INTRODUCTION AND MOTIVATION
In nuclear power generation, one of the serious problems is the production of radioactive nuclear wastes consisting of long and intermediate-lived fission products. Developing and applying advanced technologies in the area of long-lived radioactive waste utilization and transmutation is the focus of International Atomic Energy Agency (IAEA) programs, and investigation was carried out (Maschek et al., Reference Maschek, Stanculescu and Arien2008). Among the different options, recently, much attention is focused on the possibilities of laser transmutation by highly directional γ-beams, which is generated in ultra intense laser-solid interaction (Ledingham et al., Reference Ledingham, Magill, Mckenna, Yang, Galy, Schenkel, Rebizant, Mccanny, Shimizun, Robson, Singhal, Wei, Mangles, Nilson, Krushelnick, Clarke and Norreys2003; Magill et al., Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003; Takashima et al., Reference Takashima, Hasegama, Nemoto and Kato2005; Sadighi-Bonabi et al., Reference Sadighi-Bonabi and Kokabi2006, Reference Sadighi-Bonabi, Irani, Safaie, Imani, Silatani and Zare2009a; Renner et al., Reference Renner, Juha, Krasa, Krousky, Pfeifer, Velyhan, Granja, Jakubek, Linhart, Slavicek, Vykydal, Pospisil, Kravarik, Ullschmied, Andreev, Kampfer, Uschmann and Forster2008). In the above mentioned works, a high brightness γ-beam with thresholds of about 5–11 MeV, which depends on the target parameters is required (Cowan et al., Reference Cowan, Hunt, Phillips, Wilks, Perry, Brown, Foutain, Hatchett, Johnson, Key, Parnell, Pennington, Snavely and Takahashi2000). When such a intense beam coincides with the resonance (γ, n) reaction, it can throw out a neutron from the nucleus, which will be explained in more detail later in this work. The bremsstrahlung γ-rays may also be generated by electrons from conventional reactors and accelerators where it has a wide spectrum and suffers from low conversion efficiency, which is due to poor coupling of energy by wide range spectrum of accelerated electrons. The production of suitable electron sources for producing the secondary photon beams have been an interesting subject where new ways are always searched (Mccall, Reference McCall1982; Li et al., Reference Li, Liu, Xu, Chen, Chang, Wan and Wen2009; Limin et al., 2009). Since the direction of the scattered photons is given by the incident electron beam, thus, the wide spreading of the generated γ rays was not resolved properly. Other short wavelength sources such as free electron laser that can produce narrow-band pulses (ΔE/E = 1%) with a peak power at the GW level with wavelengths down to 32 nm, but in addition, the beam size with radius of w 0 ≈ 250 µm is diffraction-limited, and suffers from overall efficiency from electron to photon of less than 0.01% (Ayvazyan et al., Reference Ayvazyan, Baboi, Bähr, Balandin, Beutner, Brandt, Bohnet, Bolzmann, Brinkmann, Brovko, Carneiro, Casalbuoni, Castellano, Castro, Catani, Chiadroni, Choroba, Cianchi, Delsim-Hashemi, Di Pirro, Dohlus, Düsterer, Edwards, Faatz, Fateev, Feldhaus, Flöttmann, Frisch, Fröhlich, Garvey, Gensch, Golubeva, Grabosch, Grigoryan, Grimm, Hahn, Han, Hartrott, Honkavaara, Hüning, Ischebeck, Jaeschke, Jablonka, Kammering, Katalev, Keitel, Khodyachykh, Kim, Kocharyan, Körfer, Kollewe, Kostin, Krämer, Krassilnikov, Kube, Lilje, Limberg, Lipka, Löhl, Luong, Magne, Menzel, Michelato, Miltchev, Minty, Möller, Monaco, Müller, Nagl, Napoly, Nicolosi, Nölle, Nũnez, Oppelt, Pagani, Paparella, Petersen, Petrosyan, Pflüger, Piot, Plönjes, Poletto, Proch, Pugachov, Rehlich, Richter, Riemann, Ross, Rossbach, Sachwitz, Saldin, Sandner, Schlarb, Schmidt, Schmitz, Schmüser, Schneider, Schneidmiller, Schreiber, Schreiber, Shabunov, Sertore, Setzer, Simrock, Sombrowski, Staykov, Steffen, Stephan, Stulle, Sytchev, Thom, Tiedtke, Tischer, Treusch, Trines, Tsakov, Vardanyan, Wanzenberg, Weiland, Weise, Wendt, Will, Winter, Wittenburg, Yurkov, Zagorodnov, Zambolin and Zapfe2006).
The first measurement of bremsstrahlung photons with energies more than 2 MeV was reported by Sherman et al. (Reference Sherman, Burnett, Enright and Turner1987) more than 20 years ago, by relativistic electrons from the interaction of pulses of 600 ps produced by CO2 laser with solid target. Niu et al. (Reference Niu, Mulser and Drska1991) reported the beam generations of three kinds of charged particles: electrons, light ions, and heavy ions. The ongoing development of ultra-intense laser techniques and by the advent of new laser systems in the production of extremely ultra-intense laser pulse through chirped pulse amplification and optical parametric chirp pulsed amplification techniques (Strickland & Morou, Reference Strickland and Morou1985; Perry, Reference Perry, Pennington, Stuart, Tietbohl, Britten, Brown, Herman, Golick, Kartz, Miller, Powell, Vergino and Yanovsky1999), with the capability of delivering light pulses of high intensities (more than 1020 Wcm−2) much attention has been devoted to the generation of high energy electrons from irradiating solid targets (Cowan et al., Reference Cowan, Perry, Key, Ditmire, Hatchett, Henry, Moody, Moran, Pennington, Phillips, Sangster, Sefcik, Singh, Snavely, Stoyer, Wilks, Young, Takahash, Dong, Fountain, Parnell, Johnson, Hunt and Kühl1999; Ledingham et al., Reference Ledingham, McKenna, McCanny, Shimizu, Yang, Robson, Zweit, Gillies, Bailey, Chimson, Clarke, Neely, Norreys, Collier, Singhal, Wei, Mangles, Nilson, Krushelnick and Zepf2004). Recently, for generating a narrow spread of beam spectrum γ-rays from other techniques such as Compton scattering of laser photons with high current accelerators, is proposed for photonuclear reactions to increase the reaction efficiency (Imasaki et al., Reference Imasaki, Li, Miyamoto, Amano, Motizuki and Asano2008).
Ultra-intense femtosecond lasers have stimulated an increasing interest in the problem of laser and matter interaction. The laser-matter interaction meets the high energy physics in laser-plasma accelerators in generating highly collimated bright X/γ-ray sources (Giulietti et al., Reference Giulietti, Galimberti, Giulietti, Gizzi, Labate and Tomassini2005; Chyla, Reference Chyla2006; Bessonov et al., Reference Bessonov, Gorbunkov, Ishkhanov, Kostryukov, Maslova, Shvedunov, Tunkin and Vinogradov2008) and the production of thick ion blocks (Glowacz, Reference Glowacz, Hora, Badziak, Jablonski, Cang and Osman2006; Yazdani et al., Reference Yazdani, Cang, Sadighi-Bonabi, Hora and Osman2009; Hora, Reference Hora2009; Azizi et al., Reference Azizi, Hora, Miley, Malekynia, Ghoranneviss and He2009; Hora et al., Reference Hora, Miley, Azizi, Malekynia, Ghoranneviss and He2009). These ultra intense lasers are also used to generate a quasi-Maxwellian and quasi-mono-energetic electron beams, and the recently introduced ellipsoidal bubble regime has demonstrated the generation of high-quality electron bunches with very high energies in relatively small energy spread (Malka & Fritzler, Reference Malka and Fritzler2004; Glinec et al., Reference Glinec, Faure, Pukhov, Kiselev, Gordienko, Mercier and Malka2005; Zobdeh et al., Reference Zobdeh, Sadighi-Bonabi and Afarideh2008; Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Navid and Zobdeh2009b, Reference Sadighi-Bonabi, Rahmatallahpor, Navid, Lotfi, Zobdeh, Reiazi, Nik and Mohamadian2009c) and electron emittance of more than the conventional accelerator (Lifschitz et al., Reference Lifschitz, Faure, Glinec, Malka and Mora2006). The propagation of such intense laser field is investigated in various plasma conditions (Sadighi-Bonabi et al., Reference Sadighi-Bonabi, Habibi and Yazdani2009d, Reference Sadighi-Bonabi, Yazdani, Habibi and Lotfi2009e). These lasers with narrow width spectrum of less than 1% may solve the problem of having low conversion efficiency from the electron beam to γ ray.
Up to now, photo transmutation of 129I has been carried out successfully in experiment (Magill et al., Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003; Ledingham et al., Reference Ledingham, Magill, Mckenna, Yang, Galy, Schenkel, Rebizant, Mccanny, Shimizun, Robson, Singhal, Wei, Mangles, Nilson, Krushelnick, Clarke and Norreys2003). Also, transmutation of 99Tc has been conducted (Galy et al., Reference Galy, Magill, Schenkel, Mckenna, Ledingham, Spencer, Mccanny, Singhal, Beg, Krushelnick, Wei, Norreys, Lancaster, Clarke and Clark2002) and no evidence for reaction was detected. This could be due to its very low (γ, n) reaction cross-section that could be below the detection limit. The purpose of the present work is to analytically investigate the possibility of ultra intense short pulse laser transmutations of 90Sr (γ, n) 89Sr.
STRONTIUM
Among the six major fission by-products of 235U, 90Sr, and 137Cs, they all have half-lives of around 30 years, and contribute significantly to the short-term radioactivity and heat load, constituting a packaging problem. Thus, removal of these isotopes from the used fuel would relax the requirement for heat dissipation in a long-term burial depository. Neutron absorption cross-section for 137Cs is 250 mbarn (Harada et al., Reference Harada, Watanabe, Sekine, Hatsukawa, Kobayashi and Kotah1990), and even less for 90Sr which is about 15.3 mbarn (Harada et al., Reference Harada, Sekine, Hatsukawa, Shigeta, Kobayashi, Ohtsuki and Katoh1994), and later very minuscule amount of 10.1 mbarn is reported (Nakamura et al., Reference Nakamura, Furutaka, Wada, Fujii, Yamana, Katoh and Harada2001). Thus, no neutron source is able to deliver a sufficiently large neutron flux to proceed with a faster transmutation than radioactive decay (Wydler et al., 2001).
Strontium has 16 major radioactive isotopes in which 90Sr has a half-life sufficiently long (28.8 years) to warrant any concern for nuclear waste management. 90Sr naturally decays to 90Y by emitting an energetic β, and 90Y decays by emitting more energetic particles (0.94 MeV ) with half-life of 64 h to 90Zr. The main external health concern for 90Sr is related to these energetic β particles from 90Y (Giammarile et al., Reference Giammarile, Mognetti and Resche2001). The internal concern also comes from the ingesting or inhaling of 90Sr by food, water, or air that has been contaminated by nuclear fallouts and accidents. Because of 90Sr similarity to calcium, it is mistakenly deposited in bones, teeth, and soft tissues of the body that cause cancers and tumors. These tumors are associated with the β particles of 90Sr decay chain (Giammarile et al., Reference Giammarile, Mognetti and Resche2001).
90Sr with a half-life of 28.8 years transmutes in a (γ, n) reaction into 89Sr, where β decays to 90Y with a half-life of 50.52 days. 89Sr, a valuable radioisotope that is analogy to calcium is concentrated in areas of high osteoblastic activity, which is used in nuclear medicine for bone cancer pain palliation (that improves the quality of life), cellular dosimetry, treatment of prostate cancer, treatment of multiple myeloma, osteoblastic therapy, and as a potential agent for the treatment of bone metastases from prostate and breast cancer (Giammarile et al., Reference Giammarile, Mognetti and Resche2001).
PHOTONEUTRON REACTIONS
One of the main goals of the nuclear waste management is the transformation of long-lived nuclides into short-lived nuclides. There are two important transmutation reactions, namely neutron capture (γ, n) reaction, and photoneutron (γ, n) reaction. The photoneutron (γ, n) cross-section is in general less compared to the typical (γ, n) reaction by a factor of e 2/ћc ≈ 10−2. Therefore, for many isotopes such as 129I(γ, n)130I reaction, this is a very useful reaction to transmute 129I with half life of 1.6 × 107 years into 130I with half life of only 12 h, or to transmute long-lived (2.3 × 106 years) 135Cs into short-lived (only 19 s) 136Cs isotope (Hatsukawa et al., Reference Hatsukawa, Shinohara, Hata, Kobayashi, Motoishi, Tanase, Katoh, Nakamura and Harada1999). However, for many nuclides such as 133Cs because of the presence of highly radioactive 137Cs isotope the processing is very expensive, difficult, and dangerous (Sadighi-Bonabi et al., Reference Sadighi-Bonabi and Kokabi2006). This is because the stable 133Cs transmutes into 134Cs and then again by absorbing the second neutron it transforms into the above mentioned dangerous long-lived (2.3 × 106 years) waste of 135Cs. This is the main problem in transformation 239Pt on a well known 239Pt nuclide with a half life of 24,000 years, which transforms into 240Pt (6,500 years), 241Pt (14 years), and 241Pt (380,000 years) by absorbing one, two, and three neutrons, respectively. As a consequence for many isotopes, the transmutation through photoneutron is more feasible than the neutron capture reactions. For strontium, the measured cross-section for neutron capture through 90Sr(γ, n)91Sr reaction is only 10.1 ± 1.3 mbarn (Nakamura et al., Reference Nakamura, Furutaka, Wada, Fujii, Yamana, Katoh and Harada2001) where, the cross-section for hotoneutron reaction of 88Sr(γ, n)87Sr is 207 mbarn. This is 20 times more than the cross-section for the neutron capture. To the best of our knowledge, the cross-section to 90Sr(γ, n)89Sr was not reported. Furthermore, more neutron capture transforms the 90Sr waste into even heavier strontium isotopes, which complicates the process (Pampin & Davis, Reference Pampin and Davis2008). The conversion efficiency for the photonuclear reaction on a nucleus with A = 100 and Z = 42 as an example of medium heavy nuclei such as stratum is 30% (Tajima & Ejiri, Reference Tajima and Ejiri2003).
Furthermore, when a nuclear is bombarded by an appropriate γ beam of certain energy, tuned to the giant resonance (GR), the cross-section is considerably increased. In this condition, protons and neutrons oscillate in opposite directions with isospin of 1/2 and −1/2, respectively, and the electric dipole resonance (GR) is the isovector dipole resonance at the giant resonance energy, where the cross-section is on the same order of magnitude as the cross-sections of major reaction channels, and is larger than most (n,γ) reactions including 90Sr (Habs et al., Reference Habs, Tajima, Schreiber, Barty, Fujiwara and Thirolf2009).
Generating neutrons in photoneutron reactions can be very useful in various applications. In addition to the well known applications of neutrons including nuclear energy, corrosion detection and space require detection of light materials such as explosives or pyrotechnic devices, in recent years, new applications of neutrons due to innovative techniques have emerged. Furthermore, in photonuclear reactions, the neutrons detection is a powerful diagnostic tool in measuring of the accurate amount of the products. Photonuclear (γ, n) reaction is used to produce cold polarized neutrons with a focusing ellipsoidal device where brilliant cold polarized micro-neutrons become available (Habs et al., Reference Habs, Tajima, Schreiber, Barty, Fujiwara and Thirolf2009). Photoneutron (γ, n) threshold energy is characteristic of the neutron binding energy in the target nuclide. For 63Cu, Ta, and 197Au, the threshold energy is almost 8, 9 and 10 MeV, respectively. This is higher than the normally required threshold energy for photofission reactions. For example, for 238U (γ, f) reaction, it is only 5 MeV due to lower energy requirement of deforming into its fission nuclides (Cowan et al, Reference Cowan, Hunt, Phillips, Wilks, Perry, Brown, Foutain, Hatchett, Johnson, Key, Parnell, Pennington, Snavely and Takahashi2000).
Norreys et al. (Reference Norreys, Santala, Clark, Zepf, Watts, Beg, Krushelnick, Tatarakis, Dangor, Fang, Graham, McCanny, Singhal, Ledingham, Cresswell, Sanderson, Magill, Machacek, Wark, Allott, Kennedy and Neely1999) reported one of the first successful experimental photoneutron (γ, n) reaction by a highly directional γ-ray beam from ultra short (700 f s), ultra intense (Iλ2 = 1019 Wcm−2µm2) laser pulse on a copper target. The energy loss to bremsstrahlung scales as Z 2, so high Z material such as lead (Z = 82) was chosen as first target material to maximize the required γ-ray energy for the photonuclear reaction. Pieces of copper were placed around the lead target. The reported hotoneutron reactions 63Cu (γ, n)62Cu and 65Cu (γ, n)64Cu had threshold of about 10 MeV, and the largest reaction cross-sections being 60–70 mbarm at 15–18 MeV. In this work, a slab of tantalum is used as first target for maximum γ production and the detail is given in the next section.
THEORETICAL ANALYSIS
The procedure to estimate the number of reactions of 90Sr (γ, n) 89Sr per laser shot is nearly similar to Shkolnikov et al. (Reference Shkolnikov, Kaplan, Pukhov and Meyer-Ter-Vehn1997) approach and according to Magill et al. (Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003) technique as follows: first, the Bremsstrahlung photon spectrum in terms of a constant for the specified target type and the effective temperature of photons, is determined using experimental data available for the activation of the first target. Then, using photon spectrum and the distribution function of reaction cross-section by integrating a multiplication of them over threshold energy and upper limit energy of the reaction, the number of reactions is also calculated. Shkolnikov et al. (Reference Shkolnikov, Kaplan, Pukhov and Meyer-Ter-Vehn1997) theoretically proposed the number of electrons dN e/dE in 1 MeV at a given energy is fit fairly well to the following expression:
Norreys et al. (Reference Norreys, Santala, Clark, Zepf, Watts, Beg, Krushelnick, Tatarakis, Dangor, Fang, Graham, McCanny, Singhal, Ledingham, Cresswell, Sanderson, Magill, Machacek, Wark, Allott, Kennedy and Neely1999) tried to find an exp (−E e/k BT) fit to their experimentally obtained data above 3 MeV photon energy spectrum, where E e, k B, and T are the electron energy, the temperature and the Boltzmann's constant, respectively. One of their proposed formulas for fully relativistic electron distribution function was a Boltzmann-like distribution E e2 exp (−E e/k BT). This form of electron distribution function experimentally proved for energy region up to 10 MeV (Ledingham et al, 2000; Behrens et al., Reference Behrens, Schwoerer, Dusterer, Ambrosi, Pretzler, Karsch and Sauerbrey2003) and used in some later works (Takashima et al., Reference Takashima, Hasegama, Nemoto and Kato2005; Sadighi-Bonabi et al., Reference Sadighi-Bonabi and Kokabi2006; Reference Sadighi-Bonabi, Irani, Safaie, Imani, Silatani and Zare2009a). A similar form also was also used initially by Magill et al. (Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003) for both electron and photon distribution as following:
Where dN γ/dE γ, N 0, and k BT γ are the number of electrons per MeV, a constant and the characteristic temperature of γ photons, respectively. In the present work, we have used Eq. (2) and compared the calculated results with those obtained by the same parameters from E e2exp(−E e/k BT) similar to frequently used procedure (Ledingham et al., 2000; Takashima et al., Reference Takashima, Hasegama, Nemoto and Kato2005; Sadighi-Bonabi et al., Reference Sadighi-Bonabi and Kokabi2006; Reference Sadighi-Bonabi, Irani, Safaie, Imani, Silatani and Zare2009a). The cross section of (γ, n) reactions assumed to be Lorentzian (Norreys et al., Reference Norreys, Santala, Clark, Zepf, Watts, Beg, Krushelnick, Tatarakis, Dangor, Fang, Graham, McCanny, Singhal, Ledingham, Cresswell, Sanderson, Magill, Machacek, Wark, Allott, Kennedy and Neely1999) as,
![\sigma _{reac} = \sigma _{\max } \left[ 4\left({E_{\max } - E_\gamma \over \Gamma} \right)^2 + 1\right] ^{-1}.](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021110729665-0440:S0263034610000145_eqn3.gif?pub-status=live)
σmax is the peak cross-section at E max and Γ is the full width half maximum. Furthermore, using σmax (γ, n) cross-section, the number of reaction can be evaluated similar to Magill et al. (Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003) by the following:
where n tar is the density of target in (cm−3), d tar is its thickness of the target, E int is the upper limit energy (MeV) that is specific for any reaction and E thr is the threshold energy (MeV) to initiate the reaction. This can be calculated for any reaction and it is also available in the literature.
CALCULATIONS AND RESULTS
A primary target of tantalum with a thickness of 2-mm and the density of n tar = 5.54 × 1022 (cm−3) is irradiated by a p-polarized laser light of 1020 Wcm−2 intensity with a repetition rate of 10 Hz and the central wavelength of about 1 µm. Generated Bremsstrahlung energetic γ-rays penetrate into the secondary 1-cm thick 90Sr target with the density of n Sr = 1.81 × 1022(cm−3) to stimulate photo transmutation (γ, n). The schematic of supposed targets scenario is shown in Figure 1. As indicated in the figure, the direction of the incident ultra intense laser beam is assumed to be perpendicular to a plane parallel to the 2 mm thick tantalum target, and the emerged γ photon transmute the strontium target placed behind and parallel to the primary tantalum target. Tantalum has the advantage of high density (16.6 gcm−3), high melting point (3017°C) and high resistant against cooling. The thickness of primary target is selected according to earlier experimental works and theoretically optimized thicknesses. In the initial experiments for highly directional γ photon for photonuclear reactions, the thickness of tantalum slab was 1.75 mm (Ledingham et al., 2000). Later Magill et al. (Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003) increased the tantalum thickness to 2 mm in 129I (γ, n)128I reaction and in the later works 2 mm thickness was used (Takashima, Reference Takashima, Hasegama, Nemoto and Kato2005; Sadighi-Bonabi, Reference Sadighi-Bonabi and Kokabi2006). Based on the Berger et al. (1970) definition of Maximum bremsstrahlung efficiency, recently a simulation on the optimization of tantalum thickness for maximum bremsstrahlung photon yield is reported to be 1.88 mm for 10 MeV electron beam with an accuracy of ±4%–9% (Eshwarappa et al., 2005). Inasmuch as the mentioned inaccuracy and the fact that the maximum photon number in their report peaked more like about 2 mm (see Fig. 2 of Eshwarappa et al., 2005). Therefore, in the present work, we selected this amount as an optimum thickness for tantalum as a primary target. Furthermore, according to Berger's et al. (1970) definition, the efficient thickness of a target was selected to be half of the range of electron and also the increase in the bremsstrahlung yield with increasing the electron energy is linear (Eshwarappa et al., 2005). Generated Bremsstrahlung energetic γ-rays penetrate into the secondary 1-cm thick 90Sr target with the density of n sr = 1.81 × 1022 (cm−3) stimulate photonuclear 90Sr (γ, n) 89Sr reaction.
Schematic scenario of laser pulse and targets is presented. Ultra intense laser pulse interacts with primary tantalum target to produce the relativistic electrons on its surface. These electrons generate Bremsstrahlung γ-ray in the first target's bulk where this ray transforms90Sr into 89Sr through (γ, n) reaction in the second target.
The Lorentzian reaction cross-section of 90Sr(γ,n)89Sr with the experimental data available in Table 1. Also, the Bremsstrahlung spectrum of photons with an effective photon temperature of 1.7 MeV. E thr is the threshold energy to initiate the reaction and E int is the integral upper limit energy.
To obtain N reac numerically for any reaction, the value of some of the parameters should be determined first. The photon temperature is the same as electron temperature in the relativistic case (McCall et al., Reference McCall1982) and slightly lower in the intermediate case of several MeV (Behrens et al., Reference Behrens, Schwoerer, Dusterer, Ambrosi, Pretzler, Karsch and Sauerbrey2003). The temperature of the electrons has been estimated by Wilks et al. (Reference Wilks, Kruer, Tabak and Langdon1992) for p-polarized laser light irradiated normally on the target as,
![k_B T_\gamma \lpar MeV\rpar =0.511 \left[{\left({1+ I \lambda^2 \over 1.37 \times 10^{18}} \right)^{1 \over 2} - 1} \right]\comma \;](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021110729665-0440:S0263034610000145_eqn5.gif?pub-status=live)
I is in Wcm−2 and λ2 is in µm2. At about 1020W/cm2, the hot electrons temperature is derived from the ponderomotive force calculated to be k BT γ ≈ 3 MeV (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992), but in practice, for a 2-mm tantalum target, it is a little lower and is about 1.7 MeV/kB, and photon temperature of T γ = 1.2 MeV/k B (Magill et al., Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003). The Bremsstrahlung γ-beam generated in the tantalum target can induce a (γ, n) reaction on 181Ta itself to produce 180Ta, which decays with a probability of 86% by electron capture in 180Hf and with 14% by beta decay to 180W. In contrast to strontium the photonuclear cross-section of 181Ta is experimentally measured (Handbook on photonuclear data) and explained in Table 1. Using an equation similar to Eq. (4) for tantalum:
By using the photon temperature of T γ and E th = 7.58 MeV, σmax = 367 mbarn, and E max = 12.7 MeV from Table 1, and also the experimentally measured total number of (γ, n) reactions of 181Ta per 1020 Wcm−2 Laser shot of N reac = 160 (Magill et al., Reference Magill, Schwoeror, Ewald, Galy, Schenkel and Sauerbrey2003) the photon yield of N 0 = 6.87 × 107 per MeV is obtained.
(γ,n) reaction experimental data for 181Ta and 88Sr
To the best of our knowledge, no experimental data exist for (γ, n) reaction cross-section of 90Sr, but according to the compatibility of experimental data available for 88Sr with the result of nuclear model codes for 90Sr, the experimental data of 88Sr were used (Table 1). Calculations similar to the above mentioned procedure is done for strontium with the obtained N 0 for the primary Ta target. Figure 2 shows the reaction cross-section of 90Sr(γ, n)89Sr and the bremsstrahlung spectrum of photons. From Figure 2 by having all the parameters obtained for the above mentioned reaction, evaluated value of the number of reactions is N reac = 117 reactions per shot. This is calculated from the overlap of bremsstrahlung spectrum and the cross-section for the photonuclear reactions as indicated in Figure 2.
DISCUSSIONS
According to the 50.52 days half-life of 89Sr and N reac = 117, its activity per shot would be 1.86 × 10−5 Bq. If target is irradiated with a 10 Hz, 1020 W/cm2 laser for an hour, its activity would be 0.67 Bq. Figure 3 shows the activity of 89Sr as a function of irradiation time for several repetition rates with 1020 W/cm2. When the repetition rate increases by a factor of 100 to 1 kHz, then its activity amount would be close to 67 Bq. So increasing the repetition rate of the laser has a direct effect on the yield of the reaction that requires technical improvement of high-power lasers to attain higher rates.
Nuclear activity as a function of irradiation time for a secondary 90Sr target with a first target of tantalum for a 1020 W/cm2 laser in the repetition rates of 1, 10, 100 and 1 kHz.
In addition, increasing the effective temperature of hot electrons has a drastic influence on the total reaction activity. This can be done by increasing the intensity of the laser. The dependence of the electron temperature and the total number of electrons on the laser intensity experimentally showed to be T e ≈ I 1/2 and N e ≈ I 3/2, respectively (Gahn et al., Reference Gahn, Tsakiris, Pretzler, Witte, Thirolf, Habs, Delfin and Wahlstrom2002). For a tantalum target assumed here, increasing the intensity from 1020 W/cm2 by a factor of 10 would increase the electrons temperature to the limit that enhances the reaction yield by a factor greater than 200 to 145 Bq. By using the newly introduced laser system of 100 Hz, based on OPCPA technology (Ledingham et al., Reference Ledingham, McKenna, McCanny, Shimizu, Yang, Robson, Zweit, Gillies, Bailey, Chimson, Clarke, Neely, Norreys, Collier, Singhal, Wei, Mangles, Nilson, Krushelnick and Zepf2004), the activity can be extended to 1.45 kBq at laser intensity of 1021 Wcm−2. By increasing the electron temperature, the overlap of the cross-section with the bremsstrahlung spectrum dramatically increases where beyond an optimum intensity the overlap does not increase and it becomes constant. This optimum intensity belongs to a temperature of 16.85 MeV. The activity of reactions, also substantially depends on the produced isotope's half-life. For isotopes with several minutes half-life, activities over mega Becquerel would be attainable, just with 1020 W/cm2 intensity and 10 Hz repetition rates. This will stimulate researchers to investigate possibilities of production of short-lived isotopes through photo transmutations.
Another procedure to calculate the number of 89Sr reaction has been introduced, which is similar to 137Cs(γ, n)136Cs (Sadighi-Bonabi et al., Reference Sadighi-Bonabi and Kokabi2006). There, the bremssrahlung spectrum is obtained from relativistic electrons spectrum and the integrated-over-angle bremsstrahlung cross-section. The calculated value of the activity by that method for a laser system, similar to the one in this article, is 0.85 Bq for an hour, which is in good agreement with 0.67 Bq of the here mentioned method.
CONCLUSION
We have discussed the effect of laser intensity in laser transmutation of strontium and relativistic electrons effective temperature has been investigated. Figure 4 shows the activity of 89Sr as a function of laser intensity. As shown in Figure 4, the growth of activity by increasing the laser intensity is not linear but has a dramatic increase especially between 1020 and 1021 W/cm2. As the repetition rate is directly multiplied to the activity of the reaction by developing high repetition rate ultra intense lasers over kHz in future, more acceptable activities in mega and giga-becquerel could be achievable for lots of radioisotopes through photo transmutation productions. Radiation sources driven by ultra intense laser sources have benefits of being considerably compact and they are an interesting diagnostic tool of fast phenomena. Inasmuch as these systems generally suffer from poor wall-plug efficiency they are not suitable for transmuting bulk quantities with the present pulse repetition rate at high intensities, however, they are very suitable candidates for small quantities such as nuclear medicine radioisotopes.
The activity of 90Sr(γ,n)89Sr reaction in the laser intensities of 1020, 5 × 1020, 1021and 5 × 1021 for a 1-cm 90Sr target bombarded by the γ–rays generated in the interaction of laser pulse with a 2-mm tantalum target.
ACKNOWLEDGMENTS
The authors would like to thank the Pars Oil and Gas Company of the Ministry of Oil for partial support of this work under project No. 121PT, Also research deputy of Sharif University of Technology. We also appreciate useful comments of Professor H. Hora and Dr. E. Lotfi.
