Nuclear Reactions Target Physics STOP 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 ID: 412880
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Slide1
Radiopharmaceutical Production
Nuclear ReactionsTarget Physics
STOPSlide2
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.
ContentsNuclear ReactionQ- valuesReaction Cross Section
Stopping Power
Particle Range
Energy Straggling
Multiple ScatteringSaturation YieldsLiterature
STOPSlide3
Major Nuclear Reaction Types
γ
Target
Nucleus
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 Slide4
Nuclear Reaction Classic Model
Barrier to reactionB=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:Slide5
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 scatteringNuclear inelastic scattering with or without nucleon emissionProjectile absorption with or without nucleon emissionThere are certain probabilities for each of these pathways. The probability can be expressed as follows:
σ
i
= σ
com(Pi/ ΣPi)where, σi
= cross-section for a particular product Iσcom = cross-section for the formation of the compound nucleusPi = probability of process iΣP
i
= the sum of the probabilities of all processesSlide6
Total Excitation Energy
U = [M
A / (MA + Ma)] .Ta
+ S
a
where:
U = excitation energy MA = mass of the target nucleus
Ma = mass of the incident particle Ta = kinetic energy of the incident particle
S
a
= 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:Slide7
Q values
The probability of any particular reaction will depend on whether the reaction is exothermic or endothermicthe '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.Slide8
Q Value and Reaction ThresholdSlide9
Reaction Cross-section
where:
R is the number of nuclei formed per secondn is the target thickness in nuclei per cm2I is the incident particle flux per second and is related to beam
current
λ is the decay constant and is equal to ln2/t
1/2
t 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.Slide10
Reaction Yields
Where:
dn = number of reactions occurring in one secondI0 = number of particles incident on the target in one second
N
A
= number of target nuclei per gram
ds = thickness of the material in grams per cm2σab = 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.Slide11
Simplified Equation
where,
σ
i
= 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 irradiation
t = time of irradiation in seconds
ρ = density of the target in g/cm
3
x = thickness of the target in cm.
I = beam current in microamperesSlide12
Reaction Cross-Section
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.Slide13
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 slowsSlide14
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. Slide15
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)
The expression for the loss in energy can be given by the expression Slide16
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 equationSlide17
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).Slide18
Simple Range Calculations
Energy Range
Range Range Range(MeV
) (Simple) SRIM
Janni
WG&J
15 0.3477 0.3431 0.3430 0.3448 14 0.3077 0.3026 0.3038 0.3053 13 0.2699
0.2662 0.2668 0.2679 12 0.2344 0.2313 0.2319 0.2327 11 0.2011 0.1987 0.1992 0.1998
10
0.1702
0.1681 0.1687
0.1691
9
0.1416
0.1401 0.1405
0.1407 8 0.1155 0.1142 0.1146 0.1147 7 0.0917 0.0907 0.0910 0.0910 6 0.0705 0.0696 0.0699 0.0698 5 0.0517 0.0511 0.0513 0.0511 4 0.0357 0.0350 0.0352 0.0351 3 0.0223 0.0217 0.0219 0.0218 2 0.0118
0.0112 0.0114 0.0113 1 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 aluminumSlide19
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.Slide20
Energy Straggling
where
z = projectile atomic number (
amu
)
Z = absorber atomic number (
amu
)
A = absorber atomic mass number (
amu
)
x = particle path length (g/cm
2
)
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 centimeterSlide21
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 particleMultiple scattering in the front foil causes the beam shape to enlargeThe Multiple Scattering in the target can be approximated by a simple modelSlide22
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 Zx distance traveledE energy of the particleA atomic weight of the absorberSlide23
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 AlterationSlide24
Saturation Yields
where,
R - is the number of nuclei formed per secondn - is the target thickness in nuclei per cm2I - 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 cm
2
and is a function of energy
E - is the energy of the incident particles, and
x - is the distance traveled by the particleAs 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.Slide25
Saturation Factors
(1 - e
–λ
t
)
Fraction of saturation activitySlide26
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 1211Slide27
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