Neutron Diffraction

Saturday, July 6, 2013 By Anonymous

Neutron diffraction

Definition:

The interference process which occurs when neutrons are scattered by the atoms within solids, liquids and gases.

History:

The first neutron diffraction experiments were carried out in 1945 by Ernest O. Wollan using the Graphite Reactor at Oak Ridge.

He was joined shortly thereafter (June 1946) by Clifford Shull, and together they established the basic principles of the technique, and applied it successfully to many different materials, addressing problems like the structure of ice and the microscopic arrangements of magnetic moments in materials. For this achievement Shull was awarded one half of the 1994 Nobel Prize in Physics. Wollan had died in the 1990s. (The other half of the 1994 Nobel Prize for Physics went to Bert Brockhouse for development of the inelastic scattering technique at the Chalk River facility of AECL. This also involved the invention of the triple axis spectrometer). The delay between the achieved work (1946) and the Nobel Prize awarded to Brockhouse and Shull (1994) brings them close to the record held by Ernst Ruska between his invention of the electron microscope (1933) - also in the field of particle optics - and his own Nobel Prize (1986).

Explanation:

Neutron diffraction is a form of elastic scattering where the neutrons exiting the experiment have more or less the same energy as the incident neutrons. The technique is similar to X-ray diffraction but the different type of radiation gives complementary information. A sample to be examined is placed in a beam of thermal or cold neutrons and the intensity pattern around the sample gives information of the structure of the material.

Neutrons interact with matter differently than x-rays. X-rays interact primarily with the electron cloud surrounding each atom. The contribution to the diffracted x-ray intensity is therefore larger for atoms with a large atomic number (Z) than it is for atoms with a small Z. On the other hand, neutrons interact directly with the nucleus of the atom, and the contribution to the diffracted intensity is different for each isotope; for example, regular hydrogen and deuterium contribute differently. It is also often the case that light (low Z) atoms contribute strongly to the diffracted intensity even in the presence of large Z atoms. The scattering length varies from isotope to isotope rather than linearly with the atomic number. An element like Vanadium is a strong scattered of X-rays, but its nuclei hardly scatter neutrons, which is why it often used as a container material. Non-magnetic neutron diffraction is directly sensitive to the positions of the nuclei of the atoms.

A major difference with X-rays is that the scattering is mostly due to the tiny nuclei of the atoms. That means that there is no need for an atomic form factor to describe the shape of the electron cloud of the atom and the scattering power of an atom does not fall off with the scattering angle as it does for X-rays. Diffract grams therefore can show strong well defined diffraction peaks even at high angles, particularly if the experiment is done at low temperatures. Many neutron sources are equipped with liquid helium cooling systems that allow to collect data at temperatures down to 4.2K. The superb high angle (i.e. high resolution) information means that the data can give very precise values for the atomic positions in the structure. On the other hand, Fourier maps (and to a lesser extent difference Fourier maps) derived from neutron data suffer from series termination errors, sometimes so much that the results are meaningless.

Major achievements of neutron scattering:

Polymer Conformation and Dynamics:

Neutrons have provided the most direct information on polymer conformation and the associated scaling laws, and polymer dynamics such as reptation, corroborating the Nobel prize winning theoretical concepts of Flory (1974) and DeGennes (1991).

Electron diffraction

Definition:

An effect due to the wavelike nature of electrons and observed when a narrow beam of them upon passing through a very thin layer of a material (as a metal crystal) is deflected in particular directions and if allowed to fall on a fluorescent screen produces a pattern of light and dark areas, the pattern formed by these areas being characteristic of the material traversed

Electron diffraction refers to the wave nature of electrons. However, from a technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern. This phenomenon is commonly known as the wave-particle duality, which states that the behavior of a particle of matter (in this case the incident electron) can be described by a wave. For this reason, an electron can be regarded as a wave much like sound or water waves. This technique is similar to X-ray and neutron diffraction.

History:

The de Broglie hypothesis, formulated in 1924, predicts that particles should also behave as waves. De Broglie's formula was confirmed three years later for electrons (which have a rest-mass) with the observation of electron diffraction in two independent experiments. At the University of Aberdeen George Paget Thomson passed a beam of electrons through a thin metal film and observed the predicted interference patterns. At Bell Labs Clinton Joseph Davisson and Lester Halbert Germer guided their beam through a crystalline grid. Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their work.

Theory:

Electron interaction with matter:

Unlike other types of radiation used in diffraction studies of materials, such as X-rays and neutrons, electrons are charged particles and interact with matter through the Coulomb forces. This means that the incident electrons feel the influence of both the positively charged atomic nuclei and the surrounding electrons. In comparison, X-rays interact with the spatial distribution of the valence electrons, while neutrons are scattered by the atomic nuclei through the strong nuclear forces. In addition, the magnetic moment of neutrons is non-zero, and they are therefore also scattered by magnetic fields. Because of these different forms of interaction, the three types of radiation are suitable for different studies.

Electron diffraction in a TEM:

Electron diffraction of solids is usually performed in a Transmission Electron Microscope (TEM) where the electrons pass through a thin film of the material to be studied. The resulting diffraction pattern is then observed on a fluorescent screen, recorded on photographic film, on imaging plates or using a CCD camera.

As mentioned above, the wavelength of electron accelerated in a TEM is much smaller than that of the radiation usually used for X-ray diffraction experiments. A consequence of this is that the radius of the Ewald sphere is much larger in electron diffraction experiments than in X-ray diffraction. This allows the diffraction experiment to reveal more of the two dimensional distribution of reciprocal lattice points.

Furthermore, electron lenses allows the geometry of the diffraction experiment to be varied. The conceptually simplest geometry referred to as selected area electron diffraction (SAED) is that of a parallel beam of electrons incident on the specimen, with the specimen field selected using a sub-specimen image-plane aperture. However, by converging the electrons in a cone onto the specimen, one can in effect perform a diffraction experiment over several incident angles simultaneously. This technique is called Convergent Beam Electron Diffraction (CBED) and can reveal the full three dimensional symmetry of the crystal.

In a TEM, a single crystal grain or particle may be selected for the diffraction experiments. This means that the diffraction experiments can be performed on single crystals of nanometer size, whereas other diffraction techniques would be limited to studying the diffraction from a multicrystalline or powder sample. Furthermore, electron diffraction in TEM can be combined with direct imaging of the sample, including high resolution imaging of the crystal lattice, and a range of other techniques. These include solving and refining crystal structures by electron crystallography, chemical analysis of the sample composition through energy-dispersive X-ray spectroscopy, investigations of electronic structure and bonding through electron energy loss spectroscopy, and studies of the mean inner potential through electron holography.

Use of Zones axes:

Ø Can measure angle between spots to get interplanar angles.

Ø Can use convergent beam mode to get crystal symmetry and space group.

Ø Cam measure location and movement of HOLZ lines to measure strain and/or lattice parameter.

Ø Convergent beam patterns can be used to measure sample thicknesses.

Electron diffraction is most frequently used in solid state physics and chemistry to study the crystal structure of solids. Experiments are usually performed in a transmission electron microscope (TEM), or a scanning electron microscope (SEM) as electron backscatter diffraction. In these instruments, electrons are accelerated by an electrostatic potential in order to gain the desired energy and determine their wavelength before they interact with the sample to be studied.

Electron Diffraction Patterns:

In relation to diffraction patterns it is interesting to consider three types of solid matter: single

Crystals, polycrystals and amorphous materials.

Single Crystals:

Single crystals consist of atoms arranged in an orderly lattice. Some types of crystal lattices are simple cubic, face Centre cubic (f.c.c.), and body Centre cubic (b.c.c). In general, single crystals with different crystal structures will cleave into their own characteristic geometry. You may have seen single crystals of quartz, calcite, or carbon (diamond).

Single crystals are the most ordered of the three structures. An electron beam passing through a single crystal will produce a pattern of spots. From the diffraction spots one can determine the type of crystal structure (f.c.c., b.c.c.) and the "lattice parameter" (i.e., the distance between adjacent (100) planes).

Also, the orientation of the single crystal can be determined: if the single crystal is turned or flipped, the spot diffraction pattern will rotate around the Centre beam spot in a predictable way.

Polycrystalline Materials:

Polycrystalline materials are made up of many tiny single crystals. Most common metal materials (copper pipes, nickel coins, stainless steel forks) are polycrystalline. Also, a ground-up powder sample appears polycrystalline. Any small single crystal "grain" will not in general have the same orientation as its neighbors. The single crystal grains in a polycrystalline will have a random distribution of all the possible orientations. A polycrystal, therefore, is not as ordered as a single crystal. An electron beam passing through a polycrystal will produce a diffraction pattern equivalent to that produced by a beam passing through series of single crystals of various orientations. The diffraction pattern will therefore look like a superposition of single crystal spot patterns: a series of concentric rings resulting from many spots very close together at various rotations around the centre beam spot. From the diffraction rings one can also determine the type of crystal structure and the "lattice parameter". One cannot determine the orientation of a polycrystal, since there is no single orientation and flipping or turning the polycrystal will yield the same ring pattern. As shown in fig.

Powder method

Definition:

The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal. OR

A process for identifying minerals or crystals; a small rod is coated with a powdered form of the substance and subjected to suitably modified X-rays; the pattern of diffracted rings is used for identification.

Powder diffraction:

Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials.

A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones.

A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film.

If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result. As shown in fig.

If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards. As shown in fig,

Uses:

Relative to other methods of analysis, powder diffraction allows for rapid, non-destructive analysis of multi-component mixtures without the need for extensive sample preparation.^[4] This gives laboratories around the world the ability to quickly analyze unknown materials and perform materials characterization in such fields as metallurgy, mineralogy, forensic science, archeology, condensed matter physics, and the biological and pharmaceutical sciences. Identification is performed by comparison of the diffraction pattern to a known standard or to a database such as the International Centre for Diffraction Data's Powder Diffraction File (PDF) or the Cambridge Structural Database (CSD). Advances in hardware and software, particularly improved optics and fast detectors, have dramatically improved the analytical capability of the technique, especially relative to the speed of the analysis. The fundamental physics upon which the technique is based provides high precision and accuracy in the measurement of interplanar spacings, sometimes to fractions of an Ångström, resulting in authoritative identification frequently used in patents, criminal cases and other areas of law enforcement. The ability to analyze multiphase materials also allows analysis of how materials interact in a particular matrix such as a pharmaceutical tablet, a circuit board, a mechanical weld, a geologic core sampling, cement and concrete, or a pigment found in an historic painting. The method has been historically used for the identification and classification of minerals, but it can be used for any materials, even amorphous ones, so long as a suitable reference pattern is known or can be constructed.

Phase identification:

The most widespread use of powder diffraction is in the identification and characterization of crystalline solids, each of which produces a distinctive diffraction pattern. Both the positions (corresponding to lattice spacings) and the relative intensity of the lines are indicative of a particular phase and material, providing a "fingerprint" for comparison. A multi-phase mixture, e.g. a soil sample, will show more than one pattern superposed, allowing for determination of relative concentration.

J.D. Hanawalt, an analytical chemist who worked for Dow Chemical in the 1930s, was the first to realize the analytical potential of creating a database. Today it is represented by the Powder Diffraction File (PDF) of the International Centre for Diffraction Data (formerly Joint Committee for Powder Diffraction Studies). This has been made searchable by computer through the work of global software developers and equipment manufacturers. There are now over 550,000 reference materials in the 2006 Powder Diffraction File Databases, and these databases are interfaced to a wide variety of diffraction analysis software and distributed globally. The Powder Diffraction File contains many sub files, such as minerals, metals and alloys, pharmaceuticals, forensics, excipients, superconductors, semiconductors, etc., with large collections of organic, organometallic and inorganic reference materials.

Crystal structure refinement and determination:

Crystal structure determination from powder diffraction data is extremely challenging due to the overlap of reflections in a powder experiment. A number of different methods exist for structural determination, such as simulated annealing and charge flipping. The crystal structures of known materials can be refined, i.e. as a function of temperature or pressure, using the Rietveld method. The Rietveld method is a so-called full pattern analysis technique. A crystal structure, together with instrumental and microstructural information, is used to generate a theoretical diffraction pattern that can be compared to the observed data. A least squares procedure is then used to minimize the difference between the calculated pattern and each point of the observed pattern by adjusting model parameters.

Rotating Crystal Method

Definition:

Any method of studying crystalline structures by x-ray or neutron diffraction in which a monochromatic, collimated beam of x-rays or neutrons falls on a single crystal that is rotated about an axis perpendicular to the beam.

In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic x-ray beam. A cylindrical film is placed around it and the crystal is rotated about the chosen axis. As the crystal rotates, sets of lattice planes will at some point make the correct Bragg angle for the monochromatic incident beam, and at that point a diffracted beam will be formed. The reflected beams are located on the surface of imaginary cones. When the film is laid out flat, the diffraction spots lie on horizontal lines.

The chief use of the rotating crystal method is in the determination of unknown crystal structures.

In the rotating –crystal method a single crystal is rotated about a fixed axis in a beam of monoenergetic x- rays or neutrons. The variation is the angle brings different atomic planes into position for reflection. A simple rotating-crystal x-ray camera is shown in fig.The film is mounted in a crystal cylindrical holder concentric with a rotating spindle on which the single crystal specimen is mounted. The dimensions of the crystal usually need not be greater than 1mm.The incident x-ray beam is made nearly monochromatic by a filter or by reflection from an earlier crystal. The beam is diffracted from a given crystal plane whenever in the course of rotation the value of satisfies the Bragg equation.

Beams from all planes parallel to the vertical rotation axis will be in the horizontal plane. Planes with other orientations will reflect in layer above and below the horizontal plane. Several variation of the rotating –crystal method are in common use .in oscillating through a limited angular range ,instead of being rotated through .The limited range reduces the possibility of overlapping reflections.

A simple rotating-crystal x-ray camera, with a crystal mounted on the rotating spindle.

Laue Method

The Laue method is mainly used to determine the orientation of large single crystals. White radiation is reflected from, or transmitted through, a fixed crystal.

Explanation:

In the Laue Method a single crystal is held stationary in a beam of x-ray or neutron radiation of continuous wavelength. The crystal selects out and diffracts the discrete values of lamda for which planes exist of spacing d and incidence angle satisfying the Bragg Law.

The Laue Method is convenient for the rapid determination of crystal orientation and symmetry.it is also used to study the extent of crystalline imperfection under mechanical and thermal extent.The diffracted beams form arrays of spots that lie on curves on the film. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d and involved. Each curve therefore corresponds to a different wavelength. The spots lying on any one curve are reflections from planes belonging to one zone. Laue reflections from planes of the same zone all lie on the surface of an imaginary cone whose axis is the zone axis.

Experimental:

There are two practical variants of the Laue method, the back-reflection and the transmission Laue method. You can study these below:

Back-reflection Laue:

In the back-reflection method, the film is placed between the x-ray source and the crystal. The beams which are diffracted in a backward direction are recorded.

One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on a hyperbola.

Transmission Laue:

In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal.

One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse.

References:

Ø B.D. Cullity Elements of X-ray Diffraction Addison Wesley Mass. 1978 ISBN 0-201-01174-3

Ø Feynman, Richard P. (1963). The Feynman Lectures on Physics, Vol. I. Addison-Wesley. pp. 16–10, 17–5.

Ø B. E. Warren (1969/1990) X-ray diffraction (Addison–Wesley, Reading MA/Dover, Mineola NY) ISBN 0-486-66317-5.

Ø Charles Kittle ,Introduction to Solid State Physics, Edition

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