Arbitrarily large neutron amplification in subcritical nuclear reactors
AArbitrarily large neutron amplification in subcritical nuclear reactors
Antoine Tilloy
Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
In a subcritical reactor, each neutron produces only k eff < k eff , the external source intensity, andthe output power of the reactor. In this paper, I present various possible strategies to exploit this fact,and apply them to the design of a rudimentary multi-layer system that allows to reach an arbitrarilylarge number of fissions per source neutron, while keeping k eff < k eff = 0 . e.g. radioisotope based. Introduction –
Subcritical nuclear reactors are a se-rious candidate for the incineration of minor actinidesproduced by current light water reactors. With an asymp-totic neutron multiplication k eff <
1, they allow an in-trinsically stable operation without the need for delayedneutrons, and can thus operate which much higher minoractinide content than more standard critical reactors. Butneutrons have to come from somewhere, and one[1] of thedrawbacks of such designs lies in the need for an efficientexternal neutron source. Current proposals use a spalla-tion source driven by a high current proton accelerator.To reach an industrially meaningful actinide incinerationor electricity production, the accelerator power requiredgoes well beyond what is available off-shelf.The high power need comes from the requirement ofa k eff not too close to 1 to ensure safe operation, thechoice k eff ’ .
95 usually being made in the literature[2]. At an intuitive level, this suggests that one neutroncan produce on average 0 .
95 neutrons per generation andthus a number of fissions of the order of 20 from thefirst fission induced by the spallation source. While thisrule of thumb may give a correct order of magnitude forstandard accelerator driven subcritical reactor (ADSR)designs and explains the need for a high intensity neutronsource, but it can give the wrong order of magnitude inmore complicated situations.We need to keep track of neutron generations and con-sider the fact that the multiplication per neutron k i canbe generation dependent because of the geometry of thesystem. The number of neutrons produced from a singlesource neutron is on average: N = k + k k + k k k + · · · = + ∞ X i =1 i Y j =1 k j def. = 11 − k s . (1)This N , or equivalently k s , is what we care about whenwe consider the total neutron multiplication of the system,and the output power for a given source. The number k s is the equivalent multiplication we would need to getthe same N had the multiplication k i been generationindependent. Importantly, it has nothing to do with k eff , which is defined as: k eff def. = lim i → + ∞ k i , (2)which is the asymptotic multiplication. Only k eff governsthe long time reactor kinematics and is thus relevant toassess criticality. We see that, in principle, there is notheoretical obstacle to a N arbitrarily large ( k s arbitrarilyclose to 1) with a fixed subcritical k eff provided the first k i ’s are large.In fact, some ADSR designs obtain modest amplifi-cation from the naive k eff = k s relation with a highlyenriched core near the spallation source surrounded bylow enriched blankets. However, such designs typicallyreduce the accelerator requirements only by a rather mod-erate amount.This brings a natural question: is it possible to designa core such that k eff is fixed but N is large enough tosubstantially reduce the accelerator requirements? Canmultiplication be increased to the point that a muchsmaller accelerator can be used or even that much weakerradioisotope sources can be used? Is there a physical limitto amplification due to neutron transport in existing ma-terials? Is there a practical limit with current technologyand engineering constraints? Strategies –
To increase N while keeping k eff < k eff .However, the neutron amplification would be enhancedgeometrically with the number of layers, without obviousphysical limit.Of course, perfect neutron diodes do not grow on trees,and the challenge lies in creating a genuine unidirectional-ity of the neutron current so that the criticality k eff doesnot increase (or only in a subleading way) when stackingmore subcritical layers.There are at least four conceptually different ways tocreate a one way neutron current between different parts a r X i v : . [ phy s i c s . i n s - d e t ] O c t FIG. 1.
Strategies for multi-stage amplification – Sub-critical cells separated by ideal neutron diodes act as neutronamplifiers. In practice, this could be implemented with 4distinct physical strategies. of a reactor, summarized in Fig. 1. In practice, thesedifferent methods can be combined, and I separate themonly to emphasize the physical mechanism at play.The first idea corresponds to the one that, to my knowl-edge, was considered initially, as early as 1957 [3]. Theidea is to exploit the fact that certain materials like cad-mium are almost transparent to fast neutron but have alarge capture cross section for thermal neutrons. Puttingsuch a thermal absorber followed by a moderator, oneobtains a material that can be crossed only one way byneutrons (and converts fast neutrons to slow neutrons inthe forward direction).The second option is in a way opposite to the first.Instead of capturing slow neutrons, one could selectivelycapture fast neutrons with a material like low enricheduranium that works as a neutron multiplier only in ther-mal spectrum while captures by
U dominate in fastspectrum. The idea is to put a moderator first and thenthis fast spectrum capture material to obtain a diode.This one way amplification through fast captures is likelyinsufficient to create a genuine irreversibility alone, butseems to play a role in the multiplication reached byvarious proposals.The third option is to use a threshold fissile materiallike
Np or
Am, which is fissile in fast spectrum onlyand that consequently multiplies only fast neutrons. Asbefore, this allows to construct a one way amplifier withthe addition of a moderator.The fourth option is purely geometric, and is in a waythe simplest (see [2]). The idea is to decouple stages byhaving them grow in size (or in surface) as one goes away from the source. This way, neutrons from an earlier stagecan reach the following one where they start fissions, butthese fissions create neutrons that have lower chances tohit the initial stage. An example consists in consideringconcentric fissile spheres of quickly growing radii andshrinking width. If the ratio of the radii is large enough,the total k eff of the set of spheres is the same as that ofeach of them taken individually. However, the number offissions is amplified at each layer.Let us finally mention a more recent idea, which is toexploit magnetic fields to deflect neutrons via the couplingto their spin [4]. As it is far more speculative than theprevious ones, I will not consider this option further here. Pre-amplifier exploration –
My objective is to see howa combination of the previous strategies could fare withphysically realistic materials and geometry (includingneutron loss). To this end, it seems easier to distinguishtwo levels of amplification, that would be joined togetherin a complete reactor, but that come with starkly differentconstraints. A first stage, the preamplifier, would bringan arbitrarily faint neutron flux to the intensity needed todrive a second “power” stage. This second stage, a morestandard reactor core, would be made of only a few simplelayers, with low amplification but high power density.The proper study of a modestly amplifying reactorcore is difficult but largely done in the literature [5, 6].The difficulty comes from the need to take into accountthermohydraulic constraints (which limit the local powerdensity), the evolution of reactivity, neutron economy,and simplicity constraints. Typical two layer cores exploita threshold fissile material and concentric cylinders toachieve multiplication. For realistic designs, a modestamplification can be achieved, but is not sufficient todrastically change the neutron driving requirements.To my knowledge, a many-layer preamplifier has neverbeen studied. The constraints are different from thoseof the core. To achieve massive amplification, the designshould be scalable, which a priori excludes playing witha spherical or cylindrical concentric geometry, and favorsa linear stacking of layers. On the other hand, the powerdensity would be orders of magnitude lower than theone in the core, especially in the first layers closer to the(faint) source. This in turns means that cooling, reactivityevolution, and neutron economy would be secondary con-straints. They should be taken into account ultimately,but in the spirit of a first exploration, it makes sense toneglect them to focus on neutron transport. Finally, sucha preamplifier might be useful as a standalone device, e.g. to produce intense neutron pulses (at low averagepower), or breed radioisotopes and tritium, which furthermotivates its study.Finding an optimal or even adequate design of preampli-fier given reasonable engineering and safety constraints isa non trivial task. My objective is more modest: I merelywant demonstrate that amplification is physically feasible,at least in principle. To this end, I threw everything but
FIG. 2.
Preamplifier design – The proposal is shown on a10 layer example. The preamplifier is a cylinder with 130 ×
130 cm square basis surrounded by a lead reflector. Themoderator is heavy water, the thermal absorber cadmium, thethreshold material is metallic Am, and the fissile cell MOX(43%
Pu and 57%
U for k eff = 0 . the kitchen sink (geometry), to keep a scalable designwhere an arbitrary number of layers can be stacked. Moreprecisely, I considered a stack of layers each made of 4cells containing 1) a fissile material with fission enhancedin thermal spectrum [a mixed oxide of Pu and
U], 2)a threshold fissile material [metallic
Am], 3) a thermalneutron absorber [cadmium], and 4) a moderator [heavywater]. This combines the 3 non geometric amplificationmethods mentioned before. This choice is largely arbi-trary, and it is likely that much better combinations canbe found. The precise geometry, including the reflector,is provided in Fig. 2 for a stack of 10 layers.
Results –
I carried a simple test of the previous pro-posal with the help of the Monte Carlo transport simula-tion tool openMC [7] developed at MIT.After some trial and error but no systematic optimiza-tion, it is easy to reach a scalable amplification of ’ . n + 1 than in layer n ] for k eff = 0 .
97 while keeping layerswith a reasonable cross section 130 ×
130 cm and width ’
38 cm (see Fig. 3). Let us briefly discuss the trade offsinvolved.Most of the space is taken by the heavy water mod-erator, with a thickness of 35 cm per layer. The thickerthis cell, the better the neutron flux is thermalized whichenhances the decoupling between layers. On the otherhand, a thicker cell increases the leak of thermal neutronsinto the lead reflector where they can recouple to previous F i ss i o n s p e r s o u r ce n e u t r o n Pu AmNeutron flux in the amplifier (arb. unit) 05101520
FIG. 3.
Neutron amplification for a 10-layer system –After the first layer, the number of fissions per layer is approx-imately multiplied by × . layers. Taking a larger layer cross section allows to pro-portionally reduce the surface leaks in the reflector andthus permits an increase in the moderator thickness. Thisyields an improved amplification at fixed k eff , at the priceof an overall larger amplifier. Using light water wouldreduce the thickness required to thermalize the neutronflux, but I observed a lower overall amplification becauseof the higher capture rate.The mixed oxide section is chosen rather narrow(1 . Pu). This onlyweakly exploits the second strategy of fast capture by
U. This is because the backward decoupling effectbrought by
U is out-weight by the increased couplingbetween layers allowed by a thinner fissile cell. It is stillpossible to reach some amplification (albeit weaker) byopting for a thicker section with lower
Pu content ( e.g.
Pu contents, thereactivity first decreases after a partial loss of moderator,but reincreases after a full emptying, as layers recouplebackward).The americium section is taken similar in thickness tothe fissile one (1 . i . . . . . l a y e r s l a y e r s l a y e r s l a y e r s l a y e r s k i Critical region k eff = lim i → + ∞ k i FIG. 4.
Transient neutron multiplication – The numberof neutrons produced per generation k i reaches a plateau at k ’ .
04, that lasts longer for a larger number of layers. Thesystem remains subcritical no matter the number of layers,with k eff → .
97 up to measurement errors. remaining mostly transparent to epithermal and fast ones.This implements the first strategy.The way arbitrary geometric amplification is alloweddespite fixed k eff is transparent in this proposal and followsthe argument in the introduction. Stacking more layersdoes not bring the system closer to criticality (after acertain point) as backward coupling is suppressed (seeFig. 4). However the neutron multiplication k i remainslarger than 1 for a number of generations proportionalto the number of layers, and passes below 1 only onceneutrons start leaking at the end of the amplifier. For 10layers, this happens only after about 100 generations. Discussion –
With the rudimentary amplifier I putforward, it is possible to reach an amplification as largeas one needs. Since the amplification grows geometricallyin the number of layers, one can in principle have as inputan arbitrarily faint neutron source, and even dream ofditching the accelerator driven spallation source to useradioisotopes.Naturally, this comes with severe limitations that areyet to be addressed. Assuming a realistic amplifier couldbe designed along the lines presented here (includingcladding, cooling, control systems, etc.), its safety wouldstill need to be demonstrated. Indeed, the preamplifieroperates away from its maximally reactive geometry: asufficient bending of the layers or melting of the cellscould turn the system critical. Further, one would needan efficient in situ measurement of k eff : in all ADSRsbut even more so in a multi-layer system, the outputpower can decrease as k eff increases, making subcriticalitydifficult to guarantee in practice. The development of accurate reactivity measurement techniques is an activearea of research (see e.g. [9, 10]).A more minor drawback of a subcritical system witha large preamplification, compared to standard ADSRs,is that the neutron economy is unavoidably worse, asinput neutrons are negligible in the overall balance. Thismeans, at least a priori , that thorium breeding in thermalspectrum is more difficult with such designs.I have studied the preamplifier as a standalone device,and its use to drive a reactor core would necessarily bringsome residual backward neutron coupling, slightly increas-ing the k eff of the whole. The quick saturation of the k eff as the number of preamplifying layers is increased (see Fig.4) suggests that applying the same decoupling strategyand seeing the core merely as a last, much larger layer,should be sufficient to ensure global subcriticality. Thisshould however be demonstrated in full fledged neutrontransport simulations once a more complete and realisticdesign is proposed.These limitations notwithstanding, there are interestingopen questions of a more theoretical nature. In particular,it would be helpful to know what the maximum reachableamplification is for given k eff , reactor size budget, andsafety constraints. Likewise, what is the lowest k eff thatcan be reached with scalable amplification? Multi-layersystems also present challenges for accurate Monte-Carlosimulations: naively estimating k eff and the neutron fluxeverywhere in the system becomes prohibitively costly asthe number of layers is increased, because of the transientexponential gain. There is certainly a way to do better.In the end, scalable subcritical amplification presentsgreat challenges but is physically sound. It is in princi-ple feasible with materials that exist, and with a simplegeometry. If realized, such amplifiers could become auseful building block of full fledged reactors, to produceelectricity, incinerate (some) radioactive waste, or provideon demand sources for science. As a result, multi-layeramplification likely deserves to be studied further andat a finer level of detail. I hope the present paper willstimulate this exploration.I am grateful to Sylvain David and Alexis Nuttin forhelpful comments and discussions. [1] There are many other obstacles to a large scale transmu-tation strategy of radioactive waste. The improvement ofthe neutron efficiency of subcritical reactors would onlylift a minor hurdle.[2] H. Nifenecker, O. Meplan, and S. David, Acceleratordriven subcritical reactors (CRC Press, 2003).[3] L. B. Borst, Phys. Rev. , 905 (1957).[4] F. J. Arias and G. T. Parks, Journal of Fusion Energy , 142 (2016).[5] V. V. Seliverstov, Atomic Energy , 806 (1996).[6] G. Kiselev, Nuclear Engineering and Design , 157 (2002).[7] P. K. Romano, N. E. Horelik, B. R. Herman, A. G. Nelson,B. Forget, and K. Smith, Annals of Nuclear Energy , 90(2015), joint International Conference on Supercomputingin Nuclear Applications and Monte Carlo 2013, SNA +MC 2013. Pluri- and Trans-disciplinarity, Towards NewModeling and Numerical Simulation Paradigms.[8] I used isotopically pure Pu for simplicity, as it providesthe highest amplification. But there is no objection tothe use of reactor grade plutonium, which would allowamplification as well.[9] T. Chevret, J. Lecouey, N. Marie, F. Lecolley, G. Lehaut, G. Ban, A. Billebaud, S. Chabod, X. Doligez, X. Ko-chetkov, P. Baeten, W. Uyttenhove, G. Vittiglio, J. Wage-mans, F. Mellier, V. B´ecares, and D. Villamarin, in
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