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Magnetic Frustration at Triple-Axis

Magnetic Frustration at Triple-Axis. S.-H. Lee, S. Park, P.M. Gehring NCNR . Magnetism, Neutron Scattering, Geometrical Frustration ZnCr 2 O 4 : The Most Frustrated Magnet

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Magnetic Frustration at Triple-Axis

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  1. Magnetic Frustration at Triple-Axis S.-H. Lee, S. Park, P.M. Gehring NCNR • Magnetism, Neutron Scattering, Geometrical Frustration • ZnCr2O4: The Most Frustrated Magnet How are the fluctuating spins in the Spin Liquid phase correlated with each other? How does nature respond to the ground state degeneracy? • Summary

  2. O A B Cr3+ (3d3) eg 3d t2g Free Ion Cubic Field H = -JS Si. Sj nn ZnCr2O4 Space group Fd3m Lattice of B sites : Corner-sharing tetrahedra Edge-sharing octahedra dxy, dyz, dzx

  3. Magnetic Neutron Scattering • Neutron: • Wavelength comparable with interatomic spacings • Penetrating  bulk properties are measured • has spins = ½ so interacts with atomic moments • Scattering by atomic • magnetic moments: I = (0.54)2 S (S+1) Magnetic scattering intensities can be comparable to nuclear scattering !!

  4. d2s dWdw (Q,w) Neutron Scattering Sample ki , Ei , s ? Incident neutrons Scattered neutrons kf , Ef , s measures scattering cross section as a function of Q and w

  5. Spin-Spin Correlation Function Magnetic Neutron Scattering Cross Section d2s < SR(t) SR (0) > dWdw The probability of finding a spin in a spin state SR at position R at time t when another spin at R’ at time t=0 is in a spin state SR Ordered moment Fluctuating moment / G ~ h/t 0 0 Time w Long range order Short range order k ~ 1/x Q R-R G : relaxation time k : intrinsic linewidth t : lifetime x : correlation length Fourier Transform

  6. Phase Diagram for an Ordinary Magnet F = Hmag – TS = - S J Si Sj - TS TS > | Hmag | T (K) Paramagnet QCW= JzS(S+1)/3kB | Hmag | > TS AFM or FM Ground State : Long range (anti)ferromagnetically ordered state. Low Energy Excitations : linear spin waves around the ordered state.

  7. Neutron Scattering Cross Section Paramagnetic phase: What kind of magnetic scattering signal in Q and  space would you expect? Correlation Function d2s Fourier Transform < SR(t) SR (0) > dWdw Ordered moment Fluctuating moment G ~ h/t 0 0 Time w Long range order Short range order k ~ 1/x Q R-R G : relaxation time k : intrinsic linewidth t : lifetime x : correlation length Neel phase: What kind of magnetic scattering signal in Q and  space would you expect? (1) Any Elastic signal? (2) Any Inelastic signal? If any, what kind of shape in the - and Q-space?

  8. Geometrical Frustration A simplest example: a Triangle of three antiferromagnetic Ising spins H = -JS Si. Sj AFM AFM All exchange interactions can not be satisfied. ? AFM E=-|J| A tetrahedron with four isotropic spins S Si=0 Zero energy modes in the ground state manifold Geometrical frustration leads to a large degeneracy in the ground state

  9. O A B Cr3+ (3d3) eg 3d t2g Geometrically frustrated Free Ion Cubic Field H = -JS Si. Sj nn Why ZnCr2O4 ? Space group Fd3m Lattice of B sites : Corner-sharing tetrahedra Edge-sharing octahedra Interaction between nearest neighbor Cr3+ ions through direct overlap of t2g orbitals is dominant. Best candidate for a system with dxy, dyz, dzx

  10. Phase Diagram for Geometrical Frustration F = Hmag – TS = - S J Si Sj - TS Paramagnet T (K) QCW= JzS(S+1)/3kB ? Geometrically frustrated What is the nature of the low temperature phase ?

  11. SPIN TYPE SPIN LOW T METHOD REFERENCE Value PHASE Isotropic S=1/2 SpinLiquid Exact Diag. Canals and Lacroix PRL ’98 Isotropic S= Spin Liquid MC sim. Reimers PRB ’92 Moessner, Chalker PRL ’98 Theory of spins with AFM interactions on corner-sharing tetrahedra H = -JS Si. Sj

  12. Issues • Nature of the Spin Liquid State How are the fluctuating spins in the SL phase correlated with each other? • Novel Phase Transition? How does nature respond to the ground state degeneracy? You will be able to answer the questions after the demonstration session at SPINS!

  13. A Phase Transition in ZnCr2O4 A.P. Ramirez, unpublished QCW = -390 K TN = 12.5 K Phase transition What is the nature of the phase transition?

  14. Neutron Scattering from ZnCr2O4 SHL et al., PRL 84, 3718 (2000) Paramagnetic fluctuations in frustrated AFM Local spin resonance in magneto-elastic LRO phase

  15. Keep in mind !!! d2s < SR(t) SR (0) > dWdw G ~ h/t R-R Neutron scattering is the most powerful technique to study magnetism!! Fourier Transform Ordered moment Fluctuating moment 0 0 Time w Long range order Short range order k ~ 1/x Q G : relaxation time k : intrinsic linewidth t : lifetime x : correlation length

  16. SPINS : Cold Neutron Triple-Axis Spectrometer

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