Several of your laboratory experiments show that light can exhibit the properties of either waves or particles. The wave nature is evident in the diffraction of light by a ruled grating and in the interferometer experiments. In these experiments, wavelength, phase angle, and coherence length of wave trains were investigated--all features of wave phenomena. However, the photoelectric effect cannot be explained by a wave picture of radiation. It requires a model in which light consists of discrete bundles or quanta of energy called photons. These photons behave like particles. There are other examples illustrating this dual nature of light. Generally, those experiments involving propagation of radiation, e.g. interference or diffraction, are best described by waves. Those phenomena concerned with the interaction of radiation with matter, such as absorption or scattering, are more readily explained by a particle model. Some connection between these models can be derived by using the principal of the equivalence of mass and energy introduced by Einstein in 1905 in his special theory of relativity, namely,
E= mc2
In this equation E is the total energy of a body, m is its mass, and c is the velocity of light. From the photoelectric experiment we learned that light may be considered to consist of particles called photons whose energy is
E = hf
where f is the frequency of light and h is Planck's constant. We may equate these two energies and obtain:
mc2 = hf, or
mc = hf/c = h/
where is the wavelength of the light.
Now mc is the momentum of a
photon traveling with velocity c and m is its equivalent mass. Thus the
momentum of radiation may be expressed either in terms of the wave
characteristic or by the mass and velocity of the equivalent
particle.
This dual situation with respect to radiation led de Broglie in 1925 to suggest that since nature is likely to be symmetrical, a similar duality should exist for those entities which had previously been regarded as particles. Thus, a particle such as an electron with mass m, traveling with velocity v, has a momentum mv. De Broglie stated that this particle could also behave as a wave and its momentum should equal the wave momentum, i.e.
mv = h/ , or = h/mv
It was now a question of verifying this hypothesis experimentally. If an electron is accelerated through a potential difference V, it gains a kinetic energy
where e is the electron charge and m is its mass. Substituting this value for v in the de Broglie expression for the wavelength gives
Thus it should be fairly simple to produce a beam of electrons of a known wavelength by accelerating them in a voltage V. This beam could then be used in experiments designed to demonstrate wave properties, e.g. interference or diffraction. One might try to diffract the beam of electrons from a grating. However, the spacings between the rulings in man-made gratings are of the order of several hundred nm. From Equ. (1), we find that even with an accelerating voltage as low as 100 V, the electron wavelength is only 0.12 nm. As we will see shortly, such a large difference between the grating spacing and the electron wavelength would result in an immeasurably small diffraction angle. It was recognized, however, that the spacings between atoms in a crystal were of the order of a few tenths of a nm. Thus, it might be feasible to use the parallel rows of atoms in a crystal as the "diffraction grating" for an electron beam. This possibility seemed particularly promising since it had been found that x-rays could be diffracted by crystals, and x-ray wavelengths are of the order of the wavelengths of 100 eV electrons.
Figure 1 shows some of the possible arrangements of atoms in a cubic pattern. (a) is the simple cubic form. When an atom is placed in the center of the simple cube, we get (b), the body-centered-cubic form.
Figure 1: Three cubic arrangements of atoms in a crystal. (a) simple cubic, (b) body centered-cubic, (c) face-centered cubic
Figure 2: Interplanar spacings, d, of different families of parallel planes in a cubic array of atoms.
Figure 3: Scattering of waves from a plane of atoms. Path difference for waves from adjacent atoms.
with a row of atoms in the
surface plane waves of atoms; 
is the wave front. The scattered wave makes an angle 
with the atom row;
its wavefront is 
.
Constructive interference will occur for the rays scattered from neighboring
atoms if they are in phase -- if the difference in path length is a whole
number of wavelengths. The difference in path length is 
.
Therefore 
, where m is an integer. Another condition is
that rays scattered from
successive planes also meet in phase for constructive interference. Figure 4
shows the construction for determining this condition.
Figure 4: Path difference for waves scattered from successive planes of atoms.
The difference in path length for rays traveling from planes 1 and 2 is seen to
be 
,
the extra distance traveled by the ray scattered from plane 2. This path
difference must again be an integral number of wavelengths. Therefore
These conditions can be satisfied simultaneously if = . In that
case m = 0 for the first condition and
for the second condition.
This relation was developed by Bragg in 1912 to explain the diffraction of
x-rays from crystals. n is the order of the diffraction spectrum. Thus the
conditions for constructive interference are that the incident and scattered
beams make equal angles and that the
relation 
must be obeyed where d is the spacing between parallel adjacent planes of
atoms.
In 1927, Davisson and Germer at the Bell Telephone Laboratories, investigated the scattering of a beam of electrons from a nickel crystal. Figure 5 shows, schematically, the essentials of their apparatus.
Figure 5: Experimental arrangement for the Davisson-Germer electron diffraction experiment.
Electrons from the heated filament F, were accelerated through a potential
difference of order 100V to a plate P with a small diameter hole. A narrow beam
of electrons emerged from the opening in P and was incident normally on the
face of the nickel crystal C. The electrode E was connected to a sensitive
galvanometer and measured the intensity of the electrons scattered by the
nickel crystal at various angles 
. Some of these results are shown in Fig.
6. These are polar plots of the beam intensity as a function of the angle
for various accelerating voltages which correspond to the wavelengths
indicated. In each plot, a line drawn from the origin to any point on the curve
makes the angle ; the length of a line is proportional to the electron
beam intensity at that angle .
Figure 6: Polar plot of Davisson and Germer's data for the scattered electron beam intensity as a function of scattering angle for different incident electron energies.
As the voltage was increased from 40 to 68 volts, a characteristic peak gradually appears and then disappears. It reaches a maximum for electrons with an energy of 54 eV (.167nm) at an angle of 50o . Davisson and Germer concluded that this peak was due to Bragg reflection from a set of regularly spaced atomic planes within the crystal as shown in Fig. 7.
Figure 7: Illustrating the Bragg condition for electron waves scattered from planes of atoms in the nickel crystal used by Davisson and Germer.
The angle 
is measured in the experiment. The diffracting planes must be normal to the
bisector of . Therefore,
, and the angle of incidence,
The spacing between the planes involved in the diffraction is d. From x-ray measurements, the spacing D between the surface atoms was known to be 0.215nm.
Therefore,d = D sin 
= 0.215 sin (50o /2) = 0.215 sin (25o )
= 90o - 50o/2 =
65o
Then, the Bragg condition for the first order reinforcement is
The calculation from the de Broglie relation is
This close agreement convinced Davisson and Germer that they had observed the
diffraction of electron waves from the nickel crystal. In this experiment and
in a number of subsequent ones, Davisson and Germer observed that while
agreement was close between the observed and calculated diffraction data, there
was usually a small discrepancy that was larger for the low energy (longer
wavelength) electron beams. They were able to show that this discrepancy was
due to the fact that the electrons were refracted as they entered the crystal.
The index of refraction, 
,
of a crystal was greater than 1, due to the fact that the electrons gained
energy on entering the crystal. For those low energy beams where
is significant, the Bragg relation is modified to
Thus far, only single crystals have been considered. Most materials are polycrystalline. They are composed of a large number of small crystallites (single crystals) that are randomly oriented. An electron diffraction sample may be a polycrystalline thin film, thin enough so that the diffracted electrons can be transmitted through the film. The experimental arrangement shown in Fig. 8 was used by Thomson in 1927 to study the transmission of electrons through a thin film C. The transmitted electrons struck the photographic plate P as shown. The pattern recorded on the film was a series of concentric rings. This pattern arises from the polycrystalline nature of the film.
Figure 8: The experimental arrangement used by Thomson for his transmission electron diffraction research.
Figure 9(a) shows a beam of electrons of wavelength traveling from the
left and striking a plane of atoms in a crystallite. If this plane makes the
angle with the incident beam such that = 2d sin , where
d is the spacing of successive atomic planes, the beam will be diffracted into
the angle with respect to the atom plane (or the angle = 2
that the diffracted beam makes with the incident beam).
Figure 9: Showing how the randomly oriented crystallites in a polycrystalline film scatter into a cone when the Bragg condition is fulfilled by planes of atoms disposed symmetrically about the incident beam.
Now there are many randomly oriented crystallites in this film. Thus we may
expect that there will be crystallites in which this diffracting plane makes
the same angle with the beam direction but rotated around the beam in a
cone as shown in Fig. 9(b). The diffracted beams from this plane from all the
crystallites in the sample will fall on a circle whose diameter may be
determined from the cone angle and the distance from the sample to the
film or other detector. The Bragg condition becomes
n = 2d sin /2
For small, sin /2 = /2, giving
n = 2d /2 = d
Thus, in 1927 the wave nature of electrons was verified by reflection and transmission diffraction experiments. For this work Germer and Thomson were awarded the Nobel prize in 1937. De Broglie received the Nobel prize in 1929 for his basic insight on the wave nature of matter.