2.1 Specific activity of radioisotope

The specific activity (A/m) is one of the most important quality criteria for radioisotopes production in nuclear medicine research and application, and usually expressed in GBq/mg, Ci/mg (= 10⁶ mCi/g) or similar units. Here, A is the activity and m stands for the mass. The theoretical maximum specific activity for a pure radioisotope without admixture of stable isotopes is given by:

where λ₁=ln(2)/T_1/2 is the decay constant of the product isotope with the half-life T_1/2, M is the molar mass (g/mol) of the target isotope, and N_A = 6.02 × 10²³ represents Avogadro’s constant.

When an intense γ-beam transmutes a target isotope into the desired product, and considering that any other reactions (such as destruction of the product by nuclear reactions) do not interfere significantly, the achievable specific activity is given by:

Here, Y is the reaction yield for transmutation of the target into the product, the variable a is expressed as a=exp(-λ₁ t_irr), and t_irr is the irradiation time. Eq. (2) shows that to obtain a reliable reaction yield enables further accurate calculation of specific activity. For some specific cases, the daughter of the product isotope, rather than itself, is of particular interest in nuclear medicine application and the product isotope is only used as a radionuclide generator. A typical example is the ⁹⁹Mo/^99mTc generator, which is usually produced by a neutron-induced fission process, or by photo-neutron reaction and its disintegration, ¹⁰⁰Mo(γ,n)⁹⁹Mo → ^99mTc. For the latter, we derived the specific activity of the daughter radionuclide from the decay law:

where λ₂ corresponds to the decay constant of the daughter radioisotope and the variable b is given by

When the generator radioisotope has a smaller decay constant than that of the daughter radioisotope, a radioactive decay equilibrium between them could be reached. Then one could optimize the irradiation time to obtain achievable specific activity for the daughter radioisotope.

2.2 Monte Carlo simulation algorithm for photonuclear reaction

High-energy photons with appropriate energy, e.g. 10–30 MeV, can excite mainly through GDR mechanism, the target nuclide. The excited nuclide will decay primarily through emitting a nucleon (proton or neutron) or a photon. Such a process is usually designated as the photonuclear process, by which an activated daughter nuclide is generated. In order to produce medical radioisotopes, an interesting case is through ejecting a single neutron or proton, a so-called photo-neutron and photo-proton production process.

In a typical photonuclear process, reaction yield Y depends upon a number of parameters, such as number density, ρ, of target nuclei that are irridiated, threshold energy of nuclear reaction, E_th, end-point energy of γ-rays, E_max, probability of photonuclear reaction defined by reaction cross-section, ${\sigma }_{E_{\gamma }}$, and γ-ray flux density. While taking into account attenuation coefficient of γ-photons inside target, reaction yield can be written as:

where M is the molar mass (g/mol) of the target isotope, I(Eγ, r) is the γ-ray linear flux density at the surface of the target after the polar-angle integration, μ(Eγ) is the linear attenuation coefficient, R is the cylindrical target radius, and L is the target thickness with respect to the direction of γ-beam propagation. Due to the γ-ray attenuation inside the target, γ-ray flux decreases as target thickness increases. It should be noted that photons scattered inside the target may cause photonuclear reactions. However, the occurrence of such phenomenon depends basically on the radius of the target. For a target with a large radius, the change of the photon energy and of the angle should be included in the following simulation. Conversely, for a target with a relatively small radius, the effect of scattered photons that trigger photonuclear processes could be neglected, even if the target is very thick. This is attributed to the scattered photons that stray from the direction of incidence, and, as a result, they can easily escape from the target along lateral direction. In our simulation, small radius target is used and accordingly we did not consider the effect discussed.

For a certain geometry of irradiated target, we sampled a certain γ-photon with energy, E_γ,i, and position, ri, which is recorded at the surface of the target. Accordingly, the cross-section, i(E_γ,i), and the attenuation coefficient, μi(E_γ,i), were obtained using a linear interpolation method. Based on Eq. (5) we calculated the reaction probability for the i^th γ-photon at position ri interacting with the isotope inside the target as:

Further we obtained the reaction yield to be:

Here, I is the incident photon flux at the surface of the isotopic target, and N is the total sampling number. By substituting Eq. (7) into Eqs. (2) and (3), one could obtain specific activity of produced radioisotope and its daughter radionuclide as:

and

2.3 γ-beam characteristics simulation

As presented above, there are two main approaches for production of high-energy γ-beams of high intensity. One is through bremsstrahlung process based on linac and another is via the Compton backscattering of a laser light off a very brilliant and intense electron beam. While Geant4 toolkit is well defined to predict the beam characteristics stemmed from the bremsstrahlung mechanism, several laser-Compton scattering simulation codes [29-32], benchmarked against experimental data produced by worldwide CBS γ-ray facilities, have been developed. These codes are suitable to model the MeV-class γ-ray production and to study the spectral and flux density distributions of the γ-beam during its transport from the laser-electron interaction point (IP) to the target, through the CBS γ-ray beamline. These parameters will be taken as input data, to facilitate calculation of photonuclear reaction yield and specific activity of desired radioisotopes.

2.4 Photonuclear cross-section and γ-ray attenuation coefficient calculation

In order to obtain the reaction yield shown in Eq. (5), reliable photonuclear cross-sections are estimated (or measured) and γ-ray attenuation coefficient for a certain target isotope are required. While acquiring photonuclear cross-sections from EMPIRE calculation [33] and from TENDL evaluation [34], we can also cite experimental ones available in the EXFOR database [26]. If possible, a detailed comparison between them will be conducted, which enables us to select prudently photonuclear cross-sections. For incident photons covering energy from 1 keV to 20 MeV, the mass attenuation coefficients are available in Ref. [35]. Then the linear attenuation coefficient required by Eqs. (5) and (6) could be obtained readily. As the irradiated target with compound and mixture are employed, the linear attenuation coefficient is reprocessed according to its content and composition. Further, since the set of data for the photonuclear reaction and attenuation coefficient were popularly discrete, we used linear interpolation method to calculate their values for incident γ-rays with arbitrary energy.

According to above simulation and calculation algorithms, a data-based Monte Carlo simulation C++ program, integrated with Geant4, or Geant4-based γ-beam simulation code [30-32], was developed for determining photonuclear isotope production, induced by a high-intensity γ-ray beam either from linac-based bremsstralung source or from CBS-driven γ-ray facility. Since the prudent photonuclear cross-section, flexible incident photon beam, and attenuation effect inside irradiated target were fully considered, it can be easily used to investigate the effects of incident γ-beam and target geometric parameters on photonuclear yield and consequently to predict specific activity of radioisotopes. In the following, several benchmarking simulations will be given in order to check the validity of the use of the data-based simulation algorithm.

3.1 Benchmarking simulations

As presented in the Introduction, Geant4 can be used to simulate photonuclear processes. To check the validity of the photonuclear model existing in Geant4, we have first simulated photonuclear reaction cross-sections and compared them with experimental data provided by the EXFOR database [26]. For photonuclear interactions, Geant4 popularly uses the "G4PhotoNuclearProcess" class [36]. It handles inelastic photon scattering from nuclei by invoking one or more hadronic models and one or more hadronic cross sections. In our case, the Bertini intra-nuclear cascade model, i.e. the "G4CascadeInterface" class, was invoked. It was used to simulate final states for photonuclear reactions. Although the model is available for the region of incident photon energy 0 ≤ E_γ < 3.5 GeV in Geant4, it has only been tested in a energy range 60 MeV ∼ 3 GeV [37].

Simulations of photon-induced neutron production were performed for ⁹Be, ¹³³Cs, and ¹⁹⁷Au. ⁹Be is a one of the light nuclei and its target material is useful for producing photo-neutrons while using low-energy bremsstrahlung radiation because of a very low neutron separation energy (E_th ∼ 1.66 MeV). ¹³³Cs is a suitable candidate in an intermediate-mass region due to a considerable reaction cross-section. ¹⁹⁷Au is suitable for the production of large amounts of neutrons due to a high reaction cross-section. To compute photonuclear reaction cross-sections, we used a cylindrical target with 1.0 mm thick and 4.5 cm radius arbitrarily for representing a ’semi infinite geometry’. For brevity, the results obtained from the Geant4 with the "G4PhotoNuclearProcess" class are hereafter simply referred to as "G4PNP". As a comparison, many experimental data in the EXFOR database [26] were used, the data will hereafter be simply referred to as "EXFOR".

Figure 1 shows the comparison of G4PNP with EXFOR for ⁹Be(γ,n) and ⁹Be(γ,2n) reaction cross-sections in the GDR region of incident photon energy E_γ ≤ 35 MeV. Large discrepancies can be observed between the G4PNP simulations and EXFOR data for (γ,n) reaction cross-section in the GDR region. For ⁹Be(γ,n) reaction, the model of G4PNP does not reproduce the low-energy peaks of the cross-section and underestimates the experimental cross-section by up to ∼10³ in the region of incident photon energies E_γ < ∼15 MeV. Such discrepancy is attributed to an underestimation of neutron production for ⁹Be(γ,n) reaction, as discussed later. The cross-section on ⁹Be(γ,2n) underestimates the experimental value in the peak region by a factor of 10.

Fig. 1.Full-screen View

(γ,n) and (γ,2n) reaction cross-sections for ⁹Be as a function of incident photon energy. The open squares are the experimental data taken from the EXFOR database [26]. The solid lines and the dashed lines represent the results obtained from the data-based Monte Carlo calculations and Geant4 simulations using G4PhotoNuclearProcess, respectively.

A similar feature for the underestimation of cross-sections can also be seen in the case of ¹³³Cs(γ,n) and ¹⁹⁷Au(γ,n) reactions. Figs. 2(a) and (c) show that both the ¹³³Cs(γ,n) and ¹⁹⁷Au(γ,n) cross section peak values given by the G4PNP are lower than the experimental ones. The comparisons of G4PNP with EXFOR for ¹³³Cs(γ,2n) and ¹⁹⁷Au(γ,2n) reaction cross-sections in the GDR region are plotted in Figs. 2(b) and (d), respectively. While the G4PNP cross-section on ¹⁹⁷Au(γ,2n) is very close to experimental data, the value on ¹³³Cs(γ,2n) (100 mb) overestimates the experimental cross-section (74.9 ± 1.8 mb) in the peak region by a factor of 1.3.

Fig. 2.Full-screen View

(γ,n) reaction cross-sections for ¹³³Cs (a) and ¹⁹⁷Au (c), and (γ,2n) reaction cross-sections for ¹³³Cs (b) and ¹⁹⁷Au (d) as a function of incident photon energy. The open squares are the experimental data taken from the EXFOR database [26]. The solid lines and the dashed lines represent the results obtained from the data-based MC cal. and Geant4 simulations using G4PhotoNuclearProcess, respectively.

Comparisons with experimental data for photo-proton reactions are performed as well. Taking ⁴⁸Ti(γ,p)⁴⁷Sc as an example, the simulated results from G4PNP diverge from the data available in the EXFOR database (see Fig. 3). Through the above comparisons, it is found that Geant4 (v4.9.6.03) can hardly reproduce the experimental photonuclear reaction cross-sections in the GDR region, which is also verified by a similar comparison in [28]. Hence, it is necessary to develop an alternative approach for simulation of photonuclear reaction in this low energy region.

Fig. 3.Full-screen View

Simulation of reaction cross-sections on ⁴⁸Ti(γ,p) as a function of incident photon energy. The open squares are the experimental data taken from the EXFOR database [26]. The solid lines and the dashed lines demonstrate the results given by the data-based MC cal. and Geant4 simulations, respectively.

3.2 Data-based Monte Carlo simulation for photonuclear reaction

To check the validity of the developed algorithm demonstrated in Section 2, we also performed the same simulations for photonuclear reactions on ⁹Be, ¹³³Cs, and ¹⁹⁷Au. The data-based Monte Carlo calculations (simply referred to as MC cal.) for both (γ,n) and (γ,2n) reaction cross-sections of ⁹Be, ¹³³Cs, and ¹⁹⁷Au are shown in solid lines of Fig. 1 and Fig. 2. It is found that there is good agreement between the calculated results and EXFOR for photo-neutron production cross-sections. Although the predicted ⁹Be(γ,n) and ⁹Be(γ,2n) reaction cross-sections by using G4PNP differ from the experimental ones by a factor of 10–10³, the cross-sections from the MC cal. are very close to those of the EXFOR. In addition, it describes well the low energy resonance structure of the ⁹Be(γ,n) presented in Fig. 1(a). For photo-neutron cross-sections of the ¹³³Cs and ¹⁹⁷Au, the detailed comparisons show that the predictions from the data-based MC cal. are in good agreement with the experimental data. Fig. 3 shows the simulation of photo-proton cross-sections on the ⁴⁸Ti with experimental data. It demonstrates that the simulated results are in accordance with the experimental data.

To substantiate the use of the new developed code, benchmark simulations were performed for photonuclear reaction yield from the ⁹Be(γ,n) reaction. The photons used to trigger the reaction were generated from a 8.96 MeV electron beam (e-beam) impinging on a 1.88 mm tantalum target through the bremsstrahlung process [38, 39]. The photo-neutron yield with respect to the thickness of the beryllium target is shown in Fig. 4. The yields calculated using the data-based simulation code and the G4PNP are compared with the experimental data [38]. Calibrated SSNTD CR-39 films and Silver wrapped GM detectors were used for the experimental measurements. Since the experimental data taken from CR-39 films was in accordance with that from GM detector within 10% [38], the data used for the benchmarking should be reliable.

Fig. 4.Full-screen View

Photonuclear reaction yield from beryllium target as a function of the target thickness. The filled circles are the experimental data from [38]. The squares with a solid line and the triangles with a dashed line represent the photonuclear reaction yields obtained from data-based calculation and Geant4 simulation using the G4PhotoNuclearProcess, respectively.

The results taken from the data-based MC cal. were in accordance with the experimental data for different volumes of beryllium target. The calculation point at 8 cm overestimated the experimental one. This could be readily understood in that the bremsstrahlung photons emitted (and then recorded at the surface of the beryllium target) had a relatively large angular distributions. However, a parallel bremsstrahlung photon beam was assumed for simulation, to irradiate the target. It meant that the number of the bremsstrahlung photons, used to induce photon-neutron reactions, was overestimated slightly, and the photo-neutron reaction yield was enhanced accordingly. Large discrepancies between the results from the G4PNP calculations and the experimental measurements are observed in Fig. 4. The G4PNP underestimates the photo-neutron yields by three orders of magnitude since the model cannot reproduce the experimental ⁹Be(γ,n) reaction cross-section as shown in Fig. 1(a). This benchmark simulation shows that the data-based simulation code gives reasonable results for photonuclear reaction yield calculations in the GDR region. As a consequence, the validity of the data-based simulation algorithm was justified.