electron diffraction is a technique used to study matter by firing electrons at a sample and...

7/29/2019 Electron Diffraction is a Technique Used to Study Matter by Firing Electrons at a Sample and Observing the Resulti

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Electron diffraction is a technique used to study matter by firing electrons at a sample and observing theresulting interferencepattern. This phenomenon occurs due to wave-particle duality, which states that a particleof matter (in this case the incident electron) can be described as a wave. For this reason, an electron can beregarded as a wave much like sound or water waves. This technique is similar to X-rayandneutron diffraction.

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

The periodic structure of a crystalline solid acts as adiffraction grating, scattering the electrons in a predictablemanner. Working back from the observeddiffraction pattern, it may be possible to deduce the structure of thecrystal producing the diffraction pattern. However, the technique is limited by thephase problem.

Apart from the study of crystals i.e. electron crystallography, electron diffraction is also a useful technique to

study the short range order ofamorphous solids, and the geometry ofgaseous molecules.

History

The de Broglie hypothesis, formulated in 1926, predicts that particles should also behave as waves. De Broglie'sformula was confirmed three years later forelectrons (which have a rest-mass) with the observation of electrondiffraction in two independent experiments. At the University of AberdeenGeorge Paget Thomson passed abeam of electrons through a thin metal film and observed the predicted interference patterns. AtBell LabsClinton Joseph Davissonand Lester Halbert Germerguided their beam through a crystalline grid. Thomson andDavisson shared theNobel Prize for Physicsin 1937 for their work.

[edit] Theory

[edit] Electron interaction with matter

Unlike other types of radiation used in diffraction studies of materials, such asX-raysandneutrons, electronsare charged particlesand interact with matter through theCoulomb forces. This means that the incidentelectrons feel the influence of both the positively charged atomic nuclei and the surrounding electrons. Incomparison, X-rays interact with the spatial distribution of the valence electrons, while neutrons are scatteredby the atomic nuclei through thestrong nuclear forces. In addition, the magnetic moment of neutrons is non-zero, and they are therefore also scattered bymagnetic fields. Because of these different forms of interaction,

the three types of radiation are suitable for different studies.

[edit] Intensity of diffracted beams

In the kinematical approximation for electron diffraction, the intensity of a diffracted beam is given by:

Here is the wavefunction of the diffracted beam and is the so calledstructure factorwhich is given by:


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where is the scattering vector of the diffracted beam, is the position of an atom i in the unit cell, andfi is thescattering power of the atom, also called the atomic form factor. The sum is over all atoms in the unit cell.

The structure factor describes the way in which an incident beam of electrons is scattered by the atoms of acrystal unit cell, taking into account the different scattering power of the elements through the term fi. Since theatoms are spatially distributed in the unit cell, there will be a difference in phase when considering the scatteredamplitude from two atoms. This phase shift is taken into account by the exponential term in the equation.

The atomic form factor, or scattering power, of an element depends on the type of radiation considered. Becauseelectrons interact with matter though different processes than for example X-rays, the atomic form factors forthe two cases are not the same.

[edit] Wavelength of electrons

The wavelength of an electron is given by the de Broglie equation

Here h is Planck's constant andp the momentum of the electron. The electrons are accelerated in an electricpotential Uto the desired velocity:

m0 is the mass of the electron, and e is the elementary charge.The electron wavelength is then given by:

However, in an electron microscope, the accelerating potential is usually several thousand volts causing theelectron to travel at an appreciable fraction of the speed of light. An SEM may typically operate at anaccelerating potential of 10,000 volts (10 kV) giving an electron velocity approximately 20% of the speed of

light, while a typical TEM can operate at 200 kV raising the electron velocity to 70% the speed of light. Wetherefore need to take relativistic effects into account. It can be shown that the electron wavelength is thenmodified according to:

c is the speed of light. We recognize the first term in this final expression as the non-relativistic expressionderived above, while the last term is a relativistic correction factor. The wavelength of the electrons in a 10 kV


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SEM is then 12.3 x 10-12 m (12.3 pm) while in a 200 kV TEM the wavelength is 2.5 pm. In comparison thewavelength of X-rays usually used in X-ray diffraction is in the order of 100 pm (Cu k: =154 pm).

[edit] Electron diffraction in a TEM

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

[edit] Benefits

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

Furthermore, the electron lenses allows the geometry of the diffraction experiment to be varied. Theconceptually simplest geometry is that of a parallel beam of electrons incident on the specimen. However, byconverging the electrons in a cone onto the specimen, one can in effect perform a diffraction experiment overseveral 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 thediffraction experiments can be performed on single crystals of nanometer size, whereas other diffractiontechniques 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 resolutionimaging 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-dispersiveX-ray spectroscopy, investigations of electronic structure and bonding throughelectron energy lossspectroscopy, and studies of the mean inner potential through electron holography.

[edit] Practical aspects

1: Sketch of the electron beam-path in a TEM.


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2: Typical electron diffraction pattern obtained in a TEM with a parallel electron beam

Figure 1 to the right is a simple sketch of the path of a parallel beam of electrons in a TEM from just above thesample and down the column to the fluorescent screen. As the electrons pass through the sample, they arescattered by the electrostatic potential set up by the constituent elements. After the electrons have left thesample they pass through the electromagnetic objective lens. This lens acts to collect all electrons scattered

from one point of the sample in one point on the fluorescent screen, causing an image of the sample to beformed. We note that at the dashed line in the figure, electrons scattered in the same direction by the sample arecollected into a single point. This is the back focal plane of the microscope, and is where the diffraction patternis formed. By manipulating the magnetic lenses of the microscope, the diffraction pattern may be observed byprojecting it onto the screen instead of the image. An example of what a diffraction pattern obtained in this waymay look like is shown in figure 2.

If the sample is tilted with respect to the incident electron beam, one can obtain diffraction patterns from severalcrystal orientations. In this way, thereciprocal lattice of the crystal can be mapped in three dimensions. Bystudying the systematic absence of diffraction spots theBravais lattice and any screw axes and glide planespresent in the crystal structure may be determined.

[edit] Limitations

Electron diffraction in TEM is subject to several important limitations. First, the sample to be studied must beelectron transparent, meaning the sample thickness must be of the order of 100 nm or less. Careful and timeconsuming sample preparation may therefore be needed. Furthermore, many samples are vulnerable to radiationdamage caused by the incident electrons.

The study of magnetic materials is complicated by the fact that electrons are deflected in magnetic fields by theLorentz force. Although this phenomenon may be exploited to study the magnetic domains of materials byLorentz force microscopy, it may make crystal structure determination virtually impossible.

Furthermore, electron diffraction is often regarded as a qualitative technique suitable for symmetrydetermination, but too inaccurate for determination of lattice parameters and atomic positions. But there are alsoseveral examples where unknown crystal structures (both inorganic, organic and biological) have been solvedby electron crystallography. Lattice parameters of high accuracy can in fact be obtained from electrondiffraction, relative errors less than 0.1% have been demonstrated. However, the right experimental conditionsmay be difficult to obtain, and these procedures are often viewed as too time consuming and the data toodifficult to interpret. X-ray or neutron diffraction are therefore often the preferred methods for determininglattice parameters and atomic positions.


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However, the main limitation of electron diffraction in TEM remains the comparatively high level of userinteraction needed. Whereas both the execution of powder X-ray (and neutron) diffraction experiments and thedata analysis are highly automated and routinely performed, electron diffraction requires a much higher level ofuser input

X-Ray 6 of 7

We shall now consider the powder patterns from a sample crystal. The sample is known to have acubicstructure, but we don't know which one.

We remove the film strip from the Debye camera after exposure, then develop and fix it. From the stripof film we make measurements of the position of each diffraction line. From the results it is possible toassociate the sample with a particular type of cubic structure and also to determine a value for its latticeparameter.

When the film is laid flat, S1 can be measured. This is the distance along the film, from a diffractionline, to the centre of the hole for the transmitted direct beam.

For back reflections, i.e. where 2 > 90 you can measure S2 as the distance from the beam entry point.

The distance S1 corresponds to a diffraction angle of 2. The angle between the diffracted and thetransmitted beams is always 2. We know that the distance between the holes in the film, W,corresponds to a diffraction angle of = . So we can find from:

or
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We know Bragg's Law: n = 2dsin

and the equation for interplanar spacing, d, for cubic crystals is given by:

where a is the lattice parameter

this gives:

From the measurements of each arc we can now generate a table ofS1, and sin2.

If all the diffraction lines are considered, then the experimental values of sin2 should form a

pattern related to the values ofh, kand lfor the structure.

We now multiply the values of sin2 by some constant value to give nearly integer values forall the h2+ k2+ l2 values. Integer values are then assigned.

The integer values ofh2+ k2+ l2 are then equated with theirhklvalues to index each arc, using

the table shown below:


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For some structures e.g. bcc, fcc, not all planes reflect, so some of the arcs may be missing.

It is then possible to identify certain structures, in this case fcc (- the planes have hklvalues: alleven, or all odd in the table above).

For each line we can also calculate a value fora, the lattice parameter. For greater accuracy thevalue is

Neutron diffraction

Neutron diffraction is a method for the determination of the atomic and/or magnetic structure of amaterial. It can be equally well applied to study crystalline solids (see crystallography), gasses,

liquids or amorphous materials. Neutron diffraction is a form ofelastic 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 ofthermal or coldneutrons and the

intensity pattern around the sample gives information of the structure of theDescription

[edit] Principle

Neutrons are particles found in the atomic nucleus of almost all atoms, but they are bound. The techniquerequires free neutrons and these normally do not occur in nature, because they have limited life-time. In anuclear reactor, however, neutrons can be set free throughnuclei decay particularly when fissionoccurs. Allquantumparticles can exhibit wave phenomena we typically associate with light or sound.Diffraction is one ofthese phenomena; it occurs when waves encounter obstacles whose size is comparable with the wavelength. Ifthe wavelength of a quantum particle is short enough, atoms or their nuclei can serve as diffraction obstacles.When a beam of neutrons emanating from a reactor is slowed down and selected properly by their speed, theirwavelength lies near one ngstrm (0.1 nanometer), the typical separation between atoms in a solid material.Such a beam can then be used to perform a diffraction experiment. Impinging on a crystalline sample it willscatter under a limited number of well-defined angles according to the same Bragg's law that describes X-raydiffraction.


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[edit] Instrumental requirements

A neutron diffraction measurement requires a neutron source (e.g. anuclear reactororspallation source), asample (the material to be studied), and a detector. Samples sizes are large compared to those used in X-raydiffraction. The technique is therefore mostly performed aspowder diffraction. At a research reactor other

components such as crystal monochromatorsor filters may be needed to select the desired neutron wavelength.Some parts of the setup may also be movable. At a spallation source the time of flight technique is used to sortthe energies of the incident neutrons, so no monochromator is needed, just a bunch of electronics. (Higherenergy neutrons are faster - v. simple)

[edit] Nuclear scattering

Neutrons interact with matter differently than x-rays. X-rays interact primarily with theelectroncloudsurrounding each atom. The contribution to the diffracted x-ray intensity is therefore larger for atoms with alarge atomic number (Z) than it is for atoms with a small Z. On the other hand, neutrons interact directly withthe 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) atomscontribute strongly to the diffracted intensity even in the presence of large Z atoms. The scattering length variesfrom isotope to isotope rather than linearly with the atomic number. An element like Vanadiumis a strongscatterer 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 meansthat there is no need for a atomic form factorto describe the shape of the electron cloud of the atom and thescattering power of an atom does not fall off with the scattering angle as it does for X-rays. Diffractogramstherefore can show strong well defined diffraction peaks even at high angles, particularly if the experiment isdone at low temperatures. Many neutron sources are equipped with liquid helium cooling systems that allow to

collect data at temperatures down to 4.2 K. The superb high angle (i.e. high resolution) information means thatthe data can give very precise values for the atomic positions in the structure. On the other hand, Fourier maps(and to a lesser extentdifference Fourier maps) derived from neutron data suffer from series termination errors,sometimes so much that the results are meaningless.

[edit] Magnetic scattering

Although neutrons are uncharged, they carry a spin, and therefore interact with magnetic moments, includingthose arising from the electron cloud around an atom. Neutron diffraction can therefore reveal the microscopicmagnetic structure of a material[1].

Magnetic scattering does require an atomic form factoras it is caused by the much larger electron cloud aroundthe tiny nucleus. The intensity of the magnetic contribution to the diffraction peaks will therefore dwindletowards higher angles.

[edit] History

The first neutron diffraction experiments were carried out in 1945 by Ernest O. Wollan using the GraphiteReactor at Oak Ridge. He was joined shortly thereafter byClifford Shull, and together they established the basicprinciples of the technique, and applied it successfully to many different materials, addressing problems like thestructure of ice and the microscopic arrangements of magnetic moments in materials. For this achievement
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Shull was awarded one half of the 1994Nobel Prize in Physics. Wollan had passed away in the 1990s. (Theother half of the 1994 Nobel Prize for Physics went to Bert Brockhousefor development of the inelasticscattering technique at theChalk River facility ofAECL. This also involved the invention of the triple axisspectrometer). Brockhouse and Shull jointly take the somewhat dubious distinction of the longest gap betweenthe work being done (1945) and the Nobel Prize being awarded (1994).

[edit] Uses

Neutron diffraction is closely related to X-raypowder diffraction[2] . In fact the single crystal version of thetechnique is less commonly used because currently available neutron sources require relatively large samplesand large single crystals are hard or impossible to come by for most materials. Future developments, however,may well change this picture. Because the data a typically a 1D powder diffractogram they are usuallyprocessed using Rietveld refinement. In fact the latter found its origin in neutron diffraction (at Petten in theNetherlands) and was later extended for use in X-ray diffraction.

One practical application of elastic neutron scattering/diffraction is that the lattice constant ofmetals and othercrystalline materials can be very accurately measured. Together with an accurately aligned micropositioner amap of the lattice constant through the metal can be derived. This can easily be converted to the stress fieldexperienced by the material. This has been used to analyse stresses in aerospace and automotivecomponents togive just two examples. This technique has led to the development of dedicated stress diffractometers, such asthe ENGIN-X instrument at the ISIS neutron source.

[edit] Hydrogen, null-scattering and contrast variation

Neutron diffraction can be used to establish the structure of low atomic number materials like proteins andsurfactants much more easily with lower flux than at a synchrotron radiation source. This is because some low

atomic number materials have a higher cross section for neutron interaction than higher atomic weightmaterials.

One major advantage of neutron diffraction over X-ray diffraction is that the latter is rather insensitive to thepresence of hydrogen in a structure, whereas the nuclei 1H and 2H=D are strong scatterers for neutrons. Thismeans that the position of hydrogen in a crystal structure and its thermal motions can be determined far moreprecisely with neutrons. In addition the scattering lengths (structure factors in x-ray parlance) of H and D haveopposite sign, which allows contrast variation. In fact there is a particularisotope ratio for which thecontribution of the element would cancel, this is called null-scattering. In practice however it is not desirable towork with the relatively high concentration of H in such a sample. The scatter by H-nuclei has a large ineleasticcomponent and this creates a large continuous background that is more or less independent of scattering angle.

The elastic pattern typically consists of sharpBragg reflections if the sample is crystalline. They tend to drownin the inelastic background. This is even more serious when the technique is used for the study of liquidstructure. Nevertheless, by preparing samples with different isotope ratios it is possible to vary the scatteringcontrast enough to highlight one element in an otherwise complicated structure. The variation of other elementsis possible but usually rather expensive. Hydrogen is inexpensive and particularly interesting because it plays anexceptionally large role in biochemical structures and is difficult to study structurally in other ways.
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X-Ray 3 of 7In 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 themonochromatic 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, thediffraction spots lie on horizontal lines.

Explore the rotating crystal method by clicking on the start button repeatedly.

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

X-Ray 4 of 7The Laue method is mainly used to determine the orientation of large single crystals. White radiation isreflected from, or transmitted through, a fixed crystal.

The diffracted beams form arrays of spots, that lie on curves on the film. The Bragg angle is fixed forevery set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelengthfrom the white radiation that satisfies the Bragg law for the values ofdand involved. Each curvetherefore corresponds to a different wavelength. The spots lying on any one curve are reflections fromplanes belonging to one zone. Laue reflections from planes of the same zone all lie on the surface of animaginary cone whose axis is the zone axis.

Experimental

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

Back-reflection Laue

In the back-reflection method, the film is placedbetween the x-ray source and the crystal. The beams

which are diffracted in a backward direction arerecorded.

One side of the cone of Laue reflections is defined bythe transmitted beam. The film intersects the cone, withthe diffraction spots generally lying on an hyperbola.
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Transmission Laue

In the transmission Laue method, the film is placedbehind the crystal to record beams which aretransmitted through the crystal.

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

Crystal orientation is determined from the position of the spots. Each spot can be indexed, i.e. attributedto a particular plane, using special charts. The Greninger chart is used for back-reflection patterns and the

Leonhardt chart for transmission patterns.

The Laue technique can also be used to assess crystal perfection from the size and shape of the spots. Ifthe crystal has been bent or twisted in anyway, the spots become distorted and smeared out.

X-Ray 5 of 7


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

If amonochromatic x-ray beam isdirected at a single crystal, then

only one or two diffracted beamsmay result.

If the sample consists of some tensof randomly orientated singlecrystals, the diffracted beams areseen to lie on the surface of severalcones. The cones may emerge inall directions, forwards andbackwards.

A sample of some hundreds ofcrystals (i.e. a powdered sample)show that the diffracted beamsform continuous cones.

A circle of film is used to recordthe diffraction pattern as shown.Each cone intersects the filmgiving diffraction lines. The linesare seen as arcs on the film.

For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the corrBragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the
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appearance of a continuous line. If the crystal is not ground finely enough, the diffraction lines appear speckled

This arrangement is achieved practically in the Debye Scherrer camera illustrated here...

X ray methods

Diffraction can occur wheneverBragg's lawissatisfied. Withmonochromatic radiation, an arsetting of a single crystal in an x-ray beam willgenerally produce any diffracted beams. Theretherefore be very little information in a single cdiffraction pattern from using monochromaticradiation.

X-ray diffraction techniques

X-ray scattering techniques are a family of non-destructive analytical techniques which reveal informationabout the crystallographic structure, chemical composition, and physical properties of materials

and thin films. These techniques are based on observing thescatteredintensityof an X-ray beam

hitting a sample as a function of incident and scattered angle, polarization, and wavelength or

energy.

X-ray diffraction finds the geometry or shape of a molecule using X-rays. X-ray diffraction techniques arebased on the elastic scattering of X-rays from structures that havelong range order. The most comprehensivedescription of scattering from crystals is given by the dynamical theory of diffraction.[1]

Single-crystal X-ray diffraction is a technique used to solve the complete structure of crystallinematerials, ranging from simple inorganic solidsto complex macromolecules, such asproteins.

Powder diffraction (XRD) is a technique used to characterize the crystallographic structure, crystallitesize (grain size), and preferred orientation in polycrystalline or powdered solid samples. Powderdiffraction is commonly used to identify unknown substances, by comparing diffraction data against adatabase maintained by the International Centre for Diffraction Data. It may also be used to characterize

This problem can be overcome by continuously varying orover a range of values, to satisfy Bragg's law. Practically this isdone by:

using a range of x-ray wavelengths (i.e. white radiation), or

by rotating the crystal or, using a powder or polycrystallinespecimen.

By selecting combinations of x-ray ranges and specimen types,discover the different techniques used in x-ray diffraction.
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heterogeneous solid mixtures to determine relative abundance of crystalline compounds and, whencoupled with lattice refinement techniques, such as Rietveld refinement, can provide structuralinformation on unknown materials. Powder diffraction is also a common method for determining strainsin crystalline materials. An effect of the finite crystallite sizes is seen as a broadening of the peaks in anX-ray diffraction as is explained by the Scherrer Equation.

Thin film diffraction and grazing incidence X-ray diffraction may be used to characterize thecrystallographic structure and preferred orientation of substrate-anchored thin films. High-resolution X-ray diffraction is used to characterize thickness, crystallographic structure, and strain

in thin epitaxial films. It employs parallel-beam optics. X-raypole figureanalysis enables one to analyze and determine the distribution of crystalline

orientations within a crystalline thin-film sample. X-ray rocking curve analysis is used to quantify grain size and mosaic spread in crystalline materials.

Scattering techniques

Elastic scattering

Materials that do not have long range order may also be studied by scattering methods that rely onelasticscattering of monochromatic X-rays.

Small angle X-ray scattering (SAXS)probes structure in the nanometer to micrometer range bymeasuring scattering intensity at scattering angles 2 close to 0.[2]

X-ray reflectivity is an analytical technique for determining thickness, roughness, and density of singlelayer and multilayer thin films.

Wide angle X-ray scattering (WAXS), a technique concentrating on scattering angles 2 larger than 5.

Inelastic scattering

When the energy and angle of the inelastically scattered X-rays are monitored scattering techniques can be usedto probe the electronic band structure of materials.

Compton scattering Resonant inelastic X-ray scattering (RIXS) X-ray Raman scattering X-ray diffraction pattern
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electron diffraction is a technique used to study matter by firing electrons at a sample and...

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