Figure references are to the second edition of Modern Physics by Serway, Moses and Moyer (Saunders, 1989).

The Rutherford and Bohr models are based on the concept that the ultimate components of matter are particles. An alternative picture began to emerge in the 1920's when Louis de Broglie suggested in his doctoral thesis (1923) that, just as electromagnetic radiation can have both wave and particle properties, the components of matter may have a wavelike as well as a particle-like character.

Thus, the formulae E = hf and p = h/lambda which apply to a photon might also apply to a material object, such as an electron. If so, the wavelength lambda associated with the electron is not a fixed quantity but will depend on the electron's speed v :

where m is the relativistic mass of the electron but will be essentially equal to the rest mass if v << c , where c is the speed of light in vacuum.

For the energy E of the electron, we might take the total relativistic energy E = m c^2 introduced by Einstein. If so, the frequency associated with the electron would be:

f = E / h = mc^2 / h ........................ (2)

Things start to look strange if we evaluate the product f (lambda) to form what is called the phase velocity of a wave Vp:

Vp = f lambda = mc^2 / mv = c^2 / v ................ (3)

Clearly, the phase velocity of the electron wave is different from the particle velocity v . Moreover, we know that v < c , in which case Eq.(3) requires that Vp > c , so the phase velocity cannot represent the motion of any signal or material object.

However, things start to make more sense when we reconsider the Bohr model of the hydrogen atom and apply Eq.(1). Bohr's postulate was that the angular momentum of the electron is in a stationary orbit is given by mvr = n(h/2(pi)), in which case Eq.(1) gives:

n lambda = h / (h/2 (pi)r) = 2 (pi)r ........................ (4)

A plausible interpretation of Eq.(4) is that the circumference 2 (pi)r of the electron orbit contains an integral number of wavelengths; in other words, the electron in a stationary state forms a standing wave, somewhat analagous to a wave on a vibrating string; see Fig. 4.2. (This is an example of a periodic boundary condition: the phase of the wave would not change after one complete revolution or or any integral number n of revolutions).

The real proof of de Broglie's concepts came from experimental data on electron diffraction. Louis de Broglie had suggested that a stream of electrons passing through a narrow aperture should exhibit measurable diffraction effects and Einstein predicted (in 1925) that a beam of atoms should behave similarly. However, the first diffraction evidence was obtained from observations of the interaction of electron beam with a crystal, which contains atoms arranged with regular spacings and can therefore act as a diffraction grating for an electron beam with suitable wavelength.

This experiment was first done in 1927 by L.A. Germer and C.J. Davisson, two employees of Bell Laboratories (in New York City) who were trying to get information about the arrangement of atoms at the surface of a nickel crystal. Their apparatus is shown schematically in Fig. 4.4.

Electrons emitted into a vacuum by thermionic emission from a heated filament were accelerated by applying a modest voltage V to an electrode containing a hole, from which emerged a beam of electrons of kinetic energy K = V e , where e is the magnitude of the electron charge. For example: if V = 54 volts, K = 54 eV = mv^2 / 2 , giving v = 4.36 x 10^6 m/s (since v<<c, our use of the non-relativistic expression for kinetic energy is justified at this low accelerating voltage). Use of Eq.(1) then gives lambda = 0.165 nm, which is comparable to the atomic spacing in many solids, so diffraction effects might be anticipated. The electron beam was allowed to strike the surface of a nickel crystal and the number of scattered electrons per second (the 'intensity') was recorded as a function of the angle (phi) between the incident and scattered beams, using a simple detector which could be moved around the point of impact. An example of Davisson and Germer's results is shown in Fig. 4.5, which shows that scattered intensity reached maximum value at a scattering angle of (phi)max = 50 deg. From the value of (phi)max, the electron wavelength can be calculated, using the following argument.

Because low-energy electrons are scattered strongly by atoms (because of electrostatic interaction with the atomic electrons and nuclei), the incident beam does not penetrate much beyond the first monolayer of atoms. Therefore the diffraction effect takes place at the surface and is similar to diffraction of visible photons from a diffraction grating, rather than the three-dimensional kind of diffraction which is responsible for Bragg-reflection of x-rays. Consequently, we are justified in considering only the scattering of incident electrons by surface atoms, as illustrated in Fig. 4.6. If the incident beam arrives perpendicular to the surface, the difference in path length for electrons scattered by two adjacent atoms (separation d , measured parallel to the surface) is AB = d sin(phi) , and must be equal to an integral number of wavelengths if the scattered waves are to arrive in phase. In other words, constructive interference occurs if:

n lambda = d sin (phi) ........................ (5)

Using (phi) = 50 deg, n = 1 and d = 0.215 nm (known from x-ray diffraction measurements), Eq.(5) gives lambda = 0.165 nm for 54eV electrons, which is

just the value we calculated earlier using Eq.(1). Therefore the Davisson and Germer experiment provides convincing proof of De Broglie's assertions.

Higher-order diffracted beams would correspond to higher values of the integer n in Eq.(5). For 54eV electrons, n = 2 implies sin(phi) = 1.53 which indicates that second-order diffraction is impossible, but with electrons of somewhat higher energy (travelling at higher speeds and having shorter wavelength) multiple orders of diffraction can occur.

It is possible to apply Eq.(5) in reverse and calculate the spacing of surface atoms, knowing the accelerating voltage of the incident electrons and therefore their wavelength, from Eq.(1). This is the basis of low-energy electron diffraction (LEED), an modern analytical technique used in surface-science and layer-growth experiments. In fact, the symmetry of arrangement of the surface atoms can be determined by examining the symmetry of the diffraction pattern, recorded on a two-dimensional detector such as a phosphor screen. usually the scattered electrons are further accelerated by a system of concentric grids in order to generate sufficient light at the screen. This technique works only in ultrahigh vacuum (UHV), since at higher pressures the surface of a solid is covered by a layer of condensed water or hydrocarbons, so the diffraction of low-energy electrons would take place mainly within this adsorbed layer.

If electrons are accelerated to much higher energies (typically 20 keV - 200 keV), they penetrate many atomic layers and may pass completely through a specimen, provided the latter is thin (< 1 micrometer) in the incident direction. In doing so, however, some of them will be diffracted away from the central (undeviated) beam. Such diffraction of electrons transmitted through a thin metal foil was first observed by G.P. Thomson (son of J.J. Thomson) in 1927, only a few months after the classic experiment of Davisson and Germer experiment and shared the 1937 Nobel prize with C.J. Davisson for experimental confirmation of the de Broglie theory.

The condition for diffraction of transmitted electrons is that they fulfill the Bragg condition for three-dimensional diffraction (Bragg reflection from atomic planes) as discussed for the case of x-rays, namely:

n lambda = 2 d sin (theta) .................. (6)

Note that d in Eq.(6) refers to the separation between atomic planes within the solid and (theta) is the angle between an atomic plane and the incident beam, so that the angle of deviation (after Bragg reflection) is 2(theta) , as in the x-ray case. Transmission diffraction is used for identifying the crystalline structure of a thin specimen and is usually carried out in an electron microscope, where the incident-electron beam can be focussed to a small (<1micrometer) diameter to achieve microdiffraction from small region of the specimen.

In order to calculate the wavelength of high-energy electrons, it is necessary to take into account the relativistic increase in electron mass and use equations derived from the Special Theory of Relativity. For example, 100keV electrons (accelerated through a potential difference of 10^5 volt) are travelling at a speed of v = 0.55c and have a relativistic mass which a factor of gamma = 1.20 higher than the rest mass. Their wavelength is 0.0037 nm (i.e. 3.7 pm) which is considerably less than a typical spacing between atomic planes, so that the deflection angles 2(theta) are only a few degrees.

Other particles (such as neutrons and alpha-particles) can be diffracted, showing that they also have a wavelike nature. So called thermal neutrons, produced with high energies in the core of a nuclear reactor but slowed down by multiple collisions in a graphite moderator, have a kinetic energy of approximately (3/2)kT = 0.0388 eV (for T=300K) corresponding to a speed of 2736 m/s. From Eq.(1), their wavelength is 0.146 nm, close to atomic dimensions, so thermal neutrons are used in diffraction experiments to investigate the structure of solids.

In fact, it is possible to assign a wavelength to any material object, such as a single atom or a whole assembly of atoms, provided it is travelling at a known speed relative to the observer. But according to Eq.(1) the wavelength becomes very small as the mass of the object becomes large; for example, a 74kg person running at a speed of 5 m/s would have a de Broglie wavelength of 1.8 x 10^-36 m, far below their physical dimensions. Therefore only objects of atomic or subatomic dimensions display observable wavelike properties such as diffraction; macroscopic objects exhibit particle-like behaviour and are well described by classical physics. This can be taken as a further example of the Correspondence Principle, the predictions of quantum physics approximating to those of classical physics under the original conditions of investigation.