Streaming Calculation uing the Point-Kernel Code RANKERN Steve CHUCAS, Ian CURL AEA Technology, Winfrith Technology Centre, Dorcheter, Doret DT2 8DH, UK RANKERN olve the gamma-ray tranport equation in generalied geometry uing the point-kernel method, with build-up factor accounting for the contribution from cattered radiation. Development to RANKERN have overcome the main limitation of the point-kernel method and the aociated build-up data; i.e. that they are baed on the calculation of the flux ariing from radiation which ha paed in a direct line from the ource to the doe-point. Thi aumption i adequate for a wide range of ytem that are compried of bulk hielding. However, it i inaccurate in ituation where the dominant contribution to the reult arie from radiation which ha undergone a number of change in direction between ource and doe-point in order to travel along weaknee in the hield. Thi i the phenomenon that ha to be modelled in treaming calculation. When the direct flight from ource to the detector i at an angle to the hield, the build-up data can be altered to take account of the obliquity. Such oblique penetration build-up factor are an approximation, which work for mall angle but break down when the angle of obliquity become large. When thi occur, intermediate point of catter between the ource and doe-point need to be included in the calculation to provide a more accurate calculation. In many cae, the mechanim of catter in a volume can be approximated by reflection at a urface, with appropriate albedo data giving the probability of reflection. Thi paper decribe how RANKERN may be ued to perform treaming calculation. Validation evidence i preented for the ue of the code in uch ituation, including treaming along multi-legged duct. KEYWORDS: Gamma ray, point-kernel, radiation treaming, build-up, oblique penetration, catter, reflection, albedo, validation, multi-legged duct I. Introduction The deign of bulk hielding i relatively traightforward. However, the penetration through the hield can preent ignificant problem for the radiation hield deigner. A part of a collaborative programme AEA Technology and BNFL have further developed the 3D RANKERN 1 point-kernel code a a robut deign tool for all gamma-ray hielding tudie. The code' powerful and flexible ource and geometry capabilitie coupled with an efficient optimiation algorithm and novel feature, uch a the automatic computation of multi-layered build up factor, have led to it widepread ue. Thi paper decribe how RANKERN deal with ituation where the conventional build-up method break down. Specifically, thee can occur when intermediate event between the ource and doe-point need to be conidered, a i the cae in penetration along multi-legged duct. To tart the analyi we firt look at the ituation when penetration i oblique to the hield. φ = SB τ 4 πr e dv, 2 V where B i the build-up factor catering for cattered radiation, and ubcript relating to energy, material etc. have been omitted for clarity. In RANKERN, build-up data are held a cubic polynomial 2 uch that 3 i B τ = A i τ, i= 0 the data varying with material and energy and only one et of build-up data being ued for a given configuration. Such point iotropic build-up data are uited to ituation uch a path A in Figure 1, when the dominant penetration path i normal to the hield geometry. II. Oblique Penetration RANKERN olve the point-kernel equation for any complexity of ource and hield configuration. It algorithm can be ummaried by conidering a ource of trength S γ/cm 3 / ditributed over a volume V. The flux φ at a ditance r following penetration through τ mfp (mean free path) of hield i given by C B A Figure 1 - Normal and Oblique Build-up
A thi penetration path become oblique to the hield, ee path B, the build-up factor i corrected 3 by RANKERN uch that ( ec βθ o 1) τ B θo, τo = B 0, τo e where B(θ o ) i the oblique build-up factor, B(0) i the build-up factor for normal incidence, β i an empirically derived angle coefficient typically 0.80<β<0.95, and the variable θ o and τ o refer to the incident obliquity and hield thickne in mfp repectively. Depending on the thickne of the hield, thi technique work uccefully up to angle of obliquity of about 25 o, after which it become increaingly inaccurate. A a rule-of-thumb, if the reult uing oblique build-up i more than a factor ten higher than that uing normal build-up, the reult i unreliable. Ultimately, ee path C, the oblique build-up treatment break down and intermediate catter event have to be treated explicitly. III. Scatter Calculation Path C in Figure 1 can be analyed by conidering two leg of the path connected by a ingle catter event. Then, the flux at the doe-point due to radiation that undergoe cattering within volume V may be written: τ τ σ φ = SBe o e dvd, 2 2 4πr o 4πr V V where r o i the penetration ditance between the ource and catter point, r i the penetration ditance between the catter and doe-point, σ i the differential cattering cro-ection, and τ o and τ are the total penetration ditance in mfp along r o and r repectively. The method can be extended to higher order of catter, and i alo applicable to kyhine tudie 1. With thi algorithm a number of factor need to be conidered over a conventional point-kernel calculation, namely the cattering cro-ection, the energy of the gamma-ray after catter, the build-up data to be applied, and the ize of the catter volume. 1. Scatter cro-ection The cattering cro-ection for the hield material are baed upon the Klein-Nihina equation and the magnitude of the cro-ection are derived uing electron denity value for the material. The code accee uch data automatically by knowing in which material the catter occur. 2. Energy after catter o 1 Eo E = Eo 1+ ( 1 coθ ), 0. 511 where E o i the ource energy and θ i the angle of catter. The value of E determine the tranmiion cro-ection and build-up data (ee below) to be ued for the econd leg of the route from ource to doe-point. 3. Build-up data Since one catter ha been treated explicitly, it may be thought that build-up need only be applied to one leg. However, a preented later, experience ha hown that thi lead to underetimation of the reult. Even though the approach i phyically peimitic, it i therefore recommended that build-up i applied to both leg of the path, and thi i taken a the default treatment by the code. 4. Size of catter volume The region of the hield where catter i to occur mut be large enough for all the important event to be included. Thi i dictated by the dominant penetration path through the ytem, which may be complicated; but in general it lead to catter volume approximately 3mfp thick in the region of the hield bordering the void or duct. Thee are pecified in RANKERN a imple bodie, e.g. boxe or cylinder, which may uefully be ubdivided into maller unit. The reaon for ubdiviion i that ince RANKERN determine the contribution to the reult from each ubdiviion and alter it ampling probability a the calculation progree, the more important ubdiviion are ampled more often and thoe giving a negligible contribution are effectively neglected. It i therefore adviable to make the catter volume larger than neceary, in the knowledge that the code will optimie the calculation itelf. IV. Reflection In ome ituation, e.g. back-catter from a wall, the catter can be repreented by reflection event ditributed over an area A. In thi cae, the catter cro-ection i replaced by the probability of reflection, or albedo β thu φ = τo τ SBe βe 2 2 dvda 4πr o 4πr A V Similar conideration to catter calculation apply regarding the ue of build-up data and the ize and The energy E of the cattered gamma-ray i given by the Compton expreion uch that
pecification of the reflection urface. Again, the energy of the reflected gamma-ray i given by the Compton expreion, but albedo data have to be defined. 1. Albedo data The value of β i derived from the o-called current doe albedo 4 26 CK θ 10 + C αd =, 1 + co θ o ecθ where C and C' are fitted parameter correponding to ingle catter and multiple catter component repectively; K(θ ) i the Klein-Nihina energy cattering coefficient for cattering angle θ, and θ o and θ are the incident and emergent polar angle repectively a meaured from the normal to the urface. Value of C and C' have been obtained by fitting Monte Carlo calculation for broad parallel beam of photon incident at variou angle to the urface of the media 5, and are tored within RANKERN for variou material and range of energy. The current doe albedo i converted to β by caling by the doe-rate converion factor before and after reflection, i.e. DEo β= α D DE, where D(E) i the doe converion factor for energy E. V. Scatter or Reflection? It i vital that the dominant penetration path through the ytem being analyed are identified, and uually eparate RANKERN calculation are performed for each path in order to ae their relative contribution. Having identified the path and determined where intermediate event are to occur, a deciion mut be made whether catter or reflection i to be ued. In ome circumtance, like Figure 1, catter i the obviou choice ince reflection in inappropriate. In other cae, however, the choice i not o traightforward. Conider the ytem illutrated in Figure 2, where ource and detector are eparated by an effectively black body. The route from ource to detector therefore only involve interaction in the wall. With the wall modelled a a reflection urface, all the important path from ource to detector are included in the calculation. Figure 2 - Reflection However, if the hield i nearer the wall, a hown in Figure 3, thi i not the cae ince ome of the path are cut off. In uch a ituation, catter within the wall ha to be included, a hown by the dahed line. Figure 3 - Single Scatter When the hield i right up againt the wall, a illutrated in Figure 4, ingle catter event are alo cut off and multiple catter i required a hown by the dotted line. Figure 4 - Multiple Scatter It i therefore very important that the analyt identifie the important catter/reflection path and ue the appropriate order and ize of catter/reflection region. VI. Validation The accuracy of RANKERN calculation that ue catter and/or reflection ha been conidered for a number of different cenario 6. Experiment in the ARCAS facility at AEA Technology Winfrith ite were performed, which conited of Co 60 ource, doimeter, and hield configuration built from aluminium, iron or lead.
1. Oblique penetration A tudy of oblique penetration through iron lab, chematically imilar to Figure 1, indicated that RANKERN prediction of doe-rate were about 20% peimitic compared with meaurement until the oblique build-up method began to break down. Thi wa the bai of the rule-of-thumb mentioned in ection II. Slab 2. Single catter or reflection An experimental configuration wa built which reembled Figure 2, with an aluminium wall and a lead hield. With a ingle event, either catter or reflection, RANKERN predicted doe-rate within 20% of meaurement. Since treating reflection i quicker than catter, becaue of numerical integration over an area rather than a volume, the former i recommended for uch ituation. Similarly, a configuration reembling Figure 3 wa contructed. A expected, a ingle reflection led to underetimation of the reult, o catter hould be ued intead. 3. Multiple catter The accuracy of a multiple catter calculation wa aeed by a configuration imilar to Figure 4. In thi cae, RANKERN prediction of doe-rate were between a factor 1.5 and 4 peimitic, thi being mainly due to the conervative approach of applying build-up to all three leg. It i worth noting that applying build-up to only the lat leg led to underprediction of a factor five, hence the recommendation to apply build-up to all leg of the calculation. In practice multi-order calculation become increaing longer to execute, and for order higher than two it i uual to reort to the more rigorou Monte Carlo code MCBEND 7. Since the two code ue the ame geometry package, tranferring from one calculational method to the other i relatively traightforward. Figure 5 - Penetration following Reflection 5. Multi-legged duct A common problem in hield deign arie from the need to provide penetration for ervice uch a ventilation, cooling, ampling, chemical doing and intrumentation into cell containing intene ource of gamma-ray ariing from fiion product. The pipe that erve thee function mut be bent to avoid direct path through the hield, which would lead to exceive external doe-rate. Often, the pipe have two bend within the hield, thu dividing them into three leg, the firt and third leg being normal to the face of the hield. The attenuation of gamma-ray along a bent pipe of given diameter in a hield of pecified thickne i determined by the offet between the firt and third leg and the angle between the econd leg and the other two. The optimiation of pipe configuration require the calculation of the doe-rate at the cold end of the pipe due to gamma-ray which have treamed along all three leg of the pipe. Thi final analyi i an example of the application of RANKERN to thi typical treaming problem, and uch a three-legged duct i illutrated in Figure 6. 4. Penetration following reflection Thi aement conidered penetration of an aluminium lab following an initial reflection event, a illutrated in Figure 5. In thi cae, RANKERN prediction of doe-rate were up to a factor two peimitic compared with that meaured. Thi i aociated with the energy of the reflected gamma-ray, which, a noted in ection IV, i given by the Compton expreion. Hence, although the value of albedo take multiple catter into account, the reflected gamma-ray pectrum doe not. Thi lead to the reflected pectrum being too hard, which in turn lead to too little material attenuation and an over-prediction of the reult. Development are in hand to improve RANKERN treatment of reflection in thi repect. Scatter bodie Figure 6 - Multi-legged Duct
The duct i cylindrical, of diameter 150mm and i lined with 6mm of teel. The three leg are 440mm, 600mm and 1375mm long repectively. The ource i a dic with ource energy of 2.25MeV, and the doe-rate at the end of the third leg of the duct i required. Firt and econd order catter bodie were poitioned at the firt and econd corner repectively a een from the ource. Thi define the route from ource to doepoint. At the end of the duct, the doe-rate predicted by RANKERN wa a factor 2.4 higher that predicted uing the MCBEND Monte Carlo code. Thi i conitent with the accuracy of multiple catter calculation decribed in ection VI.3. A part of uch an analyi, the importance of other route through the ytem would be invetigated eparately, e.g. thoe hown in Figure 7 which how firt and third order penetration route. For ome of the event, e.g. catter along the third leg of the duct, reflection could be pecified intead. undertanding of the way radiation penetrate the hield i acquired. VII. Viualiation RANKERN i accompanied by a number of graphic package 8 for viualiing the geometry in 2 (VISAGE) or 3 (VISTA-RAY) dimenion. Additionally, VISTA-WIRE, provide wire-frame drawing of the configuration - including ource bodie and catter/reflection region a illutrated in Figure 8. Thi enable eay checking of the poitioning of the geometry, ource and catter/reflection bodie. VIII. Summary RANKERN provide a rapid deign method for gamma-ray hielding, featuring the treatment of intermediate catter and reflection event between ource and doe-point which allow treaming calculation to be performed. The accuracy of uch calculation ha been aeed againt meaurement and more rigorou Monte Carlo calculation, with RANKERN generally producing afely peimitic reult. Reference Figure 7 - Route through the Duct Whether the variou route provide ignificant contribution to the reult depend on the configuration of the duct, and a part of it output RANKERN indicate the relative importance of each route and order of catter. In thi way a thorough 1 Dean, M.H.: Proc. 8th Int. Conf. on Radiation ing, p1241 (1994) 2 Capo, M.: Build-up Factor for Point Polynomial Approximation of Gamma Iotropic, APEX 510 3 Pieper, A.G. and Beach, L.A.: Stochatic Etimate of the Penetration of Gamma-ray through Slab, NRL-5981 4 Chilton, A.B. and Huddleton, C.M.: Nuc. Sci. Eng., 17, 419 (1963) 5 Davion, C.M. and Beach, L.A.: Tran. Nuc. Soc., 5, 391 (1962) 6 Chuca, S.J.: Application Guide for Streaming Calculation with RANKERN, ANSWERS/RANKERN(96)4 7 Chuca, S.J. et al: Proc. 1996 Topical Meeting, Advance and Application in Radiation Protection and ing, p751 (1996) 8 Smith, N.R. et al: Proc. 8th Int. Conf. on Radiation ing, p447 (1994) Scatter body Figure 8 - VISTA-WIRE Repreentation of a Multi-legged Duct