Magnetic Neutron Scattering

1. Magnetic Neutron Scattering Collin Broholm* Johns Hopkins University and NIST Center for Neutron Research Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering Summary +Impurities in spin-1 chains *Supported by the NSF through DMR-9453362 and DMR-0074571

2. Magnetic properties of theneutron The neutron has a dipole moment mn is 960 times smaller than the electron moment A dipole in a magnetic field has potential energy Correspondingly the field exerts a torque and a force driving the neutron parallel to high field regions

3. The transition matrix element The dipole moment of unfilled shells yield inhomog. B-field The magnetic neutron senses the field The transition matrix element in Fermi’s golden rule Magnetic scattering is as strong as nuclear scattering It is sensitive to atomic dipole moment perp. to

4. The magnetic scattering cross section Spin density spread out scattering decreases at high k The magnetic neutron scattering cross section For unspecified incident & final neutron spin states

5. Un-polarized magnetic scattering Squared form factor DW factor Polarization factor Fourier transform Spin correlation function

6. Magnetic neutron diffraction Time independent spin correlations elastic scattering Periodic magnetic structures Magnetic Bragg peaks Magnetic primitive unit cell greater than chemical P.U.C. Magnetic Brillouin zone smaller than chemical B.Z. The magnetic vector structure factor is

7. Simple cubic antiferromagnet No magnetic diffraction for Real space ‘ Reciprocal Space

8. Not so simple Heli-magnet : MnO2 c b a characterize structure and Insert into diffraction cross section to obtain

9. Understanding Inelastic Magnetic Scattering: Isolate the “interesting part” of the cross section The “scattering law” is defined as for a wide class of systems It satisfies useful sum-rules Detailed balance Total moment First moment sum-rule

10. Scattering from a quantum spin liquid Dimerized spin-1/2 system: copper nitrate

11. Why a gap in spectrum of dimerized spin system J • A spin-1/2 pair has a singlet - triplet gap: • Weak inter-dimer coupling cannot close gap • Bond alternation is relevant operator for quantum critical uniform spin chain infinitesimal bond alternation yields gap

12. Sum rules and the single mode approximation 0.5 E (meV) 0.4 0 2 4 0 2 4 6 When a coherent mode dominates the spectrum: Sum-rules link S(q) and e(q) q (p)

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

14. Spin waves in an antiferromagnet Dispersion relation

15. Continuum magnetic inelastic scattering • Inelastic scattering is not confined to disp. relations when • There is thermal ensemble of excitations present • and do not uniquely specify excited state • - electron hole pair excitations in metals • - spinon excitations in quantum magnets spinon continuum in spin-1/2 AFM chain

16. r ( ) and the magnetic susceptibility to • We convert inelastic scattering data to • Compare with bulk susceptibility data • Isolate non-trivial temperature dependence • Compare with theories ab k w S , Compare to the generalized susceptibility They are related by the fluctuation dissipation theorem

17. Polarized magnetic neutron scattering Specify the incident and final neutron spin state Non spin flip: Spin flip:

18. Polarized neutron scattering Nuclear isotope incoherent scattering Paramagnetic scattering MnF2 H//k H//k SF SF NSF NSF

19. Summary • The neutron has a small dipole moment that causes it to scatter from inhomogeneous internal fields produced by electrons • The magnetic scattering cross section is similar in magnitude to the nuclear cross section • Elastic magnetic scattering probes static magnetic structure • Inelastic magnetic scattering probes spin dynamics through • Polarized neutrons can distinguish magnetic and nuclear scattering and specific spin components

20. Low T excitations in spin-1 AFM chain Y2BaNiO5 T=10 K MARI chain ki pure • Haldane gap D=8 meV • Coherent mode • S(q,w)->0 for Q->2np

21. AKLT state for spin-1 chain • Magnets with 2S=nz have a nearest neighbor singlet covering • with full lattice symmetry. • This is exact ground state for spin projection Hamiltonian • Excited states are propagating bond triplets separated from the • ground state by an energy gap Haldane PRL 1983 Affleck, Kennedy, Lieb, and Tasaki PRL 1987

22. Impurities in Y2BaNiO5 • Mg2+on Ni2+ sites finite length chains • Ca2+ on Y3+ sites mobile bond defects Mg Ca2+ Ni Y3+ Kojima et al. (1995)

23. Zeeman resonance of chain-end spins 20 g=2.16 hw (meV) 15 0 2 4 6 8 H (Tesla) 10 I(H=9 T)-I(H=0 T) (cts. per min.) 0 -5 0 0.5 1 1.5 2

24. Form factor of chain-end spins Y2BaNi1-xMgxO5 x=4% Q-dependence reveals that resonating object is AFM. The peak resembles S(Q) for pure system. Chain end spin carry AFM spin polarization of length x back into chain

25. New excitations in Ca-doped Y2BaNiO5 Pure 9.5% Ca Y2-xCaxBaNiO5: • Ca-doping • creates states • below the gap • sub-gap states • have doubly • peaked structure • factor

26. Why a double ridge below the gap in Y2-xCaxBaNiO5 ? d q is single impurity prop. Indep. of • Charge ordering yields incommensurate spin order • Quasi-particle Quasi-hole pair excitations in Luttinger liquid • Anomalous form factor for independent spin degrees of freedom associated with each donated hole x q d µ x

27. Does dq vary with calcium concentration? dq not strongly dependent on x Double peak is single impurity effect

28. Bond Impurities in a spin-1 chain: Y2-xCaxBaNiO5 (b) Ca (c) (d) (e) (f) Y Ba (a) O Ni

29. Form-factor for FM-coupled chain-end spins A symmetric AFM droplet Ensemble of independent randomly truncated AFM droplets

30. Calcium doping Y2BaNiO5 Experimental facts: • Ca doping creates sub-gap excitations with doubly peaked structure factor and bandwidth • The structure factor is insensitive to concentration and temperature for 0.04<x<0.14 (and T<100 K) Analysis: • Ca2+ creates FM impurity bonds which nucleate AFM droplets with doubly peaked structure factor • AFM droplets interact through intervening chain forming disordered random bond 1D magnet

31. Incommensurate modulations in high TC superconductors YBa2Cu3O6.6 T=13 K E=25 meV k (rlu) h (rlu) Hayden et al. 1998

32. What sets energy scale for sub gap scattering ? hw (meV) • Possibilities: • Residual spin interactions through • Haldane state. A Random bond AFM. • Hole motion induces additional • interaction between static AFM droplets • AFM droplets move with holes: • scattering from a Luttinger liquid of • holes. 10 5 ? • How to distinguish: • Neutron scattering in an applied field • Transport measurements • Theory 0

33. Conclusions on spin-1 chain • Dilute impurities in the Haldane spin chain create sub-gap composite spin degrees of freedom. • Edge states have an AFM wave function that extends into the bulk over distances of order the Haldane length. • Holes in Y2-xCaxBaNiO5 are surrounded by AFM spin polaron with central phase shift of p • Neutron scattering can detect the structure of composite impurity spins in gapped quantum magnets. • The technique may be applicable to probe impurities in other gapped systems eg. high TC superconductors. • Microscopic details of gapped spin systems may help understand related systems where there is no direct info.