Chapter 2 ( and 3 )
• TA
Cross - Sections
• Lewis 2.1, 2.2 and 2.3
Learning Objectives
• U n d e r s ta n d d i f f e r en t t y p e s of n u c l e a r reactions
• U n d e r s t a n d c r o s s s e c t i o n b e h a v i o r for different reactions
• U n d ers t an d resonance b e h av i or an d i t s relation to the nuclear energy levels
• K now where to find nuclear data
M i c r o s c o p i c C r o s s s e c t i o n
I neutrons/cm 2 . sec
N A nuclei/cm 2
x
Consider a beam of mono-energetic neutrons of intensity I incident on a very thin material such that there are Na atoms/c m . 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."
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
Microscopic Cross section
The microscopic cross-section characterizes the probability of a neutron interaction.
cm 2 s
cm 2
R = I N A
cm 2 s
[ # ]
[ cm 2 ]
[ # ] [ # ]
I neutrons/cm 2 . sec
N A nuclei/cm 2
x
Image by MIT OpenCourseWare.
Cr oss Section - Micr oscopic
Scattering s = e + in Absorption a = + f T otal t = s + a
Numbe r o f r eaction s /nucleu s / s ( R / N A )
Number of incident neutr ons / cm 2 s 1
Incident Beam (Neutr on) on a Thick T arget
I(x)
I 0
x
x = 0
x x + dx
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Now consider the case of a thick target with an incident beam I 0 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.
x
I(x)
N A nuclei/cm 2
x = 0
x
x x + dx
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Unattentuated beam in target
x
I(x)
N A nuclei/cm 2
x = 0
x
x x + dx
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Number of tar get nuclei per cm 2 in dx is
dN A = N dx where
N = number density of the tar get nuclei in units cm -3 .
Relating reaction rate to beam intensity
The total reaction rate in dx can be defined as
dR = t IdN A = t INdx
I(x)
I 0
x
x = 0
x x + dx
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Each neutron that reacts decreases the unattenuated beam intensity, thus
-dI ( x ) = - [ I ( x + dx ) - I ( x )] = t INdx
Macroscopic cross-section
we can then solve this differential equation to get I(x)
t
dI ( x ) = - N I ( x ) dx
we can then define the macroscopic cross-section such that
I ( x ) = I 0 e -N t x
Macr oscopic cr oss section interpr etation
t
Probability per unit path length that the neutron will interact with a nucleus in the target.
exp(- t x )
Probability that a neutron will travel a distance x without making a collision.
t exp(- t x ) dx
Probability that a neutron will make its
first collision in dx after traveling a distance x.
Mean fr ee path of neutr on
x dx x p ( x ) = t dx x exp(- t x ) = 1
t
Interaction probability calculates the average distance a neutron travels before interacting with a nucleus
x = Average distance traveled by a neutron
t before making a collision
Two fundamental aspects of neutron cross sections
• K inematics of two-particle collisions
– Conservation of momentum
– C onservation of energy
• D ynam i cs o f nuc l ear reac t i ons
– P otential scattering
– C ompound nucleus formation
Cross Section (b)
H y d r o g e n x . s .
10 5
10 4
10 3
10 2
10 1
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
Ener gy (MeV)
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Potential scattering
Before
After
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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 )
A +1 *
A X 1 n
Z ( Z X ) Z 0
Radiative capture
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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 pr ocesses
Resonance elastic scattering
1 n + A X
1 n + A X
0
Z
( A +1 X ) *
1 n +
( A X ) *
0 Z
Inelastic scattering
0 Z Z
Z
A +1 X +
Radiative capture
A 1 A 2 1
Fission Z 1 X + Z 2 X + 2 - 3 0 n
Nuclear Shell Model
3 s 3/2
1 d 4
3 s 1/2 2
2 d 1 g 7/2 8
1 d 5/2 6
50
1 g
1 g 9/2 10
2 p 1/2 2
2 p 1 f 5/2 6
2 p 3/2 4
1 f
28
1 f 7/2 8
20
2 s
1 d 3/2 4
1 d 2 s 1/2 2
1 d 5/2 6
8
1 p
1 p 1/2 2
1 p 3/2 4
2
1 s 1 s 1/2 2
Radiative captur e
The figure is for 238 U at E=6.67 e V .
E 100
Incident
10
neutron
kinetic ener gy
Neutron
6.67 eV
mc 2 + M 238 c 2
238 U
92
10
100
(E) (b)
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.
C r o s s S e c t i o n ( b )
U - 2 3 8
10 5
10 4
10 3
10 2
10 1
10 0
10 -1 -9
10
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
Ener gy (MeV)
Image by MIT OpenCourseWare.
Cross Section Modeling
• Experimental data i s n ’ t available at e v e ry energy
• Quantum m e c h a n i c a l m o d e l s are u s e d t o provide cross section values around data points
• Simplest version is Single Level Breit-
W i gner
– Valid for widely spaced resonances
c
0
( E
) =
( E 0
1/2
E c
1 + y 2 , y =
1
2 ( E
_ E )
max
( E c )
2
1 max
0
c
0
= T otal line width (FWHM)
= Radiative line width
0 = T otal cross section at E = E 0
E 0 E c
Breit- W igner Formula for Resonance Capture Cross Section
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Doppler Effect
• C r o s s s e c t i o n s are f u n c t i o n s of r e l a t i v e speed between neutron and target nucleus
– Generally assumed that target is at r e s t
– Valid for smooth cross sections
– N o t v a l i d for resonances
Doppler Effect
• Resonances m u s t b e averaged o v e r a t o m velocity
– Assume target n u c l e i h a v e M a x w e l l - Boltzmann energy distribution
• A s a t o m temperature i n c r e a s e s
– Resonance becomes wider
– R esonance b ecomes s h or t e r
• Area stays approximately the same
Probability
M a x w e l l - B o l t z m a n n Distribution
V elocity
100 o K
200 o K
400 o K
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(E, T)
Doppler Effect
T 1
T 1 < T 2 < T 3
T 2
T 3
E 0
E
E
E v
V
E
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Inelastic scattering
Scattering - Inelastic
E c
Neutron in
A X *
Z
Emission
mc 2 + M A c 2
A X
(E)
Z
in
M A+1 c 2
A+1
Z X
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Inelastic scattering usually occurs for neutron ener gies above 10 ke V . The excited state decays by gamma emission.
Resonance Scattering - Elastic
E
Neutron in
Neutron out
mc 2 + M A c 2
A X
Z
s (E)
A+1 X
Z
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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 ener gy is conserved.
Double-Differential Cross-Section
v'
v
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The scattering cross-section will depend on both the energy and angle.
Scattering cross section - Double Dif ferential
d '
E ,
E' , '
s ( E , ˆ
E' , ˆ ' )
dE'd ˆ ' [ cm 2 ]
Image by MIT OpenCourseWare.
s ( E , ˆ E' , ˆ ' ) [ cm 2 / eV . sterradian ]
This characterizes neutron scattering from an incident ener gy E and direction to a final ener gy E ' in the interval dE ' and ' in a solid angle d '.
Neutron Scattering
• Lill e y 5.5.2
S l o w i n g D o w n D e c r e m e n t
• Lill e y 5.5.2
Neutron Moderators
• Lewis 3 . 3
MIT OpenCourseWare http://ocw.mit.edu
22.05 Neutron Science and Reactor Physics
Fall 2009
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