10 X-Ray Diffraction

8/9/2019 10 X-Ray Diffraction

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X-Ray Diffraction

Content of Lecture1. Diffraction2. Bragg law of light diffraction

3. X-Ray diffraction (XRD)4. Instrumentation - How x-ray is generated?5. Powder X-ray Diffraction6. Instrumentation (X-Ray diffractometer)7. How waves reveal the atomic structure of crystals8. Experimental X-Ray Diffraction Patterns


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Diffraction is a wave property of electromagnetic radiationthat causes the radiation to bend as it passes by an edge orthrough an aperture. Diffraction effects will increase as thedimension of the aperture approaches the wavelength of the

radiation.Diffraction of radiation gives rise to interference thatproduces dark and bright rings, lines, or spots, depending onthe geometry of the object causing the diffraction. Commoninterference effects for visible light are the rainbow patternproduced by an oil-film on wet pavement, and the diffraction oflight from a narrow-slit or a diffraction grating.

Diffraction MethodsWavelength will constructively interfere when it is

partially reflected between surfaces that produce a pathdifference equal to an integral number ofwavelengths. Thiscondition is described by the Bragg law (see later).

Schematic of crystal-structure determination by diffraction


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Interference of radiation between atomic planes in a crystal

sinU = opposite BC/hypotenuse AC(i.e. the basal-spacing d)d sinU = opposite BC2 d sinU = 2 opposite BC

= path difference (V-shaped blue-line in graphic)= nP

nP = 2 dsinUnPsinU = dwhere

n an integer

P wavelength of the radiationd basal-pacing between surfacesU angle between the radiation and the surfaces.

nP!2 dsinU is called the path difference


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This relation demonstrates that interference effects areobservable only when radiation interacts with physicaldimensions that are approximately the same size as thewavelength of the radiation. These interference effects are useful

for determining dimensions in crystal structures. Since thedistances between atoms or ions in a crystal is on the order of10-10m=10-8cm (=1), diffraction methods require radiation in theX-ray region of the electromagnetic spectrum, or beams ofelectrons orneutrons with a similar wavelength.

Electrons and neutrons are commonly thought of as particles,but they have wave properties with the wavelength depending onthe energy of the particles as described by the de Broglie equation.

The three diffraction methods (X-rays, electrons andneutrons) have different properties that we will not described here.Retain only that thepenetration depths of the three types of beamsare quite different (neutrons > X-rays > electrons).

X- Ray Diffraction (XRD)The wavelengths of x-rays are of the same order of

magnitude as distances between atoms in a molecule or crystal

(10-10m=1). A crystal diffracts an x-ray beam passing throughit to produce beams at specific angles depending on the x-raywavelength, the crystal orientation, and the structure of thecrystal.

X-rays are predominantly diffracted by electron density.Analysis of the diffraction angles produces an (electron densitymap) of the crystal. Since hydrogen atoms have very little electron

density, determining their positions requires extensive refinementof the diffraction pattern. Electron diffraction and neutrondiffraction are sensitive to nuclei and are often used toaccurately determine hydrogen positions.


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Instrumentation - How x-ray is generated?X-ray diffractometers consist of an x-ray generator, a

goniometer and sample holder, and an x-ray detector such asphotographic film or a movable proportional counter.

X-ray tubes generate x-rays by bombarding a metal targetwith high-energy (10-100 k eV) electrons that knock out coreelectrons. An electron in an outer shell fills the hole in theinner shell and emits an x-ray photon.

Two common targets are Mo and Cu, which have strongK(alpha) x-ray emission at 0.71073 and 1.5418, respectively.The x-rays can also be generated by decelerating electrons in atarget, or a synchrotron ring. These sources produce a continuous

spectrum of x-rays and require a crystal monochromator toselect a single wavelength.

Powder X-ray DiffractionPowders of crystalline materials diffract x-rays. The beam of

x-rays passing through a sample of randomly oriented micro-crystals produces a pattern of rings on a distant screen. Powderx-ray diffraction provides less information than single-crystal

diffraction. However, it is much simpler and faster. Powder x-raydiffraction is useful for confirming the identity of a solidmaterial and determining crystallinity and phase purity.

InstrumentationModern powder x-ray diffractometers consist of an x-ray

source, a movable sample platform, an x-ray detector, and

associated computer-controlled electronics. The sample is eitherpacked into a shallow cup-shaped holder ordeposited as a slurryonto a quatz substrate, and the sample holder spins slowlyduring the experiment to reduce sample heating. The x-raysource is usually the same as used in single-crystal Mo or Cudiffractometer. The x-ray beam is fixed and the sampleplatform rotates with respect to the beam by an angle theta (U).


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The detector rotates at twice the rate of the sample and is at anangle of2Uwith respect to the incoming x-ray beam.

X-RAY DIFFRACTOMWETER

This is a schematic of an x-ray diffraction system. The sample is mounted(normally on a glass slide) and placed on the axis of the diffractometer. The collectorrotates about that axis and collects data on the scattering intensity. This diagram showsa Cu anode. In this system, the radiation is associated along the primary K line

(wavelength of 1.54184 ). The collector then registers the intensity at 2U angles tothe sample plane. The data collected is sent to a personal computer to be analyzed.

X-ray diffraction works on the principle that x-rays form predictable diffractionpatterns when interacting with a crystalline matrix of atoms. Today, with the help ofcomputers to do the mathematical calculations involved, the diffraction patterns caneasily be converted into information about the location of atoms within thematrix. From those positions, the exact crystalline structure can be determined .


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Bragg's Law and Diffraction:Howwaves reveal the atomic structure of crystals

Applet created by Konstantin Lukin

What is Bragg's Law and Why is it Important?

Bragg's Law refers to the simple equation:nP = 2d sinU derived by the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg in 1913to explain why the cleavage faces of crystals appear to reflect X-ray beams atcertain angles of incidence (theta, U). The variable d is the distance between atomiclayers in a crystal, and the variable lambda P is the wavelength of the incident X-raybeam (see applet); n is an integer.

This observation is an example of X-ray wave interference , commonly known

as X-ray diffraction (XRD), and was direct evidence for the periodic atomic structureof crystals postulated for several centuries. The Braggs were awarded the NobelPrize in physics in 1915 for their work in determining crystal structures beginningwith NaCl, ZnS and diamond. Although Bragg's law was used to explain theinterference pattern of X-rays scattered by crystals, diffraction has been developed tostudy the structure of all states of matter with any beam, e.g., ions, electrons, neutrons,and protons, with a wavelength similar to the distance between the atomic ormolecular structures of interest .

Deriving Bragg's Law

Bragg's Law can easily be derived by considering the conditions necessary tomake the phases of the beams coincide when the incident angle equals the reflectingangle. The rays of the incident beams are always in phase and parallel up to thepoint at which the top beam strikes the top layer at atom z (Fig. 1). The second beamcontinues to the next layer where it is scattered by atom B. The second beam musttravel the extra distance AB + BC if the two beams are to continue traveling adjacentand parallel. This extra distance must be an integral (n) multiple of the wavelength(P) for the phases of the two beams to be the same:nP = AB +BC (2).

In deriving Bragg's Law using the reflection geometry and applying

trigonometry, the lower beam must travel the extra distance (AB + BC) to continuetraveling parallel and adjacent to the top beam.

Recognizing d as the hypotenuse of the right triangle Abz, we can usetrigonometry to relate d andU to the distance (AB + BC). The distance AB is oppositeU so,A B = d sinU (3).Because A B = B Ceq. (2) becomes,


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n P = 2 AB (4)Substituting eq. (3) in eq. (4) we have,n P = 2 d sinU (1)Thus, Bragg's Law has been derived. The location of the surface does not change thederivation of Bragg's Law.

Experimental Diffraction PatternsThe following figure is showing experimental x-ray diffraction pattern of cubic SiliconCarbide (SiC) using synchrotron radiation.

Text written by Paul J. SchieldsCenter for High Pressure ResearchDepartment of Earth & Space SciencesState University of New York at Stony BrookStony Brook, NY 11794-2100.

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10 X-Ray Diffraction