2. PROTON ACCELERATION FROM THIN FOIL

The proton beam parameters have been deduced from 3D simulations performed using the PIC code MANDOR (Romanov et al., Reference Romanov, Bychenkov, Rozmus, Capjack and Fedosejevs2004). We consider the interaction of a 10 J linearly polarized 100 fs laser pulse with an ultrathin plain hydric target (foil) assuming a repetition rate of 10 kHz. These laser parameters are consistent with what is expected from ICAN “dream laser” (Mourou et al. Reference Mourou, Brocklesby, Tajima and Limpert2013). By changing focal spot size and target thickness, we aim to maximize both the energy and the number of accelerated protons to maximize Tc-99m yield for a single shot when the protons from the foil are directed into a Mo-100 target. In order to boost acceleration of protons with highest current and energy, the bulk concentration of protons in the foil needs to be maximal (Robinson et al., Reference Robinson, Bell and Kingham2006). The highest proton content can be achieved with a pure hydrogen target. However, such a target requires cryogenic setup and precise thickness control and currently does not exist. Instead, ultra-thin CH₂ foils can be used. They are known as PMP (poly(4-methyl-1-pentene): (CH₃)₂ CHCH₂CH = CH₂), targets. The CH₂ foils have to be free-standing over the holes with radius of at least several hundred micrometers in a base plate (cf., Henig et al., (Reference Henig, Steinke, Schnurer, Sokollik, Horlein, Kiefer, Jung, Schreiber, Hegelich, Yan, Meyer-ter-Vehn, Tajima, Nickles, Sandner and Habs2009)) because with smaller radii, ionization of the target holder could affect the laser-plasma interaction. Such design could allow the base plate to be mounted inside the vacuum chamber on a motorized stage and fast rotated to ensure that each laser shot interacts with a fresh foil similar to what was used in the experiment (Liao et al., Reference Liao, Mordovanakis, Hou, Chang, Rever, Mourou, Nees and Galvanauskas2007), where the target was shot at a repletion rate of 1 kHz. As a case in point, we consider laser interaction with ultrathin CH₂ foil and, as a reference, we set the laser wavelength at λ = 1 µm. The hot spot size was varied from 4 µm up to 10 μm, which corresponds to laser pulse intensity decrease from 5.5 × 10²⁰ W/cm² to 8.8 × 10¹⁹ W/cm². To study the possibility of Tc-99m production with a smaller scale laser, we also considered a laser pulse with an energy of 5 J and the same 100 fs pulse duration (the intensity is 2.8 × 10²⁰ W/cm² for a 4 µm focal hot spot). The full set of the laser parameters are collected in Table 1.

Table 1. Laser parameters under considereation

The laser pulse is focused at the front side of the plasma target composed from electrons and fully ionized carbon and hydrogen. The target has an electron density of 200n _c, which corresponds to a real mass density of CH₂ (1.1 g/cm³). Varying the target thickness (l) from 20 nm to 0.5 µm, we find the optimal thickness to maximize the maximum proton energy. The optimum target thickness (l ₀) corresponds to the partially transparent regime of laser–plasma interaction (Brantov & Bychenkov, Reference Brantov and Bychenkov2013). This thickness is proportional to the square root of the laser pulse intensity and therefore decreases as the focal spot size increases (see Fig. 1).

Fig. 1. Maximum proton energy vs. target thickness for different laser pulse focal spots: d = 4 µm (small black dots, case 1), d = 6 µm (medium black dots, case 2), and d = 10 µm (large black dots, case 3); grey dots correspond to a 5 J laser pulse focused into a 4 µm hot spot (case 4).

We also compare the number of energetic protons accelerated from thin foils. In this study, the number of protons with energy exceeding 8 MeV (the threshold for the Mo-100(p,2n)Tc-99m reaction) is of interest. In general, the number of accelerated particles increases with the focal spot size and laser intensity. For a given laser pulse energy, maximizing the number of accelerated particles would correspond to some trade-off between them. But our simulations demonstrate that increased intensity dominates. By analyzing the spectra of protons accelerated from a target of optimum thickness (see the left panel in Fig. 2), we conclude that a tight laser pulse focus gives not only the maximum proton energy but also the maximum number of energetic particles. Moreover, even a less energetic pulse but with a tighter focus can be used to achieve approximately the same number of energetic protons (cf., gray and dotted curves in the left panel in Fig. 2). In fact, proton acceleration from a very thin foil is a Coulomb explosion (Fig. 2, right panel). Although this regime is less effective for proton acceleration, because of significant electron exhaustion of the entire irradiated target volume, it shows some increase of the maximum proton energy and number of energetic protons as the focal spot size increases (see the right panel in Fig. 2).

Fig. 2. Spectra of protons accelerated from the target with optimum thickness (left panel) and from a target with the thickness l = 0.02λ (right panel): d = 4 µm (solid curves), d = 6 µm (dashed curves), and d = 10 µm (dotted curves). The corresponding optimum thicknesses are l ₀ = 0.07λ, l ₀ = 0.05λ, and l = 0.02λ. The grey line shows the proton spectra for a 5 J laser pulse focused into a 4 μm hot spot (l ₀ = 0.04λ).

To summarize, when the laser interacts with a thin foil whose thickness is properly adjusted to the pulse energy and focusing optics, both maximum proton energy and maximum number of particles can be achieved. The generated proton beam is characterized by an angular spreading of ≲10^° and a wide energy distribution with a sharp cut-off at the maximum energy E _max, which is of the order of 50 to 70 MeV for about 10 J laser pulse. The number of protons with energy higher than 8 MeV is of the order of 10¹¹ particles per shot.

3. TC-99M YIELD CALCULATION

We now estimate the Tc-99m yield for the Mo-100(p,2n) Tc-99g reaction in a thick target (catcher) bombarded by laser-produced protons from a thin CH₂ foil (pitcher). A pitcher-catcher set up is typical for laser initiation of nuclear reactions (Nemoto et al., Reference Nemoto, Maksimchuk, Banerjee, Flippo, Mourou, Umstadter and Bychenkov2001; Clarke et al., 2013; Storm et al., Reference Storm, Jiang, Wertepny, Orban, Morrison, Willis, McCary, Belancourt, Snyder, Chowdhury, Bang, Gaul, Dyer, Ditmire, Freeman and Akli2013; Roth et al., Reference Roth, Jung, Falk, Guler, Deppert, Devlin, Favalli, Fernandez, Gautier, Geissel, Haight, Hamilton, Hegelich, Johnson, Merrill, Schaumann, Schoenberg, Schollmeier, Shimada, Taddeucci, Tybo, Wagner, Wender, Wilde and Wurden2013). Actually, since laser produces highly collimated protons (Cowan et al., Reference Cowan, Fuchs, Ruh, Kemp, Audebert, Roth, Stephens, Barton, Blazevic, Brambrink, Cobble, Fernandez, Gauthier, Geissel, Hegelich, Kaae, Karsch, LeSage, Letzring, Manclossi, Meyroneinc, Newkirk, Pepin and Renard-LeGalloudec2004), in the laser nuclear experiments there are a wide variety of desired distances between pitcher and catcher, 5–200 mm, in a vacuum chamber (Nemoto et al., Reference Nemoto, Maksimchuk, Banerjee, Flippo, Mourou, Umstadter and Bychenkov2001; Storm et al., Reference Storm, Jiang, Wertepny, Orban, Morrison, Willis, McCary, Belancourt, Snyder, Chowdhury, Bang, Gaul, Dyer, Ditmire, Freeman and Akli2013; Roth et al., Reference Roth, Jung, Falk, Guler, Deppert, Devlin, Favalli, Fernandez, Gautier, Geissel, Haight, Hamilton, Hegelich, Johnson, Merrill, Schaumann, Schoenberg, Schollmeier, Shimada, Taddeucci, Tybo, Wagner, Wender, Wilde and Wurden2013). For example, if an angular spreading of the accelerated multi-MeV protons is 10^°, that is the case of our simulations, almost all accelerated energetic protons will interact with the 15–30 µm radius catcher which is placed at the distance of 100–200 mm. Due to such distant positioning the catcher almost surely will be safe in regards to being disrupted by laser-produced plasma. If necessary (when the distance between the pitcher and catcher is small), additional thin metal or/and plastic films with thickness less than 50 µm can be placed behind laser target (Roth et al., Reference Roth, Jung, Falk, Guler, Deppert, Devlin, Favalli, Fernandez, Gautier, Geissel, Haight, Hamilton, Hegelich, Johnson, Merrill, Schaumann, Schoenberg, Schollmeier, Shimada, Taddeucci, Tybo, Wagner, Wender, Wilde and Wurden2013). In our calculations, we neglect small proton energy losses in possible protective screen foils. As a catcher, we assume metallic target of enriched Mo-100 of 99.54% (Isoflex). To stop protons with energy of 30–40 MeV a thickness of the Mo-100 layer should be 1.6–2.6 µm. To increase an effective thickness of the catcher it can be placed at the angle of several degrees to the proton beam axis that has been recently proposed by Benard et al. (Reference Benard, Buckley, Ruth, Zeisler, Klug, Hanemaayer, Vuckovic, Hou, Celler, Appiah, Valliant, Kovacs and Schaffer2014).

Note, that the proton beam contains protons with wide energy spread. They are moving with different velocities and therefore reach a catcher target at different instants. For example, if the catcher is at 100 mm behind the pitcher, the protons with energy of 40 MeV hit Mo-100 after 340 ps and the protons with energy of 10 MeV do it after 680 ps. Therefore, effective beam duration is about 340 ps, more than 3000 times longer than the laser pulse duration that significantly reduces proton beam power compared to monoenergetic beam. Moreover, due to greater proton spot size at the catcher, the power flux density at the catcher is also dramatically decreased. Given the laser-proton conversion efficiency 1%, that is typical for our simulations, the power flux density of the proton bunch bombarding an enriched Mo-100 target is about 10⁸ W/cm². It is very likely that such power flux density is too low to cause any damage and heating of the catcher (see, for example, the experiment by Patel et al., (Reference Patel, Mackinnon, Key, Cowan, Foord, Allen, Price, Ruhl, Springer and Stephens2003) in which proton energy flux at the target was many orders of magnitude larger). Because low-energy protons (with energy < 8 MeV) considerably contribute to the total power flux density but are useless for technetium production these protons can be excluded with magnetic selection (Busold et al., Reference Busold, Schumacher, Deppert, Brabetz, Frydrych, Kroll, Joost, Al-Omari, Blazevic, Zielbauer, Hofmann, Bagnoud, Cowan and Roth2013) before they reach the catcher. This could even stronger decrease power deposited into Mo-100, thus more preventing it from destructive effects.

The yield (number of reactions) for one proton of energy E is given by Krasnov (1974) and Celler et al. (Reference Celler, Hou, Benard and Ruth2011)

(1)$$Y_1 \lpar E\rpar ={{N_A } \over {Mm_u }} \int_0^E \, {\rm \sigma} \lpar E^{\prime}\rpar \left({dE^{\prime} \over {d{\rm \rho} x}}\right)^{-1} dE^\prime\comma \;$$

where σ (E) is the cross section (Qaim et al., 2014) in cm², dE/dρ x is the stopping power in the units MeV cm²/g, M is the target atomic mass, N _A is the Avogadro number, and m _u = 1 g/mol is the molar mass constant. For the proton spectrum derived from the simulation, the total yield Y _Σ per one laser shot can be calculated from

(2)$$Y=\int_0^\infty \, \displaystyle{{dN} \over {dE}}Y_1 \lpar E\rpar dE\, .$$

The corresponding number of radioactive nuclei at the given instant t _d after the production of Tc-99m terminates is

(3)$$Y_\Sigma=Yf\left(1 - \exp\left( - {\lambda} t_r \right) \right)\exp\left( - \lambda t_d \right)/ \lambda\, \comma \;$$

where t _r is the duration of radiation exposure, f is the laser pulse repetition rate, and the decay constant λ is inversely proportional to the half-life τ_1/2, λ = log(2)/τ_1/2. The maximum yield is (Yf/λ)[1 - exp(- λ t _r )] and is shown in Table 2. The stopping power and reaction cross-sections data have been taken from SRIM code (Ziegler, Reference Ziegler2010; Berger et al., Reference Berger, Coursey, Zucker and Chan2014; and Qaim et al., Reference Qaim, Sudar, Scholten, Koning and Coenen2014).

Table 2. The Tc-99m production versus irradiation condition and target thickness

The full set of data for the four cases discussed (see Table 1) is presented in Table 2. We calculated the number of Tc-99m atoms produced from Mo-100(p,2n) reaction per one laser shot and compared it with the other Tc isotopes: Tc-99g, Tc-98, Tc-97g, Tc-97m, and Tc-96g produced from Mo-100(p,2n), Mo-100(p,3n), Mo-100(p,4n), and Mo-100(p,5n) reactions. The table shows the ratio of Tc-99m to these impurities (see “Ratio” column).

The main issue of Tc-99m sample quality is radionuclide purities in terms of other Tc isotopes which could not be removed by chemical purification. These Tc radioisotopes are generated from nuclear interactions of the accelerated protons ether with other stable Mo isotopes presented in the nuclear target (Hou et al., Reference Hou, Celler, Grimes, Benard and Ruth2012; Lebeda et al., Reference Lebeda, van Lier, Stursa, Ralis and Zyuzin2012) or from (p,4n) and (p,5n) reaction in interaction with Mo-100. By using isoflex target with 99.54% of Mo-100 and 0.41% of Mo-98 one may greatly reduce amount of radioactive technetium isotopes other than Tc-99m which can be produced from the most dangerous Mo-(95-97) impurities (Hou et al., Reference Hou, Celler, Grimes, Benard and Ruth2012). However, the unstable Tc-97m, Tc-96m, and Tc-96g isotopes appear because protons have a rather high maximum energy E _max up to 70 MeV. Their production may result in unacceptable increase of the activities (see column 8 in Table 2). In spite of the fact that Tc-96m decays to Tc-96g in approximately 50 minutes, the activities of parasitic Tc isotopes may still be high enough for medical application. For example, the ratio of parasitic Tc isotope activities to the Tc-99m activity which reaches 7.5% for protons with E _max = 63.5 MeV (see third row in a Table 2) decreases up to 1.6% during three hours after the end-of-bombardment. The isotopes Tc-97m and Tc-96g may comprise 20% to 40% of the entire Tc-99m isotope. The percentage of these isotopes can be significantly reduced when the maximum proton energy decreases, for example, up to 2% for a proton beam with the energy cutoff at 46 MeV (see the fourth row in Table 2). In latter case, the relative activities of parasitic Tc isotopes which is just 0.12% at the end-of-bombardment drops to the 0.03% after three hours. Therefore, using of proton beam with maximum energy up to 40 Mev is more favorable for Tc-99m production. We assume that unstable Tc-100 (τ_1/2 = 15.8 s) which is produced during Mo-100(p,n) reaction decays to stable Ru and does not contribute to the final radioactivity. The Tc-99m activity after six hours irradiation is calculated for laser repetition rate of 10 kHz.

We also calculated the number of Mo-99 due to Mo-100(p,d + pn)Mo-99 reaction. It may increase Tc-99m production somewhat (Qaim et al., Reference Qaim, Sudar, Scholten, Koning and Coenen2014). Because Mo-99 has a long decay time (τ_1/2 = 66.0 h), it contributes noticeably to the Tc-99m yield only for a relatively long irradiation time (more than 12 h) and a large proton energy (in excess of 30 MeV) (Qaim et al., Reference Qaim, Sudar, Scholten, Koning and Coenen2014).