Calculation Method

Reaction via Giant Dipole Resonance

Nuclear transmutation with laser Compton scattering uses photonuclear reactions via GDR because the cross section of GDR is quite large and the total cross section is a smooth function of mass number. GDR is a collective excitation of a nucleus involving almost all nucleons, which is interpreted classically as a macroscopic oscillation of a bulk of protons against that of neutrons. The total cross section GDR, the resonance energy ER, and the width ΓR are given by [4]

where N and Z are the neutron and proton numbers, A ¼ N + Z is the mass number, and κ, which is roughly equal to 0.2 for medium nuclides, is a correction coefficient for the pion exchange.

When a target nucleus is irradiated with a photon and excited to GDR, it often forms a compound nucleus with only a small contribution of a pre-equilibrium reaction [5]. The compound nucleus is an excited state in which the energy brought

Fig. 1.1 Schematic illustration of nuclear transmutation with laser Compton scattering by the incident particle is shared among all degrees of freedom of the nucleus. The reaction cross section from an initial channel α to a final channel β proceeding through a compound nucleus state of spin J can be written by the Hauser–Feshbach formula as

where kα is the wave number in the initial channel, gJ is a statistical factor, T is a transmission coefficient, and hTi is the energy average of T. The statistical factor is

where iα and Iα are the projectile and target spins.

Calculations of reaction cross sections are performed using the nuclear model code TALYS (version 1.4) [6]. The neutron transmission coefficients are calculated via the global optical potential [7]. The gamma-ray transmission coefficients are calculated through the energy-dependent gamma-ray strength function according to Brink [8] and Axel [9]. We employed the level density given by Gilbert and Cameron [10].

Figure 1.2 shows the photonuclear reaction cross sections of 137Cs calculated using the TALYS code. In the incident photon energy B(n) � Eγ < B(2n), where B

(n) and B(2n) are the oneand two-neutron binding energies, respectively, we can see that the (γ, n) reaction mainly occurs. Because the resonance energy of GDR ER is 15–18 MeV, which is roughly equal to B(2n) for medium nuclides, about half the reactions via GDR are (γ, n) reactions, which occur at B(n) � Eγ < B(2n).

High-Energy Photons Obtained by Laser Compton Scattering

Laser Compton scattering is a method to obtain high-energy photons by laser photons backscattered off energetic GeV electrons. In the case of head-on collision

Fig. 1.2 Cross sections for 137Cs (γ, γ), 137Cs (dashed line), 137Cs (γ, n), 136Cs (solid line), and 137Cs (γ, 2n). 135Cs (dash-dot line) reactions versus incident photon energy: dotted lines represent B(n) and B(2n) of 137Cs

between relativistic electrons and laser photons, the energy of scattered photons is given by

where γ ¼ Ee/me is the Lorentz factor of the electron beam with energy Ee, me is the rest mass of the electron, EL is the energy of the laser photon, and θ is the scattering angle. From Eq. (1.4), the energy of the scattered photon is maximum at θ ¼ 0, and it depends on the energy of incident electrons and photons. The minimum energy of the scattered photon can be fixed by controlling θ with collimators.

The scattering cross section of laser Compton scattering is given by the Klein– Nishina formula:

Fig. 1.3 Calculated gamma (γ)-ray spectrum (solid line) generated by laser Compton scattering. The maximum energy, 15 MeV, was chosen to be equal to the binding energy B(2n) of 137Cs. The binding energies of B(n) and B(2n) for 137Cs are indicated by dashed lines

To have a situation in which (γ, n) reactions occur, the photon beam with energy at B(n) � Eγ < B(2n) is desired. In case of the free electron laser, we may assume/ expect to get the total photon flux Nγ � 2 x 1012/s/500 mA for Ee ¼ 1.2 GeV [1]. From Eq. (1.4), with Ee ¼ 1.2 GeV and EL ¼ 0.7 eV, we obtain the maximum photon energy of Eγ ¼ 15 MeV at θ ¼ 0, which is equal to B(2n) for 137Cs. Figure 1.3 shows the calculated γ-ray spectrum generated by laser Compton scattering using Eq. (1.5), where the total photon flux with energy from 0 to B(2n) is Nγ � 2 x 1012/s.

From Fig. 1.3, we can see that about half the total scattered photons are in B(n) � Eγ < B(2n) and contribute to generate the (γ, n) reactions for 137Cs. In contrast, for the Bremsstrahlung that is usually used to generate high-energy photons, the photon intensity decreases rapidly as the photon energy increases, and only a small part of the high-energy tail is available for (γ, n) reactions [11].

Setup of the Calculation for 137Cs

When a target nucleus X is irradiated with a photon beam with energy Eγ, it forms a compound nucleus, which releases one neutron and becomes its isotope X 0. The reaction rate of X(γ, n)X 0 at time t is given by

where σX ! X 0 (Eγ) is the reaction cross section of X(γ, n)X 0, nX(t) is the number of target nucleus per unit area at time t, and a is the attenuation factor of incident photons through a thick target. dNγ/dEγ is expressed with Eq. (1.5) and σX ! X 0 (Eγ) is calculated from Eq. (1.2) using the TALYS code.

Figure 1.4 shows a calculation in which the photon beam is generated by the laser Compton scattering of 1.2 GeV electrons and 0.7 eV laser beams. We assume that the cylindrical target of 137Cs of 1 g is irradiated with a photon beam with energy B(n) � Eγ < B(2n) within a radius r � 0.8 mm at 2 m from the interaction point. When a target of 137Cs is irradiated with photons and is excited to GDR, the (γ, γ), (γ, n) and (γ, 2n) reactions mainly occur. We consider 137Cs, 136Cs, 135Cs, and 134Cs as the isotopes generated by the transmutation. The numbers of these isotopes are expressed as

One can calculate the number of each isotopes by solving these equations with the Runge–Kutta method.

Results and Discussion

Nuclear Transmutation of 137Cs with Laser Compton Scattering

Figure 1.5 shows the dependence of the reduction of 1 g 137Cs on the photon flux Nγ ¼ 1012, 1018, 1019, 1020/s, which is calculated with this setup (Fig. 1.4). The number of 137Cs is effectively reduced with photon flux over 1018/s, that is, the number of 137Cs is reduced by 10 % for 24 h irradiation. Figure 1.6 shows the number of Cs isotopes when 1 g 137Cs is irradiated with photon flux 2 x 1012/s with the same setup. From this figure, we can see that the reduction rate of 137Cs by the transmutation, which is nearly equal to the generation rate of 136Cs, is two orders of magnitude smaller than the natural decay rate of 1 g 137Cs. Thus, the transmutation of 137Cs is not effective with photon flux 2 x 1012/s, which is maximum with present accelerator systems.

Fig. 1.4 Setup for calculation of transmutation of 137Cs

Fig. 1.5 Dependence of the reduction of 1 g 137Cs on photon flux

Comparison with Other Nuclides

Table 1.1 summarizes the generation rate of A-1X from of 1 g of a fission product

AX with photon flux 2 x 1012/s also for 129I, 135Cs, and 90Sr in addition to 137Cs.

The reduction rate, which is nearly equal to the generation rate of A-1X, does not depend significantly on nuclides because the properties of GDR are smooth functions of the mass number. From this table, we can see that the reduction rate for the transmutation of 137Cs can be similar to that of other medium nuclides.

Fig. 1.6 Number of Cs isotopes when 1 g 137Cs is irradiated with photon flux

12

Table 1.1 Generation rate of A-1X from 1 g of fission product AX with photon flux 2 x 1012/s

Target (AX)

[B (n), B (2n)] (MeV)

EGDR (MeV)

σtot (b•MeV)

GDR

N (A-1X) (/s)

129I

[8.83, 15.7]

15.3

2.25

1.24 x 1010

135Cs

[8.76, 15.7]

15.2

2.31

1.31 x 1010

137Cs

[8.28, 15.1]

15.1

2.37

1.19 x 1010

90Sr

[7.81, 14.2]

16.7

1.58

4.71 x 109

Conclusion

In this work, the effectiveness of transmutation with laser Compton scattering for reducing fission products was quantitatively investigated. The transmutation of 137Cs is effective with photon flux greater than 1018/s, which results in 10 % reduction for 24 h irradiation. However, transmutation with photon flux 2 x 1012/s, which is achievable with present maximum accelerator systems, is not effective, and the reduction rate is approximately two orders of magnitude less than the natural decay rate.

Nuclear transmutation with laser Compton scattering can transmute selectively a medium mass nuclide AX into A-1X, and its reduction rate is independent of isotopes. Because the transmutation with laser Compton scattering can almost exclusively generate desired nuclides, this method will be useful for the generation of isotopes for medicine [1].

 
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