Neutrons 101

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Neutrons 101

• Properties of Neutrons

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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 neutronsboth 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.

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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- ?e
• Mass is slightly larger than that of a proton

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Neutrons have a spin

• Spin, s, is a quantum number neutrons are
  spin-half, s1/2
• Angular momentum
• Particles with angular momentum have a magnetic
  moment, ?

Spin
Moment
Angular Momentum
s
S
m
Note Although neutral, q 0, the neutron is
made up of quarkselectrically 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.
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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
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Characterizing Neutrons By.
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Neutron Sources
Neutrons must be liberated from their bonds
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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 of Be, C,
  O. As an example, a representative a-Be neutron
  source produces 30 neutrons for every million
  a-particles.
• e.g., PuBe.

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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
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Spallation Source

• Spallationblowing chunks (p,n)
• hydride ion (H-) source ? proton accelerators ?
  targets ? moderators ? instruments

http//www.isis.rl.ac.uk/
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Moderation/Slowing-down-neutrons as particles
(gas)
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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).

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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
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Moderators the Maxwellian
Note Hot source increases the number of high-E
(v2), short-l neutrons, but does so by spreading
out the distn, 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
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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
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Waves
http//upload.wikimedia.org/wikipedia/commons/1/12
/Spherical_wave2.gif
http//upload.wikimedia.org/wikipedia/commons/5/5c
/Plane_wave.gif
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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

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Wavevector
Assumes a real amplitude

• Cross-section at a snapshot in time (t 0)
• k k 2p/l, where l distance is the between
  two wavefronts

u(x)
c.f. your handouts!
x
l
A monochromatic neutron beam is characterized by
a plane wave with a single wavevector
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Huygens-Fresnel Principle
k
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
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Ocean plane waves passing through slits
http//www.physics.gatech.edu/gcuo/UltrafastOptics
/OpticsI/lectures/OpticsI-20-Diffraction-I.ppt
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Spherical Waves

• Wave energy is 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 energy?intensity E ? Amplitude2.

Amplitude of a spherical wave ? 1/r
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Interaction Strength
Neutrons interact via the strong nuclear
force (nuclear scattering).
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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
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Scattering Length, b

• Can be positive or negative!
• A positive b can be explained simply in terms of
  an impenetrable nucleus which the n cannot enter
  D? 180.
• A negative b is due to n nucleus forming a
  compound nucleus D? 0.
• More generally, b is complex b b ib the b
  imaginary component is related to absorption and
  is frequency-dependent.

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Scattering Length, b ? Cross-section, s
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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, s).
• 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 outhousesbut
not used for neutrons.
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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).

Fast
Cold
Epithermal
Thermal
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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
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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

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..which leads to comparison to X-ray scattering
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X-rays and Neutrons

• X-rays scatter from the electron cloud (r10-10m)
  surrounding the atom
• Neutrons scatter from atomic nuclei
  (r10-14-10-15m) influenced by neutron-nuclear
  force.
• ? 2 important differences

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X-rays and Neutrons- Difference 1

• X-rays scatter from the electron cloud
  ss ? Z2.
• Neutrons scatter from atomic nuclei
  ss isotope-dependent

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X-rays and Neutrons- Difference 2

• l10-10m Å (for both neutrons and X-rays)
• X-rays scatter from the electron cloud (r10-10m)
  Å
• Neutrons scatter from atomic nuclei
  (r10-14-10-15m) fm
• ?Nuclei are point scatterers wrt l.

Four orders of magnitude Nucleus l is
as Deep-RiverPembroke EarthMoon
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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.

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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
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Summary

• Spin, charge etc
• Energy, velocity, wavelength
• Moderation
• Cross section, scattering length
• X-rays vs. neutrons