8/25/11 Neutrons • Neutron discovered by James Chadwick in 1932 Neutrons for magnetism • Used in magnetic neutron diffraction in 1949 (Clifford Shull) • Neutron mass close to proton mass Stephen Blundell • Neutron decays in ~15 minutes University of Oxford • Spin ½ 2011 School - Time-dependent phenomena in magnetism Targoviste, August 2011 • Magnetic moment 1 2 Neutrons Neutrons • Neutron has no electrical charge • Energy given by • Interaction is with atomic nuclei (strong force, short • Energies are often given in meV = 10 -3 eV range) and unpaired electrons (EM) • Wavelength given in Angstroms • Neutron-matter interaction is weak : perturbation • Useful results: theory adequate (very roughly, probability of interaction in a solid ~10 -8 , mean free path ~1 cm) • Neutron interaction with nuclei and electrons similar in magnitude • Scattering cross section with proton is huge: 82 barns. For deuterons, it is an order of magnitude smaller. [1 barn = 10 -28 m 2 ] 3 4 Moderator Neutrons • Measure distribution of neutrons scattered from sample • Interaction potential determines properties measured • Scattering must be coherent for correlations to be measured • Scattering of neutrons is very weak and so can use the Born approximation • Means scattering depends on the Fourier transform of the interaction potential, and system responds linearly 5 6 1
8/25/11 Neutrons Neutrons • The Born approximation assumes coherence • Measurement non-destructive (though they and superposition might activate the sample!) • Detected amplitude = • Bulk, not surface probe • Detected intensity = • Samples have to be big (~1 mm 3 for diffraction experiments, ~1cm 3 for inelastic measurements) • Technique expensive – need a nuclear reactor depends on relative positions with holes in it (!) or a spallation source. These of atoms 1 and 2 are not cheap. Fortunately, Europe has a number of excellent sources of neutrons. 7 8 Kinematics • Conservation of energy • Conservation of momentum • Scattering event characterized by • (i) Elastic scattering • (ii) Inelastic scattering • Both processes occur in every experiment • Can set and independently in the experiment 9 10 11 12 2
8/25/11 Nuclear scattering Nuclear scattering • Scalar potential, short-range: Rigid crystal • Scattering length b~10 -14 m • Scattering function sum over reciprocal lattice vectors sum over nuclei in unit cell • structure factor • Rigid crystal sum over nuclei in unit cell sum over reciprocal lattice vectors • N=number of unit cells • V 0 =volume of unit cell 13 14 Coherent and incoherent Magnetic neutron diffraction • b varies with isotope and spin orientation • Magnetic interaction potential energy • separate into an average value and • coherent scattering results from and gives rise to diffraction peaks • depends on spin and orbital currents • incoherent scattering results from and • depends on direction of neutron spin gives rise to an incoherent background • vector, not scalar, interaction • anisotropic scattering • derives from electronic states and so the magnetic form factor is important sum over nuclei in unit cell 15 16 Magnetic neutron diffraction Magnetic neutron diffraction • Magnetic interaction potential in Q-space Magnetically ordered crystal • Maxwell: sum over magnetic reciprocal lattice vectors • magnetic structure factor is therefore perpendicular to • Neutrons scatter from , the component of the magnetic moment perpendicular to sum over magnetically ordered nuclei in unit cell 17 18 3
8/25/11 Magnetic neutron diffraction Scattering cross section Given the symbol MnO • Neutrons scattered into various final energies and also various bits of space (solid angle), so C.G. Shull et al. (1951) deal with the double differential cross section: see also A.L. Goodwin et al. PRL 96, 047209 (2006) • This sums up to the total cross section 19 20 Example Example para‐nitro Crystal structure strongly affects magnetic coupling. ‐phenyl Only the β -phase of p -NPNN is nitronyl nitroxide ferromagnetic. One can use the polarized neutron diffraction maps to An organic ferromagnet understand the important overlaps between regions with T c =0.67 K of positive and negative spin density. 21 22 Example Polarized neutron diffraction spin density of p-NPNN Measure I(k) for neutron polarization parallel or antiparallel Zheludev et al with the magnetic field, and the sign of the interference JMMM 135 147 (1994) term can be varied, allowing the magnetic structure factor to be deduced. 23 24 4
8/25/11 Scattering cross section • Numerator contains (i) (ii) (iii) speed of scattered neutrons (iv) density of incident neutrons (v) transition probability in which the sample changes its momentum by and its energy by , i.e. • Denominator contains (i) and (ii) (iii) 25 26 Detailed balance Calculation of S ( Q , ω ) neutron energy gain neutron energy loss matrix thermal occupation element of initial state • Can look at local magnetic excitations, e.g. crystal field level spectroscopy 27 28 Calculation of S ( Q , ω ) Calculation of S ( Q , ω ) spin correlation function spin correlation function • In spin waves, neutrons scatter from deviations perpendicular to Q • Intensity decreases with magnetic form factor • Compare phonons, intensity ~ b 2 ( Q.e ) 2 29 30 5
8/25/11 Cu II = 3d 9 Partially filled 3d shell gives Partially filled 3d shell gives rise to a magnetic moment rise to a magnetic moment 31 32 S.J. Blundell, Contemp. Phys. 48, 275 (2007) S.J. Blundell, Contemp. Phys. 48, 275 (2007) KCuF 3 La 2 CuO 4 J ~0.1 eV, J c ~0.05 eV • 1D chains of Heisenberg spins • orbital ordering, JT distorUon 33 34 E. Pavarini et al., PRL 101, 266405 (2008) B. Lake et al., Nat. Mat. 4, 329 (2005) R. Coldea et al., PRL 86, 5377 (2001) CoNb 2 O 6 CoNb 2 O 6 35 36 R. Coldea et al., Science 327, 177 (2010) R. Coldea et al., Science 327, 177 (2010) 6
8/25/11 CoNb 2 O 6 CoNb 2 O 6 38 39 R. Coldea et al., Science 327, 177 (2010) R. Coldea et al., Science 327, 177 (2010) 40 41 42 43 7
8/25/11 Transfer matrix: interface propagation 44 45 The end � Neutron Larmor precession in magnetic layers 46 8
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