X-ray diffraction geometry
Setting controls sample orientation in the diffraction plane. most important for single-crystal diffraction For any poly- (or nano-) crystalline specimen, we usually set: 1
X-ray diffraction pattern from nanoparticles Features: 1. peak positions. relative peak intensities 3. peak widths
Powder diffraction standards JCPDS: Joint Committee on Powder Diffraction Standards PDF: Powder Diffraction File
Two conventions for scattering vector TEM: q 1 sin d XRD: q 4 sin d 1 k 1 k q sin k k q 4sin
Integrated intensity factors for powder Structure factor squared Multiplicity factor Lorentz polarization factor Debye-Waller (thermal) factors M Fhk L m θ B θ, B T (equivalent permutations of Miller indices) I max θ If all peaks have the same shape: I int Imax θ
f Gaussian: q exp ln, Gaussian q q q 0 q q Lorentzian: f Lorentzian q q q 1 0, 1 q q0 q Peak fitting 0 q 4sin q0 centroid q FWHM f Pearson-7: Pearson7,, q q0 q m m 1/ m q q 0 1 1 q 1 I max q FWHM
Normalization For comparison of integrated intensites, peaks need to be weighted properly: lorentzian 0.634 gaussian ln 0.939
Another important lineshape Voigt: Convolution of gaussian and lorentzian Voigt Gaussian Lorentzian q, q q, q q, q f q f q f q 0 0 1 0 f q f q dq f q f qq 1 q 1 The convolution makes this difficult to use for computation.
X-ray sources Bremstrahlung & characteristic X-rays Usually use a monochromator or energy filter
Common X-ray sources Cu-K: K1 p 3/ 1s 8.048 0.15405.0 K p 1/ 1s 8.08 0.15443 1.0 mean (nm) 0.154 Mo-K: label transition E(KeV) (nm) relative intensity K1 p 3/ 1s 17.481 0.07093.0 K p 1/ 1s 17.376 0.07135 1.0 mean (nm) 0.07107 E 140 ev nm 1.4 KeV nm Co, Ag also used
K Peak Doublets Two peaks corresponding to a single lattice spacing q 4sin 4sin d 1 1 I K1 I K.0
Fitting the K Doublet sin I A1 f q0, q,... 1 sin A f q0, q,... b b b 0 1 q sin
X-Ray Scattering by Charges An electric charge in an electric field experiences a force F qe q: charge A mass subjected to a force accelerates: a F m m: mass An accelerating charge radiates: q E 3 cr r r a points from charge to observation point
X-Ray Scattering from One Electron q e Scattering in x-y plane E y eey mc r cos E z e E z mc r I E I E y z E E E 4 y z y z e E E E E cos E mc r
Polarization Factor Time averages: E E E y z Incident beam unpolarized: I E I 1 Ey Ez E I E I 4 e 1 cos I I mc r polarization factor When the incident radiation is unpolarized, the polarization dependence of x-ray scattering causes the diffracted intensity to vary with scattering angle.
Diffraction Geometry Orient g in the plane containing k and k (Not pictured at Bragg condition.) Also assume when 0 g k
Radial Scan (/or ) sin B g rel Vary and Maintain condition 1 Excitation point M orbits about 0 Excitation error s parallel to g
Rocking Curve () fixed Vary usually set B B M orbits about 0 1 s initially perpendicular to g
Excitation Error in XRD (I) Excitation error: s k kg From diffraction geometry: k kxˆ ˆ ˆ cos ˆ k k cos xsin yˆ g g sin x y : detection angle ω : sample rotation
Excitation Error in XRD (II) cos 1 ˆ sin k ˆ k k k x yˆ k k sin sin x cos y ˆ Change Coordinates: xˆ sin xˆ cos yˆ yˆ cos xˆ sin yˆ xˆ sin xˆ cos yˆ yˆ cos xˆ sin yˆ ˆ kk k sin cos xsin y g gx s ksin cos x ˆ sin y ˆ g x ˆ
Excitation Error in XRD (III) Define an angle that measures the deviation from the Bragg condition: rel s ksin cos x ˆ sin ˆ ˆ rel y rel g x For a radial scan, rel = 0 s ksingxˆ Notice: s 0sin g k B (Bragg s Law)
int, I S s, s d d rel x y rel Geometric Factor (Ia) For a large crystal, the reciprocal lattice points are delta functions in reciprocal space. s int scattering strength I S s, s ds ds s, s x y x y x y,, S s s S s s s s s s dsds x y x x y y x y x y s, s x y //reciprocal space Experimentally, we integrate intensity over an angular range. //angle space The integral over rel occurs when a powder is used. The integral over occurs when scanning a range of., Iint S sx sx sy sy drel d sx sy dsxdsy sx, s yrel,
Geometric Factor (Ib) Related angle coordinates to reciprocal-space coordinates: ds d ds d rel xˆ k sin sin rel cos xˆ k coscos rel sin Near Bragg condition: 0 ds d rel rel rel rel yˆ yˆ B ds ksin B, cos d dsy dsx drel, d, ksin kcos yˆ k x y drel d B B, k sin B B xˆ ds ds
Geometric Factor (Ic) We can now perform the integral in reciprocal-space coordinates, weighted for angle coordinates:, int S sx sx sy sy dsxdsy I sxs ydsx dsy k sin,, s x sy sx sy B 1 s Iint k sin B So, the integrated intensity of a peak in a radial scan, summed over all sample orientations (e.g., a powder), depends on scattering angle. 1 I sin B
Ring Circumference Factor If a diffraction ring is sampled only along a short slit of fixed height, the fraction of the ring contributing to the pattern depends on scattering angle. h: slit height L: arc length L L h 1 L 1 I Rsin sin Ltot sin B tot B B
Grain Orientation Factor For a randomly oriented powder, the fraction of grains oriented correctly for diffraction depends on scattering angle. Total surface area of sphere, describing all possible grain orientations A tot 4R Surface area of annular section of sphere, describing orientations of grains at Bragg condition B ARsin 90 R R cos B B A B I cosb R A R cos 1 cos A 4 tot A tot
Combine factors: Lorentz-Polarization Factor 1 1 L θb cos θb 1cos θ B sin θ B sin θ B 1 cos θ B sin θ sin θ B B This factor is applied to the integrated intensities for powders in standard diffraction experiments. I m F L θ M θ, T B B
Scattering from a Small Crystal Assume a parallelepiped shape N 1 N 1 1 3 in1k-k a1 ink-k a in3k-k a3 E F e e e N 1 e ink-k a n0 n 0 n 0 n 0 1 3 N 1 1 ein 1 e i k-ka k-k a in k-k a sin N k-k a i k-k a 1 e 1 e sin k-k a S xs N 1 x n n0 N 1 x n1 n0 S 1 x N 1 x 1 x 1 x... x N 1 x 1 x... x N sin N1 1 sin N sin N3 3 I F k-k a k-k a k-k a sin k-k a1 sin k-k a sin k-k a3
Small Crystal Small Crystal Thin Foil Thin Foil I sinn k-k a F sink-k a sin Nsa F sin sa I g T sinc st sin N F sa sa N F sinc Nsa k =k +g+s ga n (integer) k-k ag+san sa
Approximation Near the Bragg condition: sin sin Nsa sa N N e sa I N F e Nsa
qk k g+s Particle-size broadening radial scan: s g a TEM convention: q sin dq d cos s q cos cos sa sa a Scherrer equation: Particle size: 1 FWHM: I max I I max I max L Na e e cos N a Lcos ln 0.94 K 0.94 Lcos Lcos
Some Broadening Contributions to q Influence of () on q: dq cos d cos q cos q Spread in Wavelength: dq sin d q q q Particle Size: L K L cos q L K L
More Broadening D Microstrain : 1 q D dq 1 q dd D D dd da a D a a qa q a If errors are uncorrelated: Optical Spread: q sin cos q q q q q 1 q...
LaB 6 Standard Measure Instrumental Broadening
Example: Polycrystalline Si Films plasma-enhanced CVD vs. hot-wire CVD
Debye-Scherrer Camera Primarily a powder camera Filtered or monochromatic radiation used
Debye-Scherrer Geometry Sample is usually powder Otherwise, rotate specimen about camera axis
Debye-Scherrer Optics Collimator defines beam Window used for alignment
Debye-Scherrer Patterns
Bragg-Brentano Focusing (1) If a point source (S), point detector (D) and all points on the sample (e.g., A) are on a circle, then the the angles between S and A is equal to that between D and A. S S 90 D D 90 180 S S 180 360 D D S D S D
Bragg-Brentano Focusing () r Fixed distances: source-sample sample-detector Focusing circle radius changes with R r R sin
Cut and Bent Monochromator
Laue Camera Back-Reflection Transmission
Moving Film Methods (I): Weissenberg Camera
Moving Film Methods (II): Buerger Precession Camera Sample diffraction only at intersection of ZOLZ with sphere
Precession Data Screen radius, distance, and precession angle related: sin R D
Single-Crystal Diffraction: Goniometers Eulerian Cradle Phi/Chi Two-Circle Goniometer Omega/-Theta
Goniometer Head Sample usually mounted on glass fiber
Sample Orientation q qx h M 0 qy U k 0 q z l Diffraction in x-y plane M cos sin 0 sin cos 0 0 0 1 T T cos 0 sin 0 1 0 sin 0 cos Rotate by Bragg Angle cos sin 0 T sin T cos T 0 0 0 1 cos sin 0 sin cos 0 0 0 1 qx q qy T 0 0 0
hkl Scans 1 1
Reciprocal-Space Mapping Two-Dimensional Map: I( rel, Each scan performed with rel fixed. Intensity measured at each point ( rel, ). The two directions of scanning are perpendicular.
Example Ga(As,P) on GaAs 004 map 004 RSM The data are transformed into reciprocal space (i.e., q)
Asymmetric Reflections q q x y ksincos rel ksinsin rel (-) grazing incidence (+) grazing exit
strain tensor: unstrained: h g0 B 0 k 1 B A 0 0 T Influence of Strain x 0 0 0 y 0 0 0 z strained: 1 A0 A T 1 B A 1 B h g B k 1 g0 0
Measuring Strain g cos sin g sin sin cos 0 0 g 0 h k a 0 g 1 x cossin 1 g0 g01y sinsin 1zcos g 1 0 cos sin sin cos h k g x y z a, For a particular g g 0 1 sin cos h k a a a a a a a sin cos sin
Fitting Data 1-91: GaInPAs apara =0.5873 nm aperp=0.589337 nm
Single-Crystal Substrate Restrictions Ewald Sphere h max max a sin a cos 090 001 orientation h0 plane sin B d cos sin h h h d min 90 hmax B Unobstructed d min h a max max a hmax max a x 1 a max x 0 x 1