A mathematical model to describe the alpha dose rate from a UO2 surface
AA mathematical model to describe the alpha dose ratefrom a UO surface Angus Siberry a , David Hambley b , Anna Adamska c , Ross Springell a a University of Bristol, HH Wills Physics Laboratory, Bristol, BS8 1TL, UK. b National Nuclear Laboratory Ltd, Central Laboratory, Sellafield, CA20 1PG, Cumbria, UK. c Sellafield Ltd, Sir Christopher Harding House, Whitehaven, CA28 7XY, UK.
Abstract
A model to determine the dose rate of a planar alpha-emitting surface, has been developed. Theapproach presented is a computationally efficient mathematical model using stopping range data fromthe Stopping Ranges of Ions in Matter (SRIM) software. The alpha dose rates as a function of distancefrom irradiated UO spent fuel surfaces were produced for benchmarking with previous modellingattempts. This method is able to replicate a Monte Carlo (MCNPX) study of an irradiated UO fuelsurface within 0.6 % of the resulting total dose rate and displays a similar dose profile. Keywords
Alpha Radiation, Dosimetry, UO , Radiolysis,SRIM, Nuclear Materials
1. Introduction
The role of nuclear power in its potential tocombat climate change is well-established [1]. De-spite this, due to high cost and concerns oversafety, its future as a major energy resource isuncertain [2]. A drawback of nuclear power is thecomplex waste forms that it produces, and thepotential decommissioning challenges associatedwith radioactive materials [3]. To develop a com-prehensive plan of how to deal with this wastethere must be an understanding of what couldhappen when moving, treating and storing suchwaste. In order to do this safely, predictive toolsare required to highlight the potential risks andhow to mitigate them.An important component of nuclear power pro-duction is the management of spent fuel. Whether
Email address: [email protected] (AngusSiberry) the fuel is to be reprocessed or placed in a geolog-ical disposal facility, the maintenance and assess-ment of fuel integrity during storage is cruciallyimportant. Upon exposure to water, dissolutionof the fuel matrix and a release of highly radioac-tive fission products can occur [4–6]. In manystorage practices this exposure is possible. In thecase of a geological repository, it is even expecteddue to the large time scales associated with thefuel being in one location. A detailed understand-ing of these degradation mechanisms and the con-ditions that drive them could improve the effec-tiveness of any control measures, influence facilitydesign and ultimately, reduce the cost. Develop-ing modelling tools to predict the rate of radioac-tive dose and dose profile through the fuel-waterinterface is a critical part of a wider effort to pro-vide accurate predictions of fuel dissolution ratesin the event of a containment breach.Spent nuclear fuel consists of predominantly,UO ( ≈
95 %); the remaining material is com-prised of fission products and other actinides [7].The reactivity of UO in water is so low it isalmost considered inert [8], however, if oxidisedthe uranium valence state converts from U(IV) tothe much more soluble U(VI); hence, in the pres- January 21, 2021 a r X i v : . [ nu c l - t h ] J a n nce of oxidising species, UO will corrode morerapidly leading to a faster release of the radioac-tive isotopes held within the fuel matrix [6, 9, 10].G-value is the yield of a particular species re-sulting from ionisation [7, 11]. It can be used forrelating instantaneous yields, normally of radicalspecies, or equilibrium yield of molecular species.The G-value used commonly in disposal environ-ments is the yield of molecular species, used toconvert the energy lost by ionising radiation inwater, to the number of molecules of a givenspecies produced. This is the method wherebydosimetry results can be used to determine disso-lution kinetics in chemical reaction and diffusionmodels [6, 12–14].Alpha particles can generate energetic specieswhich are able to react with each other and theirsurrounding environment [15, 16]. This processcan produce oxidising conditions near the solid-water interface, which, due to the short pene-tration depth of alpha particles in water, variesrapidly with distance from the surface. In manysenses beta and gamma radiation is more ubiq-uitous than alpha across the fuel cycle becausethey are more penetrating. However, the morerapid decay of beta and gamma emitting nuclidesresults in alpha radiation being dominant at thefuel interface when considering the timescales rel-evant to disposal environments (>1000 yrs) [17].These considerations indicate the importance ofdeveloping an accurate model of alpha radiationacross the fuel-water interface.Most radiolysis models utilise the linear energytransfer (LET) curve for calculating the dose re-ceived by a medium per decay. This is becausethe rate at which a corpuscle is stopped is equiv-alent to the rate the energy is transferred to themedium; hence, a linear energy transfer betweenthe two. The functional form of the LET is de-scribed by the following relation. LET = − dEdx (1)where E is the energy lost by the ion and x isthe length over which it is lost. High LET radia-tion refers to slower heavier ions and low refers tofast moving electrons. To model radiolysis from high LET radiation the chemical events occur sofrequently along its path we assume a uniformcylindrical region, known as the penumbra [16].In order to obtain the LET function of a singlealpha particle, the stopping powers of each ma-terial it is traversing through is required. Thestopping power of a material can be derived fromBethe-Bloch theory. Bethe-Bloch theory describes the average en-ergy lost by a charged particle due to Coulomb in-teractions between the particle and the electronsof atoms within the medium [18]. At the basis ofall stopping power models, lies the Bethe-Blochequation [19]. The equation, with additional cor-rections, does well to predict the stopping rangesof high velocity ions through a variety of me-dia [20]. The Stopping Ranges of Ions in Matter(SRIM), a program created by Ziegler and Bier-sack, contains a comprehensive database of exper-imental values to use alongside a corrected Bethe-Bloch model [21]. SRIM generates the stoppingcorrections required from compounds containingcommon elements. This process is known as thecore and bond (CAB) approach. It uses the inter-action between the traversing ion in the atomiccentres and adding the stopping from the ma-terials bonding electrons [21]. The accuracy ofthis SRIM software had been tested through manycompounds [22–25] and found to predict the stop-ping of H and He ions within 2 % at the Braggpeak [21]. The LET curve can be extracted fromthe ionisation output.
The modelling approaches used for determiningalpha dose rates can be split into two categories:analytical derivations; utilising stopping powerratios, tables and geometries [26–29], or MonteCarlo methods utilising nuclear Monte Carlo sim-ulators such as GEANT4 or MCNP [30–33]. TheMonte Carlo simulators are often computationallydemanding whereas the analytical approach oftenoversimplifies stopping power and refrains fromtreating the energy distribution of particles sepa-rately. The model described in this study utilises
January 21, 2021 igure 1: An illustration of the geometry used in the planar surface model. The x position indicates a randomlygenerated position on the dotted line, while the x position indicates the perpendicular range of the alpha particle. Thesolid black-red line denoted x’ UO and x’ H O indicates the distances travelled along the axis x’, a randomly generatedpath at an angle θ from the x axis, in UO and H O, respectively. dx and dx’ represent infinitesimal distances betweensuccessive layers in H O. the SRIM software, alongside geometrical consid-erations, in an attempt to produce a fast and ac-curate method for determining dose rates fromplanar alpha emitting surfaces, particularly UO .This study also highlights issues in dimensionalanalysis when simulating this geometry and con-tradicts a theoretical analysis by Hansson et al. [29]. The following model was built in Pythonwith the use of the math, random and Numpylibraries.
2. Methods and calculations
The most commonly built spent fuel dosimetrymodel is that of a planar surface of UO . Themaximum thickness of UO considered, is boundby the furthest distance an alpha particle cantravel through the medium, δ UO , with a givendecay energy. The water layer is bound similarly,but that of a maximum distance, δ H O . The setupof this model is illustrated in Figure 1. The illus-tration shows the dependence on the path length x (cid:48) in each medium at a given decay depth x . Tocalculate the dose as a function of distance fromthe surface, a summation of all decay paths, x (cid:48) , and the associated LET fraction deposited withineach interval dx (see Figure 3) needs to be made.The dose at a distance x from the surface for agiven decay is given as D ( x ) = eAρ H O (cid:90) x + dx x − dx dEdx (cid:48) dx (cid:48) (2)where ρ is the density of the medium, A the areaof surface in question and the factor, e, is usedto determine the dose in joules. To convert thedose into a dose rate you need the flux of ions.The flux of ions is dependent on the activity andgeometry of the source. Assuming a radioactivesurface emitting radiation in 1D, the surface fluxcan be defined as F = aρ src V src P α , (3)where a is the specific activity in Bqg − , ρ src and V src are the density and volume of the source ma-terial considered, respectively. The volume of thesource material is bound by the maximum depthan alpha can originate from in the fuel and stillcontribute to a dose at or beyond the surface, thisis denoted δ UO in Figure 1. Considering now thedirection of each decay, the average probability of January 21, 2021 n alpha escaping within V src must be considered.This is because at least 50 % of all decays will goback in the direction of the bulk. This quantitycan be defined by the summation of escape prob-abilities at each depth, x . This property will bedenoted P α and can be calculated by the follow-ing derivation.
Considering the limits of probable escape, wecan deduce that at the surface there is a 50 %chance of the alpha particle leaving the material.At the maximum depth defined by δ UO we de-duce a 0 % chance beyond this. Now assumingthat the path is straight for every decay and thatthe θ dependence is truly stochastic, we can as-sume isotropic decays of length δ UO in all direc-tions, forming a spherical ‘shell’ of probable decaypositions. As shown in the derivation by Nielsen et al . [27], the probability of the decay breaching Figure 2: An illustration showing the parameters requiredto calculate the average probability of escape. A sphere ofradius δ UO is bound by the centre point, x . The positionx is bound between the depth of δ UO denoted x min ) andthe UO surface denoted x max ) . The angle, θ , representsthe maximum angle from the x axis by which a decay tra-jectory can escape the UO surface at the starting depthx . The quantities A shell and A cap indicate the surfacearea of the sphere and cap that is exposed beyond theUO surface. the surface is given by the ratio of that sphericalshell surface area which lies beyond the surface,to the total shell surface area. An illustration ofthis is shown in Figure 2.For a cap of radius δ UO , and height ( δ UO − x ) where x is the distance from the sphere to theintersecting plane, the surface area is given by A cap = 2 πδ UO ( δ UO − x ) . (4)If the probability of alpha escape is the ratio of thecap to the decay shell, it is given by the followingrelation P α ( x ) = A cap A shell = ( δ UO − x )2 δ UO . (5)this is a linear function with an average value of0.25. The dose rate, ˙D, over a defined interval at x can be estimated using the dose of an alpha par-ticle travelling though x multiplied by the totalflux from the surface. Combining (2) and (3) thedose rate becomes ˙ D = eaρ UO δ UO P αρ H O (cid:90) x + dx x − dx dEdx (cid:48) dx (cid:48) . (6)Each α particle is considered to have a differenttrajectory (Figure 1). In order to model energydeposition as a function of perpendicular distancefrom the surface, each particle must be treatedseparately. To model dose in the unit of Gys − and setting P α = 0 . , the number of decayswithin 1 s would be n = aδ UO ρ UO (7)where n is the total number of particles emittedfrom the UO surface per second. Therefore, for adose rate (in Gy s − ) at x distance from the UO surface and a water layer width of dx, the result-ing equation becomes ˙ D = eρ H O (cid:88) n = aδ UO ρ UO (cid:90) x + dx x − dx dE n dx (cid:48) n dx (cid:48) n (8) January 21, 2021 here, x = x (cid:48) cos θ. (9)The term dE n dx (cid:48) n should considered as the LET ofthe n th particle emitted from the UO surface ata distance x (cid:48) through its trajectory (see Figure3a). When considering the LET of an alpha parti-cle the distance travelled within each medium isof particular importance. In the case of receiveddose in water the limiting factor is the energy ofthe alpha particle as it crosses the fuel-water inter-face. This will be dependent on the characteristicdecay energy and distance travelled in the UO medium. An assumption has been made that allalpha particles have a characteristic initial decayenergy for simplification within the model. Theenergy deposited per alpha will in this case bebound by the distance travelled within UO . Theeffect this has on the LET function within wa-ter is shown in Figure 3b, illustrated by the peakshift, σ . Since the Bragg curve maintains muchof its functional form, and the peak shifts in alinear fashion with distance travelled in UO , a‘base-function’ of a Bragg curve unimpeded byUO can be used to then approximate all other Bragg curves in this model. This is only valid aslong as the peak shift σ is well-understood. σ Figure 3b shows the Bragg peak shift, σ , of analpha particle with a decay energy of 5.8 MeV,that occurs with increased path distance withinthe UO medium ( x (cid:48) UO ) before crossing the inter-face into H O. This shift shows a strong negativelinear correlation with x (cid:48) UO as previously shownby Poulesquen and Jégou [34]. A line of best fitof the Bragg peak position over 12 values of x (cid:48) UO (see Figure 3b), σ is approximated by the gradientat -3.269 ± value of 0.999.Understanding this shift is key in reducing thecomplexity and computation time of this model.Instead of simulating the LET interaction at atime, as a function of depth and angle of emis-sion for each particle (through both the UO andH O medium), all that is required is the func-tional form of the Bragg curve unimpinged byUO ( x (cid:48) UO = 0 ), and the total distance travelledin UO . The Bragg curve of an alpha particlethrough water only will be referred to as the ‘base-function’ throughout.Once the peak shift and base-function of theBragg curve for an alpha particle of a given energy Figure 3: (a) Illustration of energy interval deposited at decay path displacement x (cid:48) with a width dx (cid:48) that is depositedat distance x from the interface. (b) Illustration of Bragg curve through x (cid:48) H O with peak shift denoted, σ . The quantity σ highlighted as a function of distance travelled within UO , denoted x (cid:48) UO . Both illustrations use an alpha particle witha decay energy of 5.8 MeV. January 21, 2021 igure 4: Graph showing a comparison of the results fromthe planar geometry model in the present study againstprevious literature. Due to the variety of activities chosenin the literature the fraction of maximal dose rate is used. are known, the model can be built using matricesand linear algebra in a 2D Cartesian geometry.
3. Results
Figure 4 shows the resulting dose rate in theform of a decay curve that aligns well with pre-vious theoretical studies [27, 29, 33, 35]. Thefraction of the peak dose rate is used to makethe comparison activity-independent. The simu-lation number used to run each decay and scaleto the appropriate dose used is 100,000. Resultsremain relatively similar with increased computa-tional cost after this amount as the model con-verges to within a 0.5 % fluctuation after 50,000.To determine how well the model performswith regard to the magnitude of dose, we can seea direct comparison in Figures 5 and 6. The totaldose of the model built by Tribet et al. was calcu-lated to be 23740 Gyh − , in comparison with thisstudy a value of 23880 Gyh − is in good agree-ment. A comparison of average values over 30 µ m from previous studies are shown in Table 1.As expected, the values presented in this studyand the values by Tribet et al. are in good agree-ment. In comparison to the model by Hansson etal. [29], the Bethe-Bloch LS calculation presentedby Cachoir et al. [36], and the estimation made Figure 5: Graph showing the Tribet model result and thepresent study for decays of 5.3 MeV and an activity of4.73 × Bqg − . in the SFS report [5], our results suggests a sig-nificant overestimation in average dose rates fromthese studies. When analysing the literature there seems tobe a mistaken assumption about the probabilityof escape in the Hansson paper [29]. The paperpoints out that Hosoe et al. [37] correctly statedthat 25 % of all particles escape a planar sur-face. The article goes on to mention that a pa-per by Garisto et al. [38] ‘corrected’ the assump-tion that the energy of the alpha particle was lin-ear and instead, a function of emission angle. Itgoes on to say that if the energy-dependence ofthe alpha particle travelling through the mediumwas a function of emission angle, then the escapeprobability should also follow the same angular-dependence. Garisto concluded that the “Theself-shielding by the surface layer of the fuel re-duces the total number and energy of α particlesemerging from the fuel surface by a factor of fourand seven, respectively” , making the distinctionbetween energy-dependence and particle escaping[38]. This was overlooked and instead Hanssontheorised that because the energy-dependence fol-lows a cosine function the average value of cos θ between < θ < π/ ( /π ) multiplied by the pre-viously assumed 0.25 will correctly give an escape January 21, 2021 ecay Energy Tribet et al.
SFS Report Cachoir et al.
Hansson et al. present study(Mev) (Gyh − ) (Gyh − ) (Gyh − ) (Gyh − ) (Gyh − )5.3 791 - - - 7315.8 - 1760 1680 1628 1136 Table 1: A literature comparison of the average dose rate values in water over 30 µ m from the fuel-water interface[29, 33, 36]. The studies shown that use decay energies 5.3 and 5.8 Mev use an activity of 4.73 × Bqg − and 5.6 × Bqg − , respectively. probability of /π (0.318). This was backed upby the fact that their model computationally pro-duced a similar escape probability (0.315). Un-fortunately, this result is due to a dimensionalanalysis error. The model describes a 2D geo-metrical setup. To most simply calculate the es-cape probability, one takes all the possible escapepositions limited by δ and divide by all possibleend positions (see A.1. for full derivation). Thisis the ratio of a circle’s arc (where r = δ ), withits circumference in 2D. The probability gives thefollowing relation: P = cos − ( | x | − δ ) π (10)This is a non-linear function with an average valuebetween − δ < x < of /π . If the model was set Figure 6: Graph showing the Hansson model result (solidblack line) and the present study for decays of 5.8 MeVand an activity of 5.6 × Bqg − . The dashed line showsresults from the present study using the Hansson assump-tion of P α = 1 /π and the solid red line indicating thepresent study with the corrected P α = 1 / . up in 3D the calculated value would have been aspredicted, P α = 0 . , as shown in the derivationin Section 2.2.
4. Discussion
The dose rate profile was calculated for alphaparticles with initial decay energies of 5.8 MeVand 5.3 MeV, emitted from an infinitely planarsurface, using a geometrical model in Python withthe aid of fitted data from SRIM. In this modelthe resulting distributions resemble decay curveswith the maximal dose rate at the fuel-water inter-face. The resulting decay curve for initial energiesof 5.3 MeV were compared with previous attemptsby Nielsen et al. and Tribet et al. [27, 33] (Figure4).The Nielsen model derives the alpha dosime-try rate and curve from the energy-dependence ofthe particle range formula created by Jansson andJonsson [39]. This technique has been shown tounderestimate the range compared to the SRIMsoftware by Hansson et al. [29]. A potentiallykey feature of the dose rate curve is the inter-face dose rate, as this represents the local rate ofradiolysis, and hence the generation of highly re-active radicals that could affect the corrosion ofthe surface material [17]. As this model does notcalculate a dose rate closer than 3 µ m from thesurface it could lead to an underestimation of thecorrosion rate. However, it has been shown thatusing the average values of dose rate within thealpha-irradiated volume one can simulate corro-sion kinetics in good agreement with experimen-tal data [12–14]. The role of dose rate shape onthe rate of dissolution has not been properly in-vestigated and may be of greater importance inmore complex geometries. January 21, 2021 n a report from the SFS project (an EU frame-work 6 project) an estimation was made of theinterface and total dose rate of alpha particlesemitted from a UO surface. They used an initialenergy 5.8 MeV and activity 5.6 × Bqg − [5],the same parameters used in the comparison withHansson (Figure 6). The interface dose rate esti-mate was 3120 Gyh − , which was compared to amodel using the Bethe-Bloch LS equation by Ca-choir et al. [36]. The Cachoir model greatly over-estimated the dose rate, while the report alludedto an interface dose rate that Hansson closely pre-dicts. Nevertheless, this is still an overestimationof the interface dose rate.The estimation uses consecutive layers of ma-terial each emitting a dose of 1022 Gyh − in thedirection of the interface. It then multiplies thetotal fuel layers contributing to a dose beyond thesurface then divides by the total water layers re-ceiving the dose. This oversimplification fails toconsider attenuation of neighbouring fuel layersreducing the dose rate with each consecutive layergoing deeper into the fuel. Hence the interface es-timation is again an overestimation, further allud-ing to the validity of this model and the error inP α made in the Hansson report, this is supportedby the difference in average dose values shown inTable 1. The 27 % difference in total dose receiveddue to the P α correction could considerably the ef-fect of result of dissolution or radiolysis modellingbuilt on such a result.The dose rate curve by Tribet was simulated us-ing the Monte Carlo N Particle (MCNPX) trans-port code that evaluates all particle interactionsat set layer-widths and approximates between lay-ers [33]. In a comparison to the results in thisstudy (Figure 4), they produce logical values witha slight deviation from this model beyond 10 µ m.Despite the disparity in decay shape, the totaldose is in good agreement with Tribet with alarger range (Figure 5), and a lower interface doserate, due to underlying differences in the stoppingranges used for 5.3 MeV. A slight difference inthe two models is the use of UO density as 10.8gcm − ( UO ) by Tribet, instead of the 10.97gcm − used in this study. Comparing the densityeffect on the stopping range calculated by SRIM equates to an increase of 1.5 % in the stoppingrange (12.35 µ m to 12.54 µ m) due to a reductionin density. The approach presented in this studyis therefore in good agreement with the MCNPXmodel, while improving computational efficiency.The similarity in average values within an alpha-irradiated volume of thickness 30 µ m shown inTable 1 also support the argument for implemen-tation of this model over the MCNPX approach.There are significant computational challengeswhen combining dose rate calculations with achemical reaction and diffusion model. To over-come this, dose rate calculations are often signifi-cantly simplified using values determined analyti-cally [12, 14, 17, 36]. If the approach presented inthis study was combined with a chemical reactionand diffusion model one could, in comparison toprevious analytical approaches, better predict thedissolution rate of spent fuel and hydrogen releaseas a function of fuel age and repository condition.
5. Conclusions
This study presents a mathematical model,producing dose rate curves with computationalease and built on a simple geometrical approachwith the use of fitted SRIM data. The model per-forms in good agreement with Tribet but differsto the Hansson model due to their overestimationin the average probability of alpha escape. Alphadose rate models are of particular importancefor the study of spent fuel-water interface be-haviour. To better understand the importance ofdose-rate curve shape and the role it plays on thedissolution of spent fuel, a deeper understandingof the dissolution mechanisms of the fuel-waterinterface is required.
Acknowledgements
This work was supported by The Engineer-ing and Physical Sciences Research Council (EP-SRC) and the Transformative Science and En-gineering for Nuclear Decommissioning (TRAN-SCEND) consortium.
January 21, 2021 igure A.7: An illustration of the parameters used for theprobability of alpha escape function for 1-dimension.
Appendix A. Appendix
Appendix A.1. The role of dimension on P α For the 1-Dimensional system shown in FigureA.7 , the probability of P1 to P2 crossing the in-terface if it can go forwards and backwards by thedistance, δ , is the length of the path past inter-face divided by the full range of the particle, δ .This gives the following probability function for astarting position − δ < x < , P = 12 − | x | δ (A.1)it can be clearly seen that this is a linear functionwith respect to x, where the average probabilityis found at the midpoint of x , where x = − δ .Giving an average probability of escape equal to / .Using the same reasoning for the derivation in2-Dimensions we have a circle intersected by aline, as shown in Figure A.8. The probability ofthe particle crossing the line becomes the lengthof the spherical arc past the line divided by allthe possible end positions i.e the circles circum-ference. Using the formula Arclength = πθd (A.2)And substituting for x and δArclength = 4 πδ arccos( | x | δ ) . (A.3)Dividing through by the circumference of the cir-cle, the probability of escape becomes P = arccos( | x | /δ ) π . (A.4) Figure A.8: An illustration of the parameters used for theprobability of alpha escape function for 2-dimensions.
This is a non linear function, see Figure A.10, thathas an angular dependence and an average valueof /π .Lastly in a 3-Dimensional system the geometrybecomes a sphere of radius, δ , intersecting by aplane (Figure A.9). Hence, the surface area ofthe cap beyond the plane, divided by the surfacearea of the sphere itself. Using the equation A cap = 2 πδh (A.5)where h is the cap height. Substituting for x , A cap = 2 πδ ( δ − | x | ) (A.6) Figure A.9: An illustration of the parameters used for theprobability of alpha escape function for 3-dimensions.
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