In these set of slides we shall consider: scattering from an electron scattering from an atom structure factor calculations (scattering from an unit cell) the relative intensity of reflections in power patternsElements of X-Ray DiffractionB.D. Cullity & S.R. StockPrentice Hall, Upper Saddle River (2001)

Intensity of the Scattered electronsElectronAtomUnit cell (uc)Scattering by a crystalABCPolarization factorAtomic scattering factor (f)Structure factor (F)Click here to jump to structure factor calculations

Scattering by an ElectronSets electron into oscillationEmission in all directionsScattered beamsCoherent (definite phase relationship)AThe electric field (E) is the main cause for the acceleration of the electron The moving particle radiates most strongly in a direction perpendicular to its motionThe radiation will be polarized along the direction of its motion

xzrPIntensity of the scattered beam due to an electron (I) at a point Psuch that r >> For a wave oscillating in z directionFor an polarized waveThe reason we are able to neglect scattering from the protons in the nucleusThe scattered rays are also plane polarized

For an unpolarized waveE is the measure of the amplitude of the waveE2 = IntensityIPy = Intensity at point P due to EyIPz = Intensity at point P due to EzTotal Intensity at point P due to Ey & Ez

Sum of the squares of the direction cosines =1HenceIn terms of 2

Scattered beam is not unpolarized Forward and backward scattered intensity higher than at 90 Scattered intensity minute fraction of the incident intensityVery small number

Polarization factor Comes into being as we used unpolarized beam

BScattering by an AtomScattering by an atom [Atomic number, (path difference suffered by scattering from each e, )]Scattering by an atom [Z, (, )] Angle of scattering leads to path differences In the forward direction all scattered waves are in phaseAtomic Scattering Factor or Form Factor

As , path difference constructive interferenceAs , path difference destructive interferenceEquals number of electrons, say for f = 29, it should Cu or Zn+

Coherent scatteringIncoherent (Compton) scatteringZ Sin() /

BScattering by an AtomBRUSH-UP The conventional UC has lattice points as the verticesThere may or may not be atoms located at the lattice pointsThe shape of the UC is a parallelepiped (Greek paralllepipedon) in 3DThere may be additional atoms in the UC due to two reasons: The chosen UC is non-primitive The additional atoms may be part of the motif

CScattering by the Unit cell (uc) Coherent Scattering Unit Cell (UC) is representative of the crystal structure Scattered waves from various atoms in the UC interfere to create the diffraction patternThe wave scattered from the middle plane is out of phase with the ones scattered from top and bottom planes. I.e. if the green rays are in phase (path difference of ) then the red ray will be exactly out of phase with the green rays (path difference of /2).

d(h00)BRay 1 = R1Ray 2 = R2Ray 3 = R3Unit CellxMCNRBSA(h00) planea

Extending to 3DIndependent of the shape of UCNote: R1 is from corner atoms and R3 is from atoms in additional positions in UC

If atom B is different from atom A the amplitudes must be weighed by the respective atomic scattering factors (f)The resultant amplitude of all the waves scattered by all the atoms in the UC gives the scattering factor for the unit cellThe unit cell scattering factor is called the Structure Factor (F)Scattering by an unit cell = f(position of the atoms, atomic scattering factors)In complex notationStructure factor is independent of the shape and size of the unit cell!F FhklFor n atoms in the UCIf the UC distorts so do the planes in it!!Note:n is an integer

Structure factor calculationsAAtom at (0,0,0) and equivalent positions F is independent of the scattering plane (h k l)Simple Cubic All reflections are present

BAtom at (0,0,0) & (, , 0) and equivalent positions F is independent of the l indexC- centred OrthorhombicRealBoth even or both oddMixture of odd and evene.g. (001), (110), (112); (021), (022), (023)e.g. (100), (101), (102); (031), (032), (033)(h + k) even(h + k) odd

If the blue planes are scattering in phase then on C- centering the red planes will scatter out of phase (with the blue planes- as they bisect them) and hence the (210) reflection will become extinctThis analysis is consistent with the extinction rules: (h + k) odd is absent

In case of the (310) planes no new translationally equivalent planes are added on lattice centering this reflection cannot go missing.This analysis is consistent with the extinction rules: (h + k) even is present

CAtom at (0,0,0) & (, , ) and equivalent positionsBody centred OrthorhombicReal(h + k + l) even(h + k + l) odde.g. (110), (200), (211); (220), (022), (310)e.g. (100), (001), (111); (210), (032), (133)This implies that (h+k+l) even reflections are only present.The situation is identical in BCC crystals as well.

DAtom at (0,0,0) & (, , 0) and equivalent positionsFace Centred CubicReal(h, k, l) unmixed(h, k, l) mixede.g. (111), (200), (220), (333), (420)e.g. (100), (211); (210), (032), (033)(, , 0), (, 0, ), (0, , )Two odd and one even (e.g. 112); two even and one odd (e.g. 122)h,k,l all even or all odd

(h, k, l) mixede.g. (100), (211); (210), (032), (033)Mixed indicesTwo odd and one even (e.g. 112); two even and one odd (e.g. 122)Unmixed indices(h, k, l) unmixede.g. (111), (200), (220), (333), (420)All odd (e.g. 111); all even (e.g. 222)This implies that in FCConly h,k,l unmixed reflections are present.

Mixed indicesCASEhklAooeBoee

Unmixed indicesCASEhklAoooBeee

ENa+ at (0,0,0) + Face Centering Translations (, , 0), (, 0, ), (0, , ) Cl at (, 0, 0) + FCT (0, , 0), (0, 0, ), (, , )NaCl: Face Centred Cubic

Zero for mixed indices(h, k, l) mixede.g. (100), (211); (210), (032), (033)Mixed indices

Mixed indicesCASEhklAooeBoee

(h, k, l) unmixedIf (h + k + l) is evenIf (h + k + l) is odde.g. (111), (222); (133), (244)e.g. (222),(244)e.g. (111), (133)Unmixed indicesh,k,l all even or all odd

Unmixed indicesCASEhklAoooBeee

FAl at (0, 0, 0) Ni at (, , )NiAl: Simple Cubic (B2- ordered structure)SCReal(h + k + l) even(h + k + l) odde.g. (110), (200), (211), (220), (310)e.g. (100), (111), (210), (032), (133)Click here to know more about ordered structuresWhen the central atom is identical to the corner ones we have the BCC case.This implies that (h+k+l) even reflections are only present in BCC.This term is zero for BCC

Reciprocal lattice/crystal of NiAle.g. (110), (200), (211); (220), (310)e.g. (100), (111), (210), (032), (133)Click here to know more about

GAl Atom at (0,0,0) Ni atom at (, , 0) and equivalent positionsSimple Cubic (L12 ordered structure)Real(h, k, l) unmixed(h, k, l) mixede.g. (111), (200), (220), (333), (420)e.g. (100), (211); (210), (032), (033)(, , 0), (, 0, ), (0, , )Two odd and one even (e.g. 112); two even and one odd (e.g. 122)NiAlh,k,l all even or all oddClick here to know more about ordered structures

e.g. (111), (200), (220), (333), (420)e.g. (100), (211); (210), (032), (033)Reciprocal lattice/crystal of Ni3AlClick here to know more about

Presence of additional atoms/ions/molecules in the UC can alter the intensities of some of the reflections

Selection / Extinction Rules

Bravais LatticeReflections which may be presentReflections necessarily absentSimpleallNoneBody centred(h + k + l) even(h + k + l) oddFace centredh, k and l unmixedh, k and l mixedEnd centredh and k unmixed C centredh and k mixed C centred

Bravais LatticeAllowed ReflectionsSCAllBCC(h + k + l) evenFCCh, k and l unmixedDCh, k and l are all odd Or all are even & (h + k + l) divisible by 4

h2 + k2 + l2SCFCCBCCDC110021101103111111111420020020052106211211782202202202209300, 2211031031011311311311122222222221332014321321151640040040040017410, 32218411, 330411, 33019331331331

We have already noted that absolute value of intensity of a peak (which is the area under a given peak) has no significance w.r.t structure identification.The relative value of intensities of the peak gives information about the motif.One factor which determines the intensity of a hkl reflection is the structure factor.In powder patterns many other factors come into the picture as in the next slide. Some of these have origin in crystallography (like the multiplicity factor), some have origin in the powder diffraction geometry (like the Lorentz factor), some are related to the radiation (polarization factor), etc.The multiplicity factor relates to the fact that we have 8 {111} planes giving rise to single peak, while there are only 6 {100} planes (and so forth). Hence, by this very fact the intensity of the {111} planes should be more than that of the {100} planes.A brief consideration of some these factors follows. The reader may consult Cullitys book for more details.Relative intensity of peaks in powder patterns

Structure Factor (F)Multiplicity factor (p)Polarization factorLorentz factorRelative Intensity of diffraction lines in a powder patternAbsorption factorTemperature factorScattering from UC (Atomic Scattering Factor is included in this term)Number of equivalent scattering planesEffect of wave polarizationCombination of 3 geometric factorsSpecimen absorptionThermal diffuse scattering

Multiplicity factor* Altered in crystals with lower symmetryActually only 3 planes ! (as (hkl) (h k l)

LatticeIndexMultiplicityPlanesCubic(with highest symmetry)(100)6[(100) (010) (001)] ( 2 for negatives)(110)12[(110) (101) (011), (110) (101) (011)] ( 2 for negatives)(111)12[(111) (111) (111) (111)] ( 2 for negatives)(210)24*(210) 3! Ways, (210) 3! Ways, (210) 3! Ways, (210) 3! Ways(211)24(211) 3 ways, (211) 3! ways, (211) 3 ways(321)48*Tetragonal(with highest symmetry)(100)4[(100) (010)] ( 2 for negatives)(110)4[(110) (110)] ( 2 for negatives)(111)8[(111) (111) (111) (111)] ( 2 for negatives)(210)8*(210) = 2 Ways, (210) = 2 Ways, (210) = 2 Ways, (210) = 2 Ways(211)16[Same as for (210) = 8] 2 (as l can be +1 or 1)(321)16*Same as above (as last digit is anyhow not permuted)

* Altered in crystals with lower symmetry (of the same crystal class)Multiplicity factor

Cubichklhhlhk0hh0hhhh0048*2424*1286Hexagonalhk.lhh.lh0.lhk.0hh.0h0.000.l24*12*12*12*662Tetragonalhklhhlh0lhk0hh0h0000l16*888*442Orthorhombichklhk0h0l0klh000k000l8444222Monoclinichklh0l0k0422Triclinichkl2

Polarization factorXRD pattern from PoloniumClick here for detailsExample of effect of Polarization factor on power patternThe polarization factor and the Lorentz factor both have dependence (only) can be combined into the L-P factor. Note that the LP factor has a dip in middle 2 values (with large values at low and high angles). [Fig.1].The signature of this factor is evident in the powder diffraction pattern from Po (simple cubic) as shown in Fig.2.Fig.2Fig.1Lorentz factor

Chart2

27.0459231361

14.4357581985

8.7301960306

5.7735026919

4.1447411324

3.2547157257

2.8284271247

2.731030765

2.9021789848

3.3333333333

4.0709572662

5.2541862933

7.2469332629

11.1810424728

22.7744492324

Bragg Angle (q, degrees)

Lorentz-Polarization factor

Sheet1

um

0-1

0.1-1.1111111111

0.2-1.25

0.3-1.4285714286

0.4-1.6666666667

0.5-2

0.6-2.5

0.7-3.3333333333

0.8-5

0.9-10

1

1.110

1.25

1.33.3333333333

1.42.5

1.52

1.61.6666666667

1.71.4285714286

1.81.25

1.91.1111111111

21

2.10.9090909091

2.20.8333333333

2.30.7692307692

2.40.7142857143

2.50.6666666667

2.60.625

Sheet1

u

m

u versus m

Sheet2

thetalorentz-polarization factor

50.0872664626260.3131458925

100.174532925263.4108429005

150.261799387827.0459231361

200.349065850414.4357581985

250.4363323138.7301960306

300.52359877565.7735026919

350.61086523824.1447411324

400.69813170083.2547157257

450.78539816342.8284271247

500.8726646262.731030765

550.95993108862.9021789848

601.04719755123.3333333333

651.13446401384.0709572662

701.22173047645.2541862933

751.3089969397.2469332629

801.396263401611.1810424728

851.483529864222.7744492324

901.570796326832649104555238100

95

900

Sheet2

Bragg Angle (q, degrees)

Lorentz-Polarization factor

Sheet3

Intensity of powder pattern lines (ignoring Temperature & Absorption factors) Valid for Debye-Scherrer geometry I Relative Integrated Intensity F Structure factor p Multiplicity factor POINTSAs one is interested in relative (integrated) intensities of the lines constant factors are omitted: Volume of specimen me , e (1/dectector radius)We have assumed random orientation of crystals in a material with Texture relative intensities are modified. So if a powder pattern has to be compared (both in intensity and angle of peaks) with standard JCPDS/ICDD files, then the effect of texture has to be removed first. Texture can be removed by: (i) making a fine powder and then annealing the same (if required) or (ii) rotating the sample to all orientations (practically how this can be done is not explained here!).I is really diffracted energy (as Intensity is Energy/area/time).Ignoring Temperature & Absorption factors is valid for lines close-by in pattern.