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The influence of water absorption in samples for neutron capture cross section measurements
324
Nuclear Instruments and Methods in Physics Research A282 (1989) 324-328 North-Holland, Amsterdam
TIIE INFLUENCE OF WATER ABSORPTION IN SAMPLES FOR NEUTRON CAPTURE CROSS SECTION MEASUREMENTS Motoharu MIZUMOTO and Masayoshi SUGIMOTO Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken, Japan
Sample-related corrections for average neutron capture cross section measurements are complicated in the keV region due to the resonance structure. In particular, light mass nuclei present in chemical compounds of rare earth materials make corrections for neutron multiple-scattering and self-shielding difficult. Moreover, samples of chemical compounds such as oxides are hygroscopic . A Monte Carlo method has been developed by taking into account the effects caused by neutron slowing down into the resonance region due to the scattering from hydrogen and oxygen atoms. The validity of the calculated corrections has been investigated by comparing experimental data of oxide and metallic samples. The calculation method and related problems will be discussed . 1. Introduction Neutron capture cross section measurements of fission product nuclei have been carried out systematically at the JAERI (Japan Atomic Energy Research Institute) electron linear accelerator during the last years [1]. In neutron-capture experiments, the sample-related corrections such as self-shielding and multiple-scattering corrections are always very important. For fission product nuclei in the keV region, samples of chemical compounds such as oxides, carbonates, or nitrides are often used . Among them, oxide samples are hygroscopic. Due to hydrogen atoms in the water, neutrons are easily slowed down by multiple scattering to the low energy region where the capture cross section values are usually high, and to the resolved resonance region where the capture cross-section fluctuates . This effect causes a drastic rise in the observed capture yields. A proper correction is needed to obtain accurate cross section values . In the course of 149Sm capture cross section analysis [2], we realized the serious problem of the water absorp149SM203 was found to tion in the oxide sample . Our absorb 5.2% of water. The estimated correction to the cross section was 3% at 3 keV to 17% at 100 keV . Such an example is also emphasized in fig. 1 for the case of a tantalum sample (Ta205); in the figure the data for the metallic and the oxide sample are compared. The water, absorbed in the oxide sample, is 10% in weight . Although the number of tantalum atoms for both samples is about the same, effective cross sections obtained from the oxide sample are almost twice those of the metallic sample if no correction is applied. The energy dependence of the cross section for the metallic sample is clearly steeper than that for the oxide sample . Macklin [3] observed a 4.9% water absorption on the bromine capture sample of Na s1 Br which formed a 0168-9002/89/$03 .50 C Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
crystal dihydrate. The lithium chloride they used [4] had degenerated to a puddle of clear liquid . He pointed out that special chemical procedures are required to prepare a definite atomic ratio for metals such as iron and uranium which form several definite oxides. It is difficult to analyze for oxygen, although absorbed water is usually pumped away in vacuum . The enriched samples purchased from the Isotope Division, Oak Ridge National Laboratory, are not routinely dried before sale. Macklin and Winters [5] recently had some experience with samples of enriched tellurium isotopes . All 122 Te the samples had gained in weight, particularly the 124 and Te . For these isotopes, the weight had increased by 16% and 12%, respectively, eight months after these samples had been prepared from amorphous elemental powder. As the neutron capture measurements were made in vacuum at room temperature, absorbed mois-
Effective Copture Cross Section of To
10 5
Oxide 0.00692 ats/b
n b 0.5
Meta I 0 .00596 ats/b
5
10
Neutron
Energy
50
( keV)
100
Fig. 1. Effective capture cross sections of tantalum . In the oxide sample, 10% in weight of water is absorbed.
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples ture was considered not to be significant . Vacuum bakeout of the 122Te sample at 398 K, however, was not effective in reducing the weight . Many of the samples they use are made by pressing powders . The best of these achieve - 85% of the crystal density, which still leaves lots of surface area to absorb water and to oxidize .
325
The sample thickness correction can be expressed by the ratio of the total capture yield YY to the yield Yo for an infinitesimally thin sample : C= YY/Yo,
(1)
where YY
=YY +Y, +Y3 +
(2)
YY being the yield for a neutron which is captured after
2. Sample thickness correction The sample scattering corrections in the neutron beam experiments were originally derived by Schmitt [6] for the neutron capture experiments in the keV range. He assumed that the neutron scattering cross section in this energy region is nonresonant (smooth) and incident neutrons are well collimated on a disc sample. He gave the correction coefficients both analytically and graphically. Dresner [7] first derived the approximations for resonance self-shielding and multiple-scattering corrections in infinite slab geometry, by taking into account the narrow resonance structure. Then, Macklin [8] gave relatively simple analytical expressions approximating Dresner's results, and also expressions for the disc sample geometry. Dresner and Macklin used several assumptions that are generally valid only at low energies and for thin i samples . However, at higher energies and for thick samples, their formulae become unreliable. Froehner [9] developed a Monte Carlo code to calculate corrections for the unresolved resonance region by generating realistic cross section distributions. His calculation requires relatively long computation time and does not treat the recoil energy loss due to the neutron multiple scattering accurately . This might cause errors when it is applied to chemical compound samples and used to correct scattering by light mass elements which occur in the case of impurities such as water in the sample .
3 . Present calculation method In the present work, sample thickness correction for neutron capture in the keV energy region is developed for chemical compound samples by a semi-Monte-Carlo method . The primary capture yield from the sample is calculated in an analytical way which is the same as the Dresner-Macklin method. Our method also includes the effects of the neutron energy degradation due to neutron scattering from hydrogen and oxygen by using a Monte Carlo calculation. The averaged capture yield is expressed by using the narrow resonance approximation for a single level Breit-Wigner resonance formula, and the interference term between the resonance and potential scattering is ignored.
interacting i times in the sample. Such sample correction can be divided into two parts by following Dresner's method : (3)
C=C I +CZ . C l is the self-shielding correction term, Cl =
Y1 ,/Yo,
and Cz is the multiple-scattering correction term, Cz = (YZ + Y3 + . . . )/ yo,
(5)
The effective capture yields for the sample with finite thickness are decreased by the self-shielding and increased by multiple-scattering if they are compared to the yields for very thin samples. The quantities Cl and CZ can be evaluated only by the calculation . For a very thin sample, the capture yield averaged over a resonance is expressed as follows : Yo = fdEJO (E)( f nay (E') dE'/D , ) Yo =n (f aY
dE/D
.e . (6)
. )
There, JO(E) is the incident neutron flux which can be assumed constant over a single resonance, aY is the capture cross section, n is a path length for scattered neutrons in atoms/b, and D is the average level spacing . For the first interaction of the neutron with the sample, the capture yield can be expressed as follows : Yl = ( f (1 - e-no , )(aY/at)
dE/D,,
and the second interaction is sequentially expressed by y2=~ f (1-e na') X [(1 -
e - ") (aY'/ut')
I dE/D,.
(8)
The factor (1 - e "a`)(an/at) is the source term for neutrons scattered from the first collision. The integrations in formulae (6) and (7) are performed by taking into account a single level Breit-Wigner formula with Doppler broadening, resulting in : Yo = (ir/2) aFY /D, Yy
= (TY/D)
f {1-exp(-a(4,+,8))}~/(ip+ß) dx IX. FISSIONING TARGETS
326
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples
and Y1= (Ty/D)F(a,
(10)
P, 11).
Here the total cross section, at, is expressed by oo tp + spot + aD, the capture cross section, ay , is expressed by ao ~Ty/T, and tD(x, ri) is a usual Doppler broadening shape function
f exp { - .h2(x-Y)2)/(1+y2) dy .
0.5
Ty/D is the strength of the radiative capture, a = n a0 (= 4mnA?gr./T), a0 is the peak cross section, = ( apot + OD )/a0 (amt is a potential scattering cross section of the sample and OD is a dilution cross section of all other constituents), x=2(E-E0)/T, 0
and q=T/2A (d is the Doppler width). The primary capture yields at the incident neutron energy are then estimated from a numerical calculation of this factor F. For the multiple scattering correction, however, neutrons lose their energies successively and are captured at the low energy resonance . The values of the factor F have to be calculated each time and the scattering probabilities at different energies have also to be calculated. In our Monte Carlo method, the factor F is calculated numerically in advance for the various sets of the values of a, ß, and rq, and is tabulated in the computer memory . Fig. 2 shows the calculated results for the factor F as a function of a. As seen in the figure, it is convenient to arrange F for the set of values of aß rather than ß . The value aß = n(aP,,,t + U D ) is independent of T . For the Monte Carlo calculation, data are interpolated by using these tabulated values . In the thin sample limit, F can be written as follows: F= (m/2)a{1 -exp(-aß) }/aß . (11)
D/f Fig . 3 .Mapof f(a=6 .4, 1?, DI T),T=exp [- a((it/2)(T/D)f +/3) .
Macklin approximated the function F as follows under the condition of a << 1 using /3, .R (see eq. (6) in ref. F=(ir/2)a{1-exp(-L) }/L,
(12)
with L = aß + a,ß/( 2 ßcorr ) .
In order to carry out the sequential scattering, one has to estimate the probability for the next scattering by using the scattering yields . In the present calculation, the average neutron transmission T in the sample is calculated first and then used to estimate the scattering yields. The average scattering yields are expressed by the difference between the average transmission yields and capture yields as follows : Yn=1-T-Yl ,
(13)
T=
(14)
fexp(-a(tp+,ß))dE/D,
thus T= (T/D)
fDIrexp(-ao) dx exp(-aß),
(15)
i.e. T=exp[ -a{( ,ff/2)(T/D)f(a, q, D/T)+ß }] .
0.1
1
10
a=n9o Fig. 2 . Calculation of F(a, ß, ~l =1).
100
(16)
The function f was numerically calculated in the vicinity of the resonance and tabulated in the same way as function F of eq. (12). Fig. 3 shows the map of f values as a function of D/T for various values of q with a fixed a value of 6.4. The factor f approaches to unity for small values of D/T where many resonances overlap, and for small rl values where the Doppler broadening becomes large. In these regions, the average transmission T can be simply expressed by exp(- n a T ). As to the a values, f decreases as the a values increase.
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples I Y0 . Yr
( F )
I
With Width Fluctuation
Collision Point T) w(s)ds = e-s / ( 1 ds,
0 < s
Select Interacting Nucleus Isotropic Scattering In CMS Constant O pot, a D 2 Yr
Collision
The contribution from the p-wave resonances are also taken into account by adding the smooth cross section of the p-wave contribution : (17)
aYr=1 =2m2X2y_gj(rnt1EPT(E)FY) (J l(F jFPl(E) +FY )} .
(18)
YY = (TY/D)F(a, ß', q) where
+{ 1- exp( - naT)/naT)na,t-l,
(19)
exp(-naT) = T= exp[ -a {(m/2)(T/D) f+ ß') ], with iß' =,ß + aTy- 1/00 The schematic diagram of the Monte Carlo calculation is shown in fig. 4. The capture yields Yo and YY are first calculated by taking into account the width fluctuation correction due to the Porter-Thomas distribution using the method developed by Macklin. The position of the collision was then calculated by the usual Monte Carlo sampling method : w(s)ds=e-s/(1-T)ds,
0
s = a{(iT/2)(T/D)f(a, rf, D/T)+,ß } .
neutron histories and averaging the resulting capture probabilities from the 1st, 2nd, 3rd, - - - iterations one can find the total capture yields YY. At the same time, the averaged scattering yield is computed and predicted successively to the survival rate . Calculation will be terminated when the weight becomes less than a cutoff value. Termination is also made in the case that the neutron has slowed down below the cutoff energy . A particular attention needs to be paid for sampling of the next collision point along the paths of scattering neutrons. 4. Results
Fig. 4. A schematic diagram of the Monte Carlo calculation.
aY - (aoY'FY/rit=o + aYtat >
327
(16)
The neutron interacting with the nucleus and the successive scattering angle is also sampled. The weights are then calculated by using the capture and scattering yields computed in advance. The scattering angles determine the energy loss of the neutron. The cross sections, yields, and other necessary quantities such as a, ß, rt corresponding to each new energy are obtained as described previously . Simulating a sufficient number of
The calculated correction factors C (sum of selfshielding and multiple scattering) are shown in fig. 5 for the metallic and oxide samples of tantalum . In the oxide sample, a water absorption of 10% in weight was assumed. The correction factor indicated by "analytical" in the figure means that the calculation was carried out by ignoring the energy degradation due to the scattering at light mass nuclei. The quantities needed for the calculation such as the average neutron width, the radiation width, and the level spacing are adopted from the values published in BNL-325 [10]. While the correction factors for the metallic sample are only -5% in the low energy region due to self-shielding and 5% in the high energy region due to multiple scattering, those for the oxide sample are 15% in the low keV region attaining more than 100% in the 100 keV region . The effective and corrected capture cross sections are shown in fig. 6 together with the evaluated values of JENDL-2 data (Japanese Evaluated Nuclear Data Library-2) [111 . As seen in the figure, the correction factors especially for the oxide sample are obtained
Sample Thickness Correctionfor To
Oxide, Monte Corlo
ô
2.0
Oxide, Analytical
ô V
e
Metal, Monte Carlo L
i
IIIIII
5
10
Neutron
i
i
i Iiiiil 100 50
Energy
500
( keV )
Fig. 5. Thickness correction factor for the tantalum sample. The data points are calculated both analytically [81 and by our Monte Carlo method . IX. FISSIONING TARGETS
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples
328
Capture Cross Section of To -
ing the influence of the water absorption for capture cross section measurements . The contribution of the operation staff of the linac is greatly acknowledged .
fD dr -- (effective)
Cry (corrected) -I- JENDL-2
a
References
b
Neutron
Energy (keV)
Fig. 6. The capture cross sections corrected for the sample thickness and water absorption . The lines show the evaluated values of JENDL-2. quite successfully . The evaluated value from the JENDL-2 are also in good agreement with our data . Acknowledgement
The authors wish to thank Dr. Macklin for providing us with the information about his experiences concern-
[2] [3] [4] [5] [6] [7] [8] [9]
[10] [11]
M. Mizumoto, Y. Nakajima, M. Ohkubo, M. Sugimoto, Y. Furuta and Y. Kawarasaki, Proc . of the NEANDC Topical Conf. on Measurements and Evaluations of Nuclear Data and Decay Heat for Fission Products, Tokai, 1984, JAERI-M 84-182 (1984) p. 75 . M. Mizumoto, Nucl. Phys . A357 (1981) 90 . R.L. Macklin, Nucl. Sci. Eng. 99 (1988) 133. R.L. Macklin, Nucl. Sci. Eng. 29 (1984) 1996 . R.L. Macklin, private communication (1988). H.W. Schmitt, Sample Scattering Corrections in Neutron Beam Experiments, ORNL-2833 (1960) . L. Dresner, Nucl . Instr. and Meth. 16 (1962) 176. R.L. Macklin, Nucl. Instr. and Meth. 26 (1964) 213. F.H . Froehner, SESH - A Fortran IV Code for Calculating the Self-shielding and Multiple Scattering Effects for Neutron Cross Section Data Interpretation in the Unresolved Resonance Region, GA-8380 (1968) . S.F . Mughabghab, Neutron Cross Sections, vol. 1, Neutron Resonance Parameters and Thermal Cross Sections (Academic Press, New York, 1984) . T. Nakagawa, JENDL-2, Summary of JENDL-2 General Purpose File, JAERI-M 84-103 (1984) .
Nuclear Instruments and Methods in Physics Research A282 (1989) 324-328 North-Holland, Amsterdam
TIIE INFLUENCE OF WATER ABSORPTION IN SAMPLES FOR NEUTRON CAPTURE CROSS SECTION MEASUREMENTS Motoharu MIZUMOTO and Masayoshi SUGIMOTO Japan Atomic Energy Research Institute, Tokai-mura, Naka-gun, Ibaraki-ken, Japan
Sample-related corrections for average neutron capture cross section measurements are complicated in the keV region due to the resonance structure. In particular, light mass nuclei present in chemical compounds of rare earth materials make corrections for neutron multiple-scattering and self-shielding difficult. Moreover, samples of chemical compounds such as oxides are hygroscopic . A Monte Carlo method has been developed by taking into account the effects caused by neutron slowing down into the resonance region due to the scattering from hydrogen and oxygen atoms. The validity of the calculated corrections has been investigated by comparing experimental data of oxide and metallic samples. The calculation method and related problems will be discussed . 1. Introduction Neutron capture cross section measurements of fission product nuclei have been carried out systematically at the JAERI (Japan Atomic Energy Research Institute) electron linear accelerator during the last years [1]. In neutron-capture experiments, the sample-related corrections such as self-shielding and multiple-scattering corrections are always very important. For fission product nuclei in the keV region, samples of chemical compounds such as oxides, carbonates, or nitrides are often used . Among them, oxide samples are hygroscopic. Due to hydrogen atoms in the water, neutrons are easily slowed down by multiple scattering to the low energy region where the capture cross section values are usually high, and to the resolved resonance region where the capture cross-section fluctuates . This effect causes a drastic rise in the observed capture yields. A proper correction is needed to obtain accurate cross section values . In the course of 149Sm capture cross section analysis [2], we realized the serious problem of the water absorp149SM203 was found to tion in the oxide sample . Our absorb 5.2% of water. The estimated correction to the cross section was 3% at 3 keV to 17% at 100 keV . Such an example is also emphasized in fig. 1 for the case of a tantalum sample (Ta205); in the figure the data for the metallic and the oxide sample are compared. The water, absorbed in the oxide sample, is 10% in weight . Although the number of tantalum atoms for both samples is about the same, effective cross sections obtained from the oxide sample are almost twice those of the metallic sample if no correction is applied. The energy dependence of the cross section for the metallic sample is clearly steeper than that for the oxide sample . Macklin [3] observed a 4.9% water absorption on the bromine capture sample of Na s1 Br which formed a 0168-9002/89/$03 .50 C Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
crystal dihydrate. The lithium chloride they used [4] had degenerated to a puddle of clear liquid . He pointed out that special chemical procedures are required to prepare a definite atomic ratio for metals such as iron and uranium which form several definite oxides. It is difficult to analyze for oxygen, although absorbed water is usually pumped away in vacuum . The enriched samples purchased from the Isotope Division, Oak Ridge National Laboratory, are not routinely dried before sale. Macklin and Winters [5] recently had some experience with samples of enriched tellurium isotopes . All 122 Te the samples had gained in weight, particularly the 124 and Te . For these isotopes, the weight had increased by 16% and 12%, respectively, eight months after these samples had been prepared from amorphous elemental powder. As the neutron capture measurements were made in vacuum at room temperature, absorbed mois-
Effective Copture Cross Section of To
10 5
Oxide 0.00692 ats/b
n b 0.5
Meta I 0 .00596 ats/b
5
10
Neutron
Energy
50
( keV)
100
Fig. 1. Effective capture cross sections of tantalum . In the oxide sample, 10% in weight of water is absorbed.
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples ture was considered not to be significant . Vacuum bakeout of the 122Te sample at 398 K, however, was not effective in reducing the weight . Many of the samples they use are made by pressing powders . The best of these achieve - 85% of the crystal density, which still leaves lots of surface area to absorb water and to oxidize .
325
The sample thickness correction can be expressed by the ratio of the total capture yield YY to the yield Yo for an infinitesimally thin sample : C= YY/Yo,
(1)
where YY
=YY +Y, +Y3 +
(2)
YY being the yield for a neutron which is captured after
2. Sample thickness correction The sample scattering corrections in the neutron beam experiments were originally derived by Schmitt [6] for the neutron capture experiments in the keV range. He assumed that the neutron scattering cross section in this energy region is nonresonant (smooth) and incident neutrons are well collimated on a disc sample. He gave the correction coefficients both analytically and graphically. Dresner [7] first derived the approximations for resonance self-shielding and multiple-scattering corrections in infinite slab geometry, by taking into account the narrow resonance structure. Then, Macklin [8] gave relatively simple analytical expressions approximating Dresner's results, and also expressions for the disc sample geometry. Dresner and Macklin used several assumptions that are generally valid only at low energies and for thin i samples . However, at higher energies and for thick samples, their formulae become unreliable. Froehner [9] developed a Monte Carlo code to calculate corrections for the unresolved resonance region by generating realistic cross section distributions. His calculation requires relatively long computation time and does not treat the recoil energy loss due to the neutron multiple scattering accurately . This might cause errors when it is applied to chemical compound samples and used to correct scattering by light mass elements which occur in the case of impurities such as water in the sample .
3 . Present calculation method In the present work, sample thickness correction for neutron capture in the keV energy region is developed for chemical compound samples by a semi-Monte-Carlo method . The primary capture yield from the sample is calculated in an analytical way which is the same as the Dresner-Macklin method. Our method also includes the effects of the neutron energy degradation due to neutron scattering from hydrogen and oxygen by using a Monte Carlo calculation. The averaged capture yield is expressed by using the narrow resonance approximation for a single level Breit-Wigner resonance formula, and the interference term between the resonance and potential scattering is ignored.
interacting i times in the sample. Such sample correction can be divided into two parts by following Dresner's method : (3)
C=C I +CZ . C l is the self-shielding correction term, Cl =
Y1 ,/Yo,
and Cz is the multiple-scattering correction term, Cz = (YZ + Y3 + . . . )/ yo,
(5)
The effective capture yields for the sample with finite thickness are decreased by the self-shielding and increased by multiple-scattering if they are compared to the yields for very thin samples. The quantities Cl and CZ can be evaluated only by the calculation . For a very thin sample, the capture yield averaged over a resonance is expressed as follows : Yo = fdEJO (E)( f nay (E') dE'/D , ) Yo =n (f aY
dE/D
.e . (6)
. )
There, JO(E) is the incident neutron flux which can be assumed constant over a single resonance, aY is the capture cross section, n is a path length for scattered neutrons in atoms/b, and D is the average level spacing . For the first interaction of the neutron with the sample, the capture yield can be expressed as follows : Yl = ( f (1 - e-no , )(aY/at)
dE/D,,
and the second interaction is sequentially expressed by y2=~ f (1-e na') X [(1 -
e - ") (aY'/ut')
I dE/D,.
(8)
The factor (1 - e "a`)(an/at) is the source term for neutrons scattered from the first collision. The integrations in formulae (6) and (7) are performed by taking into account a single level Breit-Wigner formula with Doppler broadening, resulting in : Yo = (ir/2) aFY /D, Yy
= (TY/D)
f {1-exp(-a(4,+,8))}~/(ip+ß) dx IX. FISSIONING TARGETS
326
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples
and Y1= (Ty/D)F(a,
(10)
P, 11).
Here the total cross section, at, is expressed by oo tp + spot + aD, the capture cross section, ay , is expressed by ao ~Ty/T, and tD(x, ri) is a usual Doppler broadening shape function
f exp { - .h2(x-Y)2)/(1+y2) dy .
0.5
Ty/D is the strength of the radiative capture, a = n a0 (= 4mnA?gr./T), a0 is the peak cross section, = ( apot + OD )/a0 (amt is a potential scattering cross section of the sample and OD is a dilution cross section of all other constituents), x=2(E-E0)/T, 0
and q=T/2A (d is the Doppler width). The primary capture yields at the incident neutron energy are then estimated from a numerical calculation of this factor F. For the multiple scattering correction, however, neutrons lose their energies successively and are captured at the low energy resonance . The values of the factor F have to be calculated each time and the scattering probabilities at different energies have also to be calculated. In our Monte Carlo method, the factor F is calculated numerically in advance for the various sets of the values of a, ß, and rq, and is tabulated in the computer memory . Fig. 2 shows the calculated results for the factor F as a function of a. As seen in the figure, it is convenient to arrange F for the set of values of aß rather than ß . The value aß = n(aP,,,t + U D ) is independent of T . For the Monte Carlo calculation, data are interpolated by using these tabulated values . In the thin sample limit, F can be written as follows: F= (m/2)a{1 -exp(-aß) }/aß . (11)
D/f Fig . 3 .Mapof f(a=6 .4, 1?, DI T),T=exp [- a((it/2)(T/D)f +/3) .
Macklin approximated the function F as follows under the condition of a << 1 using /3, .R (see eq. (6) in ref. F=(ir/2)a{1-exp(-L) }/L,
(12)
with L = aß + a,ß/( 2 ßcorr ) .
In order to carry out the sequential scattering, one has to estimate the probability for the next scattering by using the scattering yields . In the present calculation, the average neutron transmission T in the sample is calculated first and then used to estimate the scattering yields. The average scattering yields are expressed by the difference between the average transmission yields and capture yields as follows : Yn=1-T-Yl ,
(13)
T=
(14)
fexp(-a(tp+,ß))dE/D,
thus T= (T/D)
fDIrexp(-ao) dx exp(-aß),
(15)
i.e. T=exp[ -a{( ,ff/2)(T/D)f(a, q, D/T)+ß }] .
0.1
1
10
a=n9o Fig. 2 . Calculation of F(a, ß, ~l =1).
100
(16)
The function f was numerically calculated in the vicinity of the resonance and tabulated in the same way as function F of eq. (12). Fig. 3 shows the map of f values as a function of D/T for various values of q with a fixed a value of 6.4. The factor f approaches to unity for small values of D/T where many resonances overlap, and for small rl values where the Doppler broadening becomes large. In these regions, the average transmission T can be simply expressed by exp(- n a T ). As to the a values, f decreases as the a values increase.
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples I Y0 . Yr
( F )
I
With Width Fluctuation
Collision Point T) w(s)ds = e-s / ( 1 ds,
0 < s
Select Interacting Nucleus Isotropic Scattering In CMS Constant O pot, a D 2 Yr
Collision
The contribution from the p-wave resonances are also taken into account by adding the smooth cross section of the p-wave contribution : (17)
aYr=1 =2m2X2y_gj(rnt1EPT(E)FY) (J l(F jFPl(E) +FY )} .
(18)
YY = (TY/D)F(a, ß', q) where
+{ 1- exp( - naT)/naT)na,t-l,
(19)
exp(-naT) = T= exp[ -a {(m/2)(T/D) f+ ß') ], with iß' =,ß + aTy- 1/00 The schematic diagram of the Monte Carlo calculation is shown in fig. 4. The capture yields Yo and YY are first calculated by taking into account the width fluctuation correction due to the Porter-Thomas distribution using the method developed by Macklin. The position of the collision was then calculated by the usual Monte Carlo sampling method : w(s)ds=e-s/(1-T)ds,
0
s = a{(iT/2)(T/D)f(a, rf, D/T)+,ß } .
neutron histories and averaging the resulting capture probabilities from the 1st, 2nd, 3rd, - - - iterations one can find the total capture yields YY. At the same time, the averaged scattering yield is computed and predicted successively to the survival rate . Calculation will be terminated when the weight becomes less than a cutoff value. Termination is also made in the case that the neutron has slowed down below the cutoff energy . A particular attention needs to be paid for sampling of the next collision point along the paths of scattering neutrons. 4. Results
Fig. 4. A schematic diagram of the Monte Carlo calculation.
aY - (aoY'FY/rit=o + aYtat >
327
(16)
The neutron interacting with the nucleus and the successive scattering angle is also sampled. The weights are then calculated by using the capture and scattering yields computed in advance. The scattering angles determine the energy loss of the neutron. The cross sections, yields, and other necessary quantities such as a, ß, rt corresponding to each new energy are obtained as described previously . Simulating a sufficient number of
The calculated correction factors C (sum of selfshielding and multiple scattering) are shown in fig. 5 for the metallic and oxide samples of tantalum . In the oxide sample, a water absorption of 10% in weight was assumed. The correction factor indicated by "analytical" in the figure means that the calculation was carried out by ignoring the energy degradation due to the scattering at light mass nuclei. The quantities needed for the calculation such as the average neutron width, the radiation width, and the level spacing are adopted from the values published in BNL-325 [10]. While the correction factors for the metallic sample are only -5% in the low energy region due to self-shielding and 5% in the high energy region due to multiple scattering, those for the oxide sample are 15% in the low keV region attaining more than 100% in the 100 keV region . The effective and corrected capture cross sections are shown in fig. 6 together with the evaluated values of JENDL-2 data (Japanese Evaluated Nuclear Data Library-2) [111 . As seen in the figure, the correction factors especially for the oxide sample are obtained
Sample Thickness Correctionfor To
Oxide, Monte Corlo
ô
2.0
Oxide, Analytical
ô V
e
Metal, Monte Carlo L
i
IIIIII
5
10
Neutron
i
i
i Iiiiil 100 50
Energy
500
( keV )
Fig. 5. Thickness correction factor for the tantalum sample. The data points are calculated both analytically [81 and by our Monte Carlo method . IX. FISSIONING TARGETS
M. Mizumoto, M. Sugimoto / Influence of water absorption in samples
328
Capture Cross Section of To -
ing the influence of the water absorption for capture cross section measurements . The contribution of the operation staff of the linac is greatly acknowledged .
fD dr -- (effective)
Cry (corrected) -I- JENDL-2
a
References
b
Neutron
Energy (keV)
Fig. 6. The capture cross sections corrected for the sample thickness and water absorption . The lines show the evaluated values of JENDL-2. quite successfully . The evaluated value from the JENDL-2 are also in good agreement with our data . Acknowledgement
The authors wish to thank Dr. Macklin for providing us with the information about his experiences concern-
[2] [3] [4] [5] [6] [7] [8] [9]
[10] [11]
M. Mizumoto, Y. Nakajima, M. Ohkubo, M. Sugimoto, Y. Furuta and Y. Kawarasaki, Proc . of the NEANDC Topical Conf. on Measurements and Evaluations of Nuclear Data and Decay Heat for Fission Products, Tokai, 1984, JAERI-M 84-182 (1984) p. 75 . M. Mizumoto, Nucl. Phys . A357 (1981) 90 . R.L. Macklin, Nucl. Sci. Eng. 99 (1988) 133. R.L. Macklin, Nucl. Sci. Eng. 29 (1984) 1996 . R.L. Macklin, private communication (1988). H.W. Schmitt, Sample Scattering Corrections in Neutron Beam Experiments, ORNL-2833 (1960) . L. Dresner, Nucl . Instr. and Meth. 16 (1962) 176. R.L. Macklin, Nucl. Instr. and Meth. 26 (1964) 213. F.H . Froehner, SESH - A Fortran IV Code for Calculating the Self-shielding and Multiple Scattering Effects for Neutron Cross Section Data Interpretation in the Unresolved Resonance Region, GA-8380 (1968) . S.F . Mughabghab, Neutron Cross Sections, vol. 1, Neutron Resonance Parameters and Thermal Cross Sections (Academic Press, New York, 1984) . T. Nakagawa, JENDL-2, Summary of JENDL-2 General Purpose File, JAERI-M 84-103 (1984) .