Neutron interaction with matter
1) Introduction
2) Elastic scattering of neutrons
3) Inelastic scattering of neutrons
4) Neutron capture
5) Other nuclear reactions
6) Spallation reactions, hadron shower
Important cross sections of nuclear interactions
Mostly neutron loses only part of energy
Introduction
Neutron has not electric charge → interaction only by strong nuclear interaction
Magnetic moment of neutron → interaction by electromagnetic interaction, mostly
negligible influence
Different energy ranges of neutrons:
Ultracold: E < 10-6 eV
Cold and very cold: E = (10-6 eV – 0,0005 eV)
Thermal neutrons – (0,002 eV – 0,5 eV) neutrons are in thermal equilibrium with
neighborhood, Maxwell distribution of velocities, for 20oC is the most probable
velocity v = 2200 m/s → E = 0,0253 eV dN 4N
v



dv
v e
2
v
3
0
π
 v v0
2
σ(v)  σ 0
0
v
Epithermál neutrons and resonance neutrons: E = (0,005 eV – 1000 eV)
Cadmium threshold: ~ 0,5 eV - with higher energy pass through 1 mm of Cd
Slow neutrons:
E < 0,3 eV
Fast neutrons:
E = (0,3 eV – 20 MeV)
Neutrons with high energies: E = (20 MeV – 100 MeV)
Relativistic neutrons: 0,1 – 10 GeV
Ultrarelativistic neutrons: E > 10 GeV
Elastic scattering of neutrons
Most frequent process used for kinetic energy decreasing (moderation) of neutrons
Moderation – process of set of independent elastic collisions of neutron on nuclei
Usage of nucleus reflected during scattering for neutron energy determination
Maximal transferred energy (nonrelativistic case of head-head collision):
MCL:
pn0 = pA - pn
ECL: En0KIN = EAKIN + EnKIN  pn02/2mn = pA2/2mA + pn2/2mn
MCL: pn2 = pA2 – 2pApn0 + pn02 
mApn2 =
ECL:
mApn2 = - mnpA2
mApA2 – 2mApApn0 + mApn02
+ mApn02
We subtract equation:
0 = mApA2 + mnpA2 – 2mApApn0  mApA + mnpA = 2mApn0
The heavier nucleus the lower energy can
neutron transferred to it:
pA 
2m A pn0
m A  mn
 EA 
4m Amn En0
 m A  m n 2

4 A  En0
 A  1 2
Nucleon number A
Usage of hydrogen (θ – neutron scattering angle, ψ – proton reflection angle) mp = mn:
pn = pn0·cosθ  En = En0·cos2θ
pn = pn0·sinψ  En = En0
·sin2ψ
For nucleus:
EA 
4 A  E n0
 A  1
2
ψ = π/2-θ
cos  CM  1  2  cos 
2
cos 
2
EA 
pp = pn0·sinθ  Ep = En0·sin2θ
pp = pn0·cosψ  Ep = En0·cos2ψ
2 A  En0
 A  1 2
1  cos  CM 
Reflection angle φ
Elastic scattering: in our case
particle 1 – neutron
particle 2 – proton, generally nucleus
Dependency of energy
transferred to proton on
reflected angle
Small expose with derivation of relation between laboratory and centre of mass angles:
Laboratory
coordinate
system
Centre of mass
coordinate
system
Derivation of relation between scattering angles at centre of mass and laboratory coordinate systems:
Relation between velocity components to the direction of beam particle motion is:
~
v 1 cos   v CM  ~
v 1 cos 
~
v 1 cos   ~
v 1 cos   v CM
Relation between velocity components to the direction perpendicular to beam particle motion:
~
~v  sin ~
sin 
1
tan  

~
~v  cos ~  v
cos    v CM ~v 1 
1
CM
Ratio of these relations leads to:
For elastic scattering is valid:
v
m
  ~CM  1
v 1
m2
~
cos   
cos  
~
2 1 2
(1  2 cos    )
Equation can be rewrite to:
m1
m2

mn
mp
1
derive!
cos ~  1
2
and then
cos  
2
and required relation is valid:
~
(1  2 cos   1)

1
2
cos ~  1  cos ~  2 cos
2
 1
~
2
2
2
2
cos   2 cos   1  2 sin   1  2 (1  cos  )  1  1  2 cos 
~
v 1 sin   ~
v 1 sin 
Angular distribution of scattering neutrons at centre of mass coordinate system:
P ( CM ) d  CM  2 sin  CM
 S  CM

S
d  CM
Relation between angular distribution and energy distribution:
We introduce and express distribution of transferred energy:
P ( CM ) d  CM  P  E A dE A
P  E A   2 sin  CM
  CM  d  CM
S
dE A
We determine appropriate differential dEA:
EA 
2 A  E n0
 A  1
2
1  cos  CM 
dE
A

2 A  E n0
 A  12
sin  CM d  CM
Introduce for dEA:
(1  A )  S ( CM ) 
2
P(E A ) 
S
A
E n0
 σS(θCM) = σS/(4π)
σS(θCM) - isotropy
(it is valid approximately for protons up to En0 < 10 MeV)
(1  A )  S ( CM ) 
2
P(E A ) 
A
Efficiency ε is given:
S
E n0

1 (1  A )
4
  1 e
A
2
1
E n0
 N j S d
Energy distribution of
reflected protons for En0 < 10
MeV
Coherent scattering – diffraction on lattice
Magnitude of energy neither momentum and wave length of neutrons are not changed
Diffraction of neutrons on crystal lattice is used
n·λ = 2d·sin Θ
Mentioning: Bragg law:
λ
En << mnc2
hc
2m n c E n  E
2
2
n
hc
λ
2m n c
1
2
En
= 0,0288 eV½∙nm for En in [eV]
Lattice constants are in the order 0,1 – 1 nm → Neutron energy in the orders
of meV up to eV
E [eV]
0,001
0,005
0,01
0,1
1
10
100
1000
λ [nm]
0,91
0,41
0,29
0,091
0,029
0,0091
0,0029
0,00091
Inelastic neutron scattering
Competitive process to elastic scattering on nuclei heavier than proton
Part of energy is transformed to excitation → accuracy of energy determination is
given by their fate
Its proportion increases with increasing energy
Nuclear reactions of neutrons
Neutron capture:
(n,γ)
Thermal region
Resonance region
High values of cross sections for
low energy neutrons
Exothermic reactions
Released energy allows
detection
Cross section of reaction 139La(n,γ)140La
157Gd(n,γ)
– for thermal neutrons cross section is biggest σ ~ 255 000 barn
průřez
[barn]
section
Cross účinný
1
0.9
0.8
Bi-208
Bi-207
0.7
Threshold reactions
Bi(n,Xn)Bi
Bi-206
0.6
0.5
Bi-205
Bi-204
0.4
Bi-203
0.3
Bi-202
0.2
0.1
Bi-201
0
0
20
40
60
80
100
energie [MeV]
Energy
[MeV]
Examples of threshold
reactions:
197Au(n,2n)196Au
197Au(n,4n)194Au
27Al(n,α)24Na
Cross section [barn]
Reaction (n, 2n), (n,3n), ... Endothermic (threshold) reactions
Energy [MeV]
Reactions (n,d), (n,t), (n,α) ...
Reactions used for detection of low energy neutrons (exoenergy):
(two particle decay of compound nucleus at rest, nonrelativistic approximation)
E N + EP = Q
mNvN = mPvP →
10B(n,α)7Li
2m N E N 
2m P E P  E N 
mP
mN
EP
EP 
mN
mP  mN
Q
Q = 2,792 and 2,310 MeV, Eα = MeV, ELi = MeV σth = 3840 b 1/v up to 1 keV
6Li(n,α)3H
Q = 4,78 MeV, Eα = 2,05 MeV, EH = 2,73 MeV σth = 940 b 1/v up to 10 keV
3He(n,p)3H
Q = 0,764 MeV, Ep = 0,573 MeV, EH = 0,191 MeV σth = 5330 b 1/v up to 2 keV
Reactions used for detection of fast neutrons – threshold reactions
Induced fission:
(n,f)
Induced by low energy neutrons (thermal): 233U, 235U, 239Pu
Exothermic with very high Q ~ 200 MeV
Induced by fast neutrons: 238U, 237Np, 232Th
Induced by „relativistic“ neutrons: 208Pb
High energies E > 0,1 GeV → reaction of protons and neutrons are similar
Bi(n,2n)Bi
0.1
Spallation reactions, hadron shower
Same behavior as for protons and nuclei
Cross section
účinný
průřez [rel.u]
[rel.j.]
Interaction of realativistic and
ultrarelativistic neutrons
protons
0.08
neutrons
0.06
0.04
0.02
0
0
500
Energy [MeV]
1000
1500