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Nuclear Reactions Target Physics STOP

Target Physics Contents Nuclear Reaction Q- values
Reaction Cross Section Stopping Power Particle Range Energy Straggling Multiple Scattering Saturation Yields Literature 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. STOP

Major Nuclear Reaction Types
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 γ Neutron reaction with the nucleus Target Nucleus Proton reaction with the nucleus with several nucleons emitted

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: 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
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

Total Excitation Energy
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: 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 >0 mass to energy (exothermic)
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 >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.

Q Value and Reaction Threshold

Reaction Cross-section
The rate of any particular reaction is given by the following expression with the variables as defined below. 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 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 ʃ is the integral from the initial to final energy of the incident particle along its path

Reaction Yields The rate of a particular reaction can also be written in the following equation. 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 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.

Simplified Equation where,
σi = cross-section for a process in millibarns for the interval in question A = 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/cm3 x = thickness of the target in cm. I = beam current in microamperes

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.

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 Particle Path with more scattering as the particle slows Energy Deposition Penetration into the target material Bragg Peak

Stopping Power 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. Stopping power S(E) = - dE/dx- where E is the particle energy (MeV) 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.

Stopping Power The expression for the loss in energy can be given by the expression 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 This expression can be simplified to the following equation by substitution the values of the physical constants into the equation 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)

Range of charged Particles
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. 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).

Simple Range Calculations
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 Energy Range Range Range Range (MeV) (Simple) SRIM Janni WG&J

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.

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 where z = projectile atomic number (amu) Z = absorber atomic number (amu) A = absorber atomic mass number (amu) x = particle path length (g/cm2)

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

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

Beam Profile Alteration
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 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

Saturation Yields 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. 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 (1 - e –λt) Fraction of saturation activity

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