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## Neutrons 101

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**Neutrons 101**Properties of Neutrons**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.**Instability of free neutron and mass**• Free neutrons are unstable; they undergo b-decay, lifetime ~ 885.7 ± 0.8 s. • They cannot be stored for long free! • n0 → p+ + e− + νe • Mass is slightly larger than that of a proton**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! mL ms**Binding energy of the nuclei ~MeV**Neutron Sources Neutrons must be liberated from their bonds**a-particles with light elements**Discovery of the Neutron 1930 Walther Bothe and H. Becker found that a-particles emitted from Po fell on certain light elements a highly penetrating radiation was produced: (a, n). 1932 Irène Joliot-Curie and Frédéric Joliot showed that if this unknown radiation fell on hydrogenous compounds it ejected very high-energy protons (n, p). 1932 James Chadwick showed that the g-ray hypothesis was untenable and that the new radiation was uncharged particles of approximately the mass of the proton. • Neutrons are produced when a-particles hit several low-Z isotopes including those ofBe, C, O. As an example, a representative a-Be neutron source produces~30 neutrons for every million a-particles. • e.g., PuBe.**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**Spallation Source**• Spallation=“blowing chunks” (p,n) • hydride ion (H-) source proton accelerators targets moderators instruments http://www.isis.rl.ac.uk/**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).**Moderators**• Light nuclei + low absorption. • Elastic* collisions between the nucleus and the neutron transfer energy. • Moderated neutrons take on the average kinetic energy of the moderator, set by its T. 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.* How many collisions are necessary to moderate a 2MeV fission neutron to a 1eV neutron? ~16 for light water, which take place in about 30 cm of travel. Simon Steinmann, Raul Roque: Creative Commons Attribution ShareAlike 2.5**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 • Wavenumber: we will meet wavevector shortly 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. • Physical solution • General form • where k is the wavevector, t time, w angular frequency, assuming a real amplitude, A**Wavevector**• Cross-section at a snapshot in time (t = 0) • |k| = k= 2p/l, where l distance is the between two wavefronts Assumes a real amplitude u(x) c.f. your handouts! x l A monochromatic neutron beam is characterized by a plane wave with a single wavevector**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 Waves**• Wave energyis conserved as wave propagates • Energy of the wavefront spreads (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**Interaction Strength**Neutrons interact via the strong nuclear force (nuclear scattering).**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**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 materials: • Moderators e.g. H thermalize fast neutrons • Attenuators: e.g. H • strong scatterers - like a diffusing screen (pearl light bulb) • 2) 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**Coherent & Incoherent Scattering**• Scattering nucleus at a given position in a crystal may be either: (i) different isotope (ii) different nuclear spin state [(iii) different element (diffuse scattering)] • Mean measure of expected value - coherent scattering – interference effects – average structure – Bragg diffraction • Std deviation measure of dispersion - incoherent scattering – “spin”/“isotopic” – single particle dynamics**X-rays and Neutrons**• X-rays scatter from the electron cloud (r~10-10m) surrounding the atom • Neutrons scatter from atomic nuclei (r~10-14-10-15m) influenced byneutron-nuclear force. 2 important differences**X-rays and Neutrons- Difference 1**• X-rays scatter from the electron cloud: ss Z2. • Neutrons scatter from atomic nuclei: ss ~isotope-dependent**X-rays and Neutrons- Difference 2**• l~10-10m [Å] (for both neutrons and X-rays) • X-rays scatter from the electron cloud (r~10-10m) [Å] • Neutrons scatter from atomic nuclei(r~10-14-10-15m) [fm] Nuclei are point scatterers wrt l. Four orders of magnitude: Nucleus: l is as Deep-River—Pembroke: Earth—Moon**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.**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**Summary**• Spin, charge etc • Energy, velocity, wavelength • Moderation • Cross section, scattering length • X-rays vs. neutrons