The Spallation ReactionEdit
Spallation of an accelerated proton beam is the way it is commonly conceived that a neutron source will be provided to sustain the fission chain reaction of subcritical reactors. Typically it is assumed that the proton beam will be accelerated to energy 1 GeV, the reasons for this are described in Section 6.2. The mechanics of the spallation reaction for protons at energy ~ 1 GeV are described presently.
At 1 GeV a proton travels at a velocity, $ v/c = \beta = 0.8579 $. This means that when a proton passes through a stationary nucleus (the spallation target) the time window over which the proton may interact with each nucleon in the target nucleus is ~10$ - $23 s. The average kinetic energy of a nucleon inside a nucleus is ~25 MeV and it has a mass-energy of ~940 MeV / c2. This then means that during the 10$ - $23 s interaction time window the relative motion of the nucleons inside the nucleus is ~0.1 fm, which is <1 % of the circumference of any given nuclide. Because of this, during spallation it can be approximated that there is no internal motion within the target nucleus. Excluding the Coulomb force, the incoming proton will only interact with the nucleons that are within range of the strong nuclear force, which is ~1.2 fm (the radius of a nucleon). The state of the nucleons outside of the range of the strong nuclear force is not affected by the proton. For decreasing proton beam energy, and therefore velocity, the approximation of an internally static target nucleus becomes invalid and the spallation reaction becomes more complex.
Spallation is usually described in two steps. First there is an abrasive phase followed by an ablation phase as the nucleus “cools down” following the reaction proper. During the abrasion phase collisions between the incoming proton and nucleons in the nucleus result in the direct ejection of nucleons either individually or as product nuclei. The timescale over which this abrasion takes place is ~10$ - $23 s. The remaining nucleons from the initial nucleus still occupy the same orbits that they were in prior to the interaction beginning. The transmutation of the nucleus however often means that these orbits are highly excited states for the product nuclide that the nucleus has now been transmuted into. The nucleons that are in excited states decay via one of several pathways (e.g. $gamma; decay, internal conversion, electron capture, etc.). In some cases nucleons are in states that are of such high excitation energy that they decay via direct proton or neutron emission. The emission of protons and neutrons via this mechanism forms the ablation phase of the spallation reaction. The timescale for the decays varies by orders of magnitude, the phase is typically characterised as occurring over 10$ – $23-10$ – $16 s. Particles from abrasion typically have orders of magnitude more energy than those emitted through ablation. The “cocktail” of product fragments from a spallation reaction commonly cause further spallation reactions with other nuclei in the target material. These reactions are for a wide range for energies and involve a wide range of projectile nuclei, not just protons.
Spallation is at times termed “fragmentation”. There is no physical difference between the two reactions, usually when the reaction is discussed in terms of fragmentation it is the products of the projectile nucleus that are of most interest, rather than the products of the target. To indicate the end products one might expect from spallation/fragmentation, the figure to the right shows the cross sections of a handful of product nuclei from the fragmentation of a 1 GeV / nucleon 208Pb beam on a thin natural Cu target [de Jong et al. 1998]. Only elements in the range $ Z = 83 $ - $ 71 $ have been displayed in the figure, appreciable quantities of all elements with less protons and/or neutrons than each colliding nucleus are produced. The presence of bismuth shows that it is possible for nucleon exchange to take place between the colliding particles, rather than a fragmentation reaction. There is significant variation in the production cross section of different isotopes of an element.
de Jong, M., Schmidt, K. H., Blank, B., Böchstiegel, C., Brohm, T., Clerec, H. G., Czajkowski, S., Dornik, M., Geissel, H., Grewe, A., Hanelt, E., Heinz, A., Irnich, H., Junghans, A.R., Magel, A., Münzenburg, G., Nickel, F., Pfützner, M., Piechaczek, A., Schidenberger, C., Schwab, W., Steinhäuser, S., Sümmerer, K., Trinder, W., Voss, B. and Ziegler, C., 1998, “Fragmentation Cross Sections of Relativistic 208Pb Projectiles” Nuclear Physics A, 628, pp. 479 492
Neutron Yield from SpallationEdit
Any practical sized target for an ADSR will need to be thick such that each primary beam proton will induce a cascade of multiple generations of spallation reactions, the situation has similarities to that of a cosmic ray entering and interacting with the atmosphere. Using an example target material of natural lead, 60 cm long and with a variable diameter, simulations have demonstrated that approximately only a quarter/third of the total number of neutrons produced (not the number that escape the target) are created directly as a result of spallation of the primary beam proton, the remainder are produced in reactions between secondary particles and the target.
Of concern to sustaining the fission reaction of an ADSR is the number of neutrons that escape the target material, and also the energy distribution of those neutrons. The figures on the right show simulation results for the number of neutrons emitted and the energy distribution of those neutrons, this data is taken from simulations with the Monte Carlo Neutron Particle (MCNP) code [Bungau et al. 2008]. In the simulation the target was natural Pb, of length 60 cm, diameter 20 cm (a range of diameters were considered for the absolute neutron intensity) and it was subject to a 1 GeV beam of protons.
Depending on the target geometry and beam energy, per primary proton entering the target one can expect ~15-35 neutrons to be emitted out of it. The neutron yield increases for increasing target radii up to a diameter of ~100 cm and the benefits of increasing the beam energy saturates at a little over 1 GeV. Most primary and secondary neutrons that escape the target have energy < 4 MeV, however both have a long tail to their distributions that reaches many hundreds of MeV. The mean energies of primary and secondary neutrons have been found to be 21.7 and 5.4 MeV, respectively.
Bungau, C., Barlow, R., Bungau, A. and Cywinski, R., 2009, “Neutron Spallation Studies for an Accelerator Driven Subcritical Reactor”, In proceedings of the 23rd Particle Accelerator Conference 4th – 8th May 2009, Vancouver, British Columbia, Canada