PHYS 545 Solid State Physics Lecture 14-15 Theory of Scattering & Diffraction: Static and Dynamic Structure Factor ZM Chap. 2; SS Chap 14; KT Chap 2. Spring 2017, Purdue University 1 Prof. Yong P. Chen Understanding Diffractions Light (e.g., X-ray) Electron Neutron…. • Elastic • Inelastic (transfer energy/measure e.g., phonon…) 2 Quantum Theory of Diffraction g is reciprocal lattice vector 3 Different types of diffractions (beams) see different kinds of potential V (sensitive to different information/order) • Neutrons • Spin-polarized neutrons • Electrons • Light (X-rays).. (neutron scattering, measure [besides phonons] magnetic order, magnons…) 4 “atomic factor” (static) “structure factor” 5 Structure factor (S) can be defined more generally for any systems… even liquid/glass..etc. • scattering/diffraction amplitude is proportional to S (S scattering amplitude) Static Structure Factor Dynamic Structure Factor (elastic scattering amplitude) (inelastic scattering amplitude) S(K,): For ideal crystal S(K)=K,g (double) Fourier transform --- peaks at reciprocal lattice points of P(R,t) (spatial & temporal correlation function) If there is atom located at r=0 and time 0, what is probability of finding another atom at R and t? Radial distribution function (spatial correlation function) 6 7 · · · 1 · Total S.F. S.F. of the (simpler) underlying lattice “Structural factor of the basis” [ rj: position of j th atom in the basis (unit cell) fj: atomic form factor, same if all atoms are same] Bcc lattice (as s.c. lattice with 2-atom basis) fcc lattice (as s.c. lattice with 4-atom basis) S=4f (if all v’s are odd or all are even) S=0 (otherwise) 8 KT Chap2 9 (relevant in power XRD) [intensity] Note: there are many other factors affecting intensity… we focus on position (appearance of peaks) here 10 bcc (-Fe) Missing (210) Missing (111) 11 Example (SS Chap 14) alternative way to write Bragg condition (“absorb n into d”): fcc lattice g=2/dn, where dn= d/n =2dn sin Reciprocal lattice vector g=(2h/a, 2k/a, 2l/a) 12 Diffraction by crystal with lattice vibrations (phonons) Introducing deviation from ideal crystal positions… Structure factor (square of this): (expansion) (higher order phonon processes etc.) (idea structure factor) (phonon scattering) (Debye-Waller factor) 13 14 Dynamic case/Inelastic scattering Note: elastic scattering, no q involved, vq=0; each q involved in inelastic scattering has its vq; • Desire low-energy wave scattering (to probe inelastic scattering eg phonons) [if wave energy >> phonon energy; “lattice looks static” at time scale of wave, mostly measure elastic scattering] 15 Debye-Waller factor High T: (zero point motion plays a role) 16 KT Appendix.A 17 xm~.2 18 Phonon-phonon interaction (due to anharmonicity) Enabling process such as: Other phenomena where anharmonicity of lattice potential is important: • Thermal expansion • Thermal conduction 19.