Neutrons 101
Properties of Neutrons
Canadian Powder Diffraction Workshop June 2010 UQTR
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, mean 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 S   s( s  1)
• Particles with angular momentum have a magnetic moment, 
q
g
S
2m
Spin
Angular Momentum
Moment
s
S

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
mL
m
• Neutron and electron moments can interacts– neutrons are sensitive to
magnetic moments in solids.
• Get additional magnetic diffraction peaks from the lattice of ordered
spins (beyond our course).
Characterizing Neutrons By….
1
meV
cm-1
THz
K
Å
ms-1
E
k

T

v
E
meV
1
0.1240
4.136
8.616e-2
81.807
5.227e-6
Linear
Reciprocal
Square-reciprocal
Root
Root-reciprocal
Square
2
h
E
2m2
Neutron Conversion Factors
k
T


-1
cm
THz
K
Å
8.006
0.2418
11.605
9.045
1
0.02998
1.439
25.68
33.336
1
48.00
4.447
-2
0.6949
2.083e
1
30.81
659.8
19.78
949.4
1
-5
-6
-5
4.216e
1.265e
6.066e
3956
Key
E = 4.136 v
1 THz  4.136 meV
-1
1 Å  3956 ms
v = 3956/
1 Å  19.78 THz
 = 19.78/2
1 meV  437.4 ms-1
v = 437.4E
1 meV  9.045 Å
= 4.447/ v
-1
-6
E = (m/2)v2
1 ms  5.227e meV
E  h
v
ms-1
437.4
154.05
889.5
128.4
3956
1
2 2
1 2
k
E

mv
E  kBT
E
2
2m
Neutron Sources
Neutrons must be liberated from their bonds
Binding energy of the nuclei ~MeV
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.
http://upload.wikimedia.org/wikipedia/commons/9/9a/Fission_chain_reaction.svg
• This is around 7700 times slower!
Spallation Source
• Spallation=“blowing chunks” (p,n)
• hydride ion (H-) source  proton accelerators
 targets  moderators  instruments
http://www.isis.rl.ac.uk/
Moderation/Slowing-down
-neutrons as particles (“gas”)
Maxwellian
• Distribution of velocities of particles as f(T)
– neutrons behave like a gas.
• Maxwell-Boltzmann distribution-the most
probable speed distribution in a collisionallydominated system consisting of a large number of
non-interacting particles.
– describes the neutron spectrum to a good approximation
(ignoring -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.
E  kBT
1 2
E  mv
2
Moderators
• Moderated neutrons take on the average kinetic
energy of the moderator, set by its T.
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.
Reactor simulator
0.264
0.283
0.304
0.330
0.360
0.440
0.495
0.565
lambda (Angstrom)
0.659
0.791
0.989
1.319
1.978
3.956
--
h2
E
2m2
0.396
Moderators & the Maxwellian
0.0016
0.0014
Maxwellian Distribution
0.0012
0.0010
Cold Source H2 20K
NRU D2O 333K
Hot Source Graphite 2303K
0.0008
0.0006
Note:
Hot source increases
the number of
high-E (v2), short-
neutrons, but does so
by spreading out the
dist’n, thereby
reducing the flux at
any ,(or v, or E, ….).
0.0004
0.0002
0.0000
0
2000
4000
6000
8000
velocity (m/s)
10000
12000
14000
1 2
E  mv
2
Cold source reduces
the spread to only very
long  and increases
the flux at those 
Wave-Particle
Duality
Neutrons have a
wavelength
• de Broglie hypothesis: all matter has a wave-like nature
• Neutrons have an associated wavelength, , diffract and
have wave-like properties
h
E  h ;  
mv
k
2

Strictly
“angular”
wavenumber
2k 2
E
2m

r
h ~ Planck' s constant; m ~ mass; v ~ velocity; mv ~ momentum;
 ~ wavelengt h;  ~ frequency; k ~ wavenumber
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.
k
Huygens-Fresnel Principle
http://upload.wikimedia.org/wikipedia/commons/a/a4/Christiaan_Huygens-painting.jpeg
Christiaan Huygens
1629-1695
Plane wave passing through a 4-slit:
Note secondary spherical wave sources
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.
A classical, very simple way of seeing the relationship
between plane wave (beams) and spherical waves
(scattering from individual particles)
Ocean plane waves passing
through slits
http://www.physics.gatech.edu/gcuo/UltrafastOptics/OpticsI/lectures/OpticsI-20-Diffraction-I.ppt
Scattering lengths
and cross-sections
Spherical Waves
• Wave energy is conserved as wave propagates
• Energy of the wavefront spreads (radiates) out
over the spherical surface area, 4r2.
 Energy/unit area decreases as 1/r2.
• Since energyintensity E  Amplitude2.
Amplitude of a spherical wave  1/r
Interaction Strength
Neutrons interact via the strong nuclear force
(nuclear scattering).
What is a
scattering length?
10-15m
Spherical wave
• Nucleus is a point with respect to .
10-10m
• 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
b
  exp(ikr )
r
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.
Scattering Length, b  Cross-section, s
 ( r ) ( r ) * defines a probability density
of finding neutron at r from the nucleus
The surface area of
a sphere at radius, r
b
  exp(ikr )
r
4r 2
s  4r  *  4r   4 b
2
2
2
Not forgetting our identities:
exp( ikr)  cos( kr)  i sin( kr)
cos2 (kr)  sin 2 (kr)  1
2
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….
Cold
Thermal
Epithermal
Energy dependence of
cross sections
Fast
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).
An absorber: 113Cd
Shielding materials:
Fast
Resonances
Epithermal
Cold
Thermal
Good neutron shielding
1)
Moderators e.g. H
thermalize fast
neutrons
2)
Attenuators: e.g. H
strong scatterers like a diffusing
screen (pearl light
bulb)
2) Thermal absorbers
Cd, 10B, Gd (6Li)
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
bco  E (b)  b   xi bi
binc  Var(b)  E (b 2 )  E (b) 2 
 xi bi  ( xi bi 
2
2
..which leads to comparison to Xray scattering
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.

f (Q)   r ( r ) expi Q rdr
0
X-ray form factor
X-ray beam: Incident
plane wave (in phase)
Path difference: therefore, X-rays have a form
factor that causes amplitude to die off as a
function of 2q
Form factors
Bigger object (cation),
faster drop off
Bigger angle: greater path
difference, more drop off
X-rays
• Nucleus is infinitesimal point wrt
neutron wavelength
• No destructive interference
• Isotopic dependence
Neutrons
Scattering lengths and cross-sections
s  4r 2 *  4r 2   4 b
2
2
Summary
•
•
•
•
•
Spin, charge etc
Energy, velocity, wavelength
Moderation
Cross section, scattering length
X-rays vs. neutrons