cyclo target targetry target physics
TRANSCRIPT
Target Physics
• 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.
Contents• Nuclear Reaction• Q- values• Reaction Cross Section• Stopping Power• Particle Range• Energy Straggling• Multiple Scattering• Saturation Yields• Literature
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Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
STOP
Major Nuclear Reaction Types
γ
TargetNucleus
Proton reaction with the nucleus with several nucleons emitted
Neutron reaction with the nucleus
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
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Nuclear Reaction Classic Model
Barrier to reaction
B=Zze2/Rwhere: Z and z = the atomic numbers of the two
speciese2 = the electric charge, squaredR = the separation of the two species in cm.
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:
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Projectile/Target Processes
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
There are certain probabilities for each of these pathways. The probability can be expressed as follows:
σi = σcom(Pi/ ΣPi)
• 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
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Total Excitation Energy
U = [MA / (MA + Ma)] .Ta + Sa
where:U = excitation energyMA = mass of the target nucleusMa = mass of the incident particleTa = kinetic energy of the incident particleSa = binding energy of the incident particle in the compound nucleus
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:
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
STOP
Q valuesThe 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>0 mass to energy (exothermic)
Q-value<0 energy to mass (endothermic)
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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Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Q Value and Reaction Threshold
d + 14N 16O t
16O
Q value Threshold
20.7 M eV 0 M ev
15O
γ
5.1 M eV 0 M ev
12Cα 13.6 M eV 0 M ev
13N
n
-4.3 M eV 4.9 M ev
14N -2.2 M eV 2.5 M ev
n+pd + 14N 16O
t
16O
Q value Threshold
20.7 M eV 0 M ev
15O
γ
5.1 M eV 0 M ev
12Cα 13.6 M eV 0 M ev
13N
n
-4.3 M eV 4.9 M ev
14N -2.2 M eV 2.5 M ev
n+p
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Reaction Cross-section
where:R is the number of nuclei formed per secondn is the target thickness in nuclei per cm2
I is the incident particle flux per second and is related to beam currentλ is the decay constant and is equal to ln2/t1/2t is the irradiation time in secondsσ 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 particleʃ is the integral from the initial to final energy of the incident particle along its path
The rate of any particular reaction is given by the following expression with the variables as defined below.
0
/)()1(
E
E
t
s
dEdxdEEenIR
dtdn
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Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
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Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Reaction Yields
dn I N dsA ab 0
Where: dn = number of reactions occurring in one secondI0 = number of particles incident on the target in one secondNA = number of target nuclei per gramds = thickness of the material in grams per cm2
σab = cross-section expressed in units of cm2
The rate of a particular reaction can also be written in the following equation.
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.
Radiopharmaceutical Production
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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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Simplified Equation
where, σi = cross-section for a process in millibarns for the interval in questionA = the atomic mass of the target material (AMU)Ni = number of nuclei created during the irradiationt = time of irradiation in secondsρ = density of the target in g/cm3
x = thickness of the target in cm.I = beam current in microamperes
xItANx i
i
1010678.2
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Reaction Cross-SectionThe 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.
Radiopharmaceutical Production
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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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Bragg Peak
Energy Deposition
Bragg Peak
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
Penetration into the target material
Particle Path with more scattering as the particle slows
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Stopping Power
Stopping power S(E) = - dE/dx-
where E is the particle energy (MeV)x is the distance traveled (cm)
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.
The stopping power depends on the characteristics of the incident particle, the target material, the energy and the chemical form of the target.
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Stopping Power
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)
)2ln()4( 2
02
0
42
IVm
AVmNZez
dxdE
The expression for the loss in energy can be given by the expression
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Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Stopping Power
where: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)
This expression can be simplified to the following equation by substitution the values of the physical constants into the equation
)2195ln(144 2
IE
AEZz
dxdE
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Range of charged Particles
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)
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.
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).
E
IEZz
AERE
max
0 2 )2195ln(144
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Simple Range Calculations
Energy Range Range Range Range(MeV) (Simple) SRIM Janni WG&J15 0.3477 0.3431 0.3430 0.344814 0.3077 0.3026 0.3038 0.305313 0.2699 0.2662 0.2668 0.267912 0.2344 0.2313 0.2319 0.232711 0.2011 0.1987 0.1992 0.199810 0.1702 0.1681 0.1687 0.16919 0.1416 0.1401 0.1405 0.14078 0.1155 0.1142 0.1146 0.11477 0.0917 0.0907 0.0910 0.09106 0.0705 0.0696 0.0699 0.06985 0.0517 0.0511 0.0513 0.05114 0.0357 0.0350 0.0352 0.03513 0.0223 0.0217 0.0219 0.02182 0.0118 0.0112 0.0114 0.01131 0.0044 0.0039 0.0040 0.0039
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
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Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Energy Straggling• 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.
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Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Energy Straggling
wherez = projectile atomic number (amu)Z = absorber atomic number (amu)A = absorber atomic mass number (amu)x = particle path length (g/cm2)
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
2/1
395.0
xAZz
Radiopharmaceutical Production
Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Multiple Scattering in Gas Targets• 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
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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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Multiple Scattering in Gas Targets 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
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Nuclear Reaction
Q- values
Reaction Cross Section
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Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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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.
• 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
• Beam spot shape was measured by irradiating a copper foil and imaging it with a phosphor plate imaging system.
Calculated beam profile
Measured beam profile
Beam Profile Alteration
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8
644 ug/cm 2800 ug/cm 2970 ug/cm 21200 ug/cm 2
Beam intensity versus radius, sam e total beam current
0
0.5
1
1.5
2
2.5
3
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5radius
Intensity
200 ug/cm 2̂400 ug/cm 2̂600 ug/cm 2̂800 ug/cm 2̂1000 ug/cm 2̂1200 ug/cm 2̂
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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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Saturation Yields
where, R - is the number of nuclei formed per secondn - is the target thickness in nuclei per cm2
I - is the incident particle flux per second and is related to beam currentλ - is the decay constant and is equal to ln2/t1/2t - is the irradiation time in secondsσ(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
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.
0
/)()1(
E
E
t
s
dEdxdEEenIR
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Saturation Factors
Saturation Factors(1 - e-
t)
0.000.200.400.600.801.001.20
0 2 4 6 8 10( Irrad. tim e/ half-life)
SF(1 - e –
λt)Fr
acti
on of
satu
rati
on a
ctivit
y
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Target PhysicsContents
Nuclear Reaction
Q- values
Reaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple Scattering
Saturation Yields
Literature
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Literature• 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