- Email: [email protected]
Cross section model of water for neutron scattering study and neutron source design
ARTICLE IN PRESS
Physica B 350 (2004) e655–e658
Cross section model of water for neutron scattering study and neutron source design Yoshinobu Edura*, Nobuhiro Morishima Department of Nuclear Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan
Abstract The motions of water molecules in liquid are described in terms of a jump diffusion mechanism and a rotational diffusive process, together with an intermolecular vibration and a hindered rotation. Intramolecular bending and stretching motions are also included. This permits us to use the present model for an interpretation of neutron scattering results in a wide range of incident energies, say, from 0:1 meV to 10 eV; at all practical temperatures. A velocity auto-correlation function, a density of states and scattering cross sections, both double-differential and total, are calculated. Good agreements with neutron-scattering and computer-simulation results are found, which makes it possible to generate a basic data library for an advanced neutron-source design. r 2004 Elsevier B.V. All rights reserved. PACS: 78.70.N; 95.30.F; 29.25.D Keywords: Water; Jump diffusion; Cross section model; Cold neutron
1. Introduction The methods of cold ðB1 meVÞ and thermal ðB25 meVÞ neutron scattering are expected to be useful in investigating structures and dynamics of soft condensed matters, bio-molecular, superconductors and other materials. For this purpose, a next-generation spallation neutron source is being constructed in Japan, USA and Europe. Highenergy ðBMeVÞ neutrons generated from a spallation target must be moderated to thermal and cold neutrons with an optimized hydrogenous material like light water ðH2 OÞ; liquid H2 and solid CH4 : We must know such moderators’ neutronic *Corresponding author. Fax: +81-75-753-5836. E-mail address: [email protected] (Y. Edura).
properties as scattering cross sections and neutron transport processes. Hence, in this paper, we develop a cross section model for neutron scattering in liquid water, which is applicable to a wide range of incident neutron energy from 0:1 meV (ultra-cold) to 10 eV (epi-thermal) and also at all practical temperatures between melting and boiling points. Another purpose of this paper is to present a molecular dynamics model which describes a hydrogen-bonding property inherent to water and consequently various relaxation processes of molecular translations and rotations in a wide time scale from B0:001 to B100 ps: A set of characteristic functions such as a velocity auto-correlation function, an intermediate scattering function and a dynamical scattering function is calculated, which
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.03.175
ARTICLE IN PRESS e656
Y. Edura, N. Morishima / Physica B 350 (2004) e655–e658
is necessary for a physical and systematic interpretation of neutron-scattering and moleculardynamics results.
behave as a free-gas of H or O at a very-short time scale of, say, 0:001 ps: This is required to represent a free-atom cross section for a neutron with a high incident energy above B1 eV:
2. Model 3. Results and discussion
2 d2 s k 0 1 H2 H ðb þ bH ¼ 2 inc ÞSinc ðQ; oÞ dO dE 0 k _ coh 2 k 0 1 O2 O ðbcoh þ bO þ ð1Þ inc ÞSinc ðQ; oÞ; k_ where _ is the Planck’s constant, k and k0 are the incident and scattering neutron’s wave numbers, X bX coh and binc are the coherent and incoherent scattering lengths, respectively. With the Gaussian X representation of Sinc ðQ; oÞ; all the molecular dynamics are described by a velocity auto-correlation function or equivalently a density of states. In view of experimental results and computer simulations in last two decades, the following characteristic pictures for water molecule dynamics are included:
*
*
*
*
A water molecule is hindered in translational motion by a hydrogen-bonds network formed intermittently with surrounding ones. Translational diffusion and vibration are repeated as a jump diffusion (JD) process [1]. The latter is such an intermolecular motion as bending and stretching modes. Rotational motions are described in terms of rotational diffusion (RD) in a large time scale [2] and librational vibrations (hindered rotation) in an intermediate time scale. Also included are intra-molecular vibrations with bending, stretching and asymmetric stretching modes.
All these dynamics are described in an explicit form of a velocity auto-correlation function and involves their changes in liquid temperature through translational and rotational diffusion coefficients, hydrogen-bond lifetime and so on. It should be also noted that these motions tend to
Fig. 1 shows the density of states, DðoÞ; as a function of energy transfer _o up to 130 meV: Experimental results [3] and molecular dynamics (MD) simulation results [4] are also shown. The proposed model can reproduce satisfactorily these results in view of the following characteristic behaviors: translational diffusion near 0 meV; hindered rotation of a molecule around 60 meV; and translational stretching and bending vibrations at B6 and B20 meV; respectively. Intramolecular vibrations with 200 and 463 meV are beyond the energy range shown. It is to be noted that a velocity auto-correlation function and an intermediate scattering function are also calculated at many different temperatures and found to be consistent with the recent MD results [5]. Fig. 2 shows the double-differential cross sections of water at 293 K for two different scattering angles of 15 and 60 with an incident neutron energy Ein ¼ 1 meV: These cross sections are composed of two components: a very high and sharp peak of quasi-elastic neutron scattering (QENS) due to the jump and rotational diffusion 0.03
Density of States D() [psec-1]
The double-differential cross section of a water molecule can be expressed using a self-scattering X function Sinc ðQ; oÞ ðX ¼ H; OÞ by incoherent approximation,
O 0.02 T = 293 K
0.01 H
0
20
40
60
80
100
120
h [meV]
Fig. 1. The density of states of H and O for water at 293 K: the present model by solid lines, the experimental results [3] by m and the MD results [4] by J; K:
ARTICLE IN PRESS Y. Edura, N. Morishima / Physica B 350 (2004) e655–e658 30
Angular Distribution [b/str/H2O]
processes and a relatively broad peak of inelastic neutron scattering (INS) arising from up-scattering by de-excitation of the intermolecular vibrations and hindered rotations. With an increase in scattering angle, the QENS peak becomes broad and small while the INS peak tends to be less significant. Fig. 3 shows the half-width at half-maximum H (HWHM) of the QENS of Sinc ðQ; oÞ as a function of squared momentum transfer Q2 at 293 K: Also shown is the experimentally estimated results [6]. The solid line is calculated with the inclusion of all
e657
Present Model Beyster [7] Lemmel [8] Ein = 1.0 meV 20
σs/4π
10
20 meV
σHs cos(θ)/π+σsO/4π 10 eV
Double-Differential Cross Section [b/eV/str/H2O]
0 30
1000
60
90
120
150
180
Scattering Angle θ [degree] 9.53E+5.0
6.93E+4.0
Fig. 4. The angular distributions of water at 293 K for Ein ¼ 1 meV; 20 meV and 10 eV; together with the experimental results [7,8].
800 600
θ = 15
θ = 60
400 200 0
Bending Stretching 0
20
Libration
20 40 40 60 80 0 Scattering Neutron Energy E [meV]
60
80
Fig. 2. The double-differential cross-sections of water at T ¼ 293 K for Ein ¼ 1:0 meV and y ¼ 15 ; 60 : 0.6
HWHM [meV]
0.5
0.4
Present Model with All Modes Only JD Model Teixeira et al. [6] Ein=3.2 meV T = 293 K
0.3
0.2
0.1
0
2
4
dynamics modes and the broken line with only the JD process. The former is larger in comparison with the latter especially for large Q2 values. This H reason is that Sinc ðQ; oÞ involves both QENS and INS components in these Q range. The broken line is similar in saturation behavior with the experimental results [6]. It should be noted that the HWHM analysis is performed successfully at many different temperatures in comparison with those on neutron scattering. Fig. 4 shows the calculated angular distributions for three different incident energies, compared with experimental results for thermal neutrons ðEin ¼ 20 meVÞ [7,8]. For a 10-eV neutron, forward scattering is significant on account of a freeatom scattering in H. On the contrary, a cold neutron scatters nearly isotropically though a backward scattering is slightly seen by up-scattering due to the de-excitation of intermolecular vibrations and librations. The result on a thermal neutron indicates a mid-behavior between the above extremes.
6
Q 2 [Å-2] H Fig. 3. The HWHM of the QENS peak of Sinc ðQ; oÞ for Ein ¼ 3:2 meV at T ¼ 293 K; together with the experimental results [6].
4. Concluding remarks It must be pointed out that the present crosssection model covers a very wide range of neutron
ARTICLE IN PRESS e658
Y. Edura, N. Morishima / Physica B 350 (2004) e655–e658
incident energy from 0:1 meV to 10 eV; i.e., from ultra-cold to epi-thermal neutron areas. Various cross-section results on neutron scattering are obtainable: besides the above, a velocity autocorrelation function, an intermediate scattering function, a mean square displacement (a width function) and total cross section are calculated. This indicates the usefulness of the model both for an interpretation of neutron-scattering and molecular-dynamics results and for generation of basic data to design an advanced neutron source. For the latter, a set of energy-averaged cross sections (group constants) is now in preparation, which
makes soon possible the combined use of those for superfluid 4 He; liquid H2 ; and solid CH4 :
References [1] [2] [3] [4] [5] [6] [7] [8]
. K.S. Singwi, A. Sjolander, Phys. Rev. 119 (1960) 863. V.F. Sears, Can. J. Phys. 44 (1966) 1299. M.-C. Bellissent-Funel, et al., Phys. Rev. E 51 (1995) 4558. G.C. Lie, E. Clementi, Phys. Rev. A 33 (1986) 2679. D.Di. Cola, et al., J. Chem. Phys. 104 (1996) 4223. J. Teixeira, et al., Phys. Rev. A 31 (1985) 1913. J.R. Beyster, Nucl. Sci. Eng. 31 (1968) 254. H.D. Lemmel, Nukleonik 7 (1965) 265.
Physica B 350 (2004) e655–e658
Cross section model of water for neutron scattering study and neutron source design Yoshinobu Edura*, Nobuhiro Morishima Department of Nuclear Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan
Abstract The motions of water molecules in liquid are described in terms of a jump diffusion mechanism and a rotational diffusive process, together with an intermolecular vibration and a hindered rotation. Intramolecular bending and stretching motions are also included. This permits us to use the present model for an interpretation of neutron scattering results in a wide range of incident energies, say, from 0:1 meV to 10 eV; at all practical temperatures. A velocity auto-correlation function, a density of states and scattering cross sections, both double-differential and total, are calculated. Good agreements with neutron-scattering and computer-simulation results are found, which makes it possible to generate a basic data library for an advanced neutron-source design. r 2004 Elsevier B.V. All rights reserved. PACS: 78.70.N; 95.30.F; 29.25.D Keywords: Water; Jump diffusion; Cross section model; Cold neutron
1. Introduction The methods of cold ðB1 meVÞ and thermal ðB25 meVÞ neutron scattering are expected to be useful in investigating structures and dynamics of soft condensed matters, bio-molecular, superconductors and other materials. For this purpose, a next-generation spallation neutron source is being constructed in Japan, USA and Europe. Highenergy ðBMeVÞ neutrons generated from a spallation target must be moderated to thermal and cold neutrons with an optimized hydrogenous material like light water ðH2 OÞ; liquid H2 and solid CH4 : We must know such moderators’ neutronic *Corresponding author. Fax: +81-75-753-5836. E-mail address: [email protected] (Y. Edura).
properties as scattering cross sections and neutron transport processes. Hence, in this paper, we develop a cross section model for neutron scattering in liquid water, which is applicable to a wide range of incident neutron energy from 0:1 meV (ultra-cold) to 10 eV (epi-thermal) and also at all practical temperatures between melting and boiling points. Another purpose of this paper is to present a molecular dynamics model which describes a hydrogen-bonding property inherent to water and consequently various relaxation processes of molecular translations and rotations in a wide time scale from B0:001 to B100 ps: A set of characteristic functions such as a velocity auto-correlation function, an intermediate scattering function and a dynamical scattering function is calculated, which
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.03.175
ARTICLE IN PRESS e656
Y. Edura, N. Morishima / Physica B 350 (2004) e655–e658
is necessary for a physical and systematic interpretation of neutron-scattering and moleculardynamics results.
behave as a free-gas of H or O at a very-short time scale of, say, 0:001 ps: This is required to represent a free-atom cross section for a neutron with a high incident energy above B1 eV:
2. Model 3. Results and discussion
2 d2 s k 0 1 H2 H ðb þ bH ¼ 2 inc ÞSinc ðQ; oÞ dO dE 0 k _ coh 2 k 0 1 O2 O ðbcoh þ bO þ ð1Þ inc ÞSinc ðQ; oÞ; k_ where _ is the Planck’s constant, k and k0 are the incident and scattering neutron’s wave numbers, X bX coh and binc are the coherent and incoherent scattering lengths, respectively. With the Gaussian X representation of Sinc ðQ; oÞ; all the molecular dynamics are described by a velocity auto-correlation function or equivalently a density of states. In view of experimental results and computer simulations in last two decades, the following characteristic pictures for water molecule dynamics are included:
*
*
*
*
A water molecule is hindered in translational motion by a hydrogen-bonds network formed intermittently with surrounding ones. Translational diffusion and vibration are repeated as a jump diffusion (JD) process [1]. The latter is such an intermolecular motion as bending and stretching modes. Rotational motions are described in terms of rotational diffusion (RD) in a large time scale [2] and librational vibrations (hindered rotation) in an intermediate time scale. Also included are intra-molecular vibrations with bending, stretching and asymmetric stretching modes.
All these dynamics are described in an explicit form of a velocity auto-correlation function and involves their changes in liquid temperature through translational and rotational diffusion coefficients, hydrogen-bond lifetime and so on. It should be also noted that these motions tend to
Fig. 1 shows the density of states, DðoÞ; as a function of energy transfer _o up to 130 meV: Experimental results [3] and molecular dynamics (MD) simulation results [4] are also shown. The proposed model can reproduce satisfactorily these results in view of the following characteristic behaviors: translational diffusion near 0 meV; hindered rotation of a molecule around 60 meV; and translational stretching and bending vibrations at B6 and B20 meV; respectively. Intramolecular vibrations with 200 and 463 meV are beyond the energy range shown. It is to be noted that a velocity auto-correlation function and an intermediate scattering function are also calculated at many different temperatures and found to be consistent with the recent MD results [5]. Fig. 2 shows the double-differential cross sections of water at 293 K for two different scattering angles of 15 and 60 with an incident neutron energy Ein ¼ 1 meV: These cross sections are composed of two components: a very high and sharp peak of quasi-elastic neutron scattering (QENS) due to the jump and rotational diffusion 0.03
Density of States D() [psec-1]
The double-differential cross section of a water molecule can be expressed using a self-scattering X function Sinc ðQ; oÞ ðX ¼ H; OÞ by incoherent approximation,
O 0.02 T = 293 K
0.01 H
0
20
40
60
80
100
120
h [meV]
Fig. 1. The density of states of H and O for water at 293 K: the present model by solid lines, the experimental results [3] by m and the MD results [4] by J; K:
ARTICLE IN PRESS Y. Edura, N. Morishima / Physica B 350 (2004) e655–e658 30
Angular Distribution [b/str/H2O]
processes and a relatively broad peak of inelastic neutron scattering (INS) arising from up-scattering by de-excitation of the intermolecular vibrations and hindered rotations. With an increase in scattering angle, the QENS peak becomes broad and small while the INS peak tends to be less significant. Fig. 3 shows the half-width at half-maximum H (HWHM) of the QENS of Sinc ðQ; oÞ as a function of squared momentum transfer Q2 at 293 K: Also shown is the experimentally estimated results [6]. The solid line is calculated with the inclusion of all
e657
Present Model Beyster [7] Lemmel [8] Ein = 1.0 meV 20
σs/4π
10
20 meV
σHs cos(θ)/π+σsO/4π 10 eV
Double-Differential Cross Section [b/eV/str/H2O]
0 30
1000
60
90
120
150
180
Scattering Angle θ [degree] 9.53E+5.0
6.93E+4.0
Fig. 4. The angular distributions of water at 293 K for Ein ¼ 1 meV; 20 meV and 10 eV; together with the experimental results [7,8].
800 600
θ = 15
θ = 60
400 200 0
Bending Stretching 0
20
Libration
20 40 40 60 80 0 Scattering Neutron Energy E [meV]
60
80
Fig. 2. The double-differential cross-sections of water at T ¼ 293 K for Ein ¼ 1:0 meV and y ¼ 15 ; 60 : 0.6
HWHM [meV]
0.5
0.4
Present Model with All Modes Only JD Model Teixeira et al. [6] Ein=3.2 meV T = 293 K
0.3
0.2
0.1
0
2
4
dynamics modes and the broken line with only the JD process. The former is larger in comparison with the latter especially for large Q2 values. This H reason is that Sinc ðQ; oÞ involves both QENS and INS components in these Q range. The broken line is similar in saturation behavior with the experimental results [6]. It should be noted that the HWHM analysis is performed successfully at many different temperatures in comparison with those on neutron scattering. Fig. 4 shows the calculated angular distributions for three different incident energies, compared with experimental results for thermal neutrons ðEin ¼ 20 meVÞ [7,8]. For a 10-eV neutron, forward scattering is significant on account of a freeatom scattering in H. On the contrary, a cold neutron scatters nearly isotropically though a backward scattering is slightly seen by up-scattering due to the de-excitation of intermolecular vibrations and librations. The result on a thermal neutron indicates a mid-behavior between the above extremes.
6
Q 2 [Å-2] H Fig. 3. The HWHM of the QENS peak of Sinc ðQ; oÞ for Ein ¼ 3:2 meV at T ¼ 293 K; together with the experimental results [6].
4. Concluding remarks It must be pointed out that the present crosssection model covers a very wide range of neutron
ARTICLE IN PRESS e658
Y. Edura, N. Morishima / Physica B 350 (2004) e655–e658
incident energy from 0:1 meV to 10 eV; i.e., from ultra-cold to epi-thermal neutron areas. Various cross-section results on neutron scattering are obtainable: besides the above, a velocity autocorrelation function, an intermediate scattering function, a mean square displacement (a width function) and total cross section are calculated. This indicates the usefulness of the model both for an interpretation of neutron-scattering and molecular-dynamics results and for generation of basic data to design an advanced neutron source. For the latter, a set of energy-averaged cross sections (group constants) is now in preparation, which
makes soon possible the combined use of those for superfluid 4 He; liquid H2 ; and solid CH4 :
References [1] [2] [3] [4] [5] [6] [7] [8]
. K.S. Singwi, A. Sjolander, Phys. Rev. 119 (1960) 863. V.F. Sears, Can. J. Phys. 44 (1966) 1299. M.-C. Bellissent-Funel, et al., Phys. Rev. E 51 (1995) 4558. G.C. Lie, E. Clementi, Phys. Rev. A 33 (1986) 2679. D.Di. Cola, et al., J. Chem. Phys. 104 (1996) 4223. J. Teixeira, et al., Phys. Rev. A 31 (1985) 1913. J.R. Beyster, Nucl. Sci. Eng. 31 (1968) 254. H.D. Lemmel, Nukleonik 7 (1965) 265.