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Basic Elements of Neutron Inelastic Scattering

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Basic Elements of Neutron Inelastic Scattering 2019 NCNR Summer School on Methods and Applications of Neutron Spectroscopy Basic Elements of Neutron Inelastic Scattering Peter M. Gehring National Institute of Standards and Technology NIST Center for Neutron Research Gaithersburg, MD USA Φs hω Outline 1. Introduction - Motivation - Scattering Probes 2. The Neutron - Production and Moderation - Wave/Particle Duality 3. Basic Elements of Neutron Scattering - The Scattering Length b - Scattering Cross Sections - Pair Correlation Functions - Coherent and Incoherent Scattering - Neutron Scattering Methods (( )) (( )) (( )) (( )) ω0 ≠ 0 4. Summary of Scattering Cross Sections (( )) (( )) (( )) (( )) - Elastic (Bragg versus Diffuse) (( )) (( )) (( )) (( )) - Quasielastic (Diffusion) - Inelastic (Phonons) ω (( )) (( )) (( )) (( )) Motivation Structure and Dynamics The most important property of any material is its underlying atomic / molecular structure (structure dictates function). Bi2Sr2CaCu2O8+δ The motions of the atoms (dynamics) are extremely important because they provide information about the interatomic potentials. An ideal method of characterization would provide detailed information about both structure and dynamics. Motivation Why Should We Scatter? Lyft Conveys information: We see something when light scatters from it. location, speed, shape Light is composed of electromagnetic waves. λ ~ 4000 Å – 7000 Å However, the details of what we can see are ultimately limited by the wavelength. Motivation The tracks of a compact disk act as a diffraction grating, producing a separation of the colors of white light when it scatters from the surface. From this one can determine the nominal distance between tracks on a CD, which is 1.6 x 10-6 meters = 16,000 Angstroms. To characterize materials we must determine the underlying structure. We do this by using d the material as a diffraction grating. Problem: Distances between atoms in materials o are of order Angstroms light is inadequate. λLight >> d ~ 4 A Moreover, most materials are opaque to light. Scattering To measure atomic structure requires a Probes probe with a λ ~ length scale of interest. X rays EM - wave Some candidates … Electrons Charged particle Neutrons Neutral particle 1929 Nobel Laureate in Physics Scattering Probes X rays: Pros and Cons If we wish only to determine relative atomic positions, then we should choose x rays almost every time. 1. Relatively cheap 2. Sources are ubiquitous easy access 3. High flux can study small samples 4. Extremely good resolution However … Scattering Probes X rays are electromagnetic radiation. Nucleus Thus they scatter from the charge density. Consequences: Electron orbitals Low-Z elements Elements with similar X rays are strongly attenuated are hard to see. atomic numbers have when passing through furnaces, very little contrast. cryostats, and samples too. Hydrogen Cobalt Nickel ?? (Z = 1) (Z = 27) (Z = 28) Scattering Probes Electrons: Pros and Cons Electrons are charged particles they see both the atomic electrons and nuclear protons at the same time. 1. Relatively cheap 2. Sources are not uncommon easy access 3. Fluxes are extremely high can study tiny crystals 4. Very small wavelengths more information However … Scattering Probes Electrons have some deficiencies too ... Requires very Radiation damage Magnetic structures are hard to thin samples. is a concern. determine because electrons are deflected by the internal magnetic fields. M Scattering Probes Neutrons: Pros and Cons Zero charge not strongly Neutrons expensive to produce attenuated by furnaces, etc. access not as easy Magnetic dipole moment Interact weakly with matter can study magnetic structures often require large samples Nuclear interaction can see low-Z Available fluxes are low elements easily like H good for the compared to other methods study of biomolecules and polymers. Nuclear interaction is simple Low energies scattering is easy to model Non-destructive probe Scattering To characterize atomic dynamics requires Probes a probe with hω ~ energy scale of interest. Neutrons: X rays: E = 81.81/λ² E = 12,398,000/λ Scattering To characterize atomic dynamics requires Probes a probe with hω ~ energy scale of interest. Phonons Neutrons: E = 81.81/λ² Scattering Neutron: A Lucky Coincidence Probes E = p²/2m = h²k²/2m = 81.81/λ² The mass of the neutron is such that one can simultaneously study structure and dynamics. Thus neutron scattering methods can directly measure the geometry of dynamic processes. The Neutron “If the neutron did not exist, it would need to be invented.” Bertram Brockhouse 1994 Nobel Laureate in Physics The Neutron “… for the discovery of the neutron.” Sir James Chadwick 1935 Nobel Laureate in Physics The Neutron -27 mn= 1.675x10 kg Q = 0 S = ½ h μn= -1.913 μN Interactions: Strong, Electro-weak, Gravity o o λ = 1 A λ = 9 A de Broglie Relation: v = 4000 m/s v = 440 m/s λ = h/p = h/m v E = 82 meV n E = 1 meV Production Free neutrons decay via the weak force. - Lifetime ~ 888 seconds (15 minutes). n p + e + νe A useful source of neutrons requires a nuclear process by which bound neutrons can be freed from n the nuclei of atoms and that is easily sustainable. There are two such processes, spallation and fission … Spallation Fission Fission t * Nuclear fission is used in power 235U + n → 236U → X +Y + 2.44n and research reactors. 92 143 [ 92 144 ] A liquid medium (D2O, or heavy water) is used to moderate the fast fission neutrons to room temperature (2 MeV 50 meV). The fission process and moderator are confined by a large containment vessel. (-mv2/2k T) Moderation Maxwellian Distribution Φ ~ v3e B Liquid Hydrogen 25 K 6 5 Heavy Water (D2O) 4 3 Hot Graphite 300 K 2 1 2000 K 0 0 5 10 15 “Fast” neutrons: v = 20,000 km/sec Neutron velocity v (km/sec) Moderation de Broglie Relation λ = h/mnv Fast Neutron, ~ 0.00002 nm V ~ 20,000,000 m/sec Thermal Neutron, ~ 0.2 nm V ~ 2,000 m/sec Cold Neutron, V ~ 200 m/sec ~ 2 nm Scattering Basics Neutron scattering experiments measure the flux Φs of neutrons scattered by a sample into a detector as a function of the change in neutron wave vector (Q) and energy (hω). Φ (Q,hω) = neutrons s sec–cm2 Momentum Energy 2 2 ki kf hk = h(2π/λ) hωn = h kn /2m 2θ hQ = hki - hkf hω = hωi - hωf The expressions for the scattered neutron flux Φs involve the positions and motions of atomic nuclei or unpaired electron spins. Contains information about Φs structure and dynamics! Scattering These “cross sections” are what Basics we measure experimentally. Consider an incident neutron beam 6 9 2 with flux Φi (~10 to 10 n/sec/cm ) We define three cross sections: and wave vector ki on a non-absorbing sample. σ = Total cross section dσ Ω = Differential cross section kf d d2σ Partial differential = dΩdEf cross section 2 Φi ki Cross What are the physical meanings Sections of these three cross sections? σ Total # of neutrons scattered per second / Φi. (Typically of order 1 barn = 10-24 cm2.) dσ Total # of neutrons scattered per second into dΩ / dΩ Φi. dΩ (Diffraction structure. Signal is summed over all energies.) d2σ Total # of neutrons scattered per second into dΩ with a final energy between Ef and dEf / dΩ dEf Φi. dΩdEf (Inelastic scattering dynamics. Small, but contains much info.) Cross What are the relative Sections sizes of the cross sections? dσ d2σ Clearly: σ = dΩ = dΩdEf dΩ dΩdEf dσ d2σ Thus: σ >> >> dΩ dΩdEf 2 dσ . d σ σ dΩ dΩdEf dσ d2σ Typically, ~ 106 x dΩ dΩdEf Elastic vs Note that both of these d2σ Inelastic cases are described by … dΩdEf dσ Elastic (ki=kf) ≠ dΩ Elastic Scattering: • Change in neutron energy = 0 sinθ = (Q/2)/k • Probes changes in momentum kf θ Q = 2ksinθ = 4πsinθ/λ only θ ki Inelastic (ki≠kf) Inelastic Scattering: • Change in neutron energy ≠ 0 • Probes changes in both k k f Q f Q momentum and energy 2θ 2θ ki ki Energy loss (hω>0) Energy gain (hω<0) Nuclear Consider the simplest case: Scattering A fixed, isolated nucleus. The scattered (final) neutron Ψf is a spherical wave: λ ∼ 2 Å ikr-ωt Ψf(r) ~ (-b/r)e QUESTIONS: 1. The scattering is elastic (ki = kf = k). Why? x 2. The scattering is isotropic. Why? The incident neutron Ψi is a plane wave: Nucleus ikx-ωt 1 fm = 10-15 m Ψi(r) ~ e (k = 2π/λ) Nuclear ANSWERS: Scattering 1. The scattering is elastic because the nucleus is fixed, so no energy can be transferred to it from the neutron (ignoring any excitations of the nucleus itself). 2. A basic result of diffraction theory states: if waves of any kind scatter from an object of a size << λ, then the scattered waves are spherically symmetric. (This is also known as s-wave scattering.) N << λ 2 V(r) = 2πh b δ(r-r ) ( mn )Σ j j j=1 ~ 10-15 m << 10-10 m Details of V(r) are unimportant! V(r) can be parametrized by a scalar b that Fermi Pseudopotential depends only the nucleus and isotope! Nuclear Calculate σ for a Scattering single, fixed nucleus: . Def. σ Φi = Total number of neutrons scattered per second by the nucleus. v = Velocity of neutrons (elastic same before and after scattering). dA v 2 Total number of neutrons scattered per second through . v dA |Ψf| = dA 2 2 2 2 Ω 2 . v dA |Ψf| = v(r dΩ)(b/r) = v b d = v 4πb = σ Φi σ Since Φ Ψ 2 σ π 2 d i = v | i| = v = 4 b Try calculating dΩ Nuclear The Neutron Scattering Length - b Scattering Units of length: Analogous to f(Q), the Varies randomly with Z and isotope b ~ 10-12 cm. x-ray form factor. Neutrons “see” atoms x rays can’t. Total Scattering Cross Section: σ = 4πb2 X rays (Q = 0) H D C N O Al Si Z Neutrons Nuclear What if many atoms are present? Scattering Scattering from Scattering from one nucleus many nuclei Scattered neutron Ψf is a spherical wave: Constructive interference The incident neutron Ψi is a plane wave: Get strong scattering in some directions, but not in others.
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