The rate of radionuclide production is dependent on a number of factors, including the magnitude of the reaction cross-section as a function of energy, the incident particle energy, the thickness of the target in nuclei per cm2, which will determine the exit particle energy, and the flux (related to beam current) of
sR =pr02(A1/3p +AT1/3)2
RADIOISOTOPE PRODUCTION
incoming particles. In the simplest case, where the cross-section is assumed to be constant, the rate of production is given by:
(1.4) The cross-section is always a function of energy, as has been shown in the previous section. If we use this more exact expression, then the equation becomes:
(1.5)
where
R is the number of nuclei formed per second;
nT is the target thickness in nuclei/cm2;
I is the incident particle flux per second and is related to the beam current;
s 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;
0
FIG. 1.5. Excitation function for the 14N(d, n)15O reaction.
R=n IT s
x is the distance travelled by the particle; and
is the integral from the initial energy to the final energy of the incident particle along its path.
As the particle passes through the target material, it loses energy due to the interactions of the particle with the electrons of the target. This is represented in the above equation by the term dE/dx (also called the stopping power).
Returning to the expression for the cross-section, it can be seen that nT is given by the following expression:
(1.6)
where
nT is the target thickness in nuclei/cm2;
AT is the atomic weight of the target material in grams;
r is the density in g/cm3; z is Avogadro’s number; and
x is the distance the particle travels through the material.
If the target material is a compound rather than a pure element, then the number of nuclei per unit area is given by the following expression:
(1.7)
where
NG is the number of target nuclei per gram;
FA is the fractional isotopic abundance;
C is the concentration by weight;
z is Avogadro’s number; and
AA is the atomic mass number of nucleus A.
E
RADIOISOTOPE PRODUCTION
The above equations lead to one of the basic facts of radioisotope production. It is not always possible to eliminate the radionuclidic impurities even with the highest isotopic enrichment and the most precise energy selection. An example of this is given in Fig. 1.6 for the production of 123I with a minimum of 124I impurity [1.8–1.11].
As can be seen from Fig. 1.6, it is not possible to eliminate the 124I impurity from the 123I because the 124I is being made at the same energy. All that can be done is to minimize the 124I impurity by choosing an energy where the production of 124I is near a minimum. In this case, proton energies higher than about 20 MeV will give a minimum of 124I impurity.
1.4.1. Saturation factor
As soon as radioisotopes have been produced, they start to decay. This leads to the following expression, where the overall rate of production is then:
(1.8) Particle energy (MeV)
Cross-section (mb)
FIG. 1.6. Nuclear reaction cross-sections for production of 123I and 124I from 124Te.
-ddn= T
Ú
d /d d -t n I EE x E N
E E
I
s( ) l
0
where
l is the decay constant and is equal to ln 2/t1/2; t is the irradiation time in seconds; and
N is the number of radioactive nuclei in the target.
The term dE/dx in the above expression is often referred to as the total stopping power. At a particular energy E, it can be represented as ST(E) in units of MeV·cm2·g–1 and is given by the following expression:
(1.9)
where
dE is the differential loss in energy; and
dx is the differential distance travelled by the particle.
The loss of energy, dE, in MeV of the particle crossing the slab, is then given by:
(1.10) where r is the density of the material in units of g/cm3,and the thickness of the slab rdx (in g/cm2) can be expressed as a function of dE:
(1.11)
If this equation is integrated, including the stopping power to account for energy loss during the transit of the particle through the target material and assuming that the beam current is the same as the particle flux (which is true only for particles with a charge of +1), then the yield of a nuclear reaction is given by:
RADIOISOTOPE PRODUCTION
The term 1 – e–lt is often referred to as the saturation factor, and accounts for the competition of the production of nuclei due to the particle reaction and the radioactive decay of the nuclei that have been produced. An example using
13N is shown in Fig. 1.7. The beam is turned on at time zero and turned off 40 min after the start of bombardment.
If an infinitely long irradiation that makes the saturation factor tend to the value 1 is assumed, then we have what is referred to as the saturation yield.
This quantity is shown as a function of energy in Fig. 1.8 for the 14N(d, n)15O reaction.
1.4.2. Nomenclature
The nomenclature for nuclear reactions as it is usually used throughout this report needs to be defined. If a 13C nucleus is irradiated with a proton beam to produce a nucleus of 13N with a neutron emitted from the compound nucleus, this reaction will be written as 13C(p, n)13N. In a similar manner, if a
20Ne nucleus is bombarded with a deuteron beam to produce a nucleus of 18F with the concomitant emission of an alpha particle, this reaction sequence will be abbreviated as 20Ne(d, a)18F.
0 5 10 15 20 25 30
0 20 40 60 80 100
Time (min)
Activity (mCi)
Beam Off
FIG. 1.7. Saturation factor of the irradiation of 13C to produce 13N, showing the produc-tion of the radionuclide and its decay after the beam has been turned off.