ELEMENTS OF NEUTRON SCATTERING
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Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
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Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
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n  p   e  ~
  0.77 MeV,   889.1 s
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Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
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Nuclear fission
In reactors, fission takes place when a fissile nucleus (always 235U or 239Pu in practice, although
there are a few others) captures a neutron, the nucleus splits into two nearly equal-mass
fragments (in large variety) that together carry about 160 MeV of kinetic energy. On average,
the process also promptly (~10-15 sec) produces about 2.5 neutrons for each fission.
Fission neutrons actually “evaporate” from the initially highly excited fragments and have average
energies of about 2 MeV. Each fission event produces a total of approximately 190 MeV of
energy: fission fragment and neutron kinetic energy, beta radiation (mostly e– because fission
fragments usually have too many neutrons), and photons (g rays), all of which appears as heat in
the reactor fuel and surroundings, and neutrino energy, which escapes. A small fraction (~
0.5%) of neutrons appear after a few seconds delay time. The delayed neutrons are essential for
reactor control. One neutron of the 2.5 goes on to cause another fission, usually (after a few
microseconds) slowing down to energies at which the fission cross-section is large. Capture in
control rods and parasitic processes absorb about 0.5 neutron per fission. This leaves about 1
neutron per fission useable for external purposes.
In round numbers, fission reactors require dissipating about 200 MeV of heat energy for each
useful neutron produced. Most (but not all) reactors used for slow-neutron scattering research
operate in a steady mode.
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f (E)  exp(1.036E) sinh( 2.29E )
E is in MeV
The energy distribution of prompt fission neutrons
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Nuclear spallation
In accelerator-driven spallation sources, highenergy particles (invariably protons of ~ 1. GeV
energy) from the accelerator impinge on a thick
target of dense, high-mass-number materials,
e.g., uranium, tungsten, tantalum, or mercury.
Here they collide, leaving highly excited nuclei.
Neutrons, protons, and pions that emerge from
collisions with sufficient energy proceed to
collide again and leave further excited nuclei.
The excited nuclei shed their energy by promptly
evaporating particles (by far, predominantly
neutrons) until there is too little left for that
process.
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spallation (approx. 40 neutrons per nucleus)
internal cascade
Thick-target yields of neutrons as a function of incident proton energy and target
material
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The energy and angular distribution of neutrons emerging from a tantalum target
irradiated by 1-GeV protons. The target is 31 cm long, 7 cm wide and 20 cm high.
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Moderators
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Paths of neutrons in a research reactor. Several fast neutrons (red) emerge from
each fission (heavy dots) and proceed to collide, losing energy in each collision
(progressively less red) until they cause another fission, disappear by non-fission
absorption (a), or travel into the reflector. In the reflector-moderator, neutrons collide
repeatedly, losing energy (less red, more green) until they come into thermodynamic
equilibrium with the moderator medium (thermal neutrons, green) and live for a long
time. Some return to the core to cause another fission. Others find their way into the
neutron beams.
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A cavity-type cold moderator. Thermal neutrons (green) enter the 25-K liquid
hydrogen from the 300-K D2O moderator. There, they collide numerous times, losing energy at
each collision (less green), and come into equilibrium with the hydrogen at 25 K (blue). The
reentrant cavity acts as a hohlraum, allowing neutrons to rattle around within, promoting
thermal equilibration but permitting cold neutrons to emerge efficiently into the neutron beams.
The diagram exaggerates the thickness of the L-H2 layer in relation to the diameter of the cavity
and omits the necessary plumbing arrangements.
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A hot-neutron source. Thermal neutrons (green) from the 300-K D2O moderator
enter the block of vacuum-insulated graphite, heated by gamma rays and fast neutrons to about
2000°C. There they collide, gaining energy in collisions (progressively yellower), and come into
thermodynamic equilibrium with the hot graphite. They emerge as hot neutrons (orange) and
travel into the neutron beams.
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Flux spectra at the source surfaces of the three moderators of the FRM-II
reactor
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Spectrum of a poisoned room-temperature polyethylene moderator at IPNS
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Brightness & Fluxes for Neutron & X-Ray Sources
brightness
(s-1m-2srad-1)
DE/E (%)
divergence
(mrad2)
flux
(s-1m-2)
neutrons
1015
2
10x10
1011
rotating
anode
1020
0.02
0.5x10
5x1014
bending
magnet
1027
0.1
0.1x5
5x1020
undulator
1033
10
0.01x0.1
1024
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Reactor ILL, Grenoble
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http://www.ill.fr/index_ill.html
The double wall of the reactor
The reactor's overhead crane
The gantry crane
The reactor operations hall
The experimental hall overhead crane
The experimental hall
Spectrometer
View into the reactor core
The reactor core (plan)
The main reactor pool
The storage pool
The heat exchanger
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next page
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IN20:
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1.
2.
3.
4.
5.
6.
Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
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•
•
•
What does it mean to “detect” a neutron?
– Need to produce some sort of measurable quantitative (countable) electrical signal
– Can’t directly “detect” slow neutrons
Need to use nuclear reactions to “convert” neutrons into charged particles
Then we can use one of the many types of charged particle detectors
– Gas proportional counters and ionization chambers
– Scintillation detectors
– Semiconductor detectors
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Nuclear Reactions for Neutron Detectors
•
•
•
•
•
•
•
n + 3He  3H + 1H + 0.764 MeV
most common
n + 6Li  4He + 3H + 4.79 MeV
n + 10B  7Li* + 4He7Li + 4He + 0.48 MeV g +2.3 MeV
(93%)
 7Li + 4He
+2.8 MeV
n + 155Gd  Gd*  g-ray spectrum  conversion electron spectrum
n + 157Gd  Gd*  g-ray spectrum  conversion electron spectrum
n + 235U  fission fragments + ~160 MeV
n + 239Pu  fission fragments + ~160 MeV
( 7%)
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Gas detectors
n 3He3H  1H 0.76 MeV
  5333

barns
1.8
~25,000 ions and electrons produced per neutron (~410-15 coulomb)
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MAPS Detector Bank
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Scintillation Detectors
n 6Li4He  3H 4.79 MeV
  940

barns
1 .8
photons per neutron
Li glass (Ce)
ZnS (Ag) - LiF
~7,000
~160,000
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GEM Detector Module
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Semiconductor Detectors
n 6Li4He  3H 4.79 MeV
  940

barns
1 .8
~1,500,000 holes and electrons produced per neutron (~2.410-13 coulomb)
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1.
2.
3.
4.
5.
6.
Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
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scattering length for neutrons
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consequences:
structure of a molecule seen by x-rays,
clouds of electron density are denoted in red,
H atoms are not detectes
structure of the same molecule seen by
neutrons, clouds of nuclear density are
denoted in blue, H atoms are detected
(purple)
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radiography (shadow projection)
neutrons (the plastic parts are
detected (H-rich))
x-rays (not sensitive to the plastic parts)
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Scattering from a set of nuclei
d
* iq .( Ri  R j )
  bi b j e
d ij
bi depends on the nucleus (isotope, orientation of spin relatively to the neutron spin etc.).
Statistical averaging yields:
bi b j
*
 (1  ij ) b
2
 ij | b |2
Coherent and incoherent parts of the cross-section
 d 

  b
 d coh
2
e
ij
iq.( Ri  R j )

2
 d 

b
 b


 d incoh
2
N
Total cross-sections
coh  4 b
2

incoh  4 b
2
 b
2

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Example:
scattering from a single isotope with the spin quantum number j. Two possible orientations of
the neutron spin exist:
• parallel spins: I  j  12 , 2I  1  2 j  2 the scattering length is b+
• antiparallel spins: I  j  12 , 2I  1  2 j the scattering length is b
probabilities of these two states are
p 
2j2
j 1
2j
j

, p 

2 j  2  2 j 2 j 1
2 j  2  2 j 2 j 1
and the average scattering lengths
 2
( j  1) b  j b
( j  1)b  jb
2
b p b p b 
, b 
2 j 1
2 j 1


 


 2
For proton with j=1/2:
b   1.041012 cm, b   4.741012 cm
p   34 , p   14

coh  1.77 barn, incoh  79.8 barn
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Various isotopes in the sample:
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Thermal Neutrons, 8 keV X-Rays, Low Energy Electrons: Absorption by Matter
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1.
2.
3.
4.
5.
6.
Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
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Magnetic Scattering
• The magnetic moment of the neutron interacts with B fields caused, for example, by unpaired
electron spins in a material
– Both spin and orbital angular momentum of electrons contribute to B
– Expressions for cross sections are more complex than for nuclear scattering
• Magnetic interactions are long range and non-central
– Nuclear and magnetic scattering have similar magnitudes
– Magnetic scattering involves a form factor – FT of electron spatial distribution
• Electrons are distributed in space over distances comparable to neutron wavelength
• Elastic magnetic scattering of neutrons can be used to probe electron distributions
– Magnetic scattering depends only on component of B perpendicular to Q
– For neutrons spin polarized along a direction z (defined by applied H field):
• Correlations involving Bz do not cause neutron spin flip
• Correlations involving Bx or By cause neutron spin flip
– Coherent & incoherent nuclear scattering affects spin polarized neutrons
• Coherent nuclear scattering is non-spin-flip
• Nuclear spin-incoherent nuclear scattering is 2/3 spin-flip
• Isotopic incoherent scattering is non-spin-flip
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Magnetic Neutron Scattering is a Powerful Tool
• In early work Shull and his collaborators:
– Provided the first direct evidence of antiferromagnetic ordering
– Confirmed the Neel model of ferrimagnetism in magnetite (Fe3O4)
– Obtained the first magnetic form factor (spatial distribution of magnetic electrons) by
measuring paramagnetic scattering in Mn compounds
– Produced polarized neutrons by Bragg reflection (where nuclear and magnetic
scattering scattering cancelled for one neutronspin state)
– Determined the distribution of magnetic moments in 3d alloys by measuring diffuse
magnetic scattering
– Measured the magnetic critical scattering at the Curie point in Fe
• More recent work using polarized neutrons has:
– Discriminated between longitudinal & transverse magnetic fluctuations
– Provided evidence of magnetic solitons in 1-d magnets
– Quantified electron spin fluctuations in correlated-electron materials
– Provided the basis for measuring slow dynamics using the neutron spin-echo
technique…..etc
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Theory of magnetic neutron scattering for pedestrians:
Interaction of the magnetic moment of neutron with the magnetic field produced by a moving electron


Interaction potential: V   μn .B


Magnetic dipole moment of the neutron: μn  g n N σ
gn  1.913 is the gyromagnetic factor of neutron,  N 

σ is the spin operator of neutron
e
2m
The magnetic field produced by a moving electron
B  BS  BL  
e ve  R
 μe  R 

rot


3
c R3
R


magnetic dipole moment of electron μe  2 B S
differential cross-section of magnetic neutron scattering
1

 d 
2
f σ . M  (q ) i

  ( g n rel )
2 B
 d 
2
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M  (r )
is the component of magnetization perpendicular to the scattering vector q
M  ( q )   d 3 r M  ( r ) e  iq . r
is its Fourier transformation
The magnetization has an orbital and spin component M  M L  M S
The spin component

M S (r )  2 B  (3) (r  rjk ) s jk
the k-th electron in the j-th atom
jk


We denote the average spin operator of atom j S j  s jk
Then we obtain (for the spin component only)
 d 
iq .r
2

  ( g n rel ) f (q) S j  e j
 d 
j
where
f ( q )   d 3 r  s ( r ) e  iq . r
2
is the magnetic scattering factor of an atom, s is the spin
density
The orbital component can be included by LS coupling:
 d 
iq .r
2

  ( g n rel ) 12 g J f (q) J j  e j
 d 
j
2
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1.
2.
3.
4.
5.
6.
Basic properties of neutrons, comparison with x-ray photons
Neutron sources
Neutron detectors
“Classical” neutron scattering
Magnetic scattering
Inelastic scattering
53
This and the following pictures are taken from the lecture notes of S.K. Sinha
ssinha@physics.ucsd.edu
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