11 Electron and Neutron Diffraction

Prof. P. N. Kotru

11.1 Electron Diffraction

In 1924, the French Physicist, Louis de Broglie suggested that matter might exhibit wavelike properties under some suitable conditions. It means that matter, like radiation, has a dual nature. According to him a moving particle of matter has always got a wave associated with it. The wavelength of the associated wave is given by an equation, popularly known as de Broglie equation

λ  = h/ m.v = h/p

This is an expression for the de Broglie wavelength, which translated in physical language means that a material particle of mass m moving with a velocity v has a wave associated with it. The associated wavelength is given by the ratio of Planck’s constant h to the momentum m.v=p of the particle.

According to relativity principle the momentum p of a photon of energy hν is given by hν/ c , where ν stands for frequency and c for the velocity of light.

P = hν /c = h/λ

Or, λ = h/p

We know that kinetic energy of the particle E = ½ mv2

Or, λ = h/m.v = h/(2mE)1/2

For particle s moving with very high velocity approaching the velocity of light c, we have to apply relativistic correction to the above equation.

The wavelengths associated with different material particles, such as electron, protons, atoms, molecules etc., can be found out on application of the relation λ = h/mv.

Considering electrons of the cathode rays, one has to use the relation:

½  mv2 = eV/300, where m is the mass of the electron, v its velocity (assumed to be far below that of light and as such in the range of non-relativistic speed),e the charge on the electron in e.s.u. and V the potential difference between the electrodes of the discharge tube expressed in volts.

When the potential difference is maintained at 150 volts, the wavelength associated with the electron is equal to 1Å. For potential difference V = 15,000 Volts, the wavelength associated with the electron turns out to be 0.1Å. These values fall within the range of wavelengths of x-rays.

11.2.1 Davisson and Germer Experiment.

De Broglie had predicted electron waves (in general particle waves) as has already been described in the above equation. Two American physicists, Davisson and Germer experimentally confirmed the same in 1927 by measuring the De Broglie wavelengths for slow electrons using diffraction methods. The experimental set-up is shown schematically in figure 11.1

Figure 11.1: The experimental arrangement for electron diffraction

E is an electron gun which produces beam of electrons. The filament T is made of tungsten which is heated so as to emit electrons as a result of thermionic emission. S1, S2 and S3 are slits to collimate the electron beam into a fine parallel pencil. A suitable electric field of pre-determined potential difference is applied by a battery V so that the electrons emitted by the heated tungsten filament get sufficiently accelerated. The fine pencil of electrons is then allowed to fall on a target N which is a single large crystal of nickel. There is an arrangement for rotating the target N about an axis parallel to the axis of the incident beam of electrons .F is a Faraday cylinder called as collector which is connected to a very sensitive galvanometer G. The collector is movable along a graduated scale CC which is circular so as to be able to receive the electrons which get reflected after hitting the target (i.e., nickel crystal) at angles ranged between 20⁰ and 90⁰. The collector F is double walled, each of which is insulated from the other. On striking the target, two types of electrons are to be taken care of. One type include those which are fast having almost the incident velocity and second type are those which are slow. The latter type are those which are secondary electrons produced as a result of collision with atoms. Since it is only the fastest electrons which are of interest, the second type of electrons being secondary and slow are stopped from entering the collector. The stoppage of the latter type of electrons in making entry into the collector is done by applying a retarding potential between the inner wall and the outer wall because of which only the fastest electrons ( i.e., of the first type ) are able to enter the collector and get detected by the sensitive galvanometer. The accelerating potential used in the experiment ranged somewhere between 30 to 600 volts and the retarding potential was only nine-tenths of the accelerating voltage. The whole set-up was completely enclosed in a highly evacuated and degassed vessel. The target used was nickel crystal which has face-centred cubic structure. It was cut parallel to the lattice plane (111), which is at right angles to one of the diagonals of the cube. The crystal was rotated about the axis as described above so as to present the crystal plane to the incident beam and the beam entering the collector as may be required by the experiment.

The experiment can be conducted in two different ways:

(a) The beam of electrons may be maintained at normal incidence on the surface of the crystal. In this arrangement the diffraction effect from the surface layer acting as a plane grating is produced. For each azimuth of the crystal, a beam of electrons is made to fall normally on the surface of a crystal. The collector is moved to different positions and the galvanometer current noted. The galvanometer current is a measure of the intensity of the diffracted beam of electrons. Dependence of intensity of diffracted beam of electrons on the angle between the incident beam and the beam entering the collector (also known as colatitude) is plotted for different voltage electrons. The curves recorded for different voltage electrons appear to be of the form as shown in figure 11.2(a-f).

Figure 11.2(a-f) : Curves showing dependence of the intensity of diffracted beam of electrons on angle between the incident beam and the beam entering the collector for different voltage electrons

The “bump” starts appearing in the curve for 44 volt electrons. The maximum development in curve is observed for 54 volts at colatitude of 500. For voltage greater than 54 volts the “bump” gradually reduces and at 68 volts it disappears and is not seen.

The “bump” in these curves provides evidence for the electron waves and experimental confirmation of De ‘Broglie concept of matter waves. The wavelength associated with 54 volt electron can be calculated using the expression:

λ  = (150/54)1/2 Å = 1.66 Å

Let us consider the diffracted beam at colatitude of 500 and assume the value of interplanar spacing d to be 2.15 Å in the equation for a plane reflection grating nλ = d sin Ө ( where n=1 , i.e., first order):

λ   = 2.15 sin 500  = 1.65 Å

(b) The beam of electrons is allowed to strike obliquely on the target. In this particular arrangement one is able to achieve diffraction effect from a space lattice. It is a situation analogous to Bragg’s x-ray diffraction and nλ = 2dsinӨ is appropriately applicable to this case.

If the glancing angles of incidence and reflection are maintained constant and the intensity of diffracted electrons (as measured by galvanometer current) is recorded for each electron velocity or accelerating voltage, curve with several sharp maxima are obtained as shown schematically in figure 11.3. The different maxima correspond to different orders. One can apply De Broglie‘s expression to calculate λ. λ can also be determined using glancing angle Ө and the grating space d , by using Bragg’s expression. The values of λ given by these two different ways are found to be in agreement.

Figure 11.3: Intensity of diffracted electrons versus accelerating voltage of electrons

11.2.2  Thomson’s Experiments

Prof. G.P.Thomson of Scotland performed experiments on electron diffraction in 1928 using high speed electrons in the range of 10,000 – 50,000 volts. The experimental set-up used by him is shown in figure 11.4.

DT is a discharge tube to generate cathode rays by using induction coil .The rays are allowed to pass through a diaphragm tube so as to obtain a narrow fine pencil of electrons. These electrons on striking a thin metallic film G of gold, aluminium or any other get diffracted. There is a provision for a photographic plate P to slide down and receive the electrons as they come out of the thin metallic film. F is a fluorescent screen which may be used for visual examination in place of photographic plate if one so desires.

Figure 11.4: Schematic diagram of experimental set-up of Thomson used for diffraction of electrons

The space FG is evacuated to a high vacuum. The high speed electrons on passing through the thin film at G get diffracted like x-rays and the effect is seen on the photographic plate in the form of symmetrical pattern of concentric rings about a central spot demonstrating that the diffraction pattern is because of electrons behaving as waves as shown in figure 11.5

Figure 11.5: Concentric rings about a central spot as a result of diffraction of electrons

Following Thomson’s experiment, several other scientists performed experiments on diffraction of electrons. For example, Kikuchi of Japan (1928) obtained electron diffraction pattern of mica using 45,000 V electrons. J.V.Hughes (1934) used electrons of speed comparable with the speed of light and proved validity of De – Broglie’s law.

The above said experiments and several others which followed them established vast scope of electron diffraction in the analysis of crystal structures.

11.3 Neutron Diffraction

Diffraction of x-rays is primarily by extranuclear electrons and as a consequence of which it gives a diffraction that increases with atomic number. Neutron diffraction is an important probe of investigation on internal structure of crystals and has the potential of providing information which is not possible for x-rays or electrons. It is, therefore, important to know about the differences in x – ray and neutron diffraction processes.

X-rays are scattered from the electrons associated with the atoms. The scattering of neutrons is mostly because of two factors:

(i) The nuclei in the crystal and

(ii) The magnetic interaction between the atoms or ions in the crystal and the neutrons, that is between the neutrons and impaired electrons in a magnetic crystal. Only in those cases whose atoms have magnetic moments, are the neutrons scattered from the electrons. For example, the transition metal atoms have magnetic moments on account of impaired electrons. Because of interaction between magnetic moment associated with the spin of the neutron and the magnetic moment of the atoms, magnetic scattering takes place.

Factor (i) is not of much significance on account of the fact that it only provides information regarding positions of the nuclei which can be found out more easily from x-ray diffraction. However, factor (ii) is extremely useful because it gives information regarding various magnetic moments of the atoms or ions that are oriented in the crystal. This information cannot be provided by x -rays. The magnetic interaction can be used to investigate magnetic structures, i.e., the location of the atoms and the orientation of the spins of the atoms.

Since neutron scattering takes place from nuclei of atoms and not from the electrons, it becomes an effective tool to determine the position of very light atoms in the crystal. For example, hydrogen atoms in a crystal cannot be detected by x-ray diffraction technique on account of the fact that x-ray scattering produced by the electrons (one for each hydrogen atom) is too weak. The situation is different when neutrons are used as probing particles. On exposure to neutrons, light atoms like hydrogen scatters relatively strongly because of which the position of hydrogen atoms get precisely located by using neutron diffraction.

On account of their magnetic moment, neutrons interact with the magnetic field of electrons of solids. Neutron diffraction techniques are extremely valuable in structural studies of magnetic crystals.

Taking the example of a non-magnetic structure of figure 11.6(a) , and the magnetic structure of figure11.6 (b) , one can notice the difference between an x-ray diffraction pattern or a neutron diffraction pattern produced by the non-magnetic structure of figure11.6(a) and the neutron diffraction pattern due to structure of figure 11.6(b) in figure 11.7(a,b) , which shows a plot of the intensity of the diffracted x-rays or neutrons I(θ) against θ .

The structure shown in figure 11.6(b) is an example of an antiferromagnet crystal. In this crystal the magnetic moments of all the atoms or ions are lined up either parallel or antiparallel to some fixed direction and arranged in such a way that neighbouring atoms or ions are antiparallel and the crystal as a whole does not possess a net magnetic moment.

Figure 11.7(a, b): Neutron diffraction intensity against (a) in case of Non-magnetic structure figure 11.6(a) and (b) Anti-ferromagnetic structure in case of 11.6(b)

MnF2 is one example of such substances which are antiferromagnetic.

Neutron diffraction experiments, though a very valuable tool, are much more complicated technically than are x-rays or electron-diffraction experiments. Neutrons are available from reactors as fission by-products and so involves a huge infrastructure. In neutron diffraction work every requirement is much bigger than in x-diffraction as for example, the sample, the source, the counter and the cost. On the other hand, the x-ray source is just an x-ray tube which is portable. The source of neutrons is a nuclear reactor which is much larger, expensive and is not portable. As such, the requirements of neutron diffraction experiments are much more larger than the x-ray diffraction set-up.

References.

1. Cracknell,A.P. : “ Crystals and Their structures”,Pergamon Press,U.K.,1969.
2. Verma,A.R. &Srivastava,O.N.: “ Crystallography For Solid State Physics”, Wiley Eastern,N.Delhi,1981.
3. Brown,F.C.: “ The Physics of Solids”,W.A.Bejamin,Inc.,N.Y.,1967.
4. Pillai,S.O.: “Solid State Physics”, New Age International (p) Ltd Publishers, New Delhi !999.
5. Franks,A. “ Applied X-rays, electrons and Ions (ed. Agarwala,B.K.), N.P.L. publication,N.Delhi.

Suggested Reading For Detailed Information on the subject.

1. Weiss ,R.J.: “ Solid State Physics For Metallurgists”, Pergamon Press, Oxford. It gives introductory discussion of the experimental techniques used in neutron diffraction work.
2. Bacon ,G.E. “ Neutron Diffraction”, Univ.Press,Oxford.It is a standard book on neutron diffraction.
3. Kothari,L.S. &Singwi,K.S. “ In Solid State Physics,Ed.Seitz,F. and Turnbull,D.,” Academic press’
4. Lonsdale,K “ Crystals and X-rays “, Bell, London
5. Bragg, W.L. “ The Crystalline State; a general survey”, Bell, London.