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## Neutrons 101

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**Neutrons 101**Properties of Neutrons Canadian Powder Diffraction Workshop May 2012 UofS**Why neutrons?**• Even if you have no particular interest in neutrons, at some point you will come across neutron diffraction studies. • Their real niche is magnetism, but they also have uses in biological applications (H/D exchange), in certain site ordering problems and in finding light atoms among heavy atoms.**What is a neutron?**• The neutron is a subatomic particle with no net electric charge. Nucleus • Neutrons are usually bound (via strong nuclear force) in atomic nuclei. Nuclei consist of protons and neutrons—both known as nucleons. • The number of protons determines the element & the number of neutrons determines the isotope, e.g. 15N and 14N have 7p and 8n and 7n respectively.**Neutrons have a spin**• Spin, s, is a quantum number: neutrons are spin-half, s=1/2 • Angular momentum • Particles with angular momentum have a magnetic moment, Spin Angular Momentum Moment s S m Note: Although neutral, q = 0, the neutron is made up of quarks—electrically charged particles. The magnetic moment of the neutron is ultimately derived from the angular momentum of spins of the individual quarks and of their orbital motions.**Electrons have a spin too.**• Orbital and spin (s = 1/2) angular momentum give rise to moments and magnetism • Neutron and electron moments can interact – neutrons are sensitive to magnetic moments in solids. • Get additional magnetic diffraction peaks from the lattice of ordered spins (well beyond our course). mL ms**So 50% of all mass is neutrons**The famous line of stability, starts off with equal numbers of protons and neutrons, but then becomes more neutron rich further down the periodic table. This is what makes an isotope stable (or not). To generate free neutrons we have to break apart stable isotopes.**Neutron Sources**Neutrons must be liberated from their bonds Neutrons are born with energies near the binding energy per nucleon ~ few MeV**And when we do liberate them, they fall apart**• Free neutrons are unstable; they undergo b-decay, mean lifetime ~ 885.7 ± 0.8 s. • They cannot be stored for long free! • n0 → p+ + e− + νe Mass of neutron is slightly larger than that of a proton**Spallation Source**• Spallation=“blowing chunks” (p,n) • hydride ion (H-) source proton accelerators targets moderators instruments http://www.isis.rl.ac.uk/**Spallation Source**http://www.isis.rl.ac.uk/**Fission Reactor**• U235 + n (thermal) • ~2 MeV neutrons produced • Fission neutrons move at ~7% of the speed of light • Moderated(thermal) neutrons move at ~8 times the speed of sound. • This is around 7700 times slower! http://upload.wikimedia.org/wikipedia/commons/9/9a/Fission_chain_reaction.svg**Maxwellian**• Distribution of velocities of particles as f(T) • neutrons behave like a gas. • Maxwell-Boltzmann distribution-the most probable speed distribution in a collisionally-dominated system consisting of a large number of non-interacting particles. • describes the neutron spectrum to a good approximation (ignoring l-dependent absorption).**Elastic Collisions**• Elastic* collisions between the nucleus and the neutron transfer energy. Simon Steinmann, Raul Roque: Creative Commons Attribution ShareAlike 2.5 An elastic collision is a collision in which the total kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision.* Good moderator nucleus = Low mass + low absorption cross-section + high scattering cross-section. (We’ll see cross-section later)**Moderators**• Moderated neutrons take on the average kinetic energy of the moderator, set by its T. How many collisions are necessary to moderate a 2 MeV fission neutron to a 1 eVneutron? ~16 for light water, which take place in about 30 cm of travel.**Moderators & the Maxwellian**Note: Hot source increases the number of high-E (v2), short-l neutrons, but does so by spreading out the dist’n, thereby reducing the flux at any l, (or v, or E, ….). Cold source reduces the spread to only very long l and increases the flux at those l**Wave-Particle Duality**Neutrons have a wavelength • de Broglie hypothesis: all matter has a wave-like nature • Neutrons have an associated wavelength, l, diffract and have wave-like properties Strictly “angular” wavenumber l r**Waves**http://upload.wikimedia.org/wikipedia/commons/5/5c/Plane_wave.gif http://upload.wikimedia.org/wikipedia/commons/1/12/Spherical_wave2.gif**Plane Waves**• A constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the wavevector, k. • Therefore: • A monochromatic beam of radiation (X-rays, neutrons etc.) can be represented as a plane wave characterized by a single wavevector (direction and l)**Spherical Waves**• Wave energyis conserved as wave propagates • Energy of the wavefrontspreads (radiates) out over the spherical surface area,4pr2. Energy/unit area decreases as 1/r2. • Since energyintensity E Amplitude2. Amplitude of a spherical wave 1/r**k**Huygens-Fresnel Principle Each point of an advancing wave front is the centre of a fresh disturbance and the source of a new train of waves. The advancing wave is the sum of all secondary waves arising from points in the medium already traversed. http://upload.wikimedia.org/wikipedia/commons/a/a4/Christiaan_Huygens-painting.jpeg Christiaan Huygens 1629-1695 A classical, very simple way of seeing the relationship between plane wave (beams) and spherical waves (scattering from individual particles) Plane wave passing through a 4l-slit: Note secondary spherical wave sources**Ocean plane waves passing through slits**http://www.physics.gatech.edu/gcuo/UltrafastOptics/OpticsI/lectures/OpticsI-20-Diffraction-I.ppt**Spherical wave**What is a scattering length? • Nucleus is a point with respect to l. • Treat the incoming monochromatic neutron beam as a plane wave of neutrons with single k • Neutrons scatter from individual nuclei (secondary source): • independently of angle as spherical waves • scattered wave amplitude 1/r • Proportionality constant: b – scattering length 10-10m 10-15m**Scattering Length, b**• Can be positive or negative! • A positiveb can be explained simply in terms of an impenetrable nucleus which the n cannot enter – D ~ 180°. • A negativeb is due to “n + nucleus” forming a compound nucleus – D ~ 0°. • More generally, b is complexb = b’+ ib”– the b” imaginary component is related to absorption and is frequency-dependent.**The surface area of**a sphere at radius, r defines a probability density of finding neutron at r from the nucleus Not forgetting our identities: Scattering Length, b Cross-section, s**Interaction Strength**Neutrons interact via the strong nuclear force (nuclear scattering).**Cross-section**U is “as big as a barn.” • The interaction probability is the likelihood of a point-projectile hitting the target area (the cross section, σ). • Each nucleus thought of as being surrounded by a a characteristic area. • Barn = 10−28 m2, ~ the cross sectional area of U. • Cross-sections for different processes: scattering, absorption, fission… • They are not constant, but energy-dependent There are also units of sheds, and outhouses…but not used for neutrons….**Energy dependence of cross sections**Note: • Resonances at high-energy • Constant plateau of scattering cross-section • Strong (1/v) dependence of absorption – related to the time spent near the nucleus (probability of capture). Cold Fast Epithermal Thermal**An absorber: 113Cd**• Shielding design for neutrons is usually “graded”: • A) Hydrogenous material to slow down the neutron and diffuse any strong beams: • Moderation e.g. H thermalize fast neutrons • Attenuators: e.g. H • strong scatterers - like a diffusing screen (pearl light bulb) • B) Thermal absorbers • Cd, 10B, Gd (6Li) Fast Resonances Good neutron shielding Thermal Cold Epithermal ENDF/B-VII Incident-Neutron Data – 60pp for 113Cd! http://t2.lanl.gov/data/neutron7.html**Nuclear versus Electromagnetic Interaction**• X-rays interact with the electrons of the atom via the electromagnetic interaction • X-rays sensitive to electronic state of atom (anomalous scattering, resonance) • X-rays scatter Z • Neutrons interact with the nucleus via the strong nuclear force. • Neutrons sensitive to isotopic composition • Magnitude of scattering only varies by ~2-3**10-12cm**5 4 3 X-ray 2 1 Neutron 1 (Sin q)/l 108cm-1 X-ray atomic form factors Low angles, little path difference High angles, greater path difference X-ray: Destructive interference is possible at high angles due to finite size of electron cloud form factor Neutron: Nucleus is orders of magnitude smaller than neutron wavelength no form factor K-atom**Form factors**Bigger object, faster drop off • Nucleus is infinitesimal point wrt • neutron wavelength • No destructive interference • Isotopic dependence Bigger angle: greater path difference, more drop off X-rays Neutrons**Comparison of Relative Scattering Powers**f Z for X-rays b more uniform in Z with some “random” variation for neutrons Note: 1. “Random” variation of neutron b’s may give good phase contrast 2. Some isotopes/ elements have negative scattering lengths**Neighbouring elements**HQ/INRS study of Ti-Ru-Fe-O ball-milled electrodes NPD XRD NPD XRD “Ti2RuFeO” Ti2RuFe**Form Factors**• The form factor, f(Q) is the Fourier Transform of the scattering density r(r) • for neutrons it is in the form of a d-function • for X-rays the electron cloud distribution.**Summary**• Spin, charge etc • Energy, velocity, wavelength • Moderation • Cross section, scattering length • X-rays vs. neutrons