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**X-RAY DIFFRACTION X- Ray Sources Diffraction: Bragg’s Law**

Crystal Structure Determination Elements of X-Ray Diffraction B.D. Cullity & S.R. Stock Prentice Hall, Upper Saddle River (2001)

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**Hence, X-rays can be used for the study of crystal structures**

For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength In crystals the typical interatomic spacing ~ 2-3 Å so the suitable radiation is X-rays Hence, X-rays can be used for the study of crystal structures Target X-rays Beam of electrons A accelerating charge radiates electromagnetic radiation

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**K K Mo Target impacted by electrons accelerated by a 35 kV potential**

Characteristic radiation → due to energy transitions in the atom K White radiation Intensity 0.2 0.6 1.0 1.4 Wavelength ()

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Target Metal Of K radiation (Å) Mo 0.71 Cu 1.54 Co 1.79 Fe 1.94 Cr 2.29

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**Incident X-rays Fluorescent X-rays Electrons Scattered X-rays**

SPECIMEN Heat Fluorescent X-rays Electrons Scattered X-rays Compton recoil Photoelectrons Coherent From bound charges Incoherent (Compton modified) From loosely bound charges Transmitted beam X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles)

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**(Darkens the background of the diffraction patterns)**

Incoherent Scattering (Compton modified) From loosely bound charges Here the particle picture of the electron & photon comes handy Electron knocked aside 2 No fixed phase relation between the incident and scattered waves Incoherent does not contribute to diffraction (Darkens the background of the diffraction patterns)

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**Fluorescent X-rays Energy levels Characteristic x-rays**

Knocked out electron from inner shell Vacuum Energy levels Characteristic x-rays (Fluorescent X-rays) (10−16s later seems like scattering!) Nucleus

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**The secondary radiation is in all directions **

A beam of X-rays directed at a crystal interacts with the electrons of the atoms in the crystal The electrons oscillate under the influence of the incoming X-Rays and become secondary sources of EM radiation The secondary radiation is in all directions The waves emitted by the electrons have the same frequency as the incoming X-rays coherent The emission will undergo constructive or destructive interference Secondary emission Incoming X-rays

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**Sets Electron cloud into oscillation Sets nucleus into oscillation **

Small effect neglected

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**Oscillating charge re-radiates In phase with the incoming x-rays**

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**d dSin The path difference between ray 1 and ray 2 = 2d Sin**

BRAGG’s EQUATION Deviation = 2 Ray 1 Ray 2 d dSin The path difference between ray 1 and ray 2 = 2d Sin For constructive interference: n = 2d Sin

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**Incident and scattered waves are in phase if**

In plane scattering is in phase Incident and scattered waves are in phase if Scattering from across planes is in phase

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**But this is still reinforced scattering and NOT reflection**

Extra path traveled by incoming waves AY These can be in phase if and only if incident = scattered Extra path traveled by scattered waves XB But this is still reinforced scattering and NOT reflection

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**Diffraction = Reinforced Coherent Scattering**

Bragg’s equation is a negative law If Bragg’s eq. is NOT satisfied NO reflection can occur If Bragg’s eq. is satisfied reflection MAY occur Diffraction = Reinforced Coherent Scattering Reflection versus Scattering Reflection Diffraction Occurs from surface Occurs throughout the bulk Takes place at any angle Takes place only at Bragg angles ~100 % of the intensity may be reflected Small fraction of intensity is diffracted X-rays can be reflected at very small angles of incidence

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**n is an integer and is the order of the reflection **

n = 2d Sin n is an integer and is the order of the reflection For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å n Sin 1 0.34 20.7º First order reflection from (110) 2 0.69 43.92º Second order reflection from (110) Also written as (220)

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**In XRD nth order reflection from (h k l) is considered as 1st order reflection from (nh nk nl)**

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**Crystal structure determination**

Many s (orientations) Powder specimen POWDER METHOD Monochromatic X-rays Single LAUE TECHNIQUE Panchromatic X-rays ROTATING CRYSTAL METHOD Varied by rotation Monochromatic X-rays

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THE POWDER METHOD

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**A B C Intensity of the Scattered electrons Scattering by a crystal**

Atom Unit cell (uc)

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**A Scattering by an Electron Coherent (definite phase relationship)**

Emission in all directions Sets electron into oscillation Coherent (definite phase relationship) Scattered beams

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**Intensity of the scattered beam due to an electron (I)**

For an polarized wave z P For a wave oscillating in z direction r x Intensity of the scattered beam due to an electron (I)

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**For an unpolarized wave**

E is the measure of the amplitude of the wave E2 = Intensityc IPy = Intensity at point P due to Ey IPz = Intensity at point P due to Ez

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** Scattered beam is not unpolarized**

Very small number Rotational symmetry about x axis + mirror symmetry about yz plane Forward and backward scattered intensity higher than at 90 Scattered intensity minute fraction of the incident intensity Polarization factor Comes into being as we used unpolarized beam

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**B Scattering by an Atom f → (Å−1) →**

Scattering 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 phase f → (Å−1) → 0.2 0.4 0.6 0.8 1.0 10 20 30 Schematic

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**Incoherent (Compton) scattering**

Coherent scattering Incoherent (Compton) scattering Z Sin() /

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**C Scattering by the Unit cell (uc)**

Coherent Scattering Unit Cell (uc) representative of the crystal structure Scattered waves from various atoms in the uc interfere to create the diffraction pattern The wave scattered from the middle plane is out of phase with the ones scattered from top and bottom planes

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**x d(h00) a B A R S B M N C Ray 1 = R1 Ray 3 = R3 Ray 2 = R2**

Unit Cell x R S Ray 2 = R2 B d(h00) a M N (h00) plane C

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**Independent of the shape of uc**

Extending to 3D Independent of the shape of uc Note: R1 is from corner atoms and R3 is from atoms in additional positions in uc

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In complex notation 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 cell The unit cell scattering factor is called the Structure Factor (F) Scattering by an unit cell = f(position of the atoms, atomic scattering factors) Structure factor is independent of the shape and size of the unit cell

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**Structure factor calculations**

Simple Cubic A Atom at (0,0,0) and equivalent positions F is independent of the scattering plane (h k l)

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**Real C- centred Orthorhombic**

Atom at (0,0,0) & (½, ½, 0) and equivalent positions C- centred Orthorhombic Real (h + k) even Both even or both odd e.g. (001), (110), (112); (021), (022), (023) Mixture of odd and even (h + k) odd e.g. (100), (101), (102); (031), (032), (033) F is independent of the ‘l’ index

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**Real Body centred Orthorhombic**

Atom at (0,0,0) & (½, ½, ½) and equivalent positions Real (h + k + l) even e.g. (110), (200), (211); (220), (022), (310) (h + k + l) odd e.g. (100), (001), (111); (210), (032), (133)

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**Real Face Centred Cubic**

Atom at (0,0,0) & (½, ½, 0) and equivalent positions Face Centred Cubic (½, ½, 0), (½, 0, ½), (0, ½, ½) Real (h, k, l) unmixed e.g. (111), (200), (220), (333), (420) (h, k, l) mixed e.g. (100), (211); (210), (032), (033) Two odd and one even (e.g. 112); two even and one odd (e.g. 122)

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E Na+ at (0,0,0) + Face Centering Translations (½, ½, 0), (½, 0, ½), (0, ½, ½) Cl− at (½, 0, 0) + FCT (0, ½, 0), (0, 0, ½), (½, ½, ½) NaCl: Face Centred Cubic

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**Zero for mixed indices If (h + k + l) is even If (h + k + l) is odd**

(h, k, l) mixed e.g. (100), (211); (210), (032), (033) (h, k, l) unmixed If (h + k + l) is even If (h + k + l) is odd Presence of additional atoms/ions/molecules in the uc (as a part of the motif ) can alter the intensities of some of the reflections

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**Relative Intensity of diffraction lines in a powder pattern**

Structure Factor (F) Scattering from uc Multiplicity factor (p) Number of equivalent scattering planes Polarization factor Effect of wave polarization Lorentz factor Combination of 3 geometric factors Absorption factor Specimen absorption Temperature factor Thermal diffuse scattering

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**Multiplicity factor Lattice Index Planes Cubic (100) 6**

[(100) (010) (001)] ( 2 for negatives) (110) 12 [(110) (101) (011), (110) (101) (011)] ( 2 for negatives) (111) 8 [(111) (111) (111) (111)] ( 2 for negatives) (210) 24 (210) = 3! Ways, (210) = 3! Ways, (210) = 3! Ways, (210) = 3! Ways, (211) 21 (321) 48 Tetragonal 4 [(100) (010)] [(110) (110)]

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Polarization factor Lorentz factor

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**Valid for Debye-Scherrer geometry I → Relative Integrated “Intensity” **

Intensity of powder pattern lines (ignoring Temperature & Absorption factors) Valid for Debye-Scherrer geometry I → Relative Integrated “Intensity” F → Structure factor p → Multiplicity factor POINTS As one is interested in relative (integrated) intensities of the lines constant factors are omitted Volume of specimen me , e (1/dectector radius) Random orientation of crystals in a with Texture intensities are modified I is really diffracted energy (as Intensity is Energy/area/time) Ignoring Temperature & Absorption factors valid for lines close-by in pattern

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**Crystal = Lattice + Motif**

In crystals based on a particular lattice the intensities of particular reflections are modified they may even go missing Diffraction Pattern Position of the Lattice points LATTICE Intensity of the diffraction spots MOTIF

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Reciprocal Lattice Properties are reciprocal to the crystal lattice The reciprocal lattice is created by interplanar spacings

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**A reciprocal lattice vector is to the corresponding real lattice plane**

The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane Planes in the crystal become lattice points in the reciprocal lattice ALTERNATE CONSTRUCTION OF THE REAL LATTICE Reciprocal lattice point represents the orientation and spacing of a set of planes

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Reciprocal Lattice The reciprocal lattice has an origin!

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**Note perpendicularity of various vectors**

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**Physics comes in from the following:**

Reciprocal lattice is the reciprocal of a primitive lattice and is purely geometrical does not deal with the intensities of the points Physics comes in from the following: For non-primitive cells ( lattices with additional points) and for crystals decorated with motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|Fhkl|2) Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity) Making of Reciprocal Crystal (Reciprocal lattice decorated with a motif of scattering power) The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment

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**SC Lattice = SC Reciprocal Lattice = SC**

Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2) SC 001 011 101 111 Lattice = SC 000 010 100 110 No missing reflections Reciprocal Lattice = SC Figures NOT to Scale

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**BCC Lattice = BCC Reciprocal Lattice = FCC 002 022 202 222 011 101 020**

000 Lattice = BCC 110 200 100 missing reflection (F = 0) 220 Reciprocal Lattice = FCC Weighing factor for each point “motif” Figures NOT to Scale

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**FCC Lattice = FCC Reciprocal Lattice = BCC 002 022 202 222 111 020 000**

200 220 100 missing reflection (F = 0) 110 missing reflection (F = 0) Weighing factor for each point “motif” Reciprocal Lattice = BCC Figures NOT to Scale

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The Ewald* Sphere The reciprocal lattice points are the values of momentum transfer for which the Bragg’s equation is satisfied For diffraction to occur the scattering vector must be equal to a reciprocal lattice vector Geometrically if the origin of reciprocal space is placed at the tip of ki then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere * Paul Peter Ewald (German physicist and crystallographer; ) See Cullity’s book: A15-4

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**K = K =g = Diffraction Vector**

Ewald Sphere The Ewald Sphere touches the reciprocal lattice (for point 41) Bragg’s equation is satisfied for 41 K = K =g = Diffraction Vector

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**Diffraction cones and the Debye-Scherrer geometry**

Film may be replaced with detector

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**Peaks or not idealized peaks broadend **

Powder diffraction pattern from Al Radiation: Cu K, = 1.54 Å 111 Note: Peaks or not idealized peaks broadend Increasing splitting of peaks with g Peaks are all not of same intensity 220 311 200 331 420 422 222 400 1 & 2 peaks resolved

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**Determination of Crystal Structure from 2 versus Intensity Data**

ratio Index 1 38.52 19.26 0.33 0.11 3 111 2 44.76 22.38 0.38 0.14 4 200 65.14 32.57 0.54 0.29 8 220 78.26 39.13 0.63 0.40 11 311 5 82.47 41.235 0.66 0.43 12 222 6 99.11 49.555 0.76 0.58 16 400 7 112.03 56.015 0.83 0.69 19 331 116.60 58.3 0.85 0.72 20 420 9 137.47 68.735 0.93 0.87 24 422

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Extinction Rules Structure Factor (F): The resultant wave scattered by all atoms of the unit cell The Structure Factor is independent of the shape and size of the unit cell; but is dependent on the position of the atoms within the cell

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**Reflections which may be present Reflections necessarily absent**

Extinction Rules Bravais Lattice Reflections which may be present Reflections necessarily absent Simple all None Body centred (h + k + l) even (h + k + l) odd Face centred h, k and l unmixed h, k and l mixed End centred h and k unmixed C centred h and k mixed C centred Bravais Lattice Allowed Reflections SC All BCC (h + k + l) even FCC h, k and l unmixed DC h, k and l are all odd Or all are even (h + k + l) divisible by 4

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**Determination of Crystal Structure from 2 versus Intensity Data**

2→ Intensity Sin Sin2 ratio

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**The ratio of (h2 + K2 + l2) derived from extinction rules**

SC 1 2 3 4 5 6 8 … BCC 7 FCC 11 12 DC 16

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**FCC 2→ Intensity Sin Sin2 ratio 1 21.5 0.366 0.134 3 2 25 0.422**

0.178 4 37 0.60 0.362 8 45 0.707 0.500 11 5 47 0.731 0.535 12 6 58 0.848 0.719 16 7 68 0.927 0.859 19 FCC

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h2 + k2 + l2 SC FCC BCC DC 1 100 2 110 3 111 4 200 5 210 6 211 7 8 220 9 300, 221 10 310 11 311 12 222 13 320 14 321 15 16 400 17 410, 322 18 411, 330 19 331

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**Consider the compound ZnS (sphalerite)**

Consider the compound ZnS (sphalerite). Sulphur atoms occupy fcc sites with zinc atoms displaced by ¼ ¼ ¼ from these sites. Click on the animation opposite to show this structure. The unit cell can be reduced to four atoms of sulphur and 4 atoms of zinc. Many important compounds adopt this structure. Examples include ZnS, GaAs, InSb, InP and (AlGa)As. Diamond also has this structure, with C atoms replacing all the Zn and S atoms. Important semiconductor materials silicon and germanium have the same structure as diamond. Structure factor calculation Consider a general unit cell for this type of structure. It can be reduced to 4 atoms of type A at 000, 0 ½ ½, ½ 0 ½, ½ ½ 0 i.e. in the fcc position and 4 atoms of type B at the sites ¼ ¼ ¼ from the A sites. This can be expressed as: The structure factors for this structure are: F = 0 if h, k, l mixed (just like fcc) F = 4(fA ± ifB) if h, k, l all odd F = 4(fA - fB) if h, k, l all even and h+ k+ l = 2n where n=odd (e.g. 200) F = 4(fA + fB) if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400)

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**Applications of XRD Bravais lattice determination**

Scattering from uc Lattice parameter determination Number of equivalent scattering planes Determination of solvus line in phase diagrams Effect of wave polarization Long range order Combination of 3 geometric factors Crystallite size and Strain Specimen absorption Temperature factor Thermal diffuse scattering

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**Crystal Intensity → Diffraction angle (2) → Intensity →**

90 180 Crystal Schematic of difference between the diffraction patterns of various phases 90 180 Diffraction angle (2) → Intensity → Monoatomic gas 90 180 Diffraction angle (2) → Intensity → Liquid / Amorphous solid 300 310

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