- Email: [email protected]
Cold neutron production in solid and liquid CH4 moderators
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
Cold neutron production in solid and liquid CH4 moderators N. Morishima*, T. Mitsuyasu Department of Nuclear Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan Received 31 July 2003; received in revised form 15 September 2003; accepted 29 September 2003
Abstract A numerical study of cold neutron production in cryogenic moderators composed of solid or liquid CH4 at temperatures ranging from 20.4 to 111:7 K is made. The study is performed by a multigroup neutron transport analysis with a set of energy-averaged cross-sections (group constants) generated by the authors. Particular attention is paid to the total and angular properties of a space–energy cold-neutron flux, especially in terms of an equilibrium energy spectrum at each temperature, a storage property of high-density cold neutrons and an anisotropic emission of cold neutrons from the moderator surface. r 2003 Elsevier B.V. All rights reserved. PACS: 28.20.G; 29.25.D; 61.25.E; 78.70.N Keywords: Cold neutron; Solid methane; Liquid methane; Neutron transport; Neutron source
1. Introduction A cryogenic moderator of solid or liquid CH4 is expected to be able to produce high-density cold neutrons in a pulsed spallation neutron source [1–4]. Among various realistic hydrogenous moderators, solid CH4 has a relatively high hydrogen-atom density which is advantageous to fast-neutron slowing down in a narrow region with a small time spread. For epithermal neutrons thus produced, there are some low-energy exchange modes, both intra- and inter-molecular, for cold neutron production. Nearly free rotations of the molecule in the solid (above 20:4 K) and liquid *Corresponding author. Tel.: +81-75-753-5836; fax: +8175-753-5836. E-mail address: [email protected] (N. Morishima).
states are especially efficient. By the rotational excitations with energy levels EJ ¼ 1:30JðJ þ 1Þ=2 meV ðJ ¼ 0; 1; 2; yÞ; down-scattering of a thermal neutron to a cold one takes place. For an intermolecular motion, low-frequency lattice vibrations in the solid and translational vibrations in the liquid may possibly contribute to the thermal neutron slowing down. Although the above neutronic advantages are present, solid CH4 is accompanied with radiation damage and hydrogen-gas formation in a high-power spallation source. Hence there is a need for significant improvement of a solid CH4 moderator, for instance, by periodic annealing to relieve radiolytically derived hydrogen [5] and/or by the combined use of light water ðH2 OÞ as a coupled moderator [6]. In order to take full advantage of solid and liquid CH4 moderators, preliminary design
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2003.09.055
ARTICLE IN PRESS N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
considerations are required. For this purpose, the present paper discusses neutronic performance of the moderators at all practical temperatures between 20.4 and 111:7 K: In particular, space– energy cold neutron fluxes in the moderators are calculated by a multigroup neutron transport analysis with a set of newly generated group constants. In Section 2, a brief description of scattering and moderating properties is given. Section 3 details the neutronic behavior of the moderators at many different temperatures. Section 4 is devoted to the concluding remarks.
70 Liquid at 111.7K
Macroscopic cross section [ 1/cm]
296
Solid at 90.7K Solid at 50K 1
10
A cross-section model for neutron scattering in solid and liquid CH4 has been developed by the authors [7]. Major features of the model are as follows: it describes short-time free rotation of the molecule and long-time isotropic rotational diffusion with a temperature-dependent relaxation constant Dr ðTÞ: The former is very efficient for thermal neutron slowing down to produce cold neutrons by the excitation of rotational levels. And the latter gives rise to a quasi-elastic scattering peak with a half-width 2Dr ðTÞ; especially at lowtemperature where it becomes very sharp and large. Other features are the inclusion of such molecular translations as very short-time free-gas like translation, short-lived vibration and longtime diffusion (only in the liquid state). As for the intramolecular vibrations, they are included representatively with two characteristic energies of 0.170 and 0:387 eV: The temperature dependence of the above inter- and intra-molecular motions is taken into account so that the model is applicable to a wide range of temperature from 20.4 to 111:7 K; i.e., between the one of order–disorder transition on solid CH4 and the boiling point of liquid CH4 : Note that the melting point is 90:7 K: It has been shown that the cross-section model is in good agreement with many experimental results, both double differential and total, at many different temperatures [8–11]. By use of the cross-section model, a set of energyaveraged cross-sections (group constants) has been generated as basic data for a cold-neutron source
Liquid at 90.7K
Σ a× 3
0
2. Scattering and moderating properties
Solid at 20.4K
10 _2 10
20.4K(S) 50K(S) 90.7K(S) 90.7K(L) 111.7K(L) _1
0
Σt 1
10 10 10 Neutron Energy [ meV]
10
2
3
10
Fig. 1. The macroscopic total cross sections St of solid and liquid CH4 at temperatures shown, together with the total absorption cross sections Sa :
design [12]. Solid CH4 at 20.4, 50 and 90:7 K and liquid CH4 at 90.7 and 111:7 K are evaluated. The results on macroscopic cross-section S ðcm1 Þ are shown in Fig. 1 where S ¼ sN with the microscopic total cross-section s ðb=CH4 Þ and the number density N of CH4 molecules at each temperature [13,14]: for solid CH4 ; N=1024 ¼ 0:0199; 0.0192 and 0:0183 cm3 at 20.4, 50 and 90:7 K; respectively; for liquid CH4 ; N=1024 ¼ 0:0170 and 0:0159 cm3 at 90.7 and 111:7 K; respectively. Shown in Fig. 1 are the total cross-sections St ¼ Ss þ Sa and the total absorption (capture) cross-sections Sa where Ss is the total scattering cross-section. The energy range from 0:1 meV to 10 eV is divided at equal logarithmic (lethargy) intervals into 80 groups. The angular distribution of scattering cross-section is represented by the Legendre expansion up to a maximum order 3, which is almost adequate for representation of forward scattering in H by an epithermal neutron.
3. Cold neutron production In order to study the neutronic property of solid and liquid CH4 moderators, a simple model in a one-dimensional bare-slab geometry is analyzed. It is sketched in Fig. 2. Vacuum boundary conditions
ARTICLE IN PRESS 297
20.4K(S) 10
0
50K(S)
2
Energy spectrum [n/cm /s/meV]
N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
Fig. 2. The cold-neutron source model of solid and liquid CH4 :
are added on either surface of the slab so that there are no incoming neutrons. Two types of external neutron sources are assumed: an isotropic plane source ð1 neutron s1 cm2 Þ located at one surface ðx ¼ 0Þ and a uniformly distributed source ð1 neutron s1 cm3 Þ throughout the moderator. Both of them emit epithermal neutrons with energies E of the group 1 ð7:94 eVpEp10 eVÞ: Neutron fluxes at x ¼ 0; D=2 and D are calculated while varying the slab width D: A neutron transport analysis is made using the one-dimensional particle transport calculation code ANISN [15] in a P3 –S8 approximation. 3.1. Equilibrium energy spectrum A set of equilibrium energy spectra for solid CH4 at 20.4, 50 and 90:7 K and liquid CH4 at 90.7 and 111:7 K is calculated. For this calculation, the slab moderator with the uniform external source of epithermal neutrons is analyzed. The energy spectra at the center ðx ¼ D=2Þ are obtained by varying D and it is found for DX15 cm that they little change in magnitude and shape. Fig. 3 shows the calculated results for D ¼ 15 cm at four different temperatures. All the energy spectra are normalized in the energy region of a 1=E component. To characterize these spectra, the following expression for a Maxwellian plus 1=E spectrum is fitted: 8 f E 2 exp½ kTEN dE > > > 0 ðkTN Þ > > < for EpEC ; ð1Þ fðEÞ dE ¼ E C EC > dE f0 EC 2 exp½ kT > > N E ðkT Þ N > > : for EXEC ;
10
_1
90.7K(S)
10
90.7K(L)
_2
111.7K(L) _3
10 _2 10
10
_1
10
0
10
1
10
2
10
3
Neutron energy [meV] Fig. 3. The equilibrium energy spectra at x ¼ D=2 ðD ¼ 15 cmÞ for solid and liquid CH4 at four different temperatures shown.
where k is the Boltzmann constant, f0 is the total flux, TN is the neutron temperature and EC is the cutoff energy. Hence a low-energy neutron gain is evaluated in terms of GN ¼ fðkTN Þ=fð1 eVÞ where fðkTN Þ is the maximum of a Maxwellian component for EpEC and fð1 eVÞ is the spectral value of fðEÞ at E ¼ 1 eV for a well-defined 1=E component. Fig. 4 shows the results on TN and GN as a function of moderator temperature T: Also shown are the experimentally estimated values of TN and GN for a cylindrically shaped pure-CH4 moderator at many different temperatures [2]. Good moderating characteristics are obvious in terms of a steady decrease in TN with T: On the other hand, there is distinct difference in GN between the calculation and the experiment. This may arise from some neutronic effects due to moderator geometry and external source condition. The ratio of the theoretical GN to the experimental one is estimated at about 2.6, 2.6 and 2.2 for 20.4, 50 and 111:7 K; respectively. Assuming that the last value for liquid CH4 arises mainly from the above neutronic effects, the former two values for solid CH4 may presumably contain a contribution from molecule-density reduction in the frozen CH4 : For instance, fðkTN Þ calculated for 87% of the original N gives a smaller GN so that the ratio under discussion takes a value of 2.2. This gives an
ARTICLE IN PRESS N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
298
fluxes ðcm2 s1 meV1 Þ at x ¼ 0; D=2 and D are calculated to compare them in magnitude and shape where D ¼ 2:1 cm for solid CH4 at 20:4 K and D ¼ 2:5 cm for liquid CH4 at 90:7 K are selected to maximize cold neutron flux at x ¼ D: Fig. 5 shows the results of energy spectra thus calculated. Good converter characteristics are obvious in view of a well-established Maxwellian component with TN very close on T: The remarkable increase of cold neutron flux in the center is due mainly to closely-spaced energy levels of molecular rotations and large scattering crosssection for cold and thermal neutrons, in addition to less leakage of neutrons outside the moderator. This indicates a storage property of high-density cold neutrons, especially in a solid CH4 at lower temperature near 20:4 K: An optimum width of the slab moderator is determined to clarify a converter characteristic from epithermal neutrons to cold ones. The total neutron fluxes at either surface are calculated by varying D from 0.5 to 5 cm: The integrated fluxes for all Ep10 meV are shown in Fig. 6 as a function of D: Both of the solid and liquid CH4 moderators indicate a similar tendency of the fluxes for a maximizing behavior at x ¼ D and a saturation one at x ¼ 0; though there is significant difference in the magnitude. Hence it results in such an optimum width as D ¼ 2:1 cm for solid
estimated value of NB0:017 1024 cm3 for the solid CH4 moderator. 3.2. Converter characteristics Basic characteristics of solid and liquid CH4 moderators converting epithermal neutrons into cold ones are studied. The slab moderator model with the isotropic plane source (at x ¼ 0) of epithermal neutrons is analyzed. The total neutron
Neutron Temperature T N [K]
100
Experimental TN Theoretical TN Experimental GN Theoretical GN
4
3 50 2 20
10 10
Neutron gain GN /1000
5
200
1
20
50
100
0 200
Moderator Temperature T [K]
10
_1
X=D/2
D=2.1cm
D=2.5cm
X=D/2
_2
10
X=0
2
Energy spectrum [n/cm /s/meV]
Fig. 4. The neutron temperatures TN and the low-energy neutron gain GN as a function of moderator temperature T: The experimental results are taken from Ref. [2].
10
10
10
10
X=0
_3
_4
X=D
_5
X=D
Solid CH4 at 20.4K
_6 _4
10
_2
10
0
10
10
2
Neutron energy [meV]
Liquid CH4 at 90.7K 4
10 10
_4
10
_2
10
0
2
10
4
10
Neutron energy [meV]
Fig. 5. The energy spectra at x ¼ 0; D=2 and D in the solid CH4 moderator at 20:4 K (left) and the liquid CH4 moderator at 90:7 K (right).
ARTICLE IN PRESS N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
0.6
Solid at 20.4K
2
Cold neutron flux [n/cm /s]
CH4 at 20:4 K and D ¼ 2:5 cm for liquid CH4 at 90:7 K: A final analysis on the converter characteristics is made to examine an effect of reduced molecule density. Namely the number density N at each temperature is reduced by 10%, 20% and 30% and the spatial distributions of cold neutron flux for all Ep10 meV are calculated. Fig. 7 shows the results on the solid CH4 moderator ðD ¼ 2:1 cmÞ at 20:4 K and the liquid CH4 moderator ðD ¼ 2:5 cmÞ at 90:7 K: The effect under discussion is very significant since a thin-slab(or small) mod-
erator is subject to the outside leakage of neutrons and also the decrease in slowing-down power being proportional to Ss : Furthermore, the cold neutron fluxes emanating from the either surface are also calculated for the same moderators. Fig. 8 shows the angular fluxes of cold neutrons for all Ep10 meV with the original values of N: There is an anisotropic angular distribution of cold neutrons around the perpendicular direction to the
Solid CH4 at 20.4K
At X =0
0.4
299
At X=D 2.1cm
Liquid CH4 at 90.7K
0.2
Liquid at 90.7K 2.5cm 0
1
2
3
4
5
Slab width D [cm]
1
2
3
4
5
Slab width D [cm]
Fig. 8. The angular fluxes of cold neutrons below 10 meV at x ¼ 0 and D for the solid and liquid CH4 moderators.
2
_10%
0.6 _10%
2
Cold Neutron Flux [n/cm /sec]
Fig. 6. The integrated cold-neutron fluxes for all Ep10 meV:
_20%
_20% _30%
0.4
_30%
1
Solid CH 4 at 20.4K
0.2
Liquid CH 4 at 90.7K
and D=2.1cm
and D=2.5cm
Plane source at X=0 0 0
1
Distance X [cm]
Plane source at X=0 2
0
0
1
2
Distance X [cm]
Fig. 7. The spatial distributions of cold neutrons below 10 meV in the solid CH4 moderator (left) and the liquid CH4 moderator (right) where the original value of N is reduced by 10%, 20% and 30%.
ARTICLE IN PRESS 300
N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
surface, i.e. at yB0 and 180 : This is due to the spatial gradient of the inside flux near the surface in view of the Fick’s law for neutron current density. With a decrease in N; the magnitudes of the emanating fluxes also reduce significantly as has been shown in Fig. 7 for the total fluxes at x ¼ 0 and D:
4. Concluding remarks Good moderating characteristics of solid CH4 as well as liquid CH4 have been demonstrated by the analysis of equilibrium energy spectra at typical temperatures. Hence high density of cold neutrons may possibly be produced especially in a solid CH4 moderator near 20:4 K; provided that the molecule density is kept almost at the equilibrium value, i.e., without large reduction due to radiation damage. It has been also shown that high intensity of cold neutron flux emanates from an adequatelydesigned solid CH4 moderator. Much more intensive beams of cold neutrons may be extracted from the center part, rather than the surface. To confirm this, a further study of the cryogenic moderator will be required, for instance, in terms of a grooved moderator [5] and/or a lowertemperature solid CH4 moderator below 20:4 K: As a powerful tool for a design assessment of practical cold neutron sources, a set of group constants on solid and liquid CH4 has been provided. By the combined use of the alreadygenerated ones on liquid 4 He; H2 and D2 [16], we may proceed to the research and development of an advanced low-energy neutron source of ultracold, very cold and cold neutrons.
Acknowledgements The authors are grateful to Mr. Yasunobu Nagaya of JAERI-Tokai for his collaboration in the neutron transport analysis and to the referee for useful comments on typical solid-CH4 moderators.
References [1] D.F.R. Mildner, R.N. Sinclair, Ann. Nucl. Energy 6 (1979) 225. [2] K. Inoue, Y. Kiyanagi, H. Iwasa, Nucl. Instr. and Meth. 192 (1982) 129. [3] E.B. Iverson, J.M. Carpenter, E.J. Hill, Physica B 241–243 (1997) 33; E.B. Iverson, J.M. Carpenter, Trans. Am. Nucl. Soc. 78 (1998) 274. [4] M. Kawai, et al., Appl. Phys. A 74 (Suppl) (2002) S43. [5] A.A. Beljakov, et al., J. Neutron Res. 3 (1996) 209. [6] G.J. Russell, et al., J. Neutron Res. 6 (1997) 33. [7] N. Morishima, Y. Sakurai, Nucl. Instr. and Meth. A 490 (2002) 527. [8] B.A. Dasannacharya, G. Venkataraman, Phys. Rev. 156 (1967) 196. [9] Y.D. Harker, R.M. Brugger, J. Chem. Phys. 42 (1965) 275. [10] W.L. Whittemore, Nucl. Sci. Eng. 18 (1964) 182. [11] Z. Rogalska, Physica 29 (1963) 491; Z. Rogalska, Acta Phys. Pol. 27 (1965) 581. [12] Y. Sakurai, T. Mitsuyasu, N. Morishima, Nucl. Instr. and Meth. A 506 (2003) 199. [13] A. Schallamach, Proc. Roy. Soc. A 171 (1939) 569. [14] R.D. McCarty, J. Hord, H.M. Roder, Selected properties of hydrogen, Engineering Design Data, NBS Monograph, USA, 1981. [15] W.W. Engle Jr., USAEC Report K-1693, 1967. [16] Y. Abe, N. Morishima, Nucl. Instr. and Meth. A 481 (2002) 414; N. Morishima, Y. Matsuo, Nucl. Instr. and Meth. A 490 (2002) 308; Y. Matsuo, N. Morishima, Y. Nagaya, Nucl. Instr. and Meth. A 496 (2003) 446.
Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
Cold neutron production in solid and liquid CH4 moderators N. Morishima*, T. Mitsuyasu Department of Nuclear Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan Received 31 July 2003; received in revised form 15 September 2003; accepted 29 September 2003
Abstract A numerical study of cold neutron production in cryogenic moderators composed of solid or liquid CH4 at temperatures ranging from 20.4 to 111:7 K is made. The study is performed by a multigroup neutron transport analysis with a set of energy-averaged cross-sections (group constants) generated by the authors. Particular attention is paid to the total and angular properties of a space–energy cold-neutron flux, especially in terms of an equilibrium energy spectrum at each temperature, a storage property of high-density cold neutrons and an anisotropic emission of cold neutrons from the moderator surface. r 2003 Elsevier B.V. All rights reserved. PACS: 28.20.G; 29.25.D; 61.25.E; 78.70.N Keywords: Cold neutron; Solid methane; Liquid methane; Neutron transport; Neutron source
1. Introduction A cryogenic moderator of solid or liquid CH4 is expected to be able to produce high-density cold neutrons in a pulsed spallation neutron source [1–4]. Among various realistic hydrogenous moderators, solid CH4 has a relatively high hydrogen-atom density which is advantageous to fast-neutron slowing down in a narrow region with a small time spread. For epithermal neutrons thus produced, there are some low-energy exchange modes, both intra- and inter-molecular, for cold neutron production. Nearly free rotations of the molecule in the solid (above 20:4 K) and liquid *Corresponding author. Tel.: +81-75-753-5836; fax: +8175-753-5836. E-mail address: [email protected] (N. Morishima).
states are especially efficient. By the rotational excitations with energy levels EJ ¼ 1:30JðJ þ 1Þ=2 meV ðJ ¼ 0; 1; 2; yÞ; down-scattering of a thermal neutron to a cold one takes place. For an intermolecular motion, low-frequency lattice vibrations in the solid and translational vibrations in the liquid may possibly contribute to the thermal neutron slowing down. Although the above neutronic advantages are present, solid CH4 is accompanied with radiation damage and hydrogen-gas formation in a high-power spallation source. Hence there is a need for significant improvement of a solid CH4 moderator, for instance, by periodic annealing to relieve radiolytically derived hydrogen [5] and/or by the combined use of light water ðH2 OÞ as a coupled moderator [6]. In order to take full advantage of solid and liquid CH4 moderators, preliminary design
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2003.09.055
ARTICLE IN PRESS N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
considerations are required. For this purpose, the present paper discusses neutronic performance of the moderators at all practical temperatures between 20.4 and 111:7 K: In particular, space– energy cold neutron fluxes in the moderators are calculated by a multigroup neutron transport analysis with a set of newly generated group constants. In Section 2, a brief description of scattering and moderating properties is given. Section 3 details the neutronic behavior of the moderators at many different temperatures. Section 4 is devoted to the concluding remarks.
70 Liquid at 111.7K
Macroscopic cross section [ 1/cm]
296
Solid at 90.7K Solid at 50K 1
10
A cross-section model for neutron scattering in solid and liquid CH4 has been developed by the authors [7]. Major features of the model are as follows: it describes short-time free rotation of the molecule and long-time isotropic rotational diffusion with a temperature-dependent relaxation constant Dr ðTÞ: The former is very efficient for thermal neutron slowing down to produce cold neutrons by the excitation of rotational levels. And the latter gives rise to a quasi-elastic scattering peak with a half-width 2Dr ðTÞ; especially at lowtemperature where it becomes very sharp and large. Other features are the inclusion of such molecular translations as very short-time free-gas like translation, short-lived vibration and longtime diffusion (only in the liquid state). As for the intramolecular vibrations, they are included representatively with two characteristic energies of 0.170 and 0:387 eV: The temperature dependence of the above inter- and intra-molecular motions is taken into account so that the model is applicable to a wide range of temperature from 20.4 to 111:7 K; i.e., between the one of order–disorder transition on solid CH4 and the boiling point of liquid CH4 : Note that the melting point is 90:7 K: It has been shown that the cross-section model is in good agreement with many experimental results, both double differential and total, at many different temperatures [8–11]. By use of the cross-section model, a set of energyaveraged cross-sections (group constants) has been generated as basic data for a cold-neutron source
Liquid at 90.7K
Σ a× 3
0
2. Scattering and moderating properties
Solid at 20.4K
10 _2 10
20.4K(S) 50K(S) 90.7K(S) 90.7K(L) 111.7K(L) _1
0
Σt 1
10 10 10 Neutron Energy [ meV]
10
2
3
10
Fig. 1. The macroscopic total cross sections St of solid and liquid CH4 at temperatures shown, together with the total absorption cross sections Sa :
design [12]. Solid CH4 at 20.4, 50 and 90:7 K and liquid CH4 at 90.7 and 111:7 K are evaluated. The results on macroscopic cross-section S ðcm1 Þ are shown in Fig. 1 where S ¼ sN with the microscopic total cross-section s ðb=CH4 Þ and the number density N of CH4 molecules at each temperature [13,14]: for solid CH4 ; N=1024 ¼ 0:0199; 0.0192 and 0:0183 cm3 at 20.4, 50 and 90:7 K; respectively; for liquid CH4 ; N=1024 ¼ 0:0170 and 0:0159 cm3 at 90.7 and 111:7 K; respectively. Shown in Fig. 1 are the total cross-sections St ¼ Ss þ Sa and the total absorption (capture) cross-sections Sa where Ss is the total scattering cross-section. The energy range from 0:1 meV to 10 eV is divided at equal logarithmic (lethargy) intervals into 80 groups. The angular distribution of scattering cross-section is represented by the Legendre expansion up to a maximum order 3, which is almost adequate for representation of forward scattering in H by an epithermal neutron.
3. Cold neutron production In order to study the neutronic property of solid and liquid CH4 moderators, a simple model in a one-dimensional bare-slab geometry is analyzed. It is sketched in Fig. 2. Vacuum boundary conditions
ARTICLE IN PRESS 297
20.4K(S) 10
0
50K(S)
2
Energy spectrum [n/cm /s/meV]
N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
Fig. 2. The cold-neutron source model of solid and liquid CH4 :
are added on either surface of the slab so that there are no incoming neutrons. Two types of external neutron sources are assumed: an isotropic plane source ð1 neutron s1 cm2 Þ located at one surface ðx ¼ 0Þ and a uniformly distributed source ð1 neutron s1 cm3 Þ throughout the moderator. Both of them emit epithermal neutrons with energies E of the group 1 ð7:94 eVpEp10 eVÞ: Neutron fluxes at x ¼ 0; D=2 and D are calculated while varying the slab width D: A neutron transport analysis is made using the one-dimensional particle transport calculation code ANISN [15] in a P3 –S8 approximation. 3.1. Equilibrium energy spectrum A set of equilibrium energy spectra for solid CH4 at 20.4, 50 and 90:7 K and liquid CH4 at 90.7 and 111:7 K is calculated. For this calculation, the slab moderator with the uniform external source of epithermal neutrons is analyzed. The energy spectra at the center ðx ¼ D=2Þ are obtained by varying D and it is found for DX15 cm that they little change in magnitude and shape. Fig. 3 shows the calculated results for D ¼ 15 cm at four different temperatures. All the energy spectra are normalized in the energy region of a 1=E component. To characterize these spectra, the following expression for a Maxwellian plus 1=E spectrum is fitted: 8 f E 2 exp½ kTEN dE > > > 0 ðkTN Þ > > < for EpEC ; ð1Þ fðEÞ dE ¼ E C EC > dE f0 EC 2 exp½ kT > > N E ðkT Þ N > > : for EXEC ;
10
_1
90.7K(S)
10
90.7K(L)
_2
111.7K(L) _3
10 _2 10
10
_1
10
0
10
1
10
2
10
3
Neutron energy [meV] Fig. 3. The equilibrium energy spectra at x ¼ D=2 ðD ¼ 15 cmÞ for solid and liquid CH4 at four different temperatures shown.
where k is the Boltzmann constant, f0 is the total flux, TN is the neutron temperature and EC is the cutoff energy. Hence a low-energy neutron gain is evaluated in terms of GN ¼ fðkTN Þ=fð1 eVÞ where fðkTN Þ is the maximum of a Maxwellian component for EpEC and fð1 eVÞ is the spectral value of fðEÞ at E ¼ 1 eV for a well-defined 1=E component. Fig. 4 shows the results on TN and GN as a function of moderator temperature T: Also shown are the experimentally estimated values of TN and GN for a cylindrically shaped pure-CH4 moderator at many different temperatures [2]. Good moderating characteristics are obvious in terms of a steady decrease in TN with T: On the other hand, there is distinct difference in GN between the calculation and the experiment. This may arise from some neutronic effects due to moderator geometry and external source condition. The ratio of the theoretical GN to the experimental one is estimated at about 2.6, 2.6 and 2.2 for 20.4, 50 and 111:7 K; respectively. Assuming that the last value for liquid CH4 arises mainly from the above neutronic effects, the former two values for solid CH4 may presumably contain a contribution from molecule-density reduction in the frozen CH4 : For instance, fðkTN Þ calculated for 87% of the original N gives a smaller GN so that the ratio under discussion takes a value of 2.2. This gives an
ARTICLE IN PRESS N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
298
fluxes ðcm2 s1 meV1 Þ at x ¼ 0; D=2 and D are calculated to compare them in magnitude and shape where D ¼ 2:1 cm for solid CH4 at 20:4 K and D ¼ 2:5 cm for liquid CH4 at 90:7 K are selected to maximize cold neutron flux at x ¼ D: Fig. 5 shows the results of energy spectra thus calculated. Good converter characteristics are obvious in view of a well-established Maxwellian component with TN very close on T: The remarkable increase of cold neutron flux in the center is due mainly to closely-spaced energy levels of molecular rotations and large scattering crosssection for cold and thermal neutrons, in addition to less leakage of neutrons outside the moderator. This indicates a storage property of high-density cold neutrons, especially in a solid CH4 at lower temperature near 20:4 K: An optimum width of the slab moderator is determined to clarify a converter characteristic from epithermal neutrons to cold ones. The total neutron fluxes at either surface are calculated by varying D from 0.5 to 5 cm: The integrated fluxes for all Ep10 meV are shown in Fig. 6 as a function of D: Both of the solid and liquid CH4 moderators indicate a similar tendency of the fluxes for a maximizing behavior at x ¼ D and a saturation one at x ¼ 0; though there is significant difference in the magnitude. Hence it results in such an optimum width as D ¼ 2:1 cm for solid
estimated value of NB0:017 1024 cm3 for the solid CH4 moderator. 3.2. Converter characteristics Basic characteristics of solid and liquid CH4 moderators converting epithermal neutrons into cold ones are studied. The slab moderator model with the isotropic plane source (at x ¼ 0) of epithermal neutrons is analyzed. The total neutron
Neutron Temperature T N [K]
100
Experimental TN Theoretical TN Experimental GN Theoretical GN
4
3 50 2 20
10 10
Neutron gain GN /1000
5
200
1
20
50
100
0 200
Moderator Temperature T [K]
10
_1
X=D/2
D=2.1cm
D=2.5cm
X=D/2
_2
10
X=0
2
Energy spectrum [n/cm /s/meV]
Fig. 4. The neutron temperatures TN and the low-energy neutron gain GN as a function of moderator temperature T: The experimental results are taken from Ref. [2].
10
10
10
10
X=0
_3
_4
X=D
_5
X=D
Solid CH4 at 20.4K
_6 _4
10
_2
10
0
10
10
2
Neutron energy [meV]
Liquid CH4 at 90.7K 4
10 10
_4
10
_2
10
0
2
10
4
10
Neutron energy [meV]
Fig. 5. The energy spectra at x ¼ 0; D=2 and D in the solid CH4 moderator at 20:4 K (left) and the liquid CH4 moderator at 90:7 K (right).
ARTICLE IN PRESS N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
0.6
Solid at 20.4K
2
Cold neutron flux [n/cm /s]
CH4 at 20:4 K and D ¼ 2:5 cm for liquid CH4 at 90:7 K: A final analysis on the converter characteristics is made to examine an effect of reduced molecule density. Namely the number density N at each temperature is reduced by 10%, 20% and 30% and the spatial distributions of cold neutron flux for all Ep10 meV are calculated. Fig. 7 shows the results on the solid CH4 moderator ðD ¼ 2:1 cmÞ at 20:4 K and the liquid CH4 moderator ðD ¼ 2:5 cmÞ at 90:7 K: The effect under discussion is very significant since a thin-slab(or small) mod-
erator is subject to the outside leakage of neutrons and also the decrease in slowing-down power being proportional to Ss : Furthermore, the cold neutron fluxes emanating from the either surface are also calculated for the same moderators. Fig. 8 shows the angular fluxes of cold neutrons for all Ep10 meV with the original values of N: There is an anisotropic angular distribution of cold neutrons around the perpendicular direction to the
Solid CH4 at 20.4K
At X =0
0.4
299
At X=D 2.1cm
Liquid CH4 at 90.7K
0.2
Liquid at 90.7K 2.5cm 0
1
2
3
4
5
Slab width D [cm]
1
2
3
4
5
Slab width D [cm]
Fig. 8. The angular fluxes of cold neutrons below 10 meV at x ¼ 0 and D for the solid and liquid CH4 moderators.
2
_10%
0.6 _10%
2
Cold Neutron Flux [n/cm /sec]
Fig. 6. The integrated cold-neutron fluxes for all Ep10 meV:
_20%
_20% _30%
0.4
_30%
1
Solid CH 4 at 20.4K
0.2
Liquid CH 4 at 90.7K
and D=2.1cm
and D=2.5cm
Plane source at X=0 0 0
1
Distance X [cm]
Plane source at X=0 2
0
0
1
2
Distance X [cm]
Fig. 7. The spatial distributions of cold neutrons below 10 meV in the solid CH4 moderator (left) and the liquid CH4 moderator (right) where the original value of N is reduced by 10%, 20% and 30%.
ARTICLE IN PRESS 300
N. Morishima, T. Mitsuyasu / Nuclear Instruments and Methods in Physics Research A 517 (2004) 295–300
surface, i.e. at yB0 and 180 : This is due to the spatial gradient of the inside flux near the surface in view of the Fick’s law for neutron current density. With a decrease in N; the magnitudes of the emanating fluxes also reduce significantly as has been shown in Fig. 7 for the total fluxes at x ¼ 0 and D:
4. Concluding remarks Good moderating characteristics of solid CH4 as well as liquid CH4 have been demonstrated by the analysis of equilibrium energy spectra at typical temperatures. Hence high density of cold neutrons may possibly be produced especially in a solid CH4 moderator near 20:4 K; provided that the molecule density is kept almost at the equilibrium value, i.e., without large reduction due to radiation damage. It has been also shown that high intensity of cold neutron flux emanates from an adequatelydesigned solid CH4 moderator. Much more intensive beams of cold neutrons may be extracted from the center part, rather than the surface. To confirm this, a further study of the cryogenic moderator will be required, for instance, in terms of a grooved moderator [5] and/or a lowertemperature solid CH4 moderator below 20:4 K: As a powerful tool for a design assessment of practical cold neutron sources, a set of group constants on solid and liquid CH4 has been provided. By the combined use of the alreadygenerated ones on liquid 4 He; H2 and D2 [16], we may proceed to the research and development of an advanced low-energy neutron source of ultracold, very cold and cold neutrons.
Acknowledgements The authors are grateful to Mr. Yasunobu Nagaya of JAERI-Tokai for his collaboration in the neutron transport analysis and to the referee for useful comments on typical solid-CH4 moderators.
References [1] D.F.R. Mildner, R.N. Sinclair, Ann. Nucl. Energy 6 (1979) 225. [2] K. Inoue, Y. Kiyanagi, H. Iwasa, Nucl. Instr. and Meth. 192 (1982) 129. [3] E.B. Iverson, J.M. Carpenter, E.J. Hill, Physica B 241–243 (1997) 33; E.B. Iverson, J.M. Carpenter, Trans. Am. Nucl. Soc. 78 (1998) 274. [4] M. Kawai, et al., Appl. Phys. A 74 (Suppl) (2002) S43. [5] A.A. Beljakov, et al., J. Neutron Res. 3 (1996) 209. [6] G.J. Russell, et al., J. Neutron Res. 6 (1997) 33. [7] N. Morishima, Y. Sakurai, Nucl. Instr. and Meth. A 490 (2002) 527. [8] B.A. Dasannacharya, G. Venkataraman, Phys. Rev. 156 (1967) 196. [9] Y.D. Harker, R.M. Brugger, J. Chem. Phys. 42 (1965) 275. [10] W.L. Whittemore, Nucl. Sci. Eng. 18 (1964) 182. [11] Z. Rogalska, Physica 29 (1963) 491; Z. Rogalska, Acta Phys. Pol. 27 (1965) 581. [12] Y. Sakurai, T. Mitsuyasu, N. Morishima, Nucl. Instr. and Meth. A 506 (2003) 199. [13] A. Schallamach, Proc. Roy. Soc. A 171 (1939) 569. [14] R.D. McCarty, J. Hord, H.M. Roder, Selected properties of hydrogen, Engineering Design Data, NBS Monograph, USA, 1981. [15] W.W. Engle Jr., USAEC Report K-1693, 1967. [16] Y. Abe, N. Morishima, Nucl. Instr. and Meth. A 481 (2002) 414; N. Morishima, Y. Matsuo, Nucl. Instr. and Meth. A 490 (2002) 308; Y. Matsuo, N. Morishima, Y. Nagaya, Nucl. Instr. and Meth. A 496 (2003) 446.