CHAPTER 3 X-RAY DIFFRACTION IN CRYSTAL X-Ray Diffraction .

1Bertha Röntgen’sHand 8 Nov, on of Waves byCrystalsX-Ray DiffractionBragg EquationX-Ray MethodsNeutron & Electron DiffractionCHAPTER 3X-RAY DIFFRACTIONIN CRYSTAL

He was awarded the Nobelprize for physics in 1901. 2X-rays were discovered in1895bytheGermanphysicist Wilhelm ConradRöntgen and were so namedbecause their nature wasunknown at the time. X-RAYWilhelm Conrad Röntgen(1845-1923)

3X ray, invisible, highly penetrating electromagnetic radiation ofmuch shorter wavelength (higher frequency) than visible light.The wavelength range for X rays is from about 10-8 m to about10-11 m, the corresponding frequency range is from about 3 1016 Hz to about 3 1019 Hz.X-RAY PROPERTIES

E λhc4λ x-ray 10-10 1A E 104 evE νλhcλcE hνElectromagnetic radiation described as having packets ofenergy, or photons. The energy of the photon is related toits frequency by the following formula:λ Wavelength , ע Frequency , c Velocity of light X-RAY ENERGY

When an electron drops to a lower orbital, it needs torelease some energy; it releases the extra energy in theform of a photon. The energy level of the photon dependson how far the electron dropped between orbitals. 5Visible light photons and X-ray photons are bothproduced by the movement of electrons in atoms.Electrons occupy different energy levels, or orbitals,around an atom's nucleus. PRODUCTION OF X-RAYS

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7AnodeEvacuated glass bulbCathodeX rays can be produced in a highly evacuated glass bulb, calledan X-ray tube, that contains essentially two electrodes—ananode made of platinum, tungsten, or another heavy metal ofhigh melting point, and a cathode. When a high voltage isapplied between the electrodes, streams of electrons (cathoderays) are accelerated from the cathode to the anode andproduce X rays as they strike the anode.X-RAY TUBE

Other electrons, which are decelerated by the periodicpotential of the metal, produce a broad spectrum of X-rayfrequencies.Depending on the diffraction experiment, either or both ofthese X-ray spectra can be used.10 8Some of these electrons excite electrons from core states inthe metal, which then recombine, producing highlymonochromatic X-rays. These are referred to ascharacteristic X-ray lines. 4X-rays can be created by bombarding a metal target withhigh energy ( 10 4 ) electrons. Monochromatic and BroadSpectrum of X-rays

X-rayRadio waves don't have enoughenergy to move electrons betweenorbitals in larger atoms, so they passthrough most stuff. X-ray photons alsopass through most things, but for theopposite reason: They have too muchenergy. 9.somethingyou won'tsee veryoften(VisibleLight)The atoms that make up your bodytissue absorb visible light photonsvery well. The energy level of thephoton fits with various energydifferencesbetweenelectronpositions. ABSORPTION OF X-RAYS

10The free electron collideswith the tungsten atom,knocking an electron out of alower orbital. A higher orbitalelectron fills the emptyposition, releasing its excessenergy as a photon.An electron in a higher orbital immediately falls to the lowerenergy level, releasing its extra energy in the form of a photon. It'sa big drop, so the photon has a high energy level; it is an X-rayphoton.Generation of X-rays (K-ShellKnockout)

The soft tissue in your body is composed of smalleratoms, and so does not absorb X-ray photons particularlywell. The calcium atoms that make up your bones aremuch larger, so they are better at absorbing X-rayphotons. 11A larger atom is more likely to absorb an X-ray photon inthis way, because larger atoms have greater energydifferences between orbitals -- the energy level moreclosely matches the energy of the photon. Smaller atoms,where the electron orbitals are separated by relatively lowjumps in energy, are less likely to absorb X-ray photons. Absorption of X-rays

Diffraction occurs with electromagneticwaves, such as light and radio waves,and also in sound waves and waterwaves.The most conceptually simple exampleof diffraction is double-slit diffraction,that’s why firstly we remember lightdiffraction. 12Diffraction is a wave phenomenon inwhich the apparent bending andspreading of waves when they meet anobstruction. DIFFRACTIONDistance d ConstantWavelength Constant(600 nm)Width b Variable(500-1500 nm)

13Thus Young’s light interferenceexperiment proves that lighthas wavelike properties.Light diffraction is caused by light bending around the edge ofan object. The interference pattern of bright and dark lines fromthe diffraction experiment can only be explained by the additivenature of waves; wave peaks can add together to make abrighter light, or a peak and a through will cancel each other outand result in darkness.LIGHT DIFFRACTION

14LIGHT INTERFERENCE

15Constructive interference isthe result of synchronizedlightwavesthataddtogether to increase the lightintensity. Destructive Đnterference.results when two out-of-phaselight waves cancel each otherout, resulting in darkness.Constructive & Destructive Waves

16Light Interference

17Solid material What happens if the beam isincident on solid material? If weconsider a crystalline material, thescattered beams may add togetherin a few directions and reinforceeach other to give diffracted beamsSingle particle To understand diffraction we alsohave to consider what happens whena wave interacts with a single particle.The particle scatters the incidentbeam uniformly in all directionsDiffraction from a particle and solid

NeutronElectron18The general princibles will be the same for each type of waves.X-rayDiffractionSolid State Physics deals how the waves are propagatedthrough such periodic structures. In this chapter we study thecrystal structure through the diffraction of photons (X-ray),nuetrons and electrons.A crystal is a periodic structure( unit cells are repeated regularly)Diffraction of Waves by Crystals

19The diffraction depends on the crystal structure and onthe wavelength.At optical wavelengths such as 5000 angstroms thesuperposition of the waves scattered elastically by theindividual atoms of a crystal results in ordinary opticalrefraction.When the wavelength of the radiation is comparablewith or smaller than the lattice constant, one can finddiffracted beams in directions quite different from theincident radiation.Diffraction of Waves by Crystals

Beam diffraction takes place only in certain specificdirections, much as light is diffracted by a grating.By measuring the directions of the diffraction and thecorresponding intensities, one obtains informationconcerning the crystal structure responsible fordiffraction. 20The structure of a crystal can be determined bystudying the diffraction pattern of a beam of radiationincident on the crystal. Diffraction of Waves by Crystals

X-ray diffraction (XRD) 21X-ray crystallography is a technique in crystallography inwhich the pattern produced by the diffraction of x-rays throughthe closely spaced lattice of atoms in a crystal is recorded andthen analyzed to reveal the nature of that lattice. X-RAY CRYSTALLOGRAPHY

We need X-rays: 22hc3Ex ray hω hυ 12.3x10eV 10λ 1x10 mhcThe wavelength of X-rays istypically 1 A , comparable to theinteratomic spacing (distancesbetween atoms or ions) in solids. X-Ray Crystallography

Information about the structure of the lines on thegrating can be obtained by measuring the relativeintensities of different ordersSimilarly, measurement of the separation of the X-raydiffraction maxima from a crystal allows us to determinethe size of the unit cell and from the intensities ofdiffracted beams one can obtain information about thearrangementof atoms within the cellcell.23A crystal behaves as a 3-D diffraction grating for x-raysIn a diffraction experiment, the spacing of lines on the gratingcan be deduced from the separation of the diffraction maximaCrystal Structure Determination

24W. L. Bragg presented a simpleexplanation of the diffracted beams from acrystal.The Bragg derivation is simple but isconvincing only since it reproduces thecorrect result.X-Ray Diffraction

English physicists Sir W.H. Braggand his son Sir W.L. Braggdeveloped a relationship in 1913 toexplain why the cleavage faces ofcrystals appear to reflect X-raybeams at certain angles of incidence(theta, θ).This observation is anexample of X-ray wave interference.Sir William Henry Bragg (1862-1942),William Lawrence Bragg (1890-1971)25"for their services in the analysis of crystal structure by means ofXrays".o 1915, the father and son were awarded the Nobel prize for physics X-Ray Diffraction & Bragg Equation

26Bragg law identifies the angles of the incidentradiation relative to the lattice planes for whichdiffraction peaks occurs.Bragg derived the condition for constructiveinterference of the X-rays scattered from a set ofparallel lattice planes.Bragg Equation

27өөW.L. Bragg considered crystals to be made up of parallelplanes of atoms. Incident waves are reflected specularly fromparallel planes of atoms in the crystal, with each plane isreflecting only a very small fraction of the radiation, like alightly silvered mirror.In mirrorlike reflection the angle of incidence is equal to theangle of reflection.BRAGG EQUATION

28The diffracted beams are found to occurwhen the reflections from planes of atomsinterfere constructively. We treatelastic scattering, in which theenergy of X-ray is not changed on reflection.Diffraction Condition

29θθ2θTotal DiffractedAngle 2θθ Incident angleθ Reflected angleλ Wavelength of X-rayWhen the X-rays strike a layer of a crystal, some of them willbe reflected. We are interested in X-rays that are in-phasewith one another. X-rays that add together constructively in xray diffraction analysis in-phase before they are reflected andafter they reflected.Bragg Equation

30DE d sin θThe line CE is equivalentto the distance betweenthe two layers (d)These two x-ray beams travel slightly different distances. Thedifference in the distances traveled is related to the distancebetween the adjacent layers.Connecting the two beams with perpendicular lines shows thedifference between the top and the bottom beams.Bragg Equation

31Constructive interference of the radiation from successiveplanes occurs when the path difference is an integralnumber of wavelenghts. This is the Bragg Law.nλ 2d sin θDE EF 2d sin θDE d sin θEF d sin θThe length DE is the same as EF, so the total distancetraveled by the bottom wave is expressed by:Bragg Law

The diffracted beams (reflections) from any set of lattice planescan only occur at particular angles pradicted by the Bragg law. 32This is why we cannot use visible light. No diffraction occurs whenthe above condition is not satisfied.nλ 2dBragg reflection can only occur for wavelength where, d is the spacing of the planes and n is the order of diffraction.2 d sin θ n λBragg Equation

33ADθ θCB2θX-rays are incident at an angle θ on one of the planesof the set.There will be constructive interference of the wavesscattered from the two successive lattice points A and B inthe plane if the distances AC and DB are equal.Scattering of X-rays from adjacentlattice points A and B

34We consider the scattering from lattice points ratherthan atoms because it is the basis of atoms associated witheach lattice point that is the true repeat unit of the crystal;The lattice point is analoque of the line on optical diffractiongrating and the basis represents the structure of the line.The diffracted wave looks as if it has been reflected from theplaneIf the scattered wave makes the same angle to the plane asthe incident waveConstructive interference of wavesscattered from the same plane

352d sinθ nλThis will be the case if the path difference forscattering off two adjacent planes is an integralnumber of wavelengthsCoherent scattering from a single plane is notsufficient to obtain a diffraction maximum. It is alsonecessary that successive planes should scatterin phaseDiffraction maximum