PDF ANALYSIS INTRODUCTION __________________________________________________ STRUCTURE OF LIQUIDS & GLASSES BY NEUTRON & X-RAY DIFFRACTION Prof Phil Salmon Department of Physics University of Bath References Champeney D C Fourier Transforms and their Physical Applications (Academic, London, 1973) Fischer H E, Barnes A C and Salmon P S Rep. Prog. Phys. 69 (2006) 233 Squires G L Introduction to the Theory of Thermal Neutron Scattering CUP, Cambridge, 1978) Whittaker D A J et al. Phys. Rev. B 98 (2018) 064205 PDF ANALYSIS DIFFRACTION __________________________________________________ DIFFRACTION • Fraunhoffer diffraction: (i) point-like single particle and (ii) assembly of point-like scattering centres (weak scattering limit) • Density functions and Fourier transforms • Diffraction from an assembly of finite-sized scattering centres • Differential scattering cross-section: The static approximation • Real-space structure • Multicomponent systems • Neutron diffraction with isotope substitution (NDIS) • Anomalous X-ray scattering (AXS) The Institut Laue-Langevin (ILL) (www.ill.fr) and the European Synchrotron Radiation Facility (ESRF) (www.esrf.fr) in Grenoble 2 PDF ANALYSIS DIFFRACTION __________________________________________________ SCATTERING BY A SINGLE PARTICLE Consider a single point-like particle at the origin and an incident plane wave To some distant point at R k1 k0 r To some λ arbitrary point from origin Incident plane wave: Φinc = φ0 exp i(k 0 • r − ω0t ) Scattered spherical wave ( λ >> particle diameter) φ A φ a − ω Φ = 0 exp i(k R − ω t + α ) ≡ 0 eik1 R e i 0 t sc R 1 0 R where a = Aeiα depends on the nature of the interaction between the incident wave and the scattering centre (α determines the phase of the scattered wave relative to that incident) 3 PDF ANALYSIS DIFFRACTION __________________________________________________ Note: For Schrödinger waves Φinc and Φsc are complex. For other waves only the real parts have physical significance. Scattering triangle 2θ = scattering angle k1 Q = k0 – k1 m k 0 = incident mom m 2θ k 1= scattered mom k0 m Change in particle’s mom (k 0 − k1) = Q Q is called the scattering vector 2 2 2 By the cosine rule Q = k0 + k1 − 2k0k1 cos2θ For elastic scattering |k0| = |k1| = 2π/λ. So 2 2 Q = 2k0 (1− cos2θ ) 2 2 = 4k0 sin θ 4π or Q = sinθ λ 4 PDF ANALYSIS DIFFRACTION __________________________________________________ If a second particle is at some position r : - ∠ BOC = β To distant O C point at R k1 k0 r A B ∠ ABO = α __________________________________________________________________ k 0 AB = r cosα = r • kˆ 0 kˆ 0 = k 0 k 1 OC = r cosβ = r • kˆ1 kˆ1 = k 1 r • k 0 r • k1 ∴ path difference AB – OC = − k 0 k1 λ = r •Q for elastic scattering 2π (phase difference) = (path difference) x 2π/λ = r • Q = Q • r φ a + • − ω ∴ Φ = 0 ei(k1 R Q r ) e i 0 t sc R i.e. second particle gives an additional phase shift of Q • r 5 PDF ANALYSIS DIFFRACTION __________________________________________________ DIFFRACTION FROM AN ASSEMBLY OF PARTICLES For an assembly of point-like scattering centres at positions ri (i = 1, 2, …., N) N φ a − ω iQ• r Φ=0 ik1 R it0 i sc ee∑ e R i =1 Ω 2 d = dS / R Hence intensity of scattered radiation in detector of area dS at R 2 2 2 R dS ∝ |Φsc| dS = |Φsc| R dΩ Define the differential scattering cross-section dσ ≡ (intensity of radn scattered into small solid dΩ angle dΩ in direction of R) / (intensity of incident radiation x dΩ) 2 2 2 N ΦΩR d iQ• r = sc = ∗ i 2 aa∑ e = ΦΩinc d i 1 So for a single point-like scattering centre at the 2 origin dσ / dΩ = aa∗: σ=∫ d Ω( dd σπ Ω=) 4 a Note: a depends on the nature of the interaction between the radn and the scattering centre 6 PDF ANALYSIS DIFFRACTION __________________________________________________ DENSITY FUNCTIONS & FOURIER TRANSFORMS Define the density function N ρδ≡− (r) ∑ ( rri ) i =1 This eqn describes a set of point delta functions at the positions of the N scattering centres s.t. ∫ ρ (r)d3 rN≡ Then ddσ Ω can be re-written in terms of the Fourier transform of ρ (r ) 2 N 2 dσ ∗∗iQ• r • = aa∑ ei = aa∫ d3 rρ ( r) eiQ r dΩ i =1 ≡ aa∗ N S( Q) The structure factor SQ( ) accounts for the effects of interference and depends on the structure of the system Note: The derivation of ddσ Ω assumes that the scattered wave is not attenuated (small sample limit), i.e. there is no beam attenuation and there are no multiple scattering events 7 PDF ANALYSIS DIFFRACTION __________________________________________________ SCATTERING CENTRES OF FINITE SIZE Consider a single atom with Z electrons distributed according to the density function ρat (r ) about the charge centre as origin. Then 2 dσ ∗ • = aad3 rρ ( r) eiQ r Ω ∫ at d 2 ≡ aa∗ f( Q) which defines the atomic form factor fQ( ) as the Fourier transform of ρat (r ) If the electron distribution is spherically symmetric then fQ( ) = fQ( ) and the integral can be made using spherical polar coordinates: - d32rr= sinθ d r d θφ d Q•= r Qrcosθ ∞ ππ2 ∴ fQ( ) = d r dθφ d eiQr cosθ ρ( rr) 2 sin θ ∫∫∫00 0 at Using x = cosθ ∞ 1 2 f( Q) = 2πρd r( r) rd x[ cos Qrx+ i sin Qrx] ∫∫01at − Hence ∞ sinQr f( Q) = 4πρd rr2 ( r) ∫0 at Qr 8 PDF ANALYSIS DIFFRACTION __________________________________________________ ∞ Limits: Q → 0, f( Q→=04) πρdrr2 ( r) = Z ∫0 at Q → ∞, fQ( →∞) =0 So for a single centre of finite size the scattering is not isotropic: it is Q-dependent E.g. for X-ray scattering from a single atom or ion dσ 2 = r2 fQ( ) dΩ e re = classical radius of an electron ( = 2.8179 × 10-15 m), and the form factor f(Q) for some selected atoms and ions (in electron units) is given by: - O: blue (Z = 8) Cl: green (Z = 17) Cl–: magenta (Z = 18) K+: red (Z = 18) 9 PDF ANALYSIS DIFFRACTION __________________________________________________ DIFFRACTION FOR AN ASSEMBLY OF FINITE-SIZED SCATTERING CENTRES Let the N atoms in the scattering system be arranged with their centres distributed according to the structural density function N ρδ≡− st (r) ∑ ( rri ) i =1 If the charge-density distribution associated with each atom is ρat (r ) then the overall charge density is given by the convolution relation ρρ(rrr) =at ( ) ⊗ ρst ( ) By using the convolution theorem it follows that 2 dσ ∗ • = aad3 rρρ( r) ⊗ ( r) eiQ r Ω ∫ at st d ∗ ∗ = aafQfQ( ) ( ) NSQst ( ) where SQst ( ) is the structure factor associated with ρst (r ) and does not depend on the nature of the scattering process For an isotopic liquid or glass, QQ→ Henceforth, we will write SQst ( ) as SQ( ) 10 PDF ANALYSIS DIFFRACTION __________________________________________________ ELASTIC & INELASTIC SCATTERING So far we have considered elastic scattering. But scattering events can be either elastic or inelastic More generally, the measured intensity will depend on both the momentum transfer Q and energy transfer E = ω for a scattering event. This intensity depends on the double differential scattering cross- section 1 d2σσk I( QE,,) ∝=1 S( QE) NEddΩ 104π k Incident energy of quantum: E0 Final energy of quantum: E1 Energy transfer: E = E0 – E1 Scattering cross section: σ 2 (e.g., σπ= 4 a for point-like scattering centres) For a glass, the dynamical structure factor S( QE, ) will have contributions from both elastic and inelastic (phonon and multi-phonon) scattering S( QE,,) = Sel ( Q)δ ( E) + Sphonon ( QE) 11 PDF ANALYSIS DIFFRACTION __________________________________________________ For example, I( QE, ) for a glass at low temperature T I( QE, ) ∼ 0 for E < 0: Quanta do not gain energy from the scattering system (it’s too cold!) I( QE, ) for E = 0: Elastic line broadened by resolution function of spectrometer. Signal related to equilibrium positions of pairs of scattering centres I( QE, ) for E > 0: finite intensity because quanta can excite system states (phonons) Note: For a liquid or gas, there is no truly elastic scattering (i.e., there is always some recoil in the scattering event) 12 PDF ANALYSIS DIFFRACTION __________________________________________________ POWDER DIFFRACTOMETER k 0 Qk= − k 01 − k1 k1 k0 Monochromator uses Bragg diffraction to select an incident wavelength λ. k0 = 2πλ, and the incident energy E0 is fixed Detector has an efficiencyε (E1 ) In the diffraction experiment, the detector integrates I( QE, ) over all energy exchanges E at constant scattering angle 2θ dσ E0 = d,Eε ( E) I( QE) ∫−∞ 1 dΩ meas 13 PDF ANALYSIS DIFFRACTION __________________________________________________ DIFFERENTIAL SCATTERING CROSS-SECTION: THE STATIC APPROXIMATION The static approximation is usually made to obtain the differential cross-section for the total scattering. Here, it is assumed that EE0 << 1 such that kk01 . Then, σσE0 d = ε k1 ∫ d,E( E1) S( QE) dΩ 4π −∞ k SA 0 σ = ε (E) SQ( ) 0 4π We can define an observation timescale τ snapshot by the time taken for an incident quantum of velocity v to travel one interatomic distance a, i.e., −10 τ snapshot av where a 10 m The condition EE0 << 1 for validity of the static approximation can be re-written as ττsnapshot min , where τ min is a characteristic time for atomic motion, e.g., the period of atomic vibration in a solid, or the relaxation time in a liquid. Typically −−13 12 τ min 10− 10 s Hence, the static approximation corresponds to a structure that is relatively static on the timescale for a quantum wavepacket to pass from one atom to the next 14 PDF ANALYSIS DIFFRACTION __________________________________________________ For X-ray diffraction, the photons travel at the speed of light −18 τ snapshot ac 10 s So the static approximation is valid For neutron diffraction, v 103 ms-1 −13 τ snapshot av 10 s So the static approximation is not valid, and inelasticity (or Placzek) corrections must be applied to the results obtained from neutron diffraction experiments Within the static approximation, the measured differential cross-section gives a more-or-less instantaneous or snapshot picture of the structure 15 PDF ANALYSIS DIFFRACTION __________________________________________________ STRUCTURE FACTOR Neutron diffraction: - 1 dσ 22 =bcoh SQ( ) + binc N dΩ SA Spin and isotope incoherence → additional term that does not depend on Q X-ray diffraction: - 1 dσ 2 2 = re fQ( ) SQ( ) N dΩ SA 16 PDF ANALYSIS DIFFRACTION __________________________________________________ REAL-SPACE STRUCTURE Subtract the forward scattering.