Neutron Scattering, correlation functions & linear response functions

1. Neutron Scattering, correlation functions &linear response functions Collin Broholm Johns Hopkins University

2. Scattering cross section &

3. Detailed Balance Sample in thermal equilibrium Low T High T 0

4. & correlation functions We measure a time and space fourier transform of the interaction potential No elastic scattering for a liquid! Equal time correlation function

5. Fluctuation Dissipation Theorem Is linear response function Time translational invariance Trace is invariant under cyclic permutation Fluctuation dissipation theorem v1 Generalized susceptibility Fluctuation dissipation theorem v2

6. First Moment Sum-rule Fourier transform Time derivative evaluated at t=0 Use detailed balance Use this result from interaction picture to Derive First moment sum-rule To learn it’s significance look at interaction potentials: Nuclear then magnetic

7. Neutron-Nuclear interaction Fermi Pseudo potential Hats indicate sample space operator FT density-density correlations For free atoms can readily compute double commutator Recoil! Impulse approximation holds generally. Use to measure momentum distribution

8. Neutron-electron spin interactions Neutron is spin-1/2 magnetic dipole g=1.913 Dipole in inhomogeneous field from sample B-Field from partially filled electron shells Interaction potential Dimensionless scattering operator Is magnetic form factor: Fourier Transf. of spin density distribution on atom ;

9. Magnetic correlation & response function time and spatial FT of 2-point correlations Linear response function Generalized susceptibility Fluctuation dissipation theorem 1st moment 0st moment

10. Weakly interacting dimers in Cu(NO3)2.2.5 H2O 0.55 G. Xu et al. PRL (2000) 0.50 ħw (meV) 0.45 0.40 0.35 0.0 0.5 1.0 1.5 Q (Å-1)

11. 0.5 E (meV) 0.4 0 2 4 0 2 4 6 “Guessing” S(qw) When a coherent mode dominates the spectrum: First moment Sum-rule links S(q) and e(q) q (p)

12. Spin waves in ferromagnet Magnon creation Magnon destruction Dispersion relation Gadolinium Magnon occupation prob.

13. Spin waves in antiferromagnet Dispersion relation

14. Two-particle continuum scattering Cu(C4H4N2)(NO3)2 Stone et al. (2003).

15. Summary • Nucl. scattering • Magnetic Scattering • Detailed balance • Fluctuation dissipation • Know your sum-rules and happy scattering!