X-ray Diffraction from Materials 008 Spring Semester Lecturer; Yang Mo Koo Monday and Wednesday 4:45~6:00
8. X-ray Diffractometers Diffractometer: a measuring instrument for analyzing the structure of a usually crystalline substance from the scattering pattern produced when a beam of radiation or particles (as X rays or neutrons) interacts with it. Goniometer: an instrument that either measures angle or allows an obect to be rotated to a precise angular position. Eulerian cradle (-circle) Diffractometer t with 4-circle goniometer
8. X-ray Diffractometers 4-circle goniometer χ φ Full circle Eulerian cradle θ ω Eulerian cradle for texture measurement
8. X-ray Diffractometers 6-circle goniometer Two circles (θ and θ) at detector side
8. X-ray Diffractometers Diffractometer with -circle goniometer Two rotation axes are ω and θ. Focusing condition reflected beam from the sample: ω θ.
8. X-ray Diffractometers Diffraction condition for a single crystal Diffraction condition for poly-crystal or powder sample
8. X-ray Diffractometers Focusing condition of reflected beam from sample surface R r R r r R sinθ
8. X-ray Diffractometers Slit system of powder diffractometer
8. X-ray Diffractometers Elements effecting diffraction peak shape g (x) h(x) f ( x ) specimen h( x ) f ( x ) g( x ) where g(x) g (x) g (x) g (x) g (x) g (x) g (x) g (x): x ray source g (x): flat g (x): vertical surface of specimen 4 divergence of 5 x 6 ray beam g4(x): transparency of x -ray in the specimen g (x): () size of reciving slit 5 g (x): misalignment 6
8. X-ray Diffractometers Monochromator
8. X-ray Diffractometers Diffractometer with 4-circle goniometer
8. X-ray Diffractometers Diffraction condition for 4-circle goniometer In order to get diffraction peak of hkl, one can move reciprocal point to diffraction plane by rotating a combination of ω, φ, and χ. ny diffraction pot can be moved by rotating two different angles such as ω and φ, φ and χ, and ω and χ. owever, rotations of all three angles are used for real experiment because of geometric restriction of the diffractometer. θ circle is used for moving detector to the reflected x-ray beam position. The angle, ω set θ for keeping focusing condition of reflected beam, and φ, and χ are adusted to move reciprocal lattice point on the diffraction plane. The coordinates system in 4-circle goiniometer (I) the coordinates system of diffractometer. ad D b D c D diffractormeter : an orthonormal system related to
8. X-ray Diffractometers where b a c D D D is parallel to the diffraction vector for positive θ and always bisects the angle (π - θ ) is in the diffraction plane, perpendiculart the x ray source when θ 0. is coincident with the main axis of o b D the instrument. is directed toward (II) the coordinate system of specimen : an orthonormal system a a b c fixed in the specimen. The axes are defined such that is coincident with when ω φ χ 0. c is always coincident with the φ axis. D
8. X-ray Diffractometers (III) rystal axes of unit cell a b c where a, b, c are lattice parameters of unit cell Orthogonal axes system X TX a where T 0 0 b cosγ b sin γ 0 c( sin c cos β c(cos α cos β cos γ ) / α + cos sin γ β cosα cos β cosγ ) / / sinγ This matrix often used for calculating g X X ( X T ) / X X
8. X-ray Diffractometers Therefore X T {( ) ( )} / T T { } / T T TX TX X T TX X ( T ) G is called metric tensor { } / T T X { X GX} / G a abcosγ ac cos β T T T abcosγ b bc cosα ca bc c cos β cosα a a ab ac T b a b b b c c a c b c c If axes system stem is orthogonal, metric tensor become unit matrix; 0 0 G 0 0 I 0 0
8. X-ray Diffractometers axes crystal Reciprocal (IV) axes crystal (IV) Reciprocal b a vectors unit are reciprocal,, where c b a c The magnitude of reciprocal vector can be estimated by metric tensor { } / T X G X X { } and T X G X X c c b c a c c b b b a b c a b a a a G T c c b c a c
8. X-ray Diffractometers (V) oordinates of a reciprocal vector The coordinates of a vector in reciprocal space in the D ( x, y,z ) ( xd, yd,zd ) ( h,k,l) such thatt a reciprocal vector h ha x + hb a D D D D D D xa + yb + + + lc y b h may be written + z z D c c D, D, and system The relationship between F D where transformation matrix and D F is defined by after rotation of specimen ω, χ, and φ
8. X-ray Diffractometers F R( φ) R( χ) R( ω) cosφ sinφ 0 0 0 cosω sinω 0 -sin φ cos φ 0 0 cosχ sinχ -sinω cosω 0 0 0 0 -sin χ cosχ 0 0 cosφcosω sinφsinωcosχ cosφsinω + sinφcosωcosχ sinφcosω cosφsinωcosχ -sinφsinω + cosφcosωcosχ sinχsinω -sinχcosω sinφsinχ cosφsinχ cosχ It follows that T X X F or X FX D since F is unitary matrix and thus T F F T D
8. X-ray Diffractometers (VII) The orientation matrix It is convinient to introduce an orientation matrix U such that U The matrix ti U (which his in general not symmetric) ti has nine independent d elements which are related to the six sell constants and three angles describing the specimen oreientation. The cell constants may be obtained, independent of the orientation, by forming the product of G - T U T from which we obtain, since G - UU T U T and its transpose Thus, by forming the product UU T I the unit cell constant t of the crystal may be obtained. T T., the reciprocal cell constants and hence
8. X-ray Diffractometers alculation of setting angles at 4-circle goiniometer If ω, χ, φ are such that the diffraction conditions are met, then D where d ( 0,d, 0 ) h Thus it follows, using rotational transformation matrix, F cosφ sinω + sinφ cos χ cosω T d sinφ sinω + cosφ cos χ cosω sin χ cosω Since U and we have U Knowing and U, we can calculate the components of and hence the setting angles T by solution of equation.
Determination of the orientation matrix 8. X-ray Diffractometers Determination of the orientation matrix Determination of U from three reflections: t any value of ω, measurement of χ, φ, and d for reflection h is a measurement of three i d d t titi hi h b d i th d t i ti f U,, h h h have we three reflections For independent quantities which may be used in the determination of U. Let us define the following matrices Specimen Reflection z y x z y x and l k h l k h z y h l k h
8. X-ray Diffractometers Then Re flection But since we have U Specimen Re flection U Re flection Specimen Specimen The matrix Specimen depends only on ω, χ, φ, and d for three reflections. The matrix specimen depends only on the indices. ence we may solve for U.
8. The Integrated X-ray Intensity of a Diffraction Peak Integrated intensity: the shade area of the diffraction peak.. Direct x-ray beam power: 0 I0. Structure factor. the number of family planes 4. Temperature factor 5. X-ray absorption by specimen 6. Lorentz factor (geometric factor) 7. Exposure time to detector 8. Polarization of x-ray
8. The Integrated X-ray Intensity of a Diffraction Peak Diffraction from a small crystal Small crystal: NNN sin Nπs a sin Nπs b sin Nπs c I P Ie F sin πs a sin πs b sin πs c where N, N, N are the number of unit cell with respect to a, b, c. Ideal diffraction experimental case, the maximum intensity becomes I P, max Ie F N N N Real case, the shape of diffraction peak - divergence of x-ray beam - defect of crystal -misalignment - other instrumental factor Measure Integrated intensity
8. The Integrated X-ray Intensity of a Diffraction Peak Integrated intensity for hkl peak - ω is angular velocity of rotating (hkl), and ω axis is parallel to (hkl) - reflected x-ray radiates into a solid angle dβdγ for S0. - S0 can be reflected by (hkl) because divergence of incident x-ray beam
8. The Integrated X-ray Intensity of a Diffraction Peak Integrated intensity of small volume of reciprocal I I P dtd where dt dα/ ω, and detecting area d R I R ω I P dαdβdγ d βd γ In order to integrate this equation, let us change dαdβdγ to reciprocal space unit as s. Δs pa + pb + pc where p, p, p are very small. Then Δ α 0 s S dα, Δs S dα, and Δs β γ S dα, The volume of the small reciprocal space associated with s. ΔV sin θ ΔV Δs Δs Δs dαdβdγ ΔV α dp a γ β λ can also be written in reciprocal unit dp b dp c V unit cell dp dp vector dp
8. The Integrated X-ray Intensity of a Diffraction Peak therefore dα dβdγ I R ω V V unit cell λ λ dpdpdp sin θ The intgrated intensity becomes unit cell sin θ The triple integral becomes I I e p dp dp F dp I e F sin N πp sin π x I p dp dp dp p sin sin sin N sin π sin N π ( s + Δ s ) a π ( s + Δs) a π ( s + Δs) b sin Nπ ( s + Δs) ( s + Δs) b sin π ( s + Δs) N πp sin N πp p sin π p sin π p dp dp dp c dp dp dp c Since the above equation exist only vicinity of Bragg peak, p, p, p are very small values. ence sinπp πp..
8. The Integrated X-ray Intensity of a Diffraction Peak With this approximatin and using the result of sin x - We have I Nx N I e R λ ωv unit cell F sin θ N N N integral from integration ti table; Let us change NNN to more practical parameters. If the volume of small crystal δv, δvvunit cellnnn. 4 I 0 e δv + cos θ I F λ 4 ω 4πε 0me c V unit cell sin θ (/sinθ) is called Lorentz factor which was driven from geometry of diffractometer. Lorentz-polarization factor is the combination of Lorentz factor and polarization factor.
8. The Integrated X-ray Intensity of a Diffraction Peak Diffraction from mosaic crystal crystal with mosaic structure does not have its atoms arranged on a perfectly regular lattice extending from one side of crystal to the other; instead, the lattice is broken up into a number of tiny small crystal, each slightly disoriented one from another. The size of these crystal is order of 00nm, while the maximum angle of disorientation between them may vary from a very small value to as much as one degree, depending on the crystal. Integrated intensity from mosaic crystal can be calculated with help of the figure; I e δv + cos θ θ e sin z 0 4 0 I F λ 4 ω 4πε 0me c Vunit cell λ μz / sinθ 0 dz δv 4 I θ ω πε μ θ 0 0 e F + cos 4 4 0me c Vunit cell sin where 0 is cross sectional area of incident beam, μ is linear absorption coefficient of mosaic crystal.
8. The Integrated X-ray Intensity of a Diffraction Peak Diffraction from polycrystal If the reflected intensity is measured by the receiving slit of which size is (h x w), solid angle, Ω along scattering vector corresponding to slit size becomes Ω hw 4R sinθ
8. The Integrated X-ray Intensity of a Diffraction Peak When the number of grains (mosaic crystals) irradiated by x-ray is N, the probability of (hkl) reflection solid angle, Ω may be (Ω/4π)mhkl. Intensity reflected from (hkl) becomes I I mosaic Ω hw mhkl Imosaic m 4π 4R sinθ 4π 4 I F λ + cos θ 0 0 e m hkl 4 ω 4πε 0me c μvunit cell 64πR sinθ sin θ hkl
8. The Integrated X-ray Intensity of a Diffraction Peak Diffraction from powder sample The number of powder which is diffracted because of angular divergence of incident x-ray dn Nm hkl Nm hkl π R cos( θ + α )d α 4πR cosθdα for small α Irradiated area of reflected beam may be written by angular divergence dβdγ; d R dβdγ Integrated intensity reflected by (hkl) becomes I I P Nm hkl R cosθdαdβdγ
8. The Integrated X-ray Intensity of a Diffraction Peak When the reflected x-ray is detected using the size of (h x w) receiving slit, I I P Nm hkl R h cosθ dαdβdγ πrπ R sin θθ omparing the other equations 4 I e F λ mhklh 0 0 + cos θ I 4 6πR 4πε me c μv unit cell sinθ sin θ 0 where ( / sinθ θ sin θ) is Lorentz factor for powder diffractio n.
8. The Integrated X-ray Intensity of a Diffraction Peak Lorentz factor Lorentz factor is related to the time it takes a point in the reciprocal to move through the Ewald sphere. If a crystal is rotating at angular velocity ω about 000, the velocity of diffracting point is ω S-S0, and the component of velocity normal to the sphere (v ), is ω S-S0 cosθ The time (t) for a point to pass through The reflecting position is proportional to / v. t v ω S S cosθ L ωsin θ ω where L / sin θ; Lorentz facor 0 ωsin θ cosθ
8. The Integrated X-ray Intensity of a Diffraction Peak Effect of temperature Effect of temperature ssume that atomic bonding motion is harmonic oscillator. The structure factor of the t l b i i N i N i N e e f e f e f s F u s r s u r s r s + π π π π ) (, 0 0, ) ( crystal becomes f f f ) ( Since u changes with temperature, and is small values, the exponential term of u can be approximated ( )... + + i i e u s u s u s π π π Time average of this equation becomes ( ) ( ) 4 4... i e u s u s u s + π π π Time average of this equation becomes ( ) - e su π If u does not have directionality, then we have ( ) cos u s u s φ u s
8. The Integrated X-ray Intensity of a Diffraction Peak If u does not have directionality, then we have ( s u ) s u cosφ s u Therefore, the time average structure factor affecting temperature can be written f,t f e π 8π πisu ( s u ) f e f e sin θ u λ f e sin θ 8π u λ,s f e B sin λ θ where and u,s B 8π u,s the component ( temperature factor), of u,s along s direction Since the diffraction peak intensity is proportional to the square of structure factor, thermal motion of atom reduces the diffraction intensity by a factor of e ( Bsin θ / λ ) Debye-Waller factor
8. The Integrated X-ray Intensity of a Diffraction Peak Debye developed the temperature factor, B as follows; 6h φ( ) B + mk Θ xx 4 where h is Plank constant, K B B Boltzmann constant, m mass of atom, Θ Debye temperture of atom, and x Θ/T. nd φ(x) is given by x ξ φ( x) dξ ξ x 0 e
8. Experimental X-ray Diffraction Procedures omework Due date: pril, 008 Solve the problems; 8., 8.4 If possible, please solve 8.7 (Extra points)