the regular three-
dimensional array of its component atoms. The waves
scattered from different parts of this structure inter-
fere with each other (Figure 5.3.1b and c), generating a
characteristic spatial pattern of high and low beam
intensities called a diffraction pattern , which can be
recorded by a roving detector or on a photographic
plate. (Mineralogists and crystallographers use diffrac-
tion, either of X-rays - electromagnetic waves - or elec-
trons, to investigate the internal structure of minerals
and other crystalline materials.) Classical physics
offers no explanation for the diffraction of electrons if
they behave solely as particles, and the phenomenon
provides firm evidence that the movements of atomic
particles like electrons are determined by an underly-
ing wave-like property.
If an electron is to be considered as a wave phenom-
enon, how can its position and motion be specified?
Figure 5.1 depicts a moving electron as a wave-pulse,
travelling in this case along the x -axis but frozen for
our inspection at some instant t , like racehorses at a
photo-finish. The electron's position at time t cannot be
defined precisely, because the wave-pulse extends
smoothly over a range of x -values. The range Δ x repre-
sents a fundamental interval of uncertainty, within
which we cannot pinpoint exactly where the electron
lies, although we know it lurks there somewhere. This
uncertainty interval, which exceeds the physical size of
the electron, is not the result of any experimental error,
but must be seen as a fundamental physical limitation
to the concept of 'position' where waves are concerned.
Heisenberg, in 1927, was the first to recognize this. He
expressed it in a quantitative form called the Uncertainty
Principle , but the details need not concern us here.
In examining the atom, therefore, we cannot look
upon electrons as minute planets, orbiting the nucleus
with precisely determined coordinates and motion.
The wave nature of the electron rules out this precise
classical image, introducing in its place a view of the
electron in which position is subject to a degree of
uncertainty. Although - as we shall see - electrons
occupy well-defined spatial domains around the
Figure 5.1 A wave train of restricted length, loosely
illustrating the wave behaviour of an electron moving in free
space. The horizontal axis is the x -direction along which the
wave is travelling. The vertical axis represents the physical
property (as yet unidentified) whose oscillation transmits
the wave. It is called the wave function , and is symbolized
by the Greek letter ψ (psi).
A familiar example of such a short wave 'pulse' is the
wave observed when a stone is dropped into a still pond:
viewed in profile, the pond wave (somewhat amplified)
would look similar to this figure. In this case the wave
function would be the vertical displacement of the pond
surface from its equilibrium (flat) state.
nucleus, the precise manner in which each electron
patrols its own territory is concealed from us by funda-
mental limits of physical perception, which must be
embraced in any model of the atom that we develop.
The mathematical certainty enshrined in the laws of
Newtonian mechanics thus gives way, on the scale of
the atom, to a statistical interpretation of particle
mechanics based on probability.
An electron that belongs to an atom is confined to a
small volume of space close to the nucleus. How does
a wave behave in these circumstances? To answer this
question it is helpful to look at waves of a more famil-
iar kind: those on a vibrating string.
When a guitar string is plucked, the lateral displace-
ment y introduced by the player's finger initiates a
rapid transverse vibration of the string, which is com-
municated as waves moving away towards each end
of the string. On a string of infinite length, each of these
disturbances would continue to propagate outwards
indefinitely (like ripples on a pond). The waves carry
the energy of the disturbance away from the point of