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

CRNL 6/20/00 Magnetic properties of the neutron The neutron has a dipole moment  n 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

CRNL 6/20/00 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

CRNL 6/20/00 The magnetic scattering cross section Spin density spread out scattering decreases at high  The magnetic neutron scattering cross section For unspecified incident & final neutron spin states

CRNL 6/20/00 Un-polarized magnetic scattering Spin correlation function Squared form factor Polarization factor Fourier transform DW factor

CRNL 6/20/00 Magnetic neutron diffraction Time independent spin correlations elastic scattering Periodic magnetic structures Magnetic Bragg peaks The magnetic vector structure factor is Magnetic primitive unit cell greater than chemical P.U.C. Magnetic Brillouin zone smaller than chemical B.Z.

CRNL 6/20/00 Simple cubic antiferromagnet Real space Reciprocal Space ‘ No magnetic diffraction for

Not so simple Heli-magnet : MnO 2 Insert into diffraction cross section to obtain and characterize structure a b c

CRNL 6/20/00 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

CRNL 6/20/00 Scattering from a quantum spin liquid Dimerized spin-1/2 system: copper nitrate

CRNL 6/20/00  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 Why a gap in spectrum of dimerized spin system J

CRNL 6/20/00 Sum rules and the single mode approximation When a coherent mode dominates the spectrum : Sum-rules link S(q) and  (q) E (meV) q (  )

CRNL 6/20/00 Spin waves in a ferromagnet Magnon creation Magnon destruction Dispersion relation Magnon occupation prob. Gadolinium

CRNL 6/20/00 Spin waves in an antiferromagnet Dispersion relation

CRNL 6/20/00 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

CRNL 6/20/00 and the magnetic susceptibility Compare to the generalized susceptibility They are related by the fluctuation dissipation theorem   ,  S We convert inelastic scattering data to Compare with bulk susceptibility data Isolate non-trivial temperature dependence Compare with theories to

CRNL 6/20/00 Polarized magnetic neutron scattering Specify the incident and final neutron spin state Non spin flip: Spin flip:

CRNL 6/20/00 Polarized neutron scattering Nuclear isotope incoherent scattering Paramagnetic scattering MnF 2 H //  SF NSF SF NSF

CRNL 6/20/00 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

CRNL 6/20/00 Low T excitations in spin-1 AFM chain Y 2 BaNiO 5 T=10 K MARI chain k i Haldane gap  =8 meV Coherent mode S(q,  )->0 for Q->2n  pure

CRNL 6/20/00 AKLT state for spin-1 chain This is exact ground state for spin projection Hamiltonian Magnets with 2S=nz have a nearest neighbor singlet covering with full lattice symmetry. 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

CRNL 6/20/00 Impurities in Y 2 BaNiO 5 Ca 2+ Y 3+ Mg 2+ on Ni 2+ sites finite length chains Ca 2+ on Y 3+ sites mobile bond defects Mg Ni Kojima et al. (1995)

CRNL 6/20/00 Zeeman resonance of chain-end spins I(H=9 T)-I(H=0 T) (cts. per min.) h  (meV) H (Tesla) g=

CRNL 6/20/00 Form factor of chain-end spins 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  back into chain Y 2 BaNi 1-x Mg x O 5 x=4%

CRNL 6/20/00 New excitations in Ca-doped Y 2 BaNiO 5 Pure 9.5% Ca Ca-doping creates states below the gap sub-gap states have doubly peaked structure factor Y 2-x Ca x BaNiO 5 :

CRNL 6/20/00 Why a double ridge below the gap in Y 2-x Ca x BaNiO 5 ?  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  q  is single impurity prop. Indep. of x

CRNL 6/20/00 Does  q vary with calcium concentration?  q not strongly dependent on x Double peak is single impurity effect

CRNL 6/20/00 (b) Ca Ba Ni Y (e) (f) (c) (d) O Bond Impurities in a spin-1 chain: Y 2-x Ca x BaNiO 5 (a)

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

CRNL 6/20/00 Calcium doping Y 2 BaNiO 5 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:  Ca 2+ 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

CRNL 6/20/00 Incommensurate modulations in high T C superconductors YBa 2 Cu 3 O 6.6 T=13 K E=25 meV h (rlu) k (rlu) Hayden et al. 1998

CRNL 6/20/00 What sets energy scale for sub gap scattering ? 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. How to distinguish: Neutron scattering in an applied field Transport measurements Theory ? h  (meV)

CRNL 6/20/00 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 Y 2-x Ca x BaNiO 5 are surrounded by AFM spin polaron with central phase shift of   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 T C superconductors.  Microscopic details of gapped spin systems may help understand related systems where there is no direct info.