||Atoms or ions in a crystal are arranged
in regular arrays as typified by the simple lattice structure shown in
Fig. 1-2. A two-dimensional display of a simple
crystal lattice showing an incoming and a reflected beam of X-rays.
This structure is repeated in the third dimension. X-rays are a form
of light with a very short wavelength. Since the spacings between the planes
of atoms in a crystal, denoted by d, are of the same order of magnitude
of the wavelength of X-rays (~10-8 cm), a beam of X-rays
reflected from the crystal will exhibit interference effects. That is,
the layers of atoms in the crystal act as a diffraction grating. The reflected
beam of X-rays will be in phase if the difference in the path length followed
by waves which strike succeeding layers in the crystal is an integral number
of wavelengths. When this occurs the reflected X-rays reinforce one another
and produce a beam of high intensity for that particular glancing angle
For some other values of the angle q, the difference
in path lengths will not be equal to an integral number of wavelengths.
The reflected waves will then be out of phase and the resulting interference
will greatly decrease the intensity of the reflected beam. The difference
in path length traversed by waves reflected by adjacent layers is 2dsinq
as indicated in the diagram. Therefore,
|which states that the
reflected beam will be intense at those angles for which the difference
in path length is equal to a whole number of wavelengths. Thus a diffraction
pattern is produced, the intensity of the reflected X-ray beam varying
with the glancing angle q.
||By using X-rays with a known wavelength
and observing the angles of maximum intensity for the reflected beam, the
spacings between the atoms in a crystal, the quantity d in equation
may be determined. For example, X-rays with a wavelength of 1.5420 C
produce an intense first-order (n = 1 in equation (6))
reflection at an angle of 21.01° when
scattered from a crystal of nickel. Determine the spacings between the
planes of nickel atoms.
||Remarkably, electrons exhibit the same kind of diffraction pattern
as do X-rays when reflected from a crystal; this provides a verification
of de Broglie's prediction. The experiment performed by Davisson and Germer
employed low energy electrons which do not penetrate the crystal.
(High energy electrons do.) In their experiment the diffraction of
the electrons was caused by the nickel atoms in the surface of the crystal.
A beam of electrons with an energy of 54 ev was directed at right angles
to a surface of a nickel crystal with d = 2.15 C.
Many electrons are reflected back, but an intense sharp reflected beam
was observed at an angle of 50° with respect to the incident beam.
Fig. 1-3. The classic experiment of Davisson and
Germer: the scattering of low energy electrons from the surface of a nickel
As indicated in Fig. 1-3
the condition for reinforcement using a plane reflection grating is
||using equation (7) with n
= 1 for the intense first-order peak. Observed at 50°, calculate the
wavelength of the electrons. Compare this experimental value for
l with that calculated using de Broglie's relationship.
||The momentum mu may be calculated
from the kinetic energy of the electrons using equation
(5) in the website.
||Even neutrons and atoms will exhibit diffraction effects when scattered
from a crystal. In 1994 Professor Brockhouse of McMaster University shared
the Nobel prize in physics with Professor Shull of MIT for their work on
the scattering of neutrons by solids and liquids. Professor Brockhouse
demonstrated how the inelastic scattering of neutrons can be used to gain
information about the motions of atoms in solids and liquids. Calculate
the velocity of neutrons which will produce a first-order reflection for
= 30° for a crystal with d = 1.5
10-8 cm. Neutrons penetrate a crystal and
hence equation (6) should be used to
determine l. The mass of the neutron is 1.66
||The neutrons obtained from an atomic reactor have high velocities.
They may be slowed down by allowing them to come into thermal equilibrium
with a cold material. This is usually done by passing them through a block
of carbon. The kinetic theory relationship between average kinetic energy
and the absolute temperature,
may be applied to the neutrons. Calculate the temperature of the carbon
block which will produce an abundant supply of neutrons with velocities
in the range required for the experiment described in (c).