Thermal neutron self-shielding factor in foils: a universal curve

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1 Proceedings of he Inernionl Conference on Reserch Recor Uilizion, Sfey, Decommissioning, Fuel nd Wse Mngemen (Snigo, Chile, -4 Nov.3) Pper IAEA-CN-/, IAEA Proceedings Series, Vienn, 5 Therml neuron self-shielding fcor in foils: universl curve I.F. Gonçlves, E. Mrinho, J. Slgdo Insiuo Tecnológico e Nucler Esrd Ncionl Scvém, Porugl Absrc. The presence of smple in he neuron field of nucler recor crees perurbion of he locl neuron fluxes. In generl, he inerpreion of he smple civion due o herml nd epiherml neurons requires he knowledge of wo prmeers: he herml neuron self-shielding fcor, G h, nd he resonnce neuron self-shielding fcor, G res. In recen works, he uhors esblished n universl curve of G res for isoled resonnces nd vrious geomeries. The presen pper dels wih he descripion of G h in foils by mens of n universl curve on he bsis of dimensionless vrible which includes he physicl, nucler nd geomericl properies of he smple. The universl curve is in very good greemen wih experimenl nd clculed vlues obined from he lierure.the sudy of oher geomeries (spheres, wires nd cylinders) is in progress.. Inroducion As is well-known, he irrdiion of smple in he neuron field of nucler recor is ffeced by he locl perurbion of he neuron fluxes produced by he smple []. In generl, he inerpreion of he smple civion due o herml nd epiherml neurons requires he knowledge of wo prmeers: he herml neuron self-shielding fcor, G h, nd he resonnce neuron self-shielding fcor, G res. In recen works [-4], he uhors esblished n universl curve of G res for isoled resonnces or groups of isoled resonnces, nd vrious geomeries of he smples (foils, wires, spheres, nd cylinders). In relion o G h, some experimenl [5-3] nd heoreicl sudies [4] hve been crried ou o deermine his fcor for foils nd wires of differen elemens. Their resuls re presened s bles or grphics of G h for given elemen nd geomery s funcion of he ypicl dimension. Recenly, Copley [5] sudied he scering effec wihin n bsorbing sphere nd, ccording o his resuls, he self-shielding fcor is obined hrough se of curves s funcion of he mcroscopic bsorpion nd scering cross-secions of he smple. The presen pper dels wih he descripion of he herml neuron self-shielding fcor in foils by mens of n universl curve. The sudy of oher geomeries (spheres, wires nd cylinders) is in progress.. Clculion For given elemen, geomery nd smple dimension, G h is clculed s he rio beween he recion res per om in he rel smple nd in similr nd infiniely dilued smple: G h ( x) = E E E E M M ( E) σ ( E) ( E) σ ( E) de de ()

2 where x is he ypicl smple dimension, M ( E) is he non-perurbed herml (mxwellin) neuron flux per uni energy inervl (inside he infiniely dilued smple), M ( E) is he perurbed herml neuron flux inside he rel smple, σ (E) designes he (n,γ) cross-secion, nd E nd E re, respecively, he lower nd he upper limis of he herml neuron specrum room emperure. The ol neuron cross-secion is doped in he clculion of he perurbed neuron flux, hus king ino ccoun he neuron scering in he smple. In 6 he clculions, he densiy for infinie diluion is ssumed o be ρ = ρ, ρ represening he densiy of he rel smple. Therml neuron self-shielding fcor in foils of very differen elemens (see Tble ) hve been clculed using he MCNP code [6]. TABLE. Physicl nd nucler properies of he sudied elemens [7, 8] Elemen A (g mol - ) ρ (g cm -3 ) σ σ s σ Al Au Cd Co Cu Eu Gd In Ir Mo Ni Pb P Rh Sc Sm T Fe σ, σ s, σ re, respecively, he ol, scering nd bsorpion microscopic cross-secions verged over he herml neuron specrum room emperure [7]. In order o elimine prciclly he effec of he neurons enering hrough he foil edge, rio of R/ (wih R cm) hs been doped, nd R being he hickness nd he rdius of he foil, respecively. 3. Resuls nd discussion As is shown in Figure, he vlue of G h depends on he physicl nd nucler properies of he meril s well s on he foil hickness.

3 G h Foil hickness (cm) Au Al Co Sc Mo Gd T Cd P Eu Cu Rh Pb Ni Ir Sm In Figure. Clculed G h s funcion of he foil hickness However, i is possible o inroduce dimensionless vrible, z, which convers h dependence ino n unique curve. The nlysis of he resuls obined in he foil (or slb) geomery hs shown h his vrible is given by: Σ s Σ z = Σ = Σ () Σ Σ where Σ, Σ s nd Σ re, respecively, he ol, scering nd bsorpion mcroscopic crosssecions verged over he herml neuron specrum, nd is he hickness of he foil. Noe h he vrible z kes ino ccoun he neuron scering in he smple. The nlysis of he resuls lso shows h sigmoid is he bes curve o be fied o he clculed G h (z) vlues. The expression of his curve is A A Gh ( z) = + A p z + z (3) where A, A, z nd p re he curve prmeers, o be djused o he clculed vlues. Noe h A is he limi of G h s z ends o zero; A is he limi of G h s z ends o infiniy; z is he inflexion poin [G h (z ) = (A +A )/]; nd p is reled wih he grdien of he curve z = z. Figure shows he clculed vlues of G h (z) nd he sigmoid (universl curve) djused o ll he poins. The vlues of he curve prmeers re: 3

4 . G h Al Au Cd Co Cu Eu Gd In Ir Mo Ni Pb P Rh Sc Sm T G h,uc Foils A = A = z =.68±.6 p =.968±.9 R = x -5 x z = Σ (Σ /Σ ).85 Figure. Clculed G h nd he universl curve (G h,uc ) A = ; A = ; z =.68 ±.6; p = (4) ± For more hn 9% of he resuls, he relive deviion beween he universl curve nd he clculed vlues is less hn 4% for z< (z domin of prcicl ineres). In Figure 3, he universl curve is compred wih experimenl nd clculed vlues of G h from he lierure. As cn be observed, he overll greemen is very good, hus confirming he vlidiy of he relions () o (4) o deermine he herml neuron self-shielding fcor in foils. The mximum relive deviion beween he universl curve nd he experimenl vlues is less hn 5%.. G h Foils Au Exp [] Au Exp [7] Au Exp [6] Co Exp [3] Cu Exp [3] Cu Exp [7] Fe Exp [3] In Exp [5] In Exp [] Au Clc [4] Co Clc [4] Cu Clc [4] Fe Clc [4] In Clc [4] G h,uc z = Σ (Σ /Σ ).85 Figure 3. Comprison of he universl curve (G h,uc ) wih experimenl nd clculed vlues from he lierure 4

5 4. Conclusion In spie of lrge differences beween he physicl nd nucler properies of he sudied elemens, n universl curve cn describe he behviour of he herml neuron self-shielding fcor for foil smples. In his curve, G h is expressed s funcion of dimensionless vrible, which kes ino ccoun he physicl nd nucler properies of he smple. The universl curve is in very good greemen wih experimenl nd clculed vlues obined from he lierure. The mximum relive deviion beween he universl curve nd he experimenl vlues is less hn 5%. REFERENCES [] BECKURTS, K.H., WIRTS, K., Neuron Physics, Springer-Verlg, 964. [] MARTINHO, E., I. F. GONÇALVES, I.F., SALGADO, J., Universl curve of epiherml neuron resonnce self-shielding fcors in foils, wires nd spheres, Applied Rdiion nd Isoopes 58 (3) (3) [3] GONÇALVES, I.F., MARTINHO, E., SALGADO, J., Exension o cylindricl smples of he universl curve of resonnce neuron self-shielding fcors, Nucler Insrumens nd Mehods (in press). [4] SALGADO, J., MARTINHO, E., GONÇALVES, I.F., Mehodology for he clculion of neuron self-shielding fcor of group of isoled resonnces, Journl of Rdionlyicl nd Nucler Chemisry (in press). [5] KLEMA, E.D., RITCHIE, R.H., Therml neuron flux mesuremens in grfie using gold nd indium foils. Physicl Review 87 (95) 67. [6] SOLA, A., Flux perurbion by deecors foils, Nucleonics 8 (3) (96) 78. [7] HASNAIN, S.A., MUSTAFA, T., BLOSSER, T.V., Therml neuron densiy perurbions by foils in wer, ORNL 393 (96). [8] EASTWOOD, T.A., WERNER, R.D., Resonnce nd herml neuron self-shielding in cobl foils nd wires, Nucler Science nd Engineering 3 (96) 385. [9] SPERNOL, A., VANNINBROUK, A., GROSSE, G., Therml flux perurbion by cobl deecors, Neuron Dosimery Symposium, Hrwell, Vol. (96) 547. [] WALKER, J.V., RANDALL, J.D., STINSON JR, R.C., Therml neuron flux perurbion due o indium foils in wer. Nucler Science nd Engineering 5 (963) 39. [] CRANE, J.L., DOERNER, R.C., Therml self-shielding nd edge effecs in bsorbing foils. Nucler Science nd Engineering 6 (963) [] TAYLOR, N.K., LINACRE, J.K., The use of cobl s n ccure herml neuron flux monior. AERE-R-4 (964). [3] CARRÉ, J.C., ROULLIER, F., VIDAL, R., Éudes des coefficiens d uobsorpion hermique, CEA Dépremen des Éudes de Piles, Service des Expériences Criiques, Rppor MIN 76 (965). [4] LOPES, M.C., Sensiiviy of self-powered neuron deecors o herml nd epiherml neurons wih muliple collision remen, PhD Thesis, Universiy of Coimbr (99). [5] COPLEY, J.R.D., Scering effecs wihin n bsorbing sphere immersed in field of neurons. Nucler Insrumens nd Mehods A37 (99) [6] BRIESMEISTER, J., (Ed.), MCNP Code, Los Almos Nionl Lborory, Repor LA- 379 (). [7] hp:// [8] Tuli, J.K., Nucler wlle crds. Nionl Nucler D Cener, Brookhven Nionl Lborory, Upon, USA (). 5