Scattering and Diffraction

Transcription

1 Scattering and Diffraction Adventures in k-space, part 1 Lenson Pellouchoud SLAC / SSL XSD summer school 7/16/018 lenson@slac.stanford.edu

2 Outline Elastic Scattering eview / overview and terminology Form factors Diffraction Crystalline arrangements Structure factors Approximations in practice

3 Elastic Scattering introduction and terms Incident light: Approximate plane-wave E-field E(r,t )=E 0 exp(i(ω t k r)) Incoming light Source wavelength: λ Wavevector k: photon momentum k = π/λ E-field frequency: ω = πc/λ Incident field accelerates the charged particle Charged particle acceleration radiates a new scattered E-field k E-field e Particle acceleration Scattered light Accelerating charged particle radiates E-fields in all transverse directions E-fields are transverse: no radiation is emitted parallel to acceleration vector Scattered field amplitude is modulated by charge, mass, and geometric factors 3

4 Elastic Scattering introduction and terms Incident light: Approximate plane-wave E-field E(r,t )=E 0 exp(i(ω t k r)) Multiple scatterers Single scatterer Multiple scatterers Scattered fields from multiple scatterers superimpose Superposition of scattered waves interference The interference pattern is related to the positions of the scatterers k e e e Position of measurement Scattered amplitude 4

5 Elastic Scattering introduction and terms Incident light: Approximate plane-wave E-field E(r,t )=E 0 exp(i(ω t k r)) Distribution of scatterers The entire distribution of charge density, ρ(r), simultaneously scatters the light Scattered fields from different parts of the distribution interfere The interference pattern is directly related to the distribution of charge k How can we use the real-space pattern to interrogate ρ(r)? ρ(r) Subject of this talk Multiple scatterers Single scatterer Position of measurement Distribution of scatterers Scattered amplitude Same distribution, different wavelength 5

6 Elastic Scattering introduction and terms Incident light: Approximate plane-wave E-field E(r,t )=E 0 exp(i(ω t k r)) k ρ(r) q Detector k -k k Detector measurement: scattered light at a chosen k Scattered light Scattering vector q = k - k describes momentum transfer Detector placement selects a scattering vector Elastic scattering: momentum transfer occurs without energy absorption k = k = k E det (t) E 0 exp(i(ω t k )) 6

7 Elastic Scattering introduction and terms Scattered field (for an electron): E det (t)= E 0 e cos(α) exp(i(ωt k)) 4 π ϵ 0 m e c r e =E 0 cos(α)exp(i(ω t k)) k e k The scattered intensity is (typically) much smaller than the incident intensity Scattered field amplitude factors: m e particle mass e particle charge 4π spherical dissipation cos(α) α is the complement of the angle between the scattering path k and the charge acceleration vector c speed of light ε 0 permittivity of vacuum r e electron radius 7

8 Elastic Scattering basic equations Scattered field from one particle with cos(α)=1: e (Not a function of q) r e E(q)= E 0 exp(i(ω t k )) Multiple particles: (assumption: distance between scatterers is much less than distance to the detector) Distribution of particles: ρ(r) (Sum over particles at positions r n ) e e r e e E(q)= E 0 exp(i(ω t k )) exp(iq r n ) n (Integrate the particle distribution over r) E(q)= E 0 r e exp(i(ω t k )) d 3 rρ(r)exp(iq r) we just found the Fourier transform of ρ(r) The scattering pattern of a charge distribution is its Fourier transform, in the space of q (elastic momentum transfer) vectors 8

9 Elastic Scattering form factors ρ(r) E=E 0 r e exp (i(ω t k )) d 3 r ρ(r)exp(i q r) This Fourier transform is the form factor of ρ(r) Form factor f(q) = F[ρ(r)] This is powerful: it is very difficult to solve or measure ρ(r), but we can interrogate it by (much easier) measurements of f(q) Atom cores have fairly repeatable ρ(r) atomic form factors are well documented Materials also have repeatable ρ(r) scattering form factors are known for many different shapes of material f (q) d 3 r ρ(r)exp(i q r) 9

10 Elastic Scattering form factors ρ(r) f (q) d 3 r ρ(r)exp(i q r) Atomic species: f (q)=z s k a k exp( s b k ) s q /(4π) Empirical parameters 10

11 Elastic Scattering form factors ρ(r) f (q) d 3 r ρ(r)exp(i q r) Spheres: f (q)= 3 sin(q r sph) q r sph cos(qr sph ) (qr sph ) 3 11

12 Scattering measurements non-dilute / condensed arrangements re E(q)= E0 exp (i ( ω t k ))f (q) E( q)= E0 re exp(i( ω t k )) [ f (q )exp (i q p ) ] p Non-dilute arrangement: condensed formation of identical scatterers Add (superimpose) the scattered amplitudes from all scattering centers in the material (scatterer location: p) Form the scattered intensity An additional pair summation appears: this is the scattering structure factor I (q)= E(q ) =E '(q) E(q ) re = E0 f (q) (exp( i q p )) (exp(i q m )) p m r 1 =N E0 e f ( q) 1+ exp(i q ( m p )) N p m p ( ) ( ) ( ) The main ingredient is the square of the form factor elative interparticle positions modulate the scattered intensity according to the structure factor S (q ) 1+ 1 exp(i q ( m p )) N p m p

13 Scattering measurements structure factors for condensed arrangements re I (q)=n E0 f ( q) S (q) ( ) S (q ) 1+ 1 exp( iq ( m p )) N p m p Non-dilute scattering structure factor Interacting particles will exhibit characteristic interparticle spacings The pair summation in S(q) is directly related to the pair correlation function g(r) (isotropic case) S (q )=1+ N 4 π r [ g( r) 1 ] 0 sin (qr ) dr qr Pair correlation functions are analytical for some simple interparticle potentials Hard sphere potential: V (r )=0, r r sph, r < r sph see plot Hard sphere structure factor: (big nasty equations) 13

14 Scattering measurements dilute / diffuse arrangements re E(q)= E0 exp (i ( ω t k ))f (q) E( q)= E0 re exp(i( ω t k )) [ f (q )exp (i q p ) ] p Dilute arrangement: dispersed formation of non-interacting identical scatterers The vast majority of scattering events will be from single scatterers The pair summation in the structure factor is assumed to average out to zero, over all pairs The structure factor reduces to 1 I (q)= E(q ) =E '(q) E(q ) re 1 =N E0 f ( q) 1+ exp(i q ( m p )) N p m p ( ) ( r N ( E ) f (q ) e 0 ) dilute scattering directly probes the square of the form factor Assumption: uncorrelated positions 1 N exp (i q ( m p)) 0 p m p

15 Scattering measurements diffuse intensity profiles re I (q)=n E0 f ( q) S (q) ( ) Dilute scattering Intensity S(q) = 1 Shown here: Directly probe the square of the form factor Guinier-Porod spheroids rg = 10A* (3/5) D=4 Identical spheres r = 10A Spheres with gaussian size distribution r0 = 10A σ=0.1 15

16 Scattering measurements condensed intensity profiles re I (q)=n E0 f ( q) S (q) ( ) Non-dilute scattering Intensity Product of form factor (squared) and structure factor Only valid for isotropic form factors Shown here: hard sphere structure factor characteristic radius: 10A volume fraction:

17 From Scattering to Diffraction scattering of crystalline arrangements re E(q)= E0 exp (i ( ω t k ))f (q) Crystalline arrangement: repeating lattice of identical charge distributions Just like scattering, superimpose amplitudes from all scattering centers in the crystal (scatterer location: p) Separate the sum over scatterers: 1. Outer crystal summation over all unit cells in the crystal (unit cell position: m). Inner form factor-weighted sum over all species in the unit cell basis: the crystalline structure factor (specie position: rn) re E (q)=e0 exp (i( ω t k )) [ f p (q)exp(i q p ) ] p r =E 0 e exp (i( ω t k )) exp(i q m ) f n (q)exp (i q r n ) m n Crystal summation Crystalline F (q) f n (q ) exp(i q r n ) structure factor: n

18 From Scattering to Diffraction scattering of crystalline arrangements re E(q)= E0 exp(i( ω t k ))F ( q) exp (iq m) m a1 Structure factor a Crystal summation Effects of crystal summation Crystal lattice vectors (spatial units of repetition): a1, a, a3 Scattering vector components in crystal basis: q1, q, q3 Analytical solution (lattice of N1, N, N3 repeating units): 3 exp(i q m )= j =1 m [ q a exp i( N j 1) j j ( ) ( sin N j sin qjaj qjaj ( ) ) ] Peaks occur at qj aj /=nπ for any integer n Bragg s Law Off-peak: crystal summation is approximately zero for large Nj Higher Nj sharper peaks; crystal defects broader peaks 18

19 From Scattering to Diffraction scattering of crystalline arrangements E( q)= E0 re exp(i( ω t k )) F ( q) exp (iq m) m Structure factor Effects of structure factor F (q) f n (q ) exp(i q r n ) Crystal summation ecall, the crystal summation leads to Bragg s Law: Peaks occur at qj aj /=nπ for all j, for any integer n Define the reciprocal lattice basis vectors: b1, b, b3 Equivalent statement of Bragg s Law: crystal summation selects q-values that satisfy qpk=πghkl, for integers h,k,l It is common to evaluate the structure factor only at reciprocal lattice points (q=πghkl) for some {h,k,l} of interest The structure factor can systematically amplify, attenuate, or eliminate a given peak, depending on its h,k,l values Different crystal structures exhibit different elimination or amplification patterns n (more about this coming up) General reciprocal lattice vector Ghkl hb1 + kb + lb3 (corresponds to lattice plane with Miller indices h,k,l) 19

20 Diffraction Profiles common practices and approximations E( q)= E0 re exp(i( ω t k ))F (q) exp (iq m) m Structure factor I (q)=c ϵ 0 E(q ) E0 r e c ϵ 0 Crystal summation Structure factor squared ( ) [ F (q h, k,l hkl ) ] Phkl (q) Broadening function centered at qhkl Typical approach: 1. Evaluate structure factor at πghkl, for some {h,k,l} of interest (simple). We generally do not have the necessary information to evaluate the crystal summation: instead, apply an empirical broadening function (Gauss, Lorentz, Voigt) (ambitious) 3. Or, if available, use physical parameters to incorporate crystal disorder ( FPA : Full Parameterization Approach) 4. Sum the (broadened) peak profiles for all {h,k,l} of interest 0

21 Diffraction Profiles common practices and approximations I (q)=c ϵ 0 E(q ) Consider: c ϵ 0( E 0 r e The broadening function creates non-zero intensities for q-values near (but not equal to) q hkl Is it a good idea to apply F(q hkl ) to the entire P hkl (q)? This tends to be okay when the form factors are slowly varying This could be a mistake when the form factors are sharply featured One safe alternative: compute the whole structure factor ) h, k,l I (q) c ϵ 0( E 0 r e Structure factor squared [ F (q hkl ) ] P hkl (q) ) [ F (q )] h,k, l P hkl (q) Broadening function centered at q hkl 1

22 Diffraction measurements Isotropic examples (powders, solutions) I (q) c ϵ 0( E 0 r e ) [ F (q )] h,k, l P hkl (q) Isotropic intensity examples fcc Aluminum powder Spherical particle close-packed superlattice with same dimensions as fcc Aluminum Spherical form factor has sharper features, which disrupt the {00} peak

23 The end thank you! Special thanks to the Tassone group at SSL! Chris Tassone, Amanda Fournier, Liheng Wu, Anthony Fong, Karsten Bruening, Tim Dunn, Ekaterina Tcareva and to the XSD summer school organizing committee: Apurva Mehta, Kevin Stone, Chris Tassone, Mike Toney eference materials: Scattering and diffraction theory: Bruce M. Clemens. Waves and Diffraction in Solids. (course reader for Stanford University Materials Science 05) Available at Stanford bookstore Computation and plotting of scattering and diffraction patterns: xrsdkit (X-ay Scattering and Diffraction toolkit) 3

24 Elastic Scattering basic experiments k k -k ρ(r) q k What do we get from the detector? Detector position selects a q value Detector has finite size: a given position integrates a small, finite range of q Detector has finite speed: the measured intensity is integrated over time We measure the integrated intensity pattern I(q), proportional to the time-averaged square of E(q) Exact (non-integrated) intensity profiles are not generally feasible to measure 4

25 Elastic Scattering basic experiments k range of q ρ(r) Area detectors Most modern experiments employ area detectors (see Tim Dunn s talk at 11:15) Essentially, thousands or millions of detectors operating simultaneously in parallel Each pixel selects a different range of q values Each pixel requires different geometric corrections Analysis is no less complicated, but the experiment happens faster and the q-range is measured concurrently 5