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=(2h/a, 2k/a, 2l/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

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