CEJP 2(2003)210{234 Thee® ective absorption cross-section ofthermal neutronsin a mediumcontaining strongly or weakly absorbingcentres Krzysztof Drozdowicz, BarbaraGaba¶ nska,Andrzej Igielski, Ewa Krynicka,Urszula Wo¶znicka ¤ TheHenryk Niewodnicza¶nski Institute ofNucle arPhysics, ul. Radzikowskiego152, 31-342 Krak¶ ow, Poland Received 24January 2003; revised 11F ebruary 2003 Abstract: Thestructure ofa heterogeneoussystem in®uences di¬usion ofthermal neutrons. Thethermal-neutron absorption in grainedmedia is considered in the paper. Asimple theory is presented for atwo-componentmedium treated asgrains embedded in the matrix oras asystem built oftwotypes ofgrains (of strongly di¬ering absorption cross-sections). Agrainparameter is de­ned asthe ratio ofthe e¬ective macroscopic absorption cross-section ofthe heterogeneousmedium to the absorption cross-section ofthe corresponding homogeneousmedium (consisting ofthe samecomponents in the sameproportions). Thegrain parameter depends onthe ratio ofthe absorption cross- sections andcontributions ofthe componentsand on the size ofgrains. Thetheoretical approachhas been veri­ed in experiments onprepared dedicated modelswhich have kept required geometricaland physical conditions (silver grains distributed regularly in Plexiglas). Thee¬ ective absorption cross-sections havebeen measuredand compared with the results ofcalculations. Avery goodagreementhas been observed. Incertain cases the di¬erences between the absorption in the heterogeneousand homogeneous media arevery signi­cant. Avalidity ofan extension ofthe theoretical modelon natural, two- component,heterogeneous mixtures hasbeen tested experimentally.Aqueous solutions of boric acid havebeen used asthe strongly absorbingcomponent. Fine- andcoarse-grained pure silicon hasbeen used asthe secondcomponent with well-de­ned thermal-neutron parameters. Small andlarge grains ofdiabase have been used asthe secondnatural component.The theoretical predictions havebeen con­rmed in these experiments. c Central EuropeanScience Journals. All rights reserved. ® Keywords: thermal neutrons,absorbing centres, e® ective absorption, grainp arameter, heterogeneity PACS(2000): 28.20.F, 28.20.G ¤ E-mail:Urszula.W [email protected] K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 211 Contents 1Introduction 212 2De¯nition of the grain parameter 213 3E®ective absorption cross-section §~ a of grains 214 eff 4E®ective absorption cross-section §a ofthe heterogeneous system 215 5Laboratory measurement ofthe macroscopicabsorption cross-section 218 6Silver-in-Plexiglasmodels as heterogeneous samples 220 6.1Design of the heterogeneous models 220 6.2Experimental results and discussion ofapossible uncertainty 221 6.3Comparison ofthe theoretical and experimentalresults obtained for the silver-in-Plexiglasmodels 223 7Rock{°uid samples 227 7.1Silicon as the arti¯cial rock material of grains 228 7.2Diabase as the natural grained rockmaterial 229 8Conclusions 231 References 233 212 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234 1Introduction The macroscopicthermal-neutron absorption cross-section §a of amedium isone of severalimportant parameters when the transport of thermal neutrons in any system is considered. The §a valuefor ahomogeneous mixture of n components can be obtained [1]from the simplerelation: n M §a = » §i=1qi§ai (1) where » isthe mass density of the mixture, qi isthe mass contribution of the i-th com- M ponent, (§qi = 1), and §ai isthe mass absorption cross-section of the i-th component (dependent on its elementalcomposition and the microscopicabsorption cross-sections ¼ j of the contributing elements,e.g. [2]). Sometimes, it isconvenient to express the contributions by the volumecontents of components: V ¿ = i (2) i V where Vi isthe volumeoccupied by the i-th component in the volume V of the sample. Then Eq.(1), stillfor the homogeneous mixture, yields: n §a = §i=1¿ i§ai (3) where §ai isthe macroscopicabsorption cross-section of the i-th component, de¯ned at the partial solid material density » i = mi/ Vi (where mi isthe mass of the i-th component inthe sample). Inthe caseof amedium that isaheterogeneous mixture, the e®ective thermal-neutron absorption can signi¯cantly di® er from that in ahomogeneous one that consists of the samecomponents inthe sameproportions. Aproblem ofthe heterogeneity resulting from the presence of grains in the sample can appear when the absorption cross-section of a rockmaterial ismeasured. The samples for aneutron experiment are usually prepared by crushing the rocks,and the possible natural heterogeneity can be increased additionally during this procedure. Finally,just an e®ective cross-section ofthe heterogeneous sample material ismeasured. Most often, neither the actual heterogeneity nor the detailed elementalcomposition isknown, and it isimpossible to do an exactneutron transport calculationfor the medium investigated. Onthe other hand, itisnecessary to introduce acorrection to the result ofthe measurement. Wepresent here acomprehensive study of the physicalproblem and its experimental implications.A simpletheory of the e®ective absorption of thermal neutrons in agrained medium isoutlined and applied to an interpretation of the pulsed measurement of the absorption cross-section on heterogeneous models (consisting of grains in amatrix, where the geometricstructure and the thermalneutron di®usion parameters are well-known). Validityof the theory isfurther tested and con¯rmed on more realisticsamples of¯ne and coarse-grained materials:arti¯ cial or natural rockgrains mixedwith a°uid absorbers. K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 213 2De¯nition of the grainparameter The material heterogeneity for the thermal-neutron transport in aconsidered volumeis understood here asmany smallregions that di®er signi¯cantly in their macroscopicchar- acteristicsof neutron di®usion. Beinglimited to atwo-component medium, we distinguish two cases.Case A:grains of material 2dispersed in homogeneous material 1,and Case B:acomplexmedium built of two types of grains (Fig.1). The volumecontribution of hom substance 2in the medium is ¿ .Then the macroscopicabsorption cross-section §a of the corresponding homogeneous, two-component medium of the absorption cross-sections §a1 and §a2 ,respectively,can beobtained from the relation: §hom = (1 ¿ )§ + ¿ § : (4) a ¡ a1 a2 The thermal-neutron absorption in grains of sizescomparable to the neutron mean free path can beno longer described by the macroscopicabsorption cross-section relevant for an in¯nite medium. Amodi¯ed, e® ective absorption cross-section of the grains, say ~ §a,isintroduced. InCase A,we assume that onlythe §a2 ismodi¯ed by the grain e®ect: § §~ , and the § remains unchanged. In Case B,both absorption cross-sections a2 ! a1 a1 are modi¯ed: § §~ and § §~ .The e®ective cross-section §eff of the entire a1 ! a1 a2 ! a1 a heterogeneous medium (the grained mixture) isstill calculated from formula (4), but with the substitutions mentioned above. Fig. 1 Two types of the materialheterogeneity .Case A:grainsof material2 dispersedin homogeneousmaterial 1. Case B:materialbuilt of twotypes ofgrains. The medium isrecognised as heterogeneous for the thermal-neutron transport when eff hom its e®ective absorption cross-section §a di®ers from the cross-section §a calculated from formula (3), validfor the homogeneous medium. The ratio eff §a G = hom (5) §a calledthe grain e®ect parameter, de¯nes the heterogeneity e®ect on the thermal-neutron absorption. The parameter G =1corresponds to the homogeneous medium. The cross- eff hom sections §a and §a can be theoreticallycalculated and experimentallymeasured. 214 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234 3E® ectiveabsorption cross-section §~ a of grains Westart from aconsideration of the thermal-neutron di®usion in amedium within the one-speed approximation. The probability that the neutron isabsorbed when it travels apath length x0 inanabsorber is x0 x §t §a x0 §t p(x0) = e¡ §a dx = (1 e¡ ) (6) §t ¡ Z0 (cf. [1]Sec.1.3), where §t isthe total cross-section: §t = §a + §s; (7) and §s isthe scattering cross-section. In the caseof asigni¯cant anisotropy of neutron scattering, the transport cross-section can beintroduced: § = (1 · ) § ; (8) tr ¡ s where · isthe averagecosine of the scattering angle.The total cross-section §t is then substituted by the thermal-neutron di®usion cross-section: § = § + § = § · § : (9) d a tr t ¡ s Instead of Eq.(6) we now have x0 x§d §a x0§d p(x0) = e¡ §adx = (1 e¡ ) (10) Z0 §d ¡ Assuming the e®ective absorption cross-section §~ a in the grain as the absorption probability per unit path length, we obtain from Eq.(10) the following de¯nition: p(d) §a d§d §~ a = = (1 e¡ ) (11) d d§d ¡ where the sizeof the grain isgiven by the averagechord length d : 4V d = g (12) Sg and Vg and Sg are the volumeand the surface of the grain, respectively[3]. ~ Thus, the e®ective absorption cross-section of the grain §a1 ,besides depending on the absorption properties of the grain substance, alsodepends on its scattering properties and the sizeof the grain. Eq.(11) leads to the following two limitingcases. When the sizeof grains d tends to zero(that is,the material becomeshomogeneous), d§ 0, the d ! absorption cross-section §~ § .In another limit, d § ,which corresponds to a a ! a d ! 1 thickabsorber, the absorption cross-section is §a §~ a = : d§d K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 215 Expressions similarto this