X-ray And Electron Diffraction

Type of physical science: Condensed matter physics

Field of study: Surfaces

Diffraction occurs when a wave encounters an aperture with a characteristic dimension that is roughly equivalent to its wavelength. Separations between crystalline planes can be determined using beams of electromagnetic radiation (in the form of X rays) or electrons of appropriate energies.

Overview

Diffraction is a wave phenomenon that occurs when a wave encounters an obstacle or aperture in its path of propagation that is of the order of the wave's periodic spatial variation (or wavelength). If a trough of uniform width is filled to a reasonable depth with water and the end of a wooden board is dipped into the water repetitively at a constant frequency, plane waves will be set up in the water propagating forward from the source of the disturbance. Wave crests will be represented by parallel lines. If two pieces of wood are inserted into the trough but do not traverse the full width of the trough, an opening is created that is smaller than the trough width, allowing the water waves to pass through and propagate beyond the barrier. If the opening at this barrier is large relative to the wave's wavelength, then the plane waves created by the disturbance can pass through the opening relatively unaltered. If the aperture is roughly the same size as the wave's wavelength, then the plane waves must diffract through the opening and the waves that pass through the opening will cease to be plane waves. Wave fronts propagating through the opening are no longer represented by parallel lines. Rather, they have a cylindrical symmetry and are represented by circular arcs of ever-increasing radius as the waves propagate away from the barrier.

Diffraction explains why a teacher lecturing in a classroom can be heard but not seen by people down the hall from the classroom if the door is open. The teacher generates sound waves when speaking. These sound waves have wavelengths that are roughly of the order of the width of the door opening and thus the sound can diffract around the corner. Light waves, on the other hand, have wavelengths that are many orders of magnitude smaller than the width of the door opening and thus travel straight through the door without alteration.

Diffraction of light can be explained in exactly the same way as the previous discussion of water waves; however, the width of the slit that light must pass through in order to diffract must be the same size as the wavelength of the incident light. A diffraction grating is a specially made slit consisting of many closely spaced grooves. Diffraction of incident light occurs as the light passes through the grating.

Materials that form compounds in regular arrays such as crystalline structures provide natural three-dimensional diffraction gratings for the portion of the electromagnetic spectrum with wavelengths roughly equal to the separation between crystal planes. Most crystalline structures have intercrystalline plane distances ranging between several individual to several tens of angstroms (10^-10 meters). Electromagnetic radiation having wavelengths of this size range are called X rays. Visible light passing through crystalline structures would not exhibit diffraction, but X rays incident on crystalline structures would.

Bragg diffraction is the diffraction of X rays or an equivalent energetic beam of particles from a crystal. A single crystal with a simple lattice structure of cubic symmetry has a unit cell composed of atoms placed at the corners of a cube of side d. In a plane, all atoms are a linear distance d away from their nearest neighbors along mutually perpendicular directions in the plane. One such plane is a distance d away from an identical plane above and below it.

The atoms in the crystal lattice act as scattering centers for incident radiation. If an incident beam of X rays strikes a crystal sample at an angle θ with respect to the first Bragg plane, the remainder of the incident ray is transmitted to the second Bragg plane. At this second plane, it suffers partial reflection at angle θ with respect to that plane. The remainder of the ray is transmitted to the third Bragg plane, and so on. All reflected rays are parallel and are scattered by an angle 2 θ from the original direction of the incident X ray. To reach a detector, the reflected ray from the second Bragg plane must travel a longer path length than the reflected ray from the first Bragg plane. The reflected ray must travel a distance d sin θ to reach the second Bragg plane and another distance d sin θ to return to the first Bragg plane. From there to the detector, reflected rays from the first and second Bragg planes travel equal distances.

For a diffraction pattern to be observed at the detector, the path length difference for two rays must be an integral number of wavelengths. Thus, the Bragg law of diffraction requires n λ = 2d sin θ, where λ is the X-ray wavelength and n = 1, 2, 3, and so on and is called the order of diffraction. To use the Laue method for crystal plane spacing determination, a crystal sample is illuminated with X rays of known wavelength. The resulting diffraction pattern is recorded and analyzed for symmetry patterns and intensity peaks.

The diffraction pattern is often recorded on a flat X-ray sensitive film. This pattern will reveal the symmetry of the crystal; a crystal with six-fold symmetry will display a diffraction pattern with six-fold symmetry on the exposed X-ray film. One-, two-, three-, four-, and six-fold symmetry occur naturally in crystals. Some quasicrystals have been produced recently that exhibit a type of five-fold symmetry. Diffraction techniques would work on these exotic materials as well. The simple Laue method has been replaced by more sophisticated analytic techniques for the determination of crystallographic planes.

Applications

In 1912, Max von Laue proposed that naturally occurring crystalline materials could be used to demonstrate diffraction effects with X rays. In a presentation to the Bavarian Academy of Sciences in Munich, Germany, von Laue elucidated a theory to explain how a periodic atomic structure could diffract beams of X rays. Experimental results accompanied the theoretical discussion. Armed with a tool to study crystalline structures, researchers firmly developed the discipline called solid-state physics.

In another area of applied physics, analysis of a resulting diffraction pattern could be used to determine experimentally the wavelength of the incident X rays if the interplanar crystal spacing was known independently. That knowledge can be obtained easily for the simpler crystalline structures. All that is required is the molecular weight and density of the substance forming the simple crystal structure, the number of atoms per unit cell, and Avogadro's number (the number of molecules in a mole of any substance).

Diffraction of electron beams by crystalline materials was discovered accidentally in 1927 by a pair of Bell Laboratory researchers, Clinton Joseph Davisson and Lester Halbert Germer, who were investigating secondary emission of electrons from a nickel sample's surface.

Secondary emission is one of several processes whereby electrons can be liberated from a solid surface. Some of the other methods are the photoelectric effect (successfully explained in 1905 by Albert Einstein), field emission, and thermionic emission. In the photoelectric effect, incident photons with sufficient energy to overcome the binding energy of an electron free up electrons with kinetic energies ranging from zero to a maximum value given by the difference between the incident photon energy and the binding energy of the atom. Field emission liberates electrons from a surface by exposing the surface to intense electric fields. Thermionic emission liberates electrons from a surface using thermal energy. In secondary emission, an energetic beam (a few tens or hundreds of electronvolts) is incident upon a solid surface and the incident electrons collide with bound atomic electrons, transferring enough kinetic energy to liberate electrons from the solid surface. This process is important for electronic applications. For certain solids and over certain incident electron energy ranges, the number of liberated electrons exceeds the number of incident electrons. This effect is used to amplify electrical signals in a photomultiplier tube.

Davisson and Germer were interested in detecting secondary emission from a polycrystalline nickel sample. The number of secondary electrons as a function of angle (defined with respect to the normal to the solid surface) was measured experimentally. The angular distribution was expected to be proportional to the cosine of that angle. A large fraction of liberated electrons was not expected at any particular angle of observation. Initial experiments verified the anticipated angular distribution of secondary electron emission. During the course of the investigations, a nickel sample was accidentally contaminated. To correct the situation, Davisson and Germer heated the nickel sample at high temperature for a prolonged time. After cooling the cleaned nickel sample, they continued the experiments; however, the experimental results now had an added feature.

In addition to the expected angular distribution of secondary electron emission, Davisson and Germer observed higher numbers of electrons at specific angles. This effect was most prevalent when the incident electron beam energy was 54 electronvolts. A peak on top of the secondary emission was observed at 50 degrees. It was clear that something in addition to secondary emission was occurring. Originally, the nickel sample was polycrystalline, meaning that the sample contained myriad microcrystals randomly oriented throughout the bulk. The heat treatment provided sufficient energy to transform the sample into a single crystal configuration, a nearly perfect (apart from impurities and point defects), face-centered cubic lattice structure.

Excess electron emission at a specific angle suggested the type of diffraction that had been demonstrated using X-ray beams. The atoms in the crystal lattice appeared to behave as scattering centers for the incident electrons. From the first-order results, an electron wavelength was determined using Bragg's law. The experimental wavelength precisely matched the de Broglie wavelength predicted by the electron beam's energy.

Diffraction of other material particles has been demonstrated in solids with crystalline structures. A thermal neutron has a kinetic temperature of approximately 300 Kelvins. If the neutron's thermal kinetic energy, 3/2 kT, is equated to 1/2 mv² or p²/2m, the neutron's momentum can be calculated. The neutron's wavelength, from the de Broglie relation, is Planck's constant divided by the momentum (h/mv).

This operation yields a wavelength of 1.4 x 10^-10 meters, or 1.4 angstroms. This wavelength is useful for diffraction studies in many solids. Specifically, 1.4 angstroms falls within the X-ray portion of the electromagnetic spectrum if the particles are photons.

Context

Wilhelm Conrad Rontgen is credited with the discovery of X-ray radiation. His work was considered so significant that he was awarded the first Nobel Prize in Physics in 1901. Von Laue first used X rays to demonstrate diffraction in crystals. Beginning in 1914 and continuing for the next three years, the Nobel Prize in Physics was awarded to researchers who investigated X-ray behavior in solids. Von Laue was granted the 1914 Nobel Prize for demonstrating X-ray diffraction. In 1915, William Henry Bragg and Lawrence Bragg shared the Nobel Prize for developing a technique to study crystal structure using X-ray radiation. Charles Glover Barkla won the 1917 Nobel Prize for his discovery of characteristic X rays of elements.

The use of X rays to ascertain crystalline structure represented the logical application of a thorough understanding of the properties of electromagnetic radiation to a specific problem.

The accidental discovery of electron diffraction in a single crystal of nickel by Davisson and Germer in 1927 represented a watershed experiment that verified the existence of wave-particle duality.

In 1924, Louis de Broglie used symmetry arguments to postulate that material particles could be assigned a wavelength. This assumption removed the asymmetry in the concept of particles and waves that was caused by the realization that certain aspects of electromagnetic radiation could be properly explained only by attributing a particle nature (photons) to electromagnetic radiation. Experimental verification of electron diffraction proved the correctness of de Broglie's hypothesis. De Broglie was awarded the Nobel Prize in Physics in 1929 for his theoretical work on the wave nature of particles. Davisson shared the 1937 Nobel Prize in Physics with George Paget Thomson for the discovery of electron diffraction in crystals.

X-ray and electron diffraction have become powerful tools in determining crystalline structures and structures of solids in general. The diffraction process also can be used with amorphous and pseudocrystalline materials to provide clues into both topological short-range ordering and quasicrystalline behavior in such materials. Whenever a solid exhibits some degree of structural symmetry, the Laue diffraction method can be used to reveal that symmetry.

Principal terms

CRYSTAL PLANE: a two-dimensional distribution of atoms having a particular symmetry

CRYSTAL STRUCTURE: atoms arranged in a periodic fashion at lattice sites; only fourteen crystal types occur in nature

DIFFRACTION: a wave phenomenon that occurs when a wave encounters an aperture with a characteristic dimension similar to the wavelength of the incident wave

ELECTRON: a fundamental particle of matter upon which resides the fundamental charge of nature (e)

ION: an atom that has either lost or gained electrons

LATTICE: a regular array of points in space with periodic locations within a three-dimensional volume

LATTICE SPACING: the distance between similar crystal planes in a lattice

X RAYS: electromagnetic radiation with wavelengths ranging from 0.01 nanometer to 10 nanometers

Bibliography

Burns, Marshall L. MODERN PHYSICS FOR SCIENCE AND ENGINEERING. San Diego: Harcourt Brace Jovanovich, 1988. A well-written account of modern physics. Although calculus-based and aimed at a college physics audience, qualitative descriptions of basic electromagnetism and quantum theory are understandable to the layperson.

Halliday, David, and Robert Resnick. FUNDAMENTALS OF PHYSICS. 3d. rev. ed. New York: John Wiley & Sons, 1988. Considered a classic textbook for investigating elementary calculus-based physics. The section on diffraction (as applied to both X-ray and electron beams) is heavily illustrated and thoroughly explained.

Myers, H. P. INTRODUCTORY SOLID STATE PHYSICS. London: Taylor & Francis, 1990. A very thorough text on the physics of the solid state. Two chapters are particularly useful in understanding crystallographic structures and the diffraction of X-ray or electron beams in solids.

Ohanian, Hans C. PHYSICS. New York: W. W. Norton, 1985. Although calculus-based, the text is not mathematically rigorous. Accessible to those with modest mathematical skills. Highly descriptive and well illustrated.

Sells, Robert L., and Richard T. Weidner. ELEMENTARY MODERN PHYSICS. Boston: Allyn & Bacon, 1980. A thorough treatment of modern physics using algebraic discourse. Covers wave-particle duality from both a theoretical and experimental viewpoint. Discusses the Davisson-Germer experiment and uses of X-ray and electron diffraction in crystallographic analyses.