Chapter 2 (and 3)
Cross-Sections
Cross
Sections
• TA
• Lewis 2.1, 2.2 and 2.3
Learning Objectives
• Understand different types of nuclear
reactions
• Understand cross section behavior for
different reactions
• U
Understand
d t d resonance behavior
b h i and
d its
it
relation to the nuclear energy levels
• Know where to find nuclear data
Microscopic Cross section
Consider a beam of mono-energetic neutrons of
intensity I incident on a very thin material such
that there are Na atoms/cm
. 2.s
The collision rate of neutrons is proportional
to the neutron beam intensity and the nuclei
density Na. The constant of proportionality
is defined as the neutron microscopic
cross-section."
I neutrons/cm2 . sec
NA nuclei/cm2
Image by MIT OpenCourseWare.
x
Image by MIT OpenCourseWare.
Microscopic Cross section
The microscopic cross-section characterizes
the probability of a neutron interaction.
R
=
[ ]
#
cm2s
σ
NA
I
[ ] [ cm s ] [ cm
cm2
#
2
I neutrons/cm2 . sec
NA nuclei/cm2
x
Image by MIT OpenCourseWare.
#
2
]
Cross Section - Microscopic
σ=
Scattering
σs = σe + σin
Absorption
σa = σγ + σf
Total
σt = σs + σa
Number of reactions /nucleus / s
Number of incident neutrons /
cm2s
=
(R / NA)
1
Incident Beam (Neutron) on a Thick Target
I(x)
I0
x
x=0
x x + dx
Image by MIT OpenCourseWare.
Now consider the case of a thick target with an
incident beam I0 for which we want to know the
unattenuated beam intensity as a function of position I(x).
Unattentuated beam in target
Taking an infinitesimally thin portion of the
target, dx, allows us to use the previous analysis
on dx between x and x + dx.
I(x)
NA nuclei/cm2
x
x=0
x x + dx
x
Images by MIT OpenCourseWare.
Unattentuated beam in target
I(x)
NA nuclei/cm2
x
x=0
x x + dx
x
Images by MIT OpenCourseWare.
Number of target nuclei per cm2 in dx is
dNA = N dx where
N = number density of the target nuclei in
units cm-3.
Relating reaction rate to beam intensity
The total reaction rate in dx can
be defined as
dR = σtIdNA = σtINdx
I(x)
I0
x
x=0
x x + dx
Each neutron that reacts decreases
the unattenuated beam intensity, thus
Image by MIT OpenCourseWare.
-dI(x) = - [I(x + dx) - I(x)] = σtINdx
Macroscopic cross-section
we can then solve this differential equation to get I(x)
dI(x)
= - Nσt I(x)
dx
we can then define the macroscopic cross-section such that
I(x) = I0e-Nσtx
Macroscopic cross section
interpretation
Σt
Probability per unit path length that the neutron will
interact with a nucleus in the target.
exp(-Σt x)
Probability that a neutron will travel a distance
x without making a collision.
Σt exp(-Σt x)dx
Probability that a neutron will make its
first collision in dx after traveling a distance x.
Mean free path of neutron
x
∫ dx x p(x) = Σt ∫ dx x exp(-Σt x) =
0
0
1
Σt
Interaction probability calculates the average distance a neutron
travels before interacting with a nucleus
x = Σt−1
Average distance traveled by a neutron
before making a collision
Two fundamental aspects of
neutron cross sections
• Kinematics of two-particle collisions
– Conservation of momentum
– Conservation of energy
• Dynamics
D
i off nuclear
l
reactions
ti
– Potential scattering
– Compound nucleus formation
Hydrogen x.s.
xs
105
104
Cross Section (b)
103
102
101
100
10-1
10-2
10-3
10-4
10-5
10-6 -9
10
10-8
10-7
10-6
10-5
10-4
10-3
Energy (MeV)
10-2
10-1
100
101
Image by MIT OpenCourseWare.
Potential scattering
Before
After
Image by MIT OpenCourseWare.
Hard sphere collision where the neutron bounces off the nucleus.
The interaction time is approximately 10-17s.
Compound nucleus formation
Before
Compound Nucleus
After
A+1
*
X
Z
A
X
Z
γ
(
A+1
X
Z
)
(
)
1
0
n
Radiative capture
Image by MIT OpenCourseWare.
Neutron penetrates the nucleus and forms a compound nucleus
(excited state). The compound nucleus regains stability by
decaying. The interaction time is approximately 10-14s.
Compound nucleus decay processes
Resonance elastic
scattering
1
n
0
1
n
0
1
n
0
+
A
X
Z
(
A+1
X
Z
A
+ZX
+
*
( ) Inelastic scattering
A
X
Z
*
)
Fission
A+1
ZX +γ
A2
A1
X+Z X
Z1
2
Radiative capture
+ 2 - 3 10 n
Nuclear Shell Model
3s
2d
1d3/2
3s1/2
1g7/2
1d5/2
4
2
8
6
1g
1g9/2 10
2p1/2 2
1f5/2 6
2p3/2 4
50
1f7/2 8
28
2s
1d
1d3/2 4
2s1/2 2
1d5/2 6
20
1p
1p1/2 2
1p3/2 4
8
1s
1s1/2 2
2
2p
1f
Radiative capture
The figure is for 238U at E=6.67 eV.
E
100
10
Incident
neutron
Neutron
kinetic energy 6.67 eV
mc2 + M238c2
238
U
92
10
100
σγ(E) (b)
γ Cascade
239
U
92
Image by MIT OpenCourseWare.
When the sum of the kinetic energy of the neutron in the CM and its
binding energy correspond to an energy level of the compound nucleus,
the neutron cross section exhibits a spike in its probability of interaction
which are called resonances.
U 238
U23
105
Cross Section (b)
104
103
102
101
100
10-1
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Energy (MeV)
Image by MIT OpenCourseWare.
Cross Section Modeling
• Experimental data isn’t
sn t available at ever
every
energy
• Quantum mechanical models are used to
provide cross section values around data
points
• Simplest version is Single Level BreitWigner
Wi
– Valid for widely spaced resonances
σγ (Ec) = σ0
σγ (Ec)
σmax
1σ
max
2
=
Γγ E0
Γ Ec
1/2
((
1 , y = 2 (Ec _ E0)
Γ
1 + y2
σ0Γγ
Γ = Total line width (FWHM)
Γ
Γγ = Radiative line width
Γ
σ0 = Total cross section at E = E0
E0
Ec
Breit-Wigner Formula for Resonance Capture Cross Section
Image by MIT OpenCourseWare.
Doppler Effect
• Cross sections are functions of relativ
relative
speed between neutron and target nucleus
– Generally assumed that target is at rest
– Valid for smooth cross sections
– Not valid for resonances
Doppler Effect
• Resonances must be averaged over atom
velocity
– Assume target nuclei have Maxwel
axwellBoltzmann energy distribution
• As atom temperature increases
– Resonance becomes wider
– Resonance becomes sh
hortter
• Area stays approximately the same
Maxwell-Boltzmann
Maxwell
Boltzmann Distribution
100oK
200oK
Probability
400oK
Velocity
Image by MIT OpenCourseWare.
Doppler Effect
σ (E, T)
σ (E )
T1
T1 < T2 < T3
T2
T3
E0
E
σ (E ) → σ v − V
(
)
E
Image by MIT OpenCourseWare.
Scattering - Inelastic
Ec
Neutron in
ring
atte
c sc
asti
A *
ZX
Inel
γ Emission
mc2 + MAc2
σin(E)
A
ZX
MA+1c2
A+1
X
Z
Image by MIT OpenCourseWare.
Inelastic scattering usually occurs for neutron energies
above 10 keV. The excited state decays by gamma emission.
Resonance Scattering - Elastic
E
Neutron in
Neutron out
mc2 + MAc2
σs(E)
A
ZX
A+1
ZX
Image by MIT OpenCourseWare.
A compound nucleus is formed by the neutron and the nucleus.
Peak and valley due to quantum mechanical interference term
characterize the cross section. Kinetic energy is conserved.
Double-Differential Cross-Section
v'
v
Image by MIT OpenCourseWare.
The scattering cross-section will depend on both the energy and angle.
Scattering cross section - Double
Differential
E', Ω'
E, Ω
dΩ'
Image by MIT OpenCourseWare.
ˆ
σs (E, Ω
ˆ ) dE'dΩ'
ˆ [cm2]
E', Ω'
σs (E, Ω̂
E', Ω̂')
[cm2 / eV . sterradian]
This characterizes neutron scattering from an incident
energy E and direction Ω to a final energy E ' in the
interval dE ' and Ω ' in a solid angle dΩ '.
Neutron Scattering
• Lilley
y 5.5.2
Slowing Down Decremen
ecrement
• Lilley
y 5.5.2
Neutron Moderators
• Lewis 3.3
3
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22.05 Neutron Science and Reactor Physics
Fall 2009
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