Nuclear Reactions

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Nuclear Reactions
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Target Physics
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•
The physics which govern the
nuclear reaction between the
incident particle and the target
material determine the how much of
a radionuclide will be produced and
how the target must be constructed.
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Major Nuclear Reaction Types
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Reactions with charged particles are often different than reactions
of the nucleus with a neutron. In the neutron reaction, a gamma is
often given off whereas in the charged particle reaction, several
nucleons may be emitted
Contents
Nuclear Reaction
γ
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Neutron reaction with the nucleus
Target
Nucleus
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Proton reaction with the nucleus with several nucleons emitted
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Nuclear Reaction
Q- values
Reaction Cross Section
Nuclear Reaction Classic Model
Barrier to reaction
As the positively charged particle approaches the nucleus,
there is an electrostatic repulsive force between the particle
and the nucleus. This is often referred to as the Coulomb
barrier and is given by the relation:
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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B=Zze2/R
where:
Z and z = the atomic numbers of the two species
e2 = the electric charge, squared
R = the separation of the two species in cm.
Projectile/Target Processes
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Nuclear Reaction
Q- values
Reaction Cross Section
As we have seen before, the following types of reactions which
may occur when the two particles approach each other and
collide.
• Electron excitation and ionization
• Nuclear elastic scattering
• Nuclear inelastic scattering with or without nucleon emission
• Projectile absorption with or without nucleon emission
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
There are certain probabilities for each of these pathways. The
probability can be expressed as follows:
σi = σcom(Pi/ ΣPi)
Literature
• where,
• σi = cross-section for a particular product I
• σcom = cross-section for the formation of the compound
nucleus
• Pi = probability of process i
• ΣPi = the sum of the probabilities of all processes
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Total Excitation Energy
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Nuclear Reaction
Q- values
Reaction Cross Section
When the incident particle combines with the target nucleus it
forms a compound nucleus which will then decay along several
channels as outlined previously. The total amount of energy in
the compound nucleus will influence the probabilities of any
particular channel. The total excitation energy of the
compound nucleus is given by the relationship:
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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U = [MA / (MA + Ma)] .Ta + Sa
where:
U = excitation energy
MA = mass of the target nucleus
Ma = mass of the incident particle
Ta = kinetic energy of the incident particle
Sa = binding energy of the incident particle in the
compound nucleus
Q values
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
The probability of any particular reaction will depend on whether
the reaction is exothermic or endothermic
• the 'Q' value of a nuclear reaction is defined as the difference
between the rest energies of the products and the reactants, ( Q
= Δmc2 )
• Negative Q values are endothermic and positive Q values are
exothermic
Particle Range
Energy Straggling
Multiple Scattering
>0
mass to energy (exothermic)
<0
energy to mass (endothermic)
Q-value
Saturation Yields
Literature
The Q value will determine the lowest energy at which a nuclear
reaction may occur. If the reaction is endothermic, the excitation
must be at least high enough to overcome this activation barrier
(This is not completely accurate since quantum mechanical
tunneling may allow the reaction to occur at lower energies).
Some examples of some potential channels for the deuteron
reaction with nitrogen-14 are shown on the following slide.
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Q Value and Reaction
Threshold
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Q value
Contents
T hres h old
Nuclear Reaction
15
Q- values
O
5.1 M eV
0 M ev
Reaction Cross Section
n
Stopping Power
α
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
d +
14
N
16
O

12
C
13.6 M eV
0 M ev
13
N
-4.3 M eV
4.9 M ev
14
N
-2.2 M eV
2.5 M ev
16
O
20.7 M eV
0 M ev
t
n+
p
Literature
γ
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Reaction Cross-section
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The rate of any particular reaction is given by the following
expression with the variables as defined below.
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
Nuclear Reaction
dt
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
where:
R
n
I
Multiple Scattering
Saturation Yields
Literature
λ
t
σ
E
x
ʃ
STOP
dn
E0
 R  nI (1  e
t
)
Es
 (E )
dE
dE / dx
is the number of nuclei formed per second
is the target thickness in nuclei per cm2
is the incident particle flux per second and is related to
beam current
is the decay constant and is equal to ln2/t1/2
is the irradiation time in seconds
is the reaction cross-section, or probability of interaction,
expressed in cm2 and is a function of energy
is the energy of the incident particles, and
is the distance traveled by the particle
is the integral from the initial to final energy of the incident
particle along its path
Reaction Yields
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
The rate of a particular reaction can also be written in the following
equation.
dn  I 0 N A ds  a b
Where:
dn = number of reactions occurring in one second
I0 = number of particles incident on the target in one second
NA = number of target nuclei per gram
ds = thickness of the material in grams per cm2
σab = cross-section expressed in units of cm2
Saturation Yields
Literature
This equation can be simplified and rearranged by incorporating the
constants in the equation and solving for the nuclear reaction cross
section. This simplified equation is given on the next slide.
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Simplified Equation
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i 
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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where,
σi =
A=
Ni =
t=
ρ=
x=
I=
2 . 678 x 10
 10
AN i
It  x
cross-section for a process in millibarns for the
interval in question
the atomic mass of the target material (AMU)
number of nuclei created during the irradiation
time of irradiation in seconds
density of the target in g/cm3
thickness of the target in cm.
beam current in microamperes
Reaction Cross-Section
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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The probability of a particular reaction as a function of energy
is the nuclear reaction cross section. The example is for the
production of fluorine-18.
Bragg Peak
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Nuclear Reaction
Q- values
As the incident particle enters the target material, the particle starts to
slow down due to collisions with electrons and nuclei. The loss of
energy as the particle slows is given off in several forms including
light and heat. This heat has to be removed by cooling the target
material during bombardment
Particle Path with more
scattering as the particle slows
Energy
Deposition
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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Penetration into the target material
Bragg Peak
Stopping Power
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The rate at which the energy of the incident particle is lost is
called the stopping power of the target material. The
stopping power is just the energy lost per unit distance.
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping power S(E) = - dE/dx-
Stopping Power
Particle Range
Energy Straggling
where
Multiple Scattering
Saturation Yields
E is the particle energy (MeV)
Literature
x is the distance traveled (cm)
The stopping power depends on the characteristics of the
incident particle, the target material, the energy and the
chemical form of the target.
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Stopping Power
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The expression for the loss in energy can be given by the expression
Target Physics
Contents
Nuclear Reaction
Q- values
 dE
dx
( 4 z e NZ )
2

4
2
m 0V A
ln(
2 m 0V
2
)
I
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
STOP
where:
z = particle atomic number (amu)
Z = absorber atomic number (amu)
e = electronic charge (esu)
mo = rest mass of the electron (MeV)
A = atomic mass number of the absorber (amu)
V = particle velocity (cm/sec)
N = Avogadro's number
I = ionization potential of the absorber (eV)
Stopping Power
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This expression can be simplified to the following equation by
substitution the values of the physical constants into the
equation
Nuclear Reaction
 dE
Q- values
Reaction Cross Section
Stopping Power
dx

144 Zz
2
ln(
2195 E
AE
)
I
Particle Range
Energy Straggling
where:
Multiple Scattering
Saturation Yields
Literature
z is the particle z (amu)
Z is the absorber Z (amu)
A is the atomic mass of the absorber (amu)
E is the energy (MeV)
I is the absorber effective ionization potential (eV)
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Range of charged Particles
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The range of the particle in the target material is just the inverse of
the stopping power as a function of the energy. It can be given by the
following expression.
E max
Contents
Nuclear Reaction
R 

0
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
AE
2
144 z Z ln(
2195 E
E
)
I
z is the particle z (amu)
Z is the absorber Z (amu)
A is the atomic mass of the absorber (amu)
E is the energy (MeV)
I is the absorber effective ionization potential (eV)
As an example we can use protons on aluminum with z=1, Z=13,
A=27 and I = 169 eV. The results of this calculation done on an
Excel spreadsheet using 0.1 MeV intervals are shown on the
next page labeled as Range (Simple).
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Simple Range Calculations
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This simplified equation can be used to calculate an approximate
particle range. This can be compared to more sophisticated
calculations as in the following table for protons on aluminum
Contents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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Energy
(MeV)
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Range
(Simple)
0.3477
0.3077
0.2699
0.2344
0.2011
0.1702
0.1416
0.1155
0.0917
0.0705
0.0517
0.0357
0.0223
0.0118
0.0044
Range
SRIM
0.3431
0.3026
0.2662
0.2313
0.1987
0.1681
0.1401
0.1142
0.0907
0.0696
0.0511
0.0350
0.0217
0.0112
0.0039
Range
Janni
0.3430
0.3038
0.2668
0.2319
0.1992
0.1687
0.1405
0.1146
0.0910
0.0699
0.0513
0.0352
0.0219
0.0114
0.0040
Range
WG&J
0.3448
0.3053
0.2679
0.2327
0.1998
0.1691
0.1407
0.1147
0.0910
0.0698
0.0511
0.0351
0.0218
0.0113
0.0039
Energy Straggling
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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•
As the particle slows down, the distribution in energy also
increases. The following graph shows the energy distribution of
a 15 MeV proton beam after it has been degraded in energy
from 200, 70 and 30 MeV. It can be seen that the beam slowed
from 200 MeV has a very broad energy distribution while the
beam slowed from 30 MeV still has a relatively narrow energy
distribution.
Energy Straggling
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
The standard deviation of the energy distribution can be given
by a relatively simple expression which is dependent only on the
atomic number and atomic weight of the target material, the
atomic number of the particle and the distance the particle has
traveled through the target in terms of the grams per square
centimeter
Z 
  0 . 395 z  x 
 A 
1/ 2
Multiple Scattering
Saturation Yields
where
Literature
z = projectile atomic number (amu)
Z = absorber atomic number (amu)
A = absorber atomic mass number (amu)
x = particle path length (g/cm2)
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Multiple Scattering in Gas Targets
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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•
•
As the particle passes through the target material, the beam
starts to spread out. This phenomenon is referred to as small
angle multiple scattering.
The magnitude of the scattering is dependent on the atomic
number of the target material and the atomic number of the
particle
Multiple scattering in the front foil causes the beam shape to
enlarge
The Multiple Scattering in the target can be approximated by a
simple model
Multiple Scattering in Gas Targets
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Nuclear Reaction
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
The scattering angle is dependent on the fraction of the energy
lost in the foil and the particular particle
 Z, z particle and absorber Z
 x distance traveled
 E energy of the particle
 A atomic weight of the absorber
Beam Profile Alteration
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Nuclear Reaction
Q- values
Reaction Cross Section
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Particle Range
Energy Straggling
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•An example of this
phenomenon is shown in
these plots where the
calculated beam profile is
compared to the measured
beam profile with
reasonable agreement.
Beam intensity versus radius, same total beam current
3
200 ug/cm^2
400 ug/cm^2
2.5
600 ug/cm^2
800 ug/cm^2
1000 ug/cm^2
1200 ug/cm^2
2
Intensity
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1.5
1
0.5
•Thicker stripper foils were
placed in the cyclotron.
The original foils were 180
ug/cm² polycrystaline
graphite. An assortment of
foils from 400 to 1200
ug/cm² were purchased
Calculated beam profile
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
radius
1 .2
1
6 4 4
u g /c m 2
8 0 0
u g /c m 2
9 7 0
u g /c m 2
1 2 00
u g /c m 2
0 .8
0 .6
0 .4
STOP
•Beam spot shape was
measured by irradiating a
copper foil and imaging it
with a phosphor plate
imaging system.
0 .2
0
0
1
2
3
4
5
6
7
Measured beam profile
8
Saturation Yields
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Nuclear Reaction
Q- values
As a nuclear reaction occurs in the cyclotron beam, the
radionuclides produced start to decay. The overall rate of
formation is given by the following equation. The term in
parentheses is known as the saturation factor. As the time of
irradiation gets longer, the rate starts to slow until at infinite
time, the rate is zero.
Reaction Cross Section
Stopping Power
Particle Range
E0
R  nI (1  e
t
)
Es
 (E )
dE
dE / dx
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
STOP
where,
R - is the number of nuclei formed per second
n - is the target thickness in nuclei per cm2
I - is the incident particle flux per second and is related to
beam current
λ - is the decay constant and is equal to ln2/t1/2
t - is the irradiation time in seconds
σ(E) - is the reaction cross-section, or probability of
interaction, expressed in cm2 and is a function of energy
E - is the energy of the incident particles, and
x - is the distance traveled by the particle
Saturation Factors
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Saturation F actors
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-–λt
t
(1
e
(1 - e ) )
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Q- values
Reaction Cross Section
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Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
SF
activity
Fraction of saturation
Nuclear Reaction
1 .2 0
1 .0 0
0 .8 0
0 .6 0
0 .4 0
0 .2 0
0 .0 0
0
2
4
6
( Irra d. time / ha lf-life )
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8
10
Literature
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•
More Information on these ideas can be found in the IAEA
Publication “Cyclotron Produced Radionuclides: Principles and
Practice” and the references in that book. “Cyclotron Produced
Radionuclides: Principles and Practice” TRS 465
•
Another IAEA publication which may be of interest is “Cyclotron
Produced Radionuclides: Physical Characteristics and
Production Methods” TRS 468
•
There is also a publication on the cross sections for a variety of
radionuclides which are useful for nuclear medicine called
“Charged particle cross-section database for medical
radioisotope production: diagnostic radioisotopes and monitor
reactions” TECDOC 1211
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
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