- Email: [email protected]
Monte Carlo calculation of epithermal neutron resonance self-shielding factors in foils of different materials
Applied Radiation and Isotopes 56 (2002) 945–951
Monte Carlo calculation of epithermal neutron resonance selfshielding factors in foils of different materials I.F. Gonc-alves*, E. Martinho, J. Salgado ! Instituto Tecnologico e Nuclear, Estrada Nacional 10, 2686-953 Sacav!em, Portugal Received 7 August 2001; received in revised form 4 January 2002; accepted 7 January 2002
Abstract Epithermal neutron resonance self-shielding factors in foils of materials used as activation detectors or as targets for radionuclide production have been calculated using the MCNP code. Two irradiation conditions have been considered: (a) foils immersed into an isotropic neutron flux and (b) foils submitted to a collimated neutron beam. The self-shielding factors, Gres ðtÞ; for gold, indium, manganese and cobalt have been compared with available published values. The selfshielding factor depends on various physical and nuclear parameters. However, it is shown that for the isotropic case and for high absorber elements submitted to a collimated beam, a dimensionless variable could be adopted that describes the self-shielding factors of different materials by quasi ‘‘universal curves’’. Gres ðtÞ for the collimated beam are always higher than those for the isotropic case. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Epithermal neutrons; Resonance self-shielding factor; MCNP code
1. Introduction
2. Methodology of calculation
Some materials used in neutron dosimetry or for radionuclide production for medical purposes exhibit resonances in the epithermal region of the reactor neutron spectrum. Consequently, the interpretation of sample activation data requires the application of resonance self-shielding factors. In a previous work (Gonc-alves et al., 2001), the resonance self-shielding factor in wires was calculated. The aims of the present work are (a) to establish a calculation method for the resonance self-shielding factor in foils by using the MCNP code (Briesmeister, 2000) and updated cross-sections (ENDF-B6), and (b) to interpret its dependence on several physical and nuclear parameters. To validate the methodology, the results are compared with available values obtained by other authors.
The resonance self-shielding factor, Gres ðtÞ; in foils of thickness t; is defined as the ratio between the reaction rates per atom in the real sample and in a similar and infinitely diluted sample. Thus
*Corresponding author. Tel.: +351-1-955-00-21; fax: +3512199-41525. E-mail address: [email protected] (I.F. Gonc-alves).
R E2 Gres ðtÞ ¼ R EE21 E1
FðEÞsng ðEÞ dE F0 ðEÞsng ðEÞ dE
;
ð1Þ
where F0 ðEÞpE 1 is the original, non-perturbed, epithermal neutron flux per unit energy interval inside the infinitely diluted sample, FðEÞ represents the perturbed epithermal neutron flux inside the real sample, sng ðEÞ designates the ðn; gÞ cross-section, and E1 and E2 are, respectively, the lower and the upper limits around the resonance energy Eres : The total neutron crosssection has been adopted in the calculation of the perturbed neutron flux FðEÞ; which takes into account the neutron scattering in the sample. In all calculations, the density for infinite dilution was assumed to be r ¼ 106 r0 ; r0 representing the density of the real sample (Gonc-alves et al., 2001).
0969-8043/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 0 2 ) 0 0 0 5 5 - 6
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
946
Table 1 Some physical and nuclear properties of the studied elements Element
Nuclide
r0 (g cm3)
y
A (g)
sðEres Þ (barn)
Eres (eV) sg
Cobalt Copper Gold Indium Manganese Rhenium
Co-59 Cu-63 Au-197 In-115 Mn-55a Mn-55b Re-185
8.9 8.96 19.3 7.3 7.3
1 0.692 1 0.957 1
58.9 63.5 197.0 114.8 54.9
21
0.374
186.2
132 579 4.91 1.46 336 1098 2.16
ss 2
8.49 10 4.12 102 2.74 104 2.80 104 46 28 2.33 104
Gres (eV) stot
3
9.52 10 5.05 102 3.4 103 1.16 103 3.24 103 1.41 103 1.22 103
4
1.04 10 9.17 102 3.08 104 2.92 104 3.29 103 1.44 103 2.45 104
Gg
Gn
G
0.47 0.485 0.124 0.072 0.435 nac 0.055
5.15 0.59 0.015 0.003 22.0 15.4 0.003
5.62 1.075 0.139 0.075 22.4 nac 0.058
a
First resonance. Second resonance. c Not available. b
For a given thickness value, t; a neutron energy dependent resonance self-shielding factor, Gt ðEÞ; can be defined as
Isotropic case Au foil r=10 mm r=5 mm r=1 mm
0.6
ð2Þ
where RRðE; r0 Þ and RRðE; 106 r0 Þ are the reaction rates for the energy E; corresponding to the densities r0 and 106 r0 ; respectively.
3. Results and discussion
0.5
Gres(t)
RRðE; r0 Þ ; Gt ðEÞ ¼ RRðE; 106 r0 Þ
0.7
0.4 0.3 0.2 0.01
The resonance self-shielding factor depends on material density and thickness and on atomic mass, natural abundance, and resonance neutron cross section. Table 1 shows the values of the density, r0 ; natural abundance, y; atomic mass, A (Tuli, 2000), resonance energy, Eres ; scattering, capture and total resonance cross-sections, ss ; sg ; stot (ENDF-B6) and the resonance widths, Gn ; Gg and GðG ¼ Gn þ Gg Þ (Mughaghab et al., 1981; Mughaghab, 1984), for the studied elements. Two irradiation conditions have been simulated: 1. Isotropic caseFcircular foils immersed into an isotropic neutron flux; 2. Collimated caseFcircular foils submitted to a collimated neutron beam perpendicular to the foil. 3.1. Isotropic case In order to study the influence of the foil edge effect, three different foils radii have been simulated (R ¼ 10; 5 and 1 mm). The resonance epithermal neutron selfshielding factor, Gres ðtÞ; for gold foils (for tX0:005 mm) as a function of the foil thickness is shown in Fig. 1. The curves show the same behaviour. For R ¼ 10 and 5 mm they are coincident. However, for R ¼ 1 mm the curve is shifted to higher values of Gres ðtÞ; being approximately, 10% higher for t > 0:05 mm: Similar results have been
0.1
Foil thickness, t (mm) Fig. 1. Resonance neutron self-shielding factor for gold foils as a function of the foil thickness, for the isotropic case.
obtained for other elements. For this reason, to avoid foil edge effects, all subsequent simulations have been performed with R ¼ 10 mm: Fig. 2 shows the energy dependent self-shielding factor of indium for different foil thickness. The minimum of the valley is observed at the resonance energy; for t ¼ 1 mm; Gres ðE Þ has its lower value. As the thickness decreases, the effect of shielding diminishes and the curves become narrower. For indium, sg > ss ; the absorption predominates over the scattering and then Gres ðEÞo1; for all energies; the curves are practically symmetrical around Eres : Similar results were obtained for gold and rhenium, which are also high neutron absorbers. On the other hand, as can be seen in Fig. 3, for manganese, which is a high neutron scatterer (sg oss ) the self-shielding factor, Gres ðEÞ > 1; for EoEres : A neutron entering in the foil with energy EEEres can suffer one or more scattering interactions before capture and it is absorbed at EoEres : The corresponding peaks are positioned at energies EoðEres GÞ: The energy of the peak decreases as the
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
947
1.6
1.0
Isotropic case t = 0.1 mm
1.4 1.2
Gres(E)
Gres(E)
0.8
0.6 Isotropic case In foil t=1 mm t=0.1 mm t=0.01 mm t=0.001 mm t=0.0001 mm
0.4
0.2
0.0 0.5
1.0
1.5
2.0
2.5
3.0
1.0 0.8 0.6 0.4 0.2 0.0 -20
3.5
-10
Au Co
Energy, E (eV) Fig. 2. Variation of the energy dependent resonance selfshielding factor of indium foils of different thicknesses with neutron energy, for the isotropic case.
0
10
20
Reduced energy, (E-Eres)/Γ In Mn(1)
Re
Cu Mn(2)
Fig. 4. Variation of the energy dependent resonance selfshielding factors of gold, indium, rhenium, copper, cobalt and manganese foils ðt ¼ 0:1 mmÞ with the reduced neutron energy, for the isotropic case.
1.8 Isotropic case Mn foil t=1 mm t=0.1 mm t=0.001 mm
1.6 1.4
Gres(E)
1.2 1.0 0.8 0.6 0.4 0.2 0.0 100
200
300
400
500
600
Energy, E (eV) Fig. 3. Variation of the energy dependent resonance selfshielding factor of manganese (first resonance) foils of different thicknesses with neutron energy, for the isotropic case.
thickness increases. The decrease of the energy is more pronounced for manganese, which has a ratio Gn =G higher than those of the other neutron scatterers, cobalt and copper. Fig. 4 shows Gres ðEÞ as a function of a ‘‘reduced energy’’, Ered ¼ ðE Eres Þ=G; for t ¼ 0:1 mm. Gold, indium and rhenium have higher capture cross-sections than the other elements: the corresponding Gres ðEÞ has pronounced valleys and the curves are wider. Similar results were obtained for other thicknesses. Fig. 5 compares the self-shielding factor, Gres ðtÞ of gold, indium, manganese and cobalt calculated in this work with published values. The agreement for gold and indium foils between the values calculated in this work and the previously published values is excellent. For manganese and cobalt, the values calculated in this work
are systematically larger than those previously published. The calculated self-shielding factor, Gres ðtÞ; as a function of the foil thickness is shown in Fig. 6. The behaviour of the curves is similar for all materials. However, the decrease of Gres ðtÞ as the thickness increases is faster when the resonance cross section and/or the density is higher (gold, indium and rhenium). Cobalt has an intermediate cross-section and its Gres ðtÞ lies among the values of the other materials. Following the results of Gonc-alves et al. (2001), an adimensional variable, z0 ; was introduced in order to convert all values into an unique curve. This variable is defined as 1=2 Gg 0 ð3Þ z ¼ Stot t G Stot being given by r Stot ¼ 0 yNA stot ðEres Þ; A
ð4Þ
where NA is the Avogadro number. Fig. 7 shows the resonance self-shielding factor as a function of z0 : In spite of the wide ranges of variation of s; r and A; it is found that all Gres values (except those of copper) can be described by an unique curve. The relative errors in the z0 and Gres values are estimated to be about 5%. In experimental work, the range of z0 is limited to 0:01oz0 o20: For z0 o0:01; the resonance self-shielding Gres E1 and it is not necessary to perform any correction. For z0 > 20; then Gres o0:15; the self-absorption in the sample is high and the reaction rate is very low. Within this range of values, the following equation was adjusted to
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
948
1.0
0.6
Isotropic case In foil This work MCL91 YAM65 BAU63
0.8
0.6
Gres(t)
0.8
Gres(t)
1.0
Isotropic case Au foil This work MCL91 YAM65 BRO64-exp BRO64-cal JEF83 MCL90
0.4
0.4
0.2 0.2 0.0 0.0 -4 10
10
-3
-2
10
-1
10
0
10
10
-4
10
-3
Foil thickness, t (mm) 1.2
1.2
1.0
1.0
0.2
Isotropic case Mn foil This work MCL91 YAM65 JEF83 MO87 KUM86
0.0 -4 10
10
-3
Gres(t)
Gres(t)
0.4
0
10
10
0.8
0.8
0.6
-1
-2
10
Foil thickness, t (mm)
0.6
0.4
0.2
-2
10
-1
10
0
10
Isotropic case Co foil This work MCL91 YAM65 EAS62 KUM86 MCL90
0.0 -4 10
10
Foil thickness, t (mm)
-3
-1
-2
0
10
10
10
Foil thickness, t (mm)
Fig. 5. Comparison of resonance self-shielding factors of gold, indium, manganese and cobalt calculated in this work with values (experimental and calculated) published by other authors, for the isotropic case. MCL90, 91 (Lopes and Molina Avila, 1990; Lopes, 1991), YAM65 (Yamamoto and Yamamoto, 1965), BRO64 (Brose, 1964), JEF83 (Jefferies et al., 1983), BAU63 (Baumann, 1963), MO87 (Mo and Ott, 1987), KUM86 (Kumpf, 1986) EAS62 (Eastwood and Werner, 1962).
the calculated points:
1.0
Gres ðlog z0 Þ ¼ 0:625 0:410 log z0 0:055ðlog z0 Þ2
0.8
0:0066ðlog z0 Þ5 0:00157ðlog z0 Þ6
ð5Þ
2
with r ¼ 0:987:
Gres(t)
þ 0:074ðlog z0 Þ3 þ þ0:0144ðlog z0 Þ4
0.6 0.4
3.2. Collimated case
0.2
Fig. 8 shows Gres ðEÞ as a function of a ‘‘reduced energy’’, Ered ¼ ðE Eres Þ=G; for t ¼ 0:1 mm: As for the isotropic case (see Fig. 3), the Gres ðEÞ of Au, In and Re have pronounced valleys and the curves are wider. Similar results were obtained for other thicknesses. The calculated self-shielding factor, Gres ðtÞ; as a function of the foil thickness is shown in Fig. 9. The behaviour of the curves corresponding to high absorber materials is similar to that of the isotropic case. However, the high scatterer materials (Mn and Co) show an initial increase of Gres ðtÞ; particularly in the Mn first resonance, which is >1, for tp0:2 mm: This effect is, certainly, due to multiple scattering into the foil.
0.0
Isotropic case Au In Re Cu Co Mn(1) Mn(2) 10
-4
10
-3
10
-2
10
-1
10
0
Foil thickness, t (mm) Fig. 6. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, cobalt and manganese on the foil thickness, for the isotropic case. The symbol Mn(i) means
Fig. 10 shows the resonance self-shielding factor of the high absorber elements and copper as a function of z0 : Within the range 0:01pz0 p30; the following equation was adjusted to the calculated points corresponding to
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951 1.2
1.0
Isotropic case Au In Re Cu Co Mn(1) Adjusted
0.6
1.0 0.8
Gres(t)
0.8
Gres(z')
949
0.4
Collimated case Au In Re Cu Co Mn(1) Mn(2)
0.6 0.4
0.2
0.2
0.0 -2 10
10
-1
10
0
10
1
10
0.0
2
10
-4
10
z'=Σtot.t.(Γγ /Γ)
1/2
-3
10
-2
10
-1
10
0
Foil thickness, t (mm)
Fig. 7. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, cobalt and manganese on the adimensional variable z0 ¼ Stot tðGg =GÞ1=2 ; for the isotropic case. The symbol Mn(1) means the first manganese resonance.
Fig. 9. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, cobalt and manganese on the foil thickness, for the collimated beam case. The symbol Mn(i) means the ith manganese resonance. 1.2
2.5
1.0 Collimated case t = 0.1 mm
0.8
Gres
2.0
Gt(E)
1.5
0.6
1.0
0.4
0.5
0.2
0.0 -25
-20
-15
-10
-5
0
5
10
15
20
25
In Mn (1)
Re
10
-2
10
-1
10
0
z'=Σtot.t.(Γγ /Γ)
Reduced energy, (E-Eres)/Γ Au Co
0.0 -3 10
Collimated case Au In Re Cu Adjusted
Cu Mn (2)
Fig. 8. Variation of the energy dependent resonance selfshielding factors of gold, indium, rhenium, copper, cobalt and manganese foils ðt ¼ 0:1 mmÞ with the reduced neutron energy, for the collimated beam case.
10
1
10
2
1/2
Fig. 10. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, on the adimensional variable z0 ¼ Stot t ðGg =GÞ1=2 ; for the collimated beam case.
points. Copper, as seen in Fig. 10, has a similar behaviour to that of the high absorber elements. the higher absorbers: 3.3. Comparison of both cases Gres ðlog z0 Þ ¼ 0:793 0:377 log z0 0:169ðlog z0 Þ2 þ þ 0:0616ðlog z0 Þ3 þ 0:0456ðlog z0 Þ4 0:00536ðlog z0 Þ5 0:0042ðlog z0 Þ6
ð6Þ
with r2 ¼ 0:997: The copper value lies below the adjusted curve. Fig. 11 shows the resonance self-shielding factor of the high scatterer elements and copper as a function of z0 : It is not possible to adjust a curve to the calculated
Fig. 12 compares the resonance self-shielding factor of gold (high absorber) and manganese (high scatterer) calculated for the isotropic and collimated cases. The values of Gres ðtÞ for the collimated case are always higher than those for the isotropic case, because the neutron pathways in the foil are, in average, greater for the isotropic flux than for the collimated beam. In, Re and Cu have a similar behaviour to that of Au, while Co is comparable to Mn.
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
950
4. Conclusions
1.2 1.1 1.0
Gres
0.9 0.8 0.7 0.6
Collimated beam Cu Co Mn(1)
0.5 0.4 0.3
10
-2
10
-1
10
z'=Σtot.t.(Γγ /Γ)
0
10
1
1/2
Fig. 11. Dependence of the resonance self-shielding factors of copper, cobalt and manganese on the adimensional variable z0 ¼ Stot t ðGg =GÞ1=2 ; for the collimated beam case. The symbol Mn(1) means the first manganese resonance.
1.2 1.0
Gres(t)
0.8 0.6 Au foil collimated isotropic
0.4
The energy dependent self-shielding factor and the self-shielding factor of gold, indium, rhenium, copper, cobalt and manganese foils of different thicknesses have been calculated. The energy dependent self-shielding factors of materials with high capture cross-sections have a different behaviour as compared to those of materials with high scattering cross-sections. In the first case, Gres ðEÞo1 in the whole energy interval; in the second case, Gres ðEÞ > 1 for EoEres because neutrons entering the foil with energy Eres can suffer one or more scattering interactions before their capture at a lower energy can occur. In the case of foils immersed into an isotropic neutron flux, a dimensionless variable can be introduced which converts the dependence of the resonance self-shielding factor on physical and nuclear parameters in a quasi ‘‘universal curve’’ valid for all materials. If the foils are irradiated in a collimated neutron beam, the thickness dependence of the resonance self-shielding factor on physical and nuclear parameters is different for high absorber and high scatterer elements. In the first case, it is also possible to adjust an unique curve to the calculated values. In the second case, the curve attains a maximum (>1) for lower thick masses and the position and value of this maximum depend on the ratio sg =ss : The values of Gres ðtÞ for the collimated case are always higher than those for the isotropic case, because the neutron pathways in the foil are, in average, larger in the case of isotropic flux than for the collimated beam.
0.2 0.0 10
-4
10
-3
10
-2
10
-1
10
0
Foil thickness, t (mm) 1.2 1.1 1.0
Gres(t)
0.9 0.8 0.7 0.6 0.5
Mn foil collimated (Mn1) isotropic (Mn1) collimated (Mn2) isotropic (Mn2)
0.4 0.3 -4 10
10
-3
10
-2
10
-1
10
0
Foil thickness, t (mm) Fig. 12. Comparison of the resonance self-shielding factors of gold (high absorber) and manganese (high scatterer) calculated for the isotropic and collimated cases.
References Baumann, N.P., 1963. Rep. DP-817. Briesmeister, J.F. (Ed.), 2000. MCNPFA general Monte Carlo n-particle transport code, Los Alamos National Laboratory report LA-13709-M. Brose, M., 1964. Zur Messung und Berechnung der Resonanzabsorption von Gold Folien. Nukleonik 6, 134. Eastwood, T.A., Werner, R.D., 1962. Resonance and thermal neutron self-shielding in cobalt foils and wires. Nucl. Sci. Eng. 13, 385–390. Gonc-alves, I.F., Martinho, E., Salgado, J., 2001. Monte Carlo calculation of epithermal neutron resonance self-shielding factors in wires of different materials. Appl. Radiat. Isot. 55/4, 447–451. Jefferies, S.M., Mac Mahon, T.D., Williams, J.G., Ahmad, A., Ryves, T.B., 1983. Analysis and evaluation of thermal and resonance neutron activation data. In: Bockhoff, K.H. (Ed.), Proceedings of the Nuclear data for Science and Technology, pp. 681–684. Kumpf, H., 1986. Self-shielding correction for the resonance activation detectors Na, Mn and Co. Nucl. Instrum. Methods A251, 193–195.
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951 Lopes, M.C., 1991. Sensitivity of self-powered neutron detectors to thermal and epithermal neutrons with multiple collision treatment, Ph.D. Thesis, University of Coimbra (in Portuguese). Lopes, M.C., Molina Avila, J., 1990. The effect of neutron flux anisotropy on the resonance self-shielding factors in foils. Kerntechnik 55 (1), 49–52. Mo, S.C., Ott, K.O., 1987. Resonance self-shielding corrections for detector foils in fast neutron spectra. Nucl. Sci. Eng. 95, 214–224.
951
Mughaghab, S.F., 1984. Neutron Cross Sections, Vol. 1, part B, Z=61–100, Academic Press, New York. Mughaghab, S.F., Divadeenam, M., Holden, N.E., 1981. Neutron Cross Sections, Vol. 1, part A, Z=1–60, Academic Press, New York. Tuli, J.K., 2000. Nuclear wallet cards, National Nuclear Data Center, Brookhaven National Laboratory, Upton, USA. Yamamoto, H., Yamamoto, K., 1965. Self-shielding factors for resonance foils. J. Nucl. Sci. Technol. 2 (10), 421.
Monte Carlo calculation of epithermal neutron resonance selfshielding factors in foils of different materials I.F. Gonc-alves*, E. Martinho, J. Salgado ! Instituto Tecnologico e Nuclear, Estrada Nacional 10, 2686-953 Sacav!em, Portugal Received 7 August 2001; received in revised form 4 January 2002; accepted 7 January 2002
Abstract Epithermal neutron resonance self-shielding factors in foils of materials used as activation detectors or as targets for radionuclide production have been calculated using the MCNP code. Two irradiation conditions have been considered: (a) foils immersed into an isotropic neutron flux and (b) foils submitted to a collimated neutron beam. The self-shielding factors, Gres ðtÞ; for gold, indium, manganese and cobalt have been compared with available published values. The selfshielding factor depends on various physical and nuclear parameters. However, it is shown that for the isotropic case and for high absorber elements submitted to a collimated beam, a dimensionless variable could be adopted that describes the self-shielding factors of different materials by quasi ‘‘universal curves’’. Gres ðtÞ for the collimated beam are always higher than those for the isotropic case. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Epithermal neutrons; Resonance self-shielding factor; MCNP code
1. Introduction
2. Methodology of calculation
Some materials used in neutron dosimetry or for radionuclide production for medical purposes exhibit resonances in the epithermal region of the reactor neutron spectrum. Consequently, the interpretation of sample activation data requires the application of resonance self-shielding factors. In a previous work (Gonc-alves et al., 2001), the resonance self-shielding factor in wires was calculated. The aims of the present work are (a) to establish a calculation method for the resonance self-shielding factor in foils by using the MCNP code (Briesmeister, 2000) and updated cross-sections (ENDF-B6), and (b) to interpret its dependence on several physical and nuclear parameters. To validate the methodology, the results are compared with available values obtained by other authors.
The resonance self-shielding factor, Gres ðtÞ; in foils of thickness t; is defined as the ratio between the reaction rates per atom in the real sample and in a similar and infinitely diluted sample. Thus
*Corresponding author. Tel.: +351-1-955-00-21; fax: +3512199-41525. E-mail address: [email protected] (I.F. Gonc-alves).
R E2 Gres ðtÞ ¼ R EE21 E1
FðEÞsng ðEÞ dE F0 ðEÞsng ðEÞ dE
;
ð1Þ
where F0 ðEÞpE 1 is the original, non-perturbed, epithermal neutron flux per unit energy interval inside the infinitely diluted sample, FðEÞ represents the perturbed epithermal neutron flux inside the real sample, sng ðEÞ designates the ðn; gÞ cross-section, and E1 and E2 are, respectively, the lower and the upper limits around the resonance energy Eres : The total neutron crosssection has been adopted in the calculation of the perturbed neutron flux FðEÞ; which takes into account the neutron scattering in the sample. In all calculations, the density for infinite dilution was assumed to be r ¼ 106 r0 ; r0 representing the density of the real sample (Gonc-alves et al., 2001).
0969-8043/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 0 2 ) 0 0 0 5 5 - 6
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
946
Table 1 Some physical and nuclear properties of the studied elements Element
Nuclide
r0 (g cm3)
y
A (g)
sðEres Þ (barn)
Eres (eV) sg
Cobalt Copper Gold Indium Manganese Rhenium
Co-59 Cu-63 Au-197 In-115 Mn-55a Mn-55b Re-185
8.9 8.96 19.3 7.3 7.3
1 0.692 1 0.957 1
58.9 63.5 197.0 114.8 54.9
21
0.374
186.2
132 579 4.91 1.46 336 1098 2.16
ss 2
8.49 10 4.12 102 2.74 104 2.80 104 46 28 2.33 104
Gres (eV) stot
3
9.52 10 5.05 102 3.4 103 1.16 103 3.24 103 1.41 103 1.22 103
4
1.04 10 9.17 102 3.08 104 2.92 104 3.29 103 1.44 103 2.45 104
Gg
Gn
G
0.47 0.485 0.124 0.072 0.435 nac 0.055
5.15 0.59 0.015 0.003 22.0 15.4 0.003
5.62 1.075 0.139 0.075 22.4 nac 0.058
a
First resonance. Second resonance. c Not available. b
For a given thickness value, t; a neutron energy dependent resonance self-shielding factor, Gt ðEÞ; can be defined as
Isotropic case Au foil r=10 mm r=5 mm r=1 mm
0.6
ð2Þ
where RRðE; r0 Þ and RRðE; 106 r0 Þ are the reaction rates for the energy E; corresponding to the densities r0 and 106 r0 ; respectively.
3. Results and discussion
0.5
Gres(t)
RRðE; r0 Þ ; Gt ðEÞ ¼ RRðE; 106 r0 Þ
0.7
0.4 0.3 0.2 0.01
The resonance self-shielding factor depends on material density and thickness and on atomic mass, natural abundance, and resonance neutron cross section. Table 1 shows the values of the density, r0 ; natural abundance, y; atomic mass, A (Tuli, 2000), resonance energy, Eres ; scattering, capture and total resonance cross-sections, ss ; sg ; stot (ENDF-B6) and the resonance widths, Gn ; Gg and GðG ¼ Gn þ Gg Þ (Mughaghab et al., 1981; Mughaghab, 1984), for the studied elements. Two irradiation conditions have been simulated: 1. Isotropic caseFcircular foils immersed into an isotropic neutron flux; 2. Collimated caseFcircular foils submitted to a collimated neutron beam perpendicular to the foil. 3.1. Isotropic case In order to study the influence of the foil edge effect, three different foils radii have been simulated (R ¼ 10; 5 and 1 mm). The resonance epithermal neutron selfshielding factor, Gres ðtÞ; for gold foils (for tX0:005 mm) as a function of the foil thickness is shown in Fig. 1. The curves show the same behaviour. For R ¼ 10 and 5 mm they are coincident. However, for R ¼ 1 mm the curve is shifted to higher values of Gres ðtÞ; being approximately, 10% higher for t > 0:05 mm: Similar results have been
0.1
Foil thickness, t (mm) Fig. 1. Resonance neutron self-shielding factor for gold foils as a function of the foil thickness, for the isotropic case.
obtained for other elements. For this reason, to avoid foil edge effects, all subsequent simulations have been performed with R ¼ 10 mm: Fig. 2 shows the energy dependent self-shielding factor of indium for different foil thickness. The minimum of the valley is observed at the resonance energy; for t ¼ 1 mm; Gres ðE Þ has its lower value. As the thickness decreases, the effect of shielding diminishes and the curves become narrower. For indium, sg > ss ; the absorption predominates over the scattering and then Gres ðEÞo1; for all energies; the curves are practically symmetrical around Eres : Similar results were obtained for gold and rhenium, which are also high neutron absorbers. On the other hand, as can be seen in Fig. 3, for manganese, which is a high neutron scatterer (sg oss ) the self-shielding factor, Gres ðEÞ > 1; for EoEres : A neutron entering in the foil with energy EEEres can suffer one or more scattering interactions before capture and it is absorbed at EoEres : The corresponding peaks are positioned at energies EoðEres GÞ: The energy of the peak decreases as the
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
947
1.6
1.0
Isotropic case t = 0.1 mm
1.4 1.2
Gres(E)
Gres(E)
0.8
0.6 Isotropic case In foil t=1 mm t=0.1 mm t=0.01 mm t=0.001 mm t=0.0001 mm
0.4
0.2
0.0 0.5
1.0
1.5
2.0
2.5
3.0
1.0 0.8 0.6 0.4 0.2 0.0 -20
3.5
-10
Au Co
Energy, E (eV) Fig. 2. Variation of the energy dependent resonance selfshielding factor of indium foils of different thicknesses with neutron energy, for the isotropic case.
0
10
20
Reduced energy, (E-Eres)/Γ In Mn(1)
Re
Cu Mn(2)
Fig. 4. Variation of the energy dependent resonance selfshielding factors of gold, indium, rhenium, copper, cobalt and manganese foils ðt ¼ 0:1 mmÞ with the reduced neutron energy, for the isotropic case.
1.8 Isotropic case Mn foil t=1 mm t=0.1 mm t=0.001 mm
1.6 1.4
Gres(E)
1.2 1.0 0.8 0.6 0.4 0.2 0.0 100
200
300
400
500
600
Energy, E (eV) Fig. 3. Variation of the energy dependent resonance selfshielding factor of manganese (first resonance) foils of different thicknesses with neutron energy, for the isotropic case.
thickness increases. The decrease of the energy is more pronounced for manganese, which has a ratio Gn =G higher than those of the other neutron scatterers, cobalt and copper. Fig. 4 shows Gres ðEÞ as a function of a ‘‘reduced energy’’, Ered ¼ ðE Eres Þ=G; for t ¼ 0:1 mm. Gold, indium and rhenium have higher capture cross-sections than the other elements: the corresponding Gres ðEÞ has pronounced valleys and the curves are wider. Similar results were obtained for other thicknesses. Fig. 5 compares the self-shielding factor, Gres ðtÞ of gold, indium, manganese and cobalt calculated in this work with published values. The agreement for gold and indium foils between the values calculated in this work and the previously published values is excellent. For manganese and cobalt, the values calculated in this work
are systematically larger than those previously published. The calculated self-shielding factor, Gres ðtÞ; as a function of the foil thickness is shown in Fig. 6. The behaviour of the curves is similar for all materials. However, the decrease of Gres ðtÞ as the thickness increases is faster when the resonance cross section and/or the density is higher (gold, indium and rhenium). Cobalt has an intermediate cross-section and its Gres ðtÞ lies among the values of the other materials. Following the results of Gonc-alves et al. (2001), an adimensional variable, z0 ; was introduced in order to convert all values into an unique curve. This variable is defined as 1=2 Gg 0 ð3Þ z ¼ Stot t G Stot being given by r Stot ¼ 0 yNA stot ðEres Þ; A
ð4Þ
where NA is the Avogadro number. Fig. 7 shows the resonance self-shielding factor as a function of z0 : In spite of the wide ranges of variation of s; r and A; it is found that all Gres values (except those of copper) can be described by an unique curve. The relative errors in the z0 and Gres values are estimated to be about 5%. In experimental work, the range of z0 is limited to 0:01oz0 o20: For z0 o0:01; the resonance self-shielding Gres E1 and it is not necessary to perform any correction. For z0 > 20; then Gres o0:15; the self-absorption in the sample is high and the reaction rate is very low. Within this range of values, the following equation was adjusted to
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
948
1.0
0.6
Isotropic case In foil This work MCL91 YAM65 BAU63
0.8
0.6
Gres(t)
0.8
Gres(t)
1.0
Isotropic case Au foil This work MCL91 YAM65 BRO64-exp BRO64-cal JEF83 MCL90
0.4
0.4
0.2 0.2 0.0 0.0 -4 10
10
-3
-2
10
-1
10
0
10
10
-4
10
-3
Foil thickness, t (mm) 1.2
1.2
1.0
1.0
0.2
Isotropic case Mn foil This work MCL91 YAM65 JEF83 MO87 KUM86
0.0 -4 10
10
-3
Gres(t)
Gres(t)
0.4
0
10
10
0.8
0.8
0.6
-1
-2
10
Foil thickness, t (mm)
0.6
0.4
0.2
-2
10
-1
10
0
10
Isotropic case Co foil This work MCL91 YAM65 EAS62 KUM86 MCL90
0.0 -4 10
10
Foil thickness, t (mm)
-3
-1
-2
0
10
10
10
Foil thickness, t (mm)
Fig. 5. Comparison of resonance self-shielding factors of gold, indium, manganese and cobalt calculated in this work with values (experimental and calculated) published by other authors, for the isotropic case. MCL90, 91 (Lopes and Molina Avila, 1990; Lopes, 1991), YAM65 (Yamamoto and Yamamoto, 1965), BRO64 (Brose, 1964), JEF83 (Jefferies et al., 1983), BAU63 (Baumann, 1963), MO87 (Mo and Ott, 1987), KUM86 (Kumpf, 1986) EAS62 (Eastwood and Werner, 1962).
the calculated points:
1.0
Gres ðlog z0 Þ ¼ 0:625 0:410 log z0 0:055ðlog z0 Þ2
0.8
0:0066ðlog z0 Þ5 0:00157ðlog z0 Þ6
ð5Þ
2
with r ¼ 0:987:
Gres(t)
þ 0:074ðlog z0 Þ3 þ þ0:0144ðlog z0 Þ4
0.6 0.4
3.2. Collimated case
0.2
Fig. 8 shows Gres ðEÞ as a function of a ‘‘reduced energy’’, Ered ¼ ðE Eres Þ=G; for t ¼ 0:1 mm: As for the isotropic case (see Fig. 3), the Gres ðEÞ of Au, In and Re have pronounced valleys and the curves are wider. Similar results were obtained for other thicknesses. The calculated self-shielding factor, Gres ðtÞ; as a function of the foil thickness is shown in Fig. 9. The behaviour of the curves corresponding to high absorber materials is similar to that of the isotropic case. However, the high scatterer materials (Mn and Co) show an initial increase of Gres ðtÞ; particularly in the Mn first resonance, which is >1, for tp0:2 mm: This effect is, certainly, due to multiple scattering into the foil.
0.0
Isotropic case Au In Re Cu Co Mn(1) Mn(2) 10
-4
10
-3
10
-2
10
-1
10
0
Foil thickness, t (mm) Fig. 6. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, cobalt and manganese on the foil thickness, for the isotropic case. The symbol Mn(i) means
Fig. 10 shows the resonance self-shielding factor of the high absorber elements and copper as a function of z0 : Within the range 0:01pz0 p30; the following equation was adjusted to the calculated points corresponding to
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951 1.2
1.0
Isotropic case Au In Re Cu Co Mn(1) Adjusted
0.6
1.0 0.8
Gres(t)
0.8
Gres(z')
949
0.4
Collimated case Au In Re Cu Co Mn(1) Mn(2)
0.6 0.4
0.2
0.2
0.0 -2 10
10
-1
10
0
10
1
10
0.0
2
10
-4
10
z'=Σtot.t.(Γγ /Γ)
1/2
-3
10
-2
10
-1
10
0
Foil thickness, t (mm)
Fig. 7. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, cobalt and manganese on the adimensional variable z0 ¼ Stot tðGg =GÞ1=2 ; for the isotropic case. The symbol Mn(1) means the first manganese resonance.
Fig. 9. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, cobalt and manganese on the foil thickness, for the collimated beam case. The symbol Mn(i) means the ith manganese resonance. 1.2
2.5
1.0 Collimated case t = 0.1 mm
0.8
Gres
2.0
Gt(E)
1.5
0.6
1.0
0.4
0.5
0.2
0.0 -25
-20
-15
-10
-5
0
5
10
15
20
25
In Mn (1)
Re
10
-2
10
-1
10
0
z'=Σtot.t.(Γγ /Γ)
Reduced energy, (E-Eres)/Γ Au Co
0.0 -3 10
Collimated case Au In Re Cu Adjusted
Cu Mn (2)
Fig. 8. Variation of the energy dependent resonance selfshielding factors of gold, indium, rhenium, copper, cobalt and manganese foils ðt ¼ 0:1 mmÞ with the reduced neutron energy, for the collimated beam case.
10
1
10
2
1/2
Fig. 10. Dependence of the resonance self-shielding factors of gold, indium, rhenium, copper, on the adimensional variable z0 ¼ Stot t ðGg =GÞ1=2 ; for the collimated beam case.
points. Copper, as seen in Fig. 10, has a similar behaviour to that of the high absorber elements. the higher absorbers: 3.3. Comparison of both cases Gres ðlog z0 Þ ¼ 0:793 0:377 log z0 0:169ðlog z0 Þ2 þ þ 0:0616ðlog z0 Þ3 þ 0:0456ðlog z0 Þ4 0:00536ðlog z0 Þ5 0:0042ðlog z0 Þ6
ð6Þ
with r2 ¼ 0:997: The copper value lies below the adjusted curve. Fig. 11 shows the resonance self-shielding factor of the high scatterer elements and copper as a function of z0 : It is not possible to adjust a curve to the calculated
Fig. 12 compares the resonance self-shielding factor of gold (high absorber) and manganese (high scatterer) calculated for the isotropic and collimated cases. The values of Gres ðtÞ for the collimated case are always higher than those for the isotropic case, because the neutron pathways in the foil are, in average, greater for the isotropic flux than for the collimated beam. In, Re and Cu have a similar behaviour to that of Au, while Co is comparable to Mn.
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951
950
4. Conclusions
1.2 1.1 1.0
Gres
0.9 0.8 0.7 0.6
Collimated beam Cu Co Mn(1)
0.5 0.4 0.3
10
-2
10
-1
10
z'=Σtot.t.(Γγ /Γ)
0
10
1
1/2
Fig. 11. Dependence of the resonance self-shielding factors of copper, cobalt and manganese on the adimensional variable z0 ¼ Stot t ðGg =GÞ1=2 ; for the collimated beam case. The symbol Mn(1) means the first manganese resonance.
1.2 1.0
Gres(t)
0.8 0.6 Au foil collimated isotropic
0.4
The energy dependent self-shielding factor and the self-shielding factor of gold, indium, rhenium, copper, cobalt and manganese foils of different thicknesses have been calculated. The energy dependent self-shielding factors of materials with high capture cross-sections have a different behaviour as compared to those of materials with high scattering cross-sections. In the first case, Gres ðEÞo1 in the whole energy interval; in the second case, Gres ðEÞ > 1 for EoEres because neutrons entering the foil with energy Eres can suffer one or more scattering interactions before their capture at a lower energy can occur. In the case of foils immersed into an isotropic neutron flux, a dimensionless variable can be introduced which converts the dependence of the resonance self-shielding factor on physical and nuclear parameters in a quasi ‘‘universal curve’’ valid for all materials. If the foils are irradiated in a collimated neutron beam, the thickness dependence of the resonance self-shielding factor on physical and nuclear parameters is different for high absorber and high scatterer elements. In the first case, it is also possible to adjust an unique curve to the calculated values. In the second case, the curve attains a maximum (>1) for lower thick masses and the position and value of this maximum depend on the ratio sg =ss : The values of Gres ðtÞ for the collimated case are always higher than those for the isotropic case, because the neutron pathways in the foil are, in average, larger in the case of isotropic flux than for the collimated beam.
0.2 0.0 10
-4
10
-3
10
-2
10
-1
10
0
Foil thickness, t (mm) 1.2 1.1 1.0
Gres(t)
0.9 0.8 0.7 0.6 0.5
Mn foil collimated (Mn1) isotropic (Mn1) collimated (Mn2) isotropic (Mn2)
0.4 0.3 -4 10
10
-3
10
-2
10
-1
10
0
Foil thickness, t (mm) Fig. 12. Comparison of the resonance self-shielding factors of gold (high absorber) and manganese (high scatterer) calculated for the isotropic and collimated cases.
References Baumann, N.P., 1963. Rep. DP-817. Briesmeister, J.F. (Ed.), 2000. MCNPFA general Monte Carlo n-particle transport code, Los Alamos National Laboratory report LA-13709-M. Brose, M., 1964. Zur Messung und Berechnung der Resonanzabsorption von Gold Folien. Nukleonik 6, 134. Eastwood, T.A., Werner, R.D., 1962. Resonance and thermal neutron self-shielding in cobalt foils and wires. Nucl. Sci. Eng. 13, 385–390. Gonc-alves, I.F., Martinho, E., Salgado, J., 2001. Monte Carlo calculation of epithermal neutron resonance self-shielding factors in wires of different materials. Appl. Radiat. Isot. 55/4, 447–451. Jefferies, S.M., Mac Mahon, T.D., Williams, J.G., Ahmad, A., Ryves, T.B., 1983. Analysis and evaluation of thermal and resonance neutron activation data. In: Bockhoff, K.H. (Ed.), Proceedings of the Nuclear data for Science and Technology, pp. 681–684. Kumpf, H., 1986. Self-shielding correction for the resonance activation detectors Na, Mn and Co. Nucl. Instrum. Methods A251, 193–195.
I.F. Gonc-alves et al. / Applied Radiation and Isotopes 56 (2002) 945–951 Lopes, M.C., 1991. Sensitivity of self-powered neutron detectors to thermal and epithermal neutrons with multiple collision treatment, Ph.D. Thesis, University of Coimbra (in Portuguese). Lopes, M.C., Molina Avila, J., 1990. The effect of neutron flux anisotropy on the resonance self-shielding factors in foils. Kerntechnik 55 (1), 49–52. Mo, S.C., Ott, K.O., 1987. Resonance self-shielding corrections for detector foils in fast neutron spectra. Nucl. Sci. Eng. 95, 214–224.
951
Mughaghab, S.F., 1984. Neutron Cross Sections, Vol. 1, part B, Z=61–100, Academic Press, New York. Mughaghab, S.F., Divadeenam, M., Holden, N.E., 1981. Neutron Cross Sections, Vol. 1, part A, Z=1–60, Academic Press, New York. Tuli, J.K., 2000. Nuclear wallet cards, National Nuclear Data Center, Brookhaven National Laboratory, Upton, USA. Yamamoto, H., Yamamoto, K., 1965. Self-shielding factors for resonance foils. J. Nucl. Sci. Technol. 2 (10), 421.