Plan 1. Introduction 2. The neutron interaction with magnetism 3. The Born Approximation 4. Elastic scattering 5. Bragg scattering 6. Neutron polarization analysis 7. Diffuse scattering INSTITUT MAX VON LAUE - PAUL LANGEVIN
Neutrons and Magnetism I. Elastic scattering of neutrons Andrew Wildes Institut Laue-Langevin, Grenoble, France 10 September 2005 INSTITUT MAX VON LAUE - PAUL LANGEVIN A.R.Wildes
No time to present: • Sources • Techniques and instrumentation • Resolution • Analysis methods INSTITUT MAX VON LAUE - PAUL LANGEVIN
What do neutrons measure? Magnetization and the size of the magnetic moment per atom the spatial distribution of magnetization Magnetic correlations in space and time magnetic structures, from Å to µ m the influence of impurities, frustration on magnetism magnetic phase transitions and critical exponents Energy states associated with magnetic electrons and dynamics eigenstates of Hamiltonians magnetic exchange integrals crystal field transitions Coupling between magnetic and chemical/structural properties superconductivity colossal- and giant magnetoresistance magnetostriction and INVAR effect INSTITUT MAX VON LAUE - PAUL LANGEVIN
Limitations of neutrons FLUX! Measurements are limited by statistics. Normally have to compromise on resolution. It is difficult to measure < 0.1 µ B per atom. Neutrons interact with all the magnetic fields in the sample, it can be difficult to separate different components. Neutrons don’t measure spatial dimensions directly, they measure in Fourier space. Conclusions are often model-dependent This is both a plus and a minus! INSTITUT MAX VON LAUE - PAUL LANGEVIN
The ‘Family Tree’ of Magnetism INSTITUT MAX VON LAUE - PAUL LANGEVIN
The properties of the neutron Neutrons are matter waves . They have a de Broglie wavelength of λ • ( λ ~ 1.8 Å is a ‘standard’ for ‘thermal’ neutrons, but experiments can use ~ 0.1 to 100 Å neutrons) They have a momentum of p = h/ λ = h k • and a kinetic energy of E = h 2 /2m λ 2 = h 2 k 2 /2m • They have a magnetic dipole moment given by − γµ σ ˆ N σ (where is the Pauli spin operator) ˆ • The kinetic energy of a thermal neutron is about the same as the energy of a lattice or magnetic vibration INSTITUT MAX VON LAUE - PAUL LANGEVIN
The neutron interaction with magnetism We must solve the Wave equation for the neutron/sample ensemble. [ ] ( ) ( ) ˆ − 2 ∇ 2 + ψ = ψ 2 M V r E h ˆ is the potential energy operator. For neutrons it is called the Fermi pseudo-potential. V ( r ) ˆ ( ) = V n r ( ) + V m r ( ) V r ( ) = 2 π h 2 ( ) ( ) b + B ˆ I ⋅ ˆ σ δ r V n r m n where ( ) = − γµ N ˆ ( ) σ ⋅ B r V m r b is the nuclear scattering length B Î is the nuclear spin B ( r ) is the magnetic induction INSTITUT MAX VON LAUE - PAUL LANGEVIN
The neutron interaction with magnetism We must solve the Wave equation for the neutron/sample ensemble. [ ] ( ) ( ) ˆ − 2 ∇ 2 + ψ = ψ 2 M V r E h ˆ is the potential energy operator. For neutrons it is called the Fermi pseudo-potential. V ( r ) ˆ ( ) = V n r ( ) + V m r ( ) V r ( ) = 2 π h 2 ( ) ( ) b + B ˆ I ⋅ ˆ σ δ r V n r m n where ( ) = − γµ N ˆ ( ) σ ⋅ B r V m r b is the nuclear scattering length B Î is the nuclear spin magnetic! B ( r ) is the magnetic field INSTITUT MAX VON LAUE - PAUL LANGEVIN
Neutron scattering Most neutron magnetic experiments are scattering experiments. Interaction with target Incident neutrons with wavelength λ , wave vector k , (|k |=2 π / λ ) spin s Scattered neutrons with wave vector k ´, spin s ´ Q Constructive and destructive interference leads to peaks in the intensity as a function of: 1) the momentum transfer: Q = k − k ´ ∆ E = ( h 2 ⁄ 2m)(| k| 2 − | k ´| 2 ) 2) the energy transfer: 3) the change in the neutron spin orientation INSTITUT MAX VON LAUE - PAUL LANGEVIN
The Born approximation Most neutron experiments are scattering experiments. The target volume is initially in state ζ . A neutron enters with wave vector k and spin s It interacts with the target. The final neutron wave vector is k ´ and spin s ´. The final target state is ζ ´. If the neutron has a plane wave function , if the interaction is weak , then the wave equation can be solved using first order perturbation theory, i.e. Fermi’s Golden Rule 2 ′ σ 2 ( ) ⎛ ⎞ d k m ( ) 2 ∑ ∑ ˆ ′ ′ ′ = ζ ζ δ ω + − ⎜ ⎟ n p p k , s , V r k , s , E E h ′ ζ ζ ζ s Ω ⋅ π 2 d d E k ⎝ 2 ⎠ h ′ ′ ζ ζ , s , s This is known as the cross-section , and gives the probability that a neutron will scatter in to a certain solid angle with a certain change in energy The two assumptions form the first Born approximation INSTITUT MAX VON LAUE - PAUL LANGEVIN
The cross-section Probabilities of initial target state ⎛ ⎞ 2 ′ d 2 σ k m n and neutron spin ∑ ∑ 2 ( ) ( ) k , s , ζ ζ ˆ d Ω⋅ d E = k , ′ ′ s , ′ δ h ω + E ζ − E ′ ⎜ ⎟ p ζ p s V r ζ ⎝ 2 π h 2 ⎠ k ζ , s ζ , ′ ′ s Conservation of energy The matrix element , which contains all the physics. Appropriate averaging over the target energy states, the positions r , and the neutron spin directions is necessary to find the measured cross-section G. L. Squires, Introduction to the theory of thermal neutron scattering , Dover Publications, New York, 1978 W. Marshall and S. W. Lovesey, Theory of thermal neutron scattering , Oxford University Press, Oxford, 1971 S. W. Lovesey, Theory of neutron scattering from condensed matter , Oxford University Press, Oxford, 1986 INSTITUT MAX VON LAUE - PAUL LANGEVIN
Elastic scattering If the incident neutron energy = the final neutron energy, the scattering is elastic . ⎛ ⎞ 2 d σ m n ∑ 2 s ˆ ( ) k , s k , ′ ′ d Ω = ⎜ ⎟ p s V r ⎝ ⎠ 2 π h 2 ζ , ′ ′ s Forget about the spins for the moment and integrate over all r : ( ) ( ) ˆ ∫ ˆ ′ ⋅ = i Q r ⋅ k V r k V r e d r Momentum transfer Q = k – k ´ The elastic cross-section is then directly proportional to the Fourier transform squared of the potential. Neutron scattering thus works in Fourier space, otherwise called reciprocal space. Elastic neutron scattering is also referred to as neutron diffraction INSTITUT MAX VON LAUE - PAUL LANGEVIN
f ( r ) e i Q ⋅ r ⋅ d r ∫ Fourier Transforms F ( Q ) = f (r) F (Q) A Delta function A A A series of Delta functions 1/a a Two Delta functions a 1/a A Gaussian A × a A a 1/a INSTITUT MAX VON LAUE - PAUL LANGEVIN
V m (r ), B ( r ) and the magnetization ( ) ( ) ∫ ∫ ⋅ ⋅ = − γµ σ ⋅ ⋅ − γµ σ ⋅ i Q r i Q r V r e d r ˆ B r e d r = ˆ B ( Q ) m N N B ( Q ) is related to the magnetization of the sample, M ( r ), through the equation: ( ) = ∫ ( ) ( ) ( ) ˆ ˆ × × ⋅ ⋅ = e i Q r B Q Q M r Q d r M Q ⊥ Neutron scattering therefore probes the components of the sample magnetization that are perpendicular to the neutron’s momentum transfer, Q . ( ) ∫ ⋅ i Q r ⋅ = V r e d r M ( Q ) ⊥ m σ d and magnetic = * M ( Q ) M ( Q ) ⊥ Ω d ⊥ Neutron scattering measures the correlations in magnetization, i.e. the influence a magnetic moment has on its neighbours. It is capable of doing this over all length scales, limited only by wavelength. INSTITUT MAX VON LAUE - PAUL LANGEVIN
Elastic scattering 2 σ ⎛ ⎞ d m ( ) 2 ∑ ˆ ′ ′ = ⎜ n ⎟ p k , s V r k , s s Ω π 2 d 2 ⎝ ⎠ h ′ ′ ζ , s ⎛ ⎞ 2 2 ˆ ˆ ˆ ∫ ∫ ∝ ⋅ ⋅ + − ⋅ ⋅ i Q r 2 i Q r V e d r ⎜ V V ⎟ z ( r ) e d r ⎝ ⎠ The contribution from deviations from the average structure: Short-range order The contribution from the average structure of the sample: Long-range order INSTITUT MAX VON LAUE - PAUL LANGEVIN
Magnetic structure determination 1. Long-range structure Crystalline structures d σ e i Q ⋅ r ⋅ d r 2 ∫ ˆ d Ω ∝ V Recall the Fourier transform from a series of delta-functions f (r) F (Q) 1/a a Bragg’s Law: 2 d sin θ = λ Leads to Magnetic Crystallography INSTITUT MAX VON LAUE - PAUL LANGEVIN
A simple example of magnetic elastic scattering MgB 2 is a superconductor below 39K, and expels all magnetic field lines (Meisner effect). Above a critical field, flux lines penetrate the sample. Scattering geometry k 2 θ Q k ´ k ´ The momentum transfer, Q , is roughly perpendicular to the flux lines, therefore all the magnetization is seen. σ d (recall ) magnetic = * M ( Q ) M ( Q ) ⊥ ⊥ Ω d INSTITUT MAX VON LAUE - PAUL LANGEVIN
A simple example of magnetic elastic scattering MgB 2 is a superconductor below 39K, and expels all magnetic field lines (Meisner effect). Above a critical field, flux lines penetrate the sample. Via Bragg’s Law 2 d sin θ = λ λ = 10 Å d = 425Å ~2 ° The reciprocal lattice has 60 ° rotational symmetry, therefore the flux line lattice is hexagonal Cubitt et. al. Phys. Rev. Lett. 91 047002 (2003) Cubitt et. al. Phys. Rev. Lett. 90 157002 (2003) INSTITUT MAX VON LAUE - PAUL LANGEVIN
Recommend
More recommend
