The Effective Absorption Cross-Section of Thermal Neutrons in a Medium

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 solidmaterial 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 ofthe materialheterogeneity .Case A:grainsof material 2 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) leadsto 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 formula appear when considering the self-shielding e®ect, when the inner parts of the absorber are partly protected from the incoming neutrons by outer absorbing layers,and neutron scattering isalso taken into account (cf. [1]Sec.11.2, [4], [5]).

eff 4E® ectiveabsorption cross-section §a ofthe heterogeneous system

eff The e®ective absorption cross-section §a of the heterogeneous system in Case A is given by the formula:

§a2 eff d2§d2 §a = (1 ¿ )§a1 + ¿ (1 e¡ ) (13) ¡ d2§d2 ¡ ~ when the de¯nition of the e®ective cross-section of the grains, §a2 ,isintroduced into formula (4). The substitution ofEqs.(4) and (13) into Eq.(5) yields:

1 e Y ¿ (S ¡ ¡ 1) + 1 G(Y; S; ¿ ) = Y ¡ (14) ¿ (S 1) + 1 ¡ where S isthe ratio ofthe absorption cross-sections of the components:

§ S = a2 (15) §a1 and

Y = d2§d2 (16) isthe averagesize of the grain expressed in the neutron di®usion mean free paths and de¯nes adimensionless sizeof the grain. The parameter G can be studied as afunction either of the dimensionless size Y of grains, or of the ratio S of the cross-sections of the contributing materials,or of the material volumecontent ¿ ,depending onwhich ofthe variablesare ¯xedas parameters. Examples of the functions de¯ned in Eq.(14) are plotted in Fig.2. The plot in Fig. 2a shows that the parameter G(Y )can attain averylow value,down to about 0.1in certain cases,which means that the e®ective cross-section can be about as low as 10 % of the valuefor the corresponding homogeneous mixture (for which Y = 0 and G = 1). Functions G(S) and G(¿ )in Figs.2b and 2care plotted for the samegrain size Y = 1. They show asaturation e®ect: that means averyweak dependence of G on S or ¿ when they attain su±ciently high values.In the examplespresented at a¯xedgrain size,this weakdependence isobserved for S 100 (the leveldepending on ¿ ) and for ¿ 0.2 ¶ ¶ (depending on S).Generally,as should beexpectedat a¯xedgrain size Y ,the degreeof the material heterogeneity for thermal neutrons (measured as 1/ G)increasesfaster when 216 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234 the di®erence between the absorption cross-sections of the grains and matrix islarger and/or the volumecontent of grains (that is,their number) ishigher.

Fig. 2 a Parameter G ofthe graine¬ ect as a function ofthe dimensionlesssize Y of the grain atxed valuesof the ratios S andvolume contents ¿ .

Fig. 2 b Parameter G ofthe graine¬ ect as a function ofthe ratio S ofthe absorptioncross- sectionsat a constantsize Y ofthe grainsand a xed volumecontents ¿ .

It isimportant to noticethat for heterogeneous media,always G < 1,that is,the e®ective absorption cross-section isalways lower than for ahomogeneous material. Until now, the theoretical analysisof the problem has been performed in the one- K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 217

Fig. 2 c Parameter G ofthe graine¬ ect as a function ofthe volumecontent ¿ of grainsat a constantsize Y ofthe grainsand xed ratios S. speed approach to the thermal-neutron di®usion. However,thermal neutrons in real experimentalconditions are characterised by an energydistribution and then average valuesare observed. For acomparison between the theoretical and experimental results, the energy-averagedneutron parameters must be used in the calculation.The grain parameter G isa function of three variables: G = G(Y , S, ¿ ).Two of them, Y and S, depend on the neutron energy,because of the energydependence of the cross-sections. The grain parameter, de¯ned by the energy-averagedcross-sections, is

§eff G (Y (E); S(E); ¿ ) = h a i (17) av §hom h a i where E isthe neutron energyand the bracket denotes the energy-averaged magnitude. h i Let the energydistribution ofthe thermal-neutron °uxin the investigated volumebe approximated by the Maxwelliandistribution:

E E=ET M (E) = 2 e¡ (18) ET with ET = kBT , where kB isthe Boltzman constant and T isthe absolute temperature. The distribution (18) isnormalized to unity and the averagethermal-neutron parameter, say P ,isde¯ned as h i

P P (E) = 1 P (E)M (E)dE (19) h i ² h i Z0 Thus, the energy-averaged,macroscopic absorption cross-sections of the mixtures are 218 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

§hom = (1 ¿ ) § (E) + ¿ § (E) (20) h a i ¡ h a1 i h a2 i In Case A we have:

§eff = (1 ¿ )§ (E) + ¿ §~ (E) h a i h ¡ a1 a2 i §a2(E) d2§d2(E) = (1 ¿ ) §a1(E) + ¿ 1 e¡ (21) ¡ h i *d § (E) ¡ + 2 d2 h i eff In Case B the energy-averaged,e® ective absorption cross-section § 00 isexpressed by: h a i

eff § 00 = (1 ¿ )§~ (E) + ¿ §~ (E) h a i h ¡ a1 a2 i §a1 (E) d1§ (E) §a2(E) d2§ (E) = (1 ¿ ) 1 e¡ d1 + ¿ 1 e¡ d2 (22) ¡ *d § (E) ¡ + *d § (E) ¡ + 1 d1 h i 2 d2 h i and the grain parameter Gav00 is

eff §a 00 G00 = h i (23) av §hom h a i The experimentalgrain parameter isde¯ ned by

§exp v§exp Gexp = h a i = h a i (24) §hom v§hom h a i h a i where the energy-averagedthermal-neutron absorption rate v§ isgiven by the relation h ai

1 v§a = v§a(E)M (E)dE = v0§a(v0) v0§a (25) h i Z0 ² and §exp or v§exp isobtained from the experiment. h a i h a i Further consideration willbe limitedto an analysisof the in°uence of the grain size on the absorption when the other parameters ( S and ¿ ) are ¯xed.

5Laboratorymeasurement of the macroscopic absorptioncross- section

Czubek’s pulsed neutron method has been used to measure the absorption cross-section of the discussed heterogeneous materials.This method has been chosen because only one sample isneeded to measure the thermal-neutron absorption cross-section §a of the material.Notice that §a isa parameter that characterizesan in¯nite medium, not a particular ¯nite sample of a¯xedsize used in the measurement. Additionally,Czubek’s method of the §a measurement isindependent ofthe scattering properties of the sample. The physicalprinciples of the §a measurement method, which uses atwo-region geometry, havebeen recapitulated in[6].The system consists of the investigated cylindricalsample of size HS = 2RS surrounded by acylindricalmoderator ( HM = 2RM )coveredwith a K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 219 cadmium shield,which assures vacuum boundary conditions for thermal neutrons. The geometry of the measurement isshown in Fig.3. The sample-moderator system is irradiated by bursts of 14 MeVneutrons that are slowed down, and the timedecay of the thermal-neutron °ux ’(t)isobserved. Pulses from a 3Hedetector are stored (during many consecutivecycles) in amultiscaler,and the decayconstant ¶ of the fundamental exponential mode of the °ux, ’(t) exp({¶ t),isdetermined from the registered curve ¹ [7].The experiment isrepeated using di®erent sizes HM of the outer moderator. In this 1 way the experimentaldependence ¶ = ¶ (xM ; RS =const)isobtained, where xM = 2 . RM

Fig. 3 The experimentalset-up.

Another curve ¶ * = ¶ *(xM ; RS =const)iscalculated based on the di®usion approximation for the thermal-neutron °ux in the two-region cylindricalsystem [8],[9]. The curve ¶ *(xM )isdependent onthe size RS of the inner sample and on the thermal-neutron parameters of the external moderator (that is,the absorption rate v§ ,the di®usion h aM i constant D0M ,and the di®usion coolingcoe± cient CM )and, therefore, they haveto be known with ahigh accuracy.The absorption rate v§ and the di®usion constant D h aM i 0M of Plexiglas,routinely used asthe outer moderator in the measurements, are wellestab- lished [10]and are quoted in [11].Instead of the constant coe±cient CM used earlier, the correction function CM¤ (¶ ¤)isnow used. It results from the di®usion coolingof the thermal-neutron energyspectrum in the two-region system,in which the inner sample is amixture of ahydrogenous and non-hydrogenous media[12]. The ordinate ¶ = v§exp X h a i 220 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

3 1 §a;Ag(v0) Density [g cm ¡ ] § (v ) [cm¡ ] a 0 §a;Plexi(v0) Silver, Ag 10.50 3.71053 195.8

Plexiglas, (C 5H8O2)n 1.176 0.01895 195.8

Table 1 Physicalcharacteristics of silverand Plexiglas.

of the intersection point of the curves ¶ = ¶ (xM ) and ¶ * = ¶ *(xM )determines the absorption cross-section §aS ofthe material of the inner sample:

§ ¶ 1=v = 0; (26) h aSi ¡ Xh i where v isthe neutron speed and the averagesare overthe thermal-neutron °ux energy distribution. The standard deviation ¼ (¶ X )of the intersection point ¶ X iscalculated using acomputer simulation method [13].

6Silver-in-Plexiglasmo delsas heterogeneous samples

6.1 Design of theheterogeneous models

Well-de¯ned models of heterogeneous materials havebeen constructed to perform afully controlled experiment to investigatehow the thermal-neutron absorption cross-section depends onthe grain sizeof the material.The models ful¯l the following assumptions: (1) They consist of two materials: I and II. (2) The di®erence between the absorption cross-sections of material I and II is signi¯- cant; the highly absorbing centres (grains) ofmaterial II are embedded inaweakly absorbing material I. (3) The grains II of regular shapes are regularly distributed inmaterial I. (4) The models di®er from eachother with respect to the grain sizeof material II, but its total mass contribution iskept constant. Plexiglashas been used asthe weaklyabsorbing material,and silverof ahigh purity has been chosen for making the highly absorbing grains. The physicalcharacteristics of silverand Plexiglasare givenin Table1. Their macroscopicabsorption cross-sections havebeen calculatedat the most probable thermal-neutron velocity v0 =2200 m/s,based on the elementalcompositions and onthe microscopicabsorption cross-sections givenin nuclear data tables [14].

Three models of aregular cylindricalshape with HS = 2RS =9cmhave been constructed. Each one consists of layersof two types: the pure Plexiglaslayers are separated by layerswith silvergrains embedded inPlexiglas.The silvergrains are regular cylinders of constant dimensions Hgr = 2Rgr,di®erent in eachmodel. Their total volumecontent in the samples iskept almost constant, about ¿ II =0.05,corresponding to the mass contribution about qII =0.32.The characteristics ofthe three silver-in-Plexiglasmodels are givenin Table2. K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 221

Symbol Size of the Number of Mass con- Volume Average Dimensionless of the grain the grains tribution content chord of size of the Model Hgr [mm] in the of silver of silver the grain grain Model q ¿ d [cm] Y II II 2 h i GR4 4 570 0.3199 0.0500 0.267 0.9563 GR6 6 168 0.3187 0.0498 0.400 1.4344 GR10 10 36 0.3169 0.0494 0.667 2.3906

Table 2 Characteristicsof the silver-in-Plexiglasmodels.

In order to construct the sample of material with regularly distributed absorbing grains, the modelhas been considered as part of an in¯nite spatial grid, in which the grains represent the latticepoints. Aregular prism has been assumed as the grid. The smallcylindrical grains are coaxialwith the entire cylindricalmodel. The verticaland horizontal cross-sections of one of the models (with the biggest grains) are shown in Fig.4. The geometricdetails of the other models can be found in [15].The vertical distance between top and bottom of two neighbouring grains isapproximately equalto the grain height Hgr .The ¯rst and the last (top and bottom) pure Plexiglasslices havethe thickness about 1/2of the grain height. The distribution of the grains in the horizontal cross-section has been selectedfrom the square or hexagonal in¯nite grid. The slicescontaining grains can be rotated with respect to eachother to observe apossible in°uence of the silverdistribution onthe e®ective absorption of the material.

6.2 Experimentalresults and discussion of apossible uncertainty

The sizeof the silvergrains in the di®erent silver-in-Plexiglasmodels varies,keeping always aconstant volumecontent ¿ II of silver.Thus, the models represent aheterogeneous material on which an e®ect of the varying grain sizeon the e®ective absorption can be measured and compared to the absorption of the homogeneous mixture. exp The ¯nal experimentalresults §a for the three investigated heterogeneous media h i exp are reported in Table3. These macroscopicabsorption cross-sections §a can be com- hom 1 h i pared with § = 0:1805cm¡ for the homogeneous mixture consisting of the same h a i components, calculatedat the silvercontent ¿ II =0.05.The comparison clearlyshows that the in°uence of the heterogeneity of the material on the e®ective, macroscopic absorption cross-section for thermal neutrons issigni¯ cant, evenin the caseof the dense grid of the smallestsilver grains, which isclosest to ahomogeneous material.

The absorption cross-section §a isdetermined from the intersection point ofthe theoreticaland experimentalcurves and, therefore, its accuracy ¼ (§a)within the theory of the measurement method depends on the accuracyboth of the experimentalpoints and of the neutron parameters used inthe calculation.The accuracy ¼ (¶ i)ofthe experimental points depends on the characteristics and possibilitiesof the experimentalset-up and 222 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

Fig. 4 Distributionof silvergrains in Model GR10.

Model v§exp §exp a a exp h expi 1 h expi 1 ¾( §a ) h exp i ¼ ( v§a ) [s¡ ] ¼ ( §a ) [cm¡ ] § [%] h i h i h a i GR4 26868 0.1083 0.18 54 0.0002 GR6 20743 0.0836 0.36 78 0.0003 GR10 14999 0.0605 0.33 59 0.0002

Table 3 Experimentalresults obtained for the silver-in-Plexiglasmodels. K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 223 on the precision of determination of the fundamental mode decayconstant ¶ from the decaycurve registered in the multiscaler.In the present experimentalseries, the determination of ¶ has been di±cult. As ¶ describes the exponential decay,its valueshould be constant, independent of ashift t0 of the analysed interval along the timeaxis, that is, ¶ (t0) = const (with astatistical accuracy).The heterogeneity of the sample leads to astronger contamination with higher modes of the pulsed thermal-neutron °ux,and makesmore di±cult the isolation of the fundamental mode. Therefore, the dependence

¶ (t0) = const has been alwayscarefully tested in the interpretation of the experiments.

Finally,the relativestandard deviations ¼ (¶ i)/¶ i havebeen no higher than 0.9%. Consider alsoan uncertainty in the sizeof the sample.Although the geometricdi- mensions HS and RS are known with averyhigh accuracy,how neutrons \see"this size can beaproblem. Namely,the models contain the silvergrain latticein Plexiglas,which isalso used as the outer moderator. In order to investigatethis possible uncertainty,a new theoretical curvehas been calculated, ¶ * = ¶ *(xM ) at 2RS increased by 0.2cm. A new experimentalresult, §exp n ,has then been obtained for eachmodel sample. (Notice h a i that the change ofthe size2 Rs alsochanges the contributions ¿ II and, inaconsequence, the cross-sections §hom ).The new experimentalgrain parameters, Gexp n, have been h a i obtained and the relativechanges, " = (Gexp n Gexp)=Gexp,havebeen calculated.They ¡ appear negligiblewhen compared with the statistical uncertainty ¼ ( § )= § of the h ai h ai measurements [15]. Aseparate problem iswhether the modelrepresents wella part ofanin¯nite, regular latticeof grains. In order to test it we havemade the following experiment for Model GR10,that is,for the sparsest lattice,where the in°uence of grains on the e®ective absorption cross-section isstrongest. The sample has been enclosedonly in acadmium shield and the decayconstant ¶ has been measured. Next,every other layerwith the silvergrains has been rotated 45ºto introduce the intentional perturbation of the lattice (Fig.5). The decayconstant has been again measured. The results are following: Regular lattice: ¶ = 27 505 s{1, ¼ (¶ ) = 68 s{1; {1 1 Perturbed lattice: ¶ = 27 567 s , ¼ (¶ ) = 85 s¡ . The measured decayconstants are in full agreement within one standard deviation. This means that the perturbation of the latticehas not changed the thermal-neutron transport in the investigated volume.This suggests alsothat any possible error in the cut ofthe sample from the in¯nite latticeis not signi¯cant. Finally,we can state that the factors considered abovedo not in°uence signi¯ cantly the experimentalresults. The valueslisted in Table3 can be treated as the measured, e®ective absorption cross-sections with errors on the levelof two standard deviations.

6.3 Comparison ofthetheoretical and experimentalresults obtained for thesilver-in-Plexiglas models

The theoretical approach presented inparagraphs 1to 4has been applied to aninterpre- tation ofthe results of the experiments made on the three silver-in-Plexiglasmodels. 224 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

Fig. 5 Perturbed latticeof grainsin Model GR10 (45 o rotatedlayers).

6.3.1Case A:Grains in the matrix

The theoretical,e® ective, macroscopic absorption cross-sections §eff ofthe three silver- h a i in-Plexiglasmodels havebeen calculatedfrom Eq.(21), and the absorption cross-section §hom of the corresponding homogeneous material from Eq.(20). The energydepen- h a i denceof the thermal-neutron, macroscopicabsorption cross-sections of Plexiglas,§ a1 (E), and of silver,§ a2 (E),isdescribed by a1/ v law.Therefore, these cross-sections are com- pletelyde¯ ned when givenat onlyone energy(cf. Table1). The di®usion cross-section

§d2,present in formula (21), contains the absorption and transport cross-sections ofsil- ver.The energy-dependent, scattering cross-section ¼ s2(E)of silver(given essentially by the free gas model formula) is,for thermal neutrons, verywell approximated by aconstant valueequal to the free-atom scattering cross-section ¼ sf2.Thus, the macroscopic {1 scattering cross-section for silveris § s2(E) = §sf2 = 0.2978 cm .The scattering cosine

· 2(E)of silveris su± ciently accurately given by the approximate relation · 2 = 2/(3A), where A isthe relative-to-neutron atomic mass of silver(cf. [1]). The calculated,energy-averaged, e® ective absorption cross-sections §eff of the three h a i models are listed in Table4. The calculations havebeen performed for the silvervol- ume content ¿ II = 0:05 at ET = 0:02526 eV,corresponding to the experiment temperature t =20ºC. The macroscopicabsorption cross-section of the homogeneous mixture consisting of the samecomponents with the samecontributions has been quoted inparagraph 6.2.

Theoretical valuesof parameter Gav ofthe grain sizee® ect have been calculatedfrom Eq.(17). The experimentalvalues Gexp of the parameter havebeen determined from the experimentaldata § exp for the three silver-in-Plexiglasmodels from Eq.(24). The h ai theoretical and experimentalresults are compared in Table4. Plots of the grain e®ect K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 225

exp eff 1 exp Gav G Model § [cm¡ ] G G ¡exp [%] h a i av G GR4 0.1164 0.64490.6000 7.48 GR6 0.0985 0.54570.4632 17.81 GR10 0.0754 0.41770.3352 24.61

Table 4 Comparisonof the theoreticaland experimental results obtained for the silver-in- Plexiglas models. parameter calculatedas afunction of the grain size Y are shown in Fig.6, where the three experimentalvalues are alsomarked. The ¯rst curve, G(Y ),has been calculatedfrom the one-speed approximation, Eq.(5) with (13) and (4), using the valuesof the cross- sections at the thermal-neutron energy ET .The second curve, Gav(Y ),has been obtained from Eq.(17) with (21) and (20). The two functions present adi®erence between the theoretical results based on the one-speed treatment and on the approach in which all energy-dependent magnitudes are averaged overthe thermal-neutron energydistribution. exp Although the curve Gav(Y )iscloser to the experimentalpoints G than the curve G(Y ), exp the prediction Gav (Y )isstill unsatisfactory for the experimentalpoints G . Two other curves inthe plot are considered in the next paragraph.

Fig. 6 Parameterof the grainsize e¬ ect as a function G(Y )ofthe grainsize (at S = 195.8, ¿ =0.05)compared to the experimentalresults G exp for three silver-in-Plexiglas models.

6.3.2Case B:Two-grain medium

Weconsider Plexiglasas amedium built ofgrains, too. In this case,the volumetreated as aPlexiglasgrain isde¯ ned by the latticepoints, without the spaceoccupied in corners 226 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234 by parts of the silvergrains. The de¯ned volumeand its surface determine from Eq.(12) the averagechord length d1 of the grain of the matrix (although such atreatment violates slightlythe assumption of the convexcurvature ofthe grain surface). Then the e®ective ~ absorption cross-section §a1 of the weaklyabsorbing material isalsocalculated from Eq. (11). The e®ective cross-section of the material isexpressed by:

eff § 00 = (1 ¿ )§~ + ¿ §~ (27) a ¡ a1 a2 according to the de¯nition in paragraph 2.

Fig. 7 Energydependence ofthe thermal-neutrontransport cross-section of Plexiglas and of silver,and the thermal-neutron® ux energydistribution.

eff The energy-averaged,e® ective cross-section §a (E) 00 can be calculatedwhen the ~ h i dependence §a1 (E)isknown. It contains not onlythe absorption cross-section §a1 (E) but alsothe energy-dependent transport cross-section §tr1(E)of Plexiglas[cf. Eqs. (11), (9) and (8)]. As in any hydrogenous material,the scattering cross-section and the averagecosine of the scattering angleare strongly dependent on the thermal-neutron energy.

The functions §s1(E) and · 1(E)havebeen obtained inthe sameway as in [16],following Granada’s model of the slow-neutron scattering kernel[17], [18], [19]. The dependence

§tr1(E)isshown in Fig.7, together with the thermal neutron distribution M (E). Addi- tionallythe silvercross-section §tr2(E)isalso plotted.

Theoretical valuesof the parameter of the grain sizee® ect Gav00 ,resulting from the present approach to the considered three silver-in-Plexiglasmodels, are speci¯ed in Table 5.

According to assumption (27), anew set of curveshas been obtained, G00(Y ) and

Gav00 (Y ),corresponding to the one-velocitytreatment [Eq.(5) with (27)] and to the energy averaging [Eq.(23) with (22)], respectively.They are plotted in Fig.6 asthe two lowest lines.In order to plot these new functions versus Y ,the lengths d1 havebeen expressed using the averagechord lengths d2 of the corresponding silvergrains, d1 = kmd2, where K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 227

exp eff 1 Ga00v Ga00v G Model § 00 [cm¡ ] G00 ¡exp [%] h a i av Gav G GR4 0.1080 0.59830.9277 - 0.28 GR6 0.0873 0.48370.8864 4.43 GR10 0.0634 0.35120.8408 4.77

Table 5 Comparisonof the resultsof the two-graintheory with the experimentaldata.

km isa constant dependent onthe sample,that is,on the particular spatial distribution of grains. The new approach, especially Gav00 (Y ),givesmuch better agreement with the experimentalresults Gexp.

7Rock{° uidsamples

When rockmaterials are investigated,the sample usually contains grains which create heterogeneity.The experiments on the arti¯cial rock material (silicon,Si) and on the natural rockmaterial (diabase) havebeen performed to examinethe in°uence of the granulation on the §a measurement. The samples havebeen prepared as mixtures of grained materials with aqueous solutions of boric acid,H 3BO3. The thermal-neutron absorption rates v§exp ofthe complexsamples havebeen mea- h a i sured with Czubek’s method, as outlined in paragraph 5.The absorption rate of the homogeneous mixture v§hom has been obtained from Eq.(26) and §hom has been h a i a calculatedfrom the following formula:

§hom = » q §M + (1 q )§M a rock rock ¡ rock fluid j k M M M = » qrock§rock + (1 qrock) (1 k)§H O + k§H BO (28) ( ¡ ¡ 2 3 3 ) h i which results from Eq.(1), and where » isthe mass density ofthe complexsample, qrock isthe mass fraction ofthe rockin the complex rock-°uid sample, k isthe mass fraction M of H3BO3 in the °uid, and §x = §a=» isthe relevant,mass absorption cross-section calculatedon the basis of the elementalcomposition and the microscopicabsorption cross- sections ([14],except for boron). The absorption cross-section of the boric acidH 3BO3 was measured [20]in our Lab to avoidan uncertainty introduced by the °uctuating isotopic ratio of natural boron. Finally,the experimental grain parameter Gexp has been obtained from Eq.(24).

The theoretical grain parameter Gav has been calculatedfrom Eq.(17), assuming Case A (Eq. 21), that is,that the grains of the rock(a weakabsorber) are embedded in the matrix (a strong absorber). 228 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

7.1 Silicon asthearti¯ cial rockmaterial of grains

Anarti¯cial rock material allows us (1) to createmore realisticexperimental conditions, that is,a sample that contains grains of a°uctuating size,spread irregularly (not preserving aregular lattice,as in the model samples), and (2) to keepexactly known thermal-neutron parameters of the contributing materials. High purity silicon(0.9999 Si) has been used. Its thermal-neutron mass macroscopic cross-sections are: M 2 1 Absorption: §a (v0) = 0:00367 0:00006 cm g¡ . M § 2 1 Scattering: § (v ) = 0:04382 0:00004 cm g¡ s 0 § ([21],[14]). The complexsamples havebeen made of two di®erent grain sizesembedded in various H3BO3 solutions. The speci¯cation ofthe samples isgiven in Table6.

Anexampleof the theoretical dependence of the grain parameter Gav for the investigated samples isshown in Fig.8 (the upper line).The parameter isplotted asafunction of the grain diameter, 2 Rgr,with the other variables¯ xedaccording to the experimental conditions. The theoretical grain parameter is,in practice,equal to unity in the considered interval of the grain size.The exactvalues for the samples are givenin Table 6.F or sample Si-07, onlythe results obtained for the lowest and highest concentrations of the H3BO3 solutions used are shown. In the calculation,the grain sizes d have been de¯ned by the averagechords relevant to the mean diameters ofthe grains used, that is, 0.6mm and 1.7mm for the grains of (0.5 to 0.7) and of (1.4 to 2.0) mm of sievemesh, respectively.

hom Sample Silicon grain Nominal Volume con- §a Gav §a h i Code size d concentration tribution of 1 = 1 Eq. (20) Eq.(17) S §a 2 1 [mm of sieve ofboric acid silicon ¿ [cm¡ ] mesh] k [wt. %]

Si-07c 0.5 0.7 1.2 0.497 12.89 0.0529 0.9998 ¥ Si-07d 0.5 0.7 3.5 0.493 32.86 0.1298 0.9999 ¥ Si-2 1.4 2.0 1.2 0.474 12.89 0.0549 0.9996 ¥ Si-2a 1.4 2.0 2.0 0.479 19.81 0.0817 0.9997 ¥ Density ofsilicon » = 2.3280 0.0002 g cm{3 Si §

Table 6 Specication of the complexsamples (mixtures of silicon grains with the H 3BO3 solutions)and the theoreticalgrain parameter.

The thermal-neutron absorption rates v§ exp of the complexsamples havebeen mea- h ai sured. The absorption rate of the homogeneous mixture v§ hom and the experimental h ai parameter Gexp has been obtained. The experimentalabsorption cross-sections of pure exp silicon, § (v0),has been evaluated from the usual procedure in Czubek’s method. The exp ratio ofthe obtained experimentalvalue to the cross-section ofsilicon,§ (v0)/§Si(v0), K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 229

Fig. 8 Theoreticalgrain parameter Gav asa function ofthe geometricgrain size and example of the experimentalresults Gexp (with the onestandard deviation marked) for the siliconand diabasecomplex samples. has been obtained. Details of the experiments and calculationprocedures can be found in [12].The results are collectedin Table7. No e®ect of heterogeneity isobserved for the complexsamples tested, with the silicongrains of about 0.6and 1.7mm, the silicon contribution ¿ 0.5,and the absorption cross-section ratios 1/S 13 and 1/S 20. º exp º exp º Both the grain parameter G for the complexsample and the ratio § (v0)/§Si(v0) for the material of interest are equalto unity within the one standard deviation,which means that the experiment has con¯rmed the theoretical predictions on the homogeneity of the prepared complexmedia.

7.2 Diabase as thenatural grained rockmaterial

Diabase samples used in the present neutron measurements were taken from the com- pact deposit exploitedby \Nied¶zwiedziaG¶ ora" Quarry (Quarries of Natural Resources,

Krzeszowicenear Krak¶ow, Poland). The absorption cross-section §a = §R of diabase (as for in¯nite, homogeneous medium) was measured inour Lab, using Czubek’s method, inanindependent experiment [22],[23] on the ¯nesamples.The scattering cross-section

§s has been calculated[21], [14] according to atypicalelemental composition ofdiabase [24].The following mass cross-sections havebeen obtained: M 2 1 Absorption: §a (v0) = 0:00890 0:00015 cm g¡ M § 2 1 Scattering: § (v ) = 0:09428 0:00011 cm g¡ . s 0 § The samematerial of granulation between 6.3and 12.8mm of sievemesh has been used to calculateand measure the grain e®ect. The complexsamples of the diabase grains and aqueous solutions ofboric acidH 3BO3 were prepared. The listof the complex 230 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

Complexsample (Si +absorber) Silicon exp hom exp exp M exp exp § (v0) Sample v§a v§a G § (v0) § (v0) h i h i §Si(v0) Code ¼ ( v§ hom) ¼ ( v§ exp) ¼ (Gexp) ¼ (§M exp) ¼ (§exp) ¼ (§exp/§ ) h ai h ai Si [s{1] [s{1] [cm2g{1] [cm{1]

Si-07c 13 115 13 050 0.995 0.00340 0.00791 0.99 51 62 0.006 0.00031 0.00072 0.09 Si-07d 32 185 32 222 1.001 0.00374 0.00870 1.02 143 178 0.007 0.00090 0.00210 0.25 Si-2 13 621 13 678 1.004 0.00389 0.00906 1.06 53 69 0.006 0.00034 0.00080 0.10 Si-2a 20 273 20 250 0.999 0.00355 0.00827 0.97 85 107 0.007 0.00055 0.00127 0.15

Table 7 Comparisonof the experimentalresults for silicon complex samples with the datafor the homogeneousmedia.

hom Sample Nominalconcen- Volumecontri- §a Gav §a h i Code tration ofboric bution ofthe 1 = 1 Eq.(20) Eq.(17) S §a2 acid solution rock ¿

Dia-a 1.3 0.572 4.8 0.0570 0.978 Dia-b 2.0 0.559 6.9 0.0783 0.985 Dia-c 2.5 0.557 8.4 0.0932 0.988 Density ofdiabase » = 2:7680 0:0002 g cm 3 R § ¡ Table 8 Specications of the complexsamples: mixtures of diabasegrains (6.3 12.8 mm of ¥ sievemesh) with the H 3BO3 solutions,and the theoreticalgrain parameter.

samples and the theoreticallycalculated grain parameters Gav are givenin Table8. The mean sizeof the diabase grain in eachsample has been estimated as the diameter 2 Rgr of an equivalentsphere, based on the number and total mass ofthe grains contained. An exampleof the theoretical grain parameter asafunction ofthe grain sizeis plotted inFig. 8(the lower line).The contributions of the components and the ratio of the absorption cross-sections havebeen assumed as for the realsample, Dia-a. The measurements of the e®ective absorption cross-sections of the complexsamples and the evaluation of the experimentalcross-section for diabase havebeen performed in the sameway as described in the previous paragraph for the siliconsamples. The results are collectedin Table9. The theoretical and experimentalgrain parameters °uctuate about 0.98.The smalldiscrepancies between them can result from the necessary approximation of the grain sizein the theoretical calculation.The heterogeneity e®ect, although weak,is clearly seen in acomparison with the results for the samples of ¯ne silicon(cf. Fig.8). The experimentalcross-section of diabase itself,obtained from K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 231

Complexsample (Diabase + absorber) Diabase exp hom exp exp Mexp exp § (v0) Sample v§a v§a G § (v0) § (v0) h i h i §R(v0) Code ¼ ( v§ hom) ¼ ( v§ exp) ¼ (Gexp) ¼ (§Mexp) ¼ (§exp) ¼ (§exp/§ ) h ai h ai Si 1 1 2 1 1 [s¡ ] [s¡ ] [cm g¡ ] [cm¡ ]

Dia-a 14 192 13 881 0.978 0.00801 0.02217 0.90 69 35 0.005 0.00016 0.00044 0.02 Dia-b 19 470 18 989 0.975 0.00749 0.02073 0.84 87 101 0.007 0.00036 0.00100 0.04 Dia-c 23 141 22 566 0.975 0.00721 0.01996 0.81 102 118 0.007 0.00044 0.00122 0.05

Table 9 Comparisonof the experimentalresults for diabase complex samples with the datafor the homogeneousmedia. the standard interpretation of the measurement (assuming ahomogeneous medium) is, however,underestimated evenup to 20 %.The valueof the underestimation depends on the ratio of the absorption cross-sections of the components in the complexsamples. The higher isthe ratio 1/ S,the lower isthe interpreted cross-section §exp of diabase, which isseen from the data in Tables8 and 9.This behaviour isin agreement with the predictions ofthe theoretical model.

8Conclusions

The theoretical approach to the thermal-neutron e®ective absorption inthe type of heterogeneous medium considered here has been veri¯ed experimentallyon dedicated model samples and onnatural rocksamples. The heterogeneity e®ect depends on few parameters: on the ratio S ofthe absorption cross sections of the components, on their scattering properties, on the grain size,and on the mass (or volume) contributions of the components. Acomparison of the in°uence of the grain sizeon the absorption indi®erent samples can be done if the sizeis not expressed geometricallybut using the neutron di®usion units. The thermal-neutron di®usion mean free path ld in agivenmaterial isde¯ ned by the macroscopicdi® usion cross section, ld = 1/§d.The grain size Y isreferred to the geometricaldimension (the averagechord length d)and to the neutron mean free path, as shown in Eq.(16). The grain sizes Y inthe particular samples are listed in Table10. For the silver-in-Plexiglas samples,the geometricalrelations are preciselyde¯ ned and it ispossible to de¯ne accuratelythe grain sizeof the matrix material.The grain sizes Y for the two components are comparable and, as found earlier(paragraph 6.3.2),the medium in this caseshould beconsidered rather as the two-grain system. For the rock{°uid samples,only the mean geometricalgrain sizeis known, the spaces between the grains (¯lled with the matrix material) are highly irregular, and the sizeof 232 K.Drozdowiczet al. / CentralEuropean Journal of Physics 2 (2003)210{234

Sample GR10GR6 GR4 Si-07 Si-2 Dia

Y 2.391.43 0.96 0:004 0:012 0:2 h igrain ¹ ¹ ¹ Y 3.963.04 1.49 0:12 0:34 2:0 h imatrix ¹ ¹ ¹ Y matrix h Yi 1.7 2.1 1.6 30 30 10 h igrain ¹ ¹ ¹ Adequate Two-grainsystem Homogeneousmedium Grains in the matrix theoretical model

Table 10 Ratioof the dimensionlessgrain sizes Y inthe particularsamples. h i the matrix grain isonly estimated. In the caseof the siliconsamples, the ratio of the

Y1 and Y2 sizesindicates that the system should be considered as the Si grains in the °uid matrix. Additionally,the geometricalsizes of grains are verysmall in comparison to the corresponding neutron mean free paths. Both the theoretical grain-in-the-matrix model and the experiment con¯rm in this casethat the samples are homogeneous from the point of viewof the neutron di®usion. The diabase grains are much greater, but their Y sizeis stillone order lower than the estimated matrix grain size.Therefore, the grain- in-the-matrix mode isstill applicable. The theoretical calculationand the experiment show averyweak grain e®ect G due to alow ratio of §a ofhighly and weaklyabsorbing components, two orders lower than in the caseof the silver-in-Plexiglas models (cf. Tables 8 and 1). The generalbehaviour isas follows: the thermal-neutron absorption cross-section of any heterogeneous medium isalways lower than ahomogeneous one consisting of the samecomponents. This e®ect can be asource of an underestimation of the absorption cross-section of the rockmaterial when coarse-grain samples are used in laboratory measurements, as ithas been proved in the paper. The opposite e®ect, an overestima- tion of the calculatedabsorption cross-section of ageologicalformation can happen if this medium istreated as homogenous material,while in fact it contains concretions of strongly di®ering absorption. Similarly,for such formation, an overestimated valuecan alsobe obtained experimentallyfrom ameasurement on its sample which isperfectly homogenised.

Acknowledgments

Wewish to thank Prof. NilsG. SjÄostrand from Dept. of Reactor Physics at the Chalmers Universityof Technology(GÄ oteborg, Sweden) for stimulating discussions on the problem. Wethank our colleagues,M.Sc. J. Dabrowska, Eng. A.Kurowski, Eng. J.Burda, and Mr. W.Janik, for their participation in the measurements. The work was partly sponsored by the State Committee for Scienti¯c Researchof Poland (Project No.9 T12B 027 16). K.Drozdowiczet al. / CentralEuropean Journal of Physics2 (2003)210{234 233

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