- Email: [email protected]
Understanding UCN production in solid D2 : The generalized density of states measured via inelastic neutron scattering
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 611 (2009) 256–258
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Understanding UCN production in solid D2 : The generalized density of states measured via inelastic neutron scattering a,e ¨ E. Gutsmiedl a,, A. Frei a, A.R. Muller , S. Paul a, M. Urban a, H. Schober b, C. Morkel c, T. Unruh d a
Technische Universit¨ at M¨ unchen, Physik Department E18, D-85747 Garching, Germany Institute Laue-Langevin, F-3802 Grenoble, France c Technische Universit¨ at M¨ unchen, Physik Department E21, D-85747 Garching, Germany d Technische Universit¨ at M¨ unchen, Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), D-85747 Garching, Germany e Technische Universit¨ at M¨ unchen, Exzellenzcluster Universe, D-85748 Garching, Germany b
a r t i c l e in f o
a b s t r a c t
Available online 6 August 2009
The dynamics of solid deuterium [H. Nielsen, et al., Phys. Rev. B 7 (1973) 1626] ðsD2 Þ is studied by means of inelastic thermal neutron scattering (coherent and incoherent scattering) at different temperatures and para–ortho ratios. In this paper, the results for the generalized density of states (GDOS) are presented and discussed. The measurements were performed at the thermal time-of-flight (TOF) instrument IN4 at the ILL Grenoble, and at the cold neutron TOF instrument TofTof at the FRM II. The GDOS contains beside the hcp phonon excitations of the sD2 the rotational transition J ¼ 0-1 and 1-2. The measured and expected intensities of these rotational excitations are strongly depended on the ortho concentration in the sD2 . Above E ¼ 10 meV there are still strong excitations, which are very likely smeared out higher energy optic phonons and multiphonon contributions. An empirical method of separation of the one- and multiphonon contributions to the GDOS will be presented and discussed. The impact of these measurements on the UCN production in sD2 will be shown in detail. & 2009 Elsevier B.V. All rights reserved.
Keywords: Neutron Phonons Ultra-cold neutrons (UCN) UCN production
1. Introduction Ultra-cold neutrons (UCN) are slow enough to be confined [2,3] in traps, which can be formed by material with a high Fermi potential or by a magnetic field (60 neV T1 ). UCN can be observed for more than 1000 s in these traps, and are excellent tools for high precision measurements, concerning the life time [4,5] of the neutrons itself, and also for determining a possible small electric dipole moment of the neutron [6] (current upper limit 1026 e cm). Improving these high precision measurements makes it necessary to develop a new generation of strong UCN sources (UCN densities up to 104 UCN cm3 ). At the moment only sources [7,8] with UCN densities up to 50–300 UCN cm3 are available. Solid deuterium is one converter material (converting thermal or cold neutrons into UCN), which seems to be capable to deliver high UCN densities [9]. The downscattering of neutrons in solid D2 is mainly done by excitation of phonons in the deuterium crystals. A precise knowledge of the phonon system in the solid D2 is the base for a clear understanding of UCN production in the solid, and also a guideline for optimization of a high UCN density deuterium source. The aim of this paper is to present new neutron
Corresponding author.
E-mail address: [email protected] (E. Gutsmiedl). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.07.082
scattering measurements on solid D2 , concerning the phonons in this solid system.
2. Experiments The neutron scattering experiments have been performed at the time-of-flight machines TofTof at the FRM II [10] and at the IN4 [11] at the ILL Grenoble. The measurements at the IN4 have been carried out at two different wavelengths of the incoming ˚ whereas a wavelength of l2:0 A˚ at neutrons (l2:2 and 1:1 A), ˚ ˚ the TofTof was used. The solid D2 was frozen out in a double cylinder shell sample holder. Typically the thickness of the sample was 2–3 mm (see Fig. 1).
3. Results and discussion 3.1. Generalized density of states (GDOS) A typical dynamical neutron scattering cross-section (integrated over all scattering angles) for two different ortho concentrations of solid D2 is shown in Fig. 2. The phonon contribution to the cross-section is dominant between 0 and 18 meV on the neutron energy loss side, on the neutron energy
ARTICLE IN PRESS E. Gutsmiedl et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 256–258
257
Fig. 1. D2 sample cell.
Fig. 3. Generalized density of states of solid D2 at T ¼ 4 K. Comparison of two ortho concentrations (95%/66%). Data from IN4 measurements.
that the incoherent response of the sample should be large enough, or a large area of scattering angle should be covered by the experiment. In our scattering experiments at the IN4 or TofTof both conditions are fulfilled. Furthermore it should be pointed out that Eq. (1) is only valid if the scattering system is in thermal equilibrium. The GDOS enhances excitation modes with higher energy compared to the appearance in the dynamical neutron cross-section Z ymax " 2 # ds d s ¼ siny dy dE dO dE ymin
Fig. 2. An example of dynamical neutron cross-section of solid D2 at T ¼ 7 K. Comparison of two ortho concentrations. Data from TofTof measurements.
gain side close to the elastic line (E0). At E7:5 meV the rotational transition J ¼ 1-0 is visible. The influence of the ortho-concentration co is clearly seen in the spectrum. The J ¼ 1-0 transition (neutron energy gain side Eo0) increases with increasing concentration ðcp ¼ 1 co ) of the para molecules. On the neutron energy loss side one can see also an enhancement of the J ¼ 0-1 transition with increasing ortho concentration. The energy transfer of this transition is E7:5 meV [12]. These experimental observations are more pronounced in the plot of the generalized density of states GDOSðEÞ. The GDOS can be extracted from the data by [13] GDOSðEÞ
‘2 2M
E0 ðnðEÞ þ 1Þ
!2
Z
E 4 ðQmax
4 Þ Qmin
ymax ymin
"
2
d s dO dE
# siny dy:
ð1Þ
data
E0 is the energy of the incoming neutron in the scattering, while E is the energy transfer on the neutron. nðEÞ is the Bose statistic distribution. Qmax and Qmin are the limits of the scattering area in momentum space, while ymin and ymax are the limits for the scattering angles. This expression is only valid in the framework of the incoherent approximation [14], which means
due to the factor E in Eq. (1) (nðEÞ þ 11 for higher energies), and is therefore a good tool for inspecting these excitations. The enhancement of the J ¼ 0-1 at ðE7:5 meVÞ transition with increasing ortho concentration can be seen clearly in Fig. 3. In the case of natural deuterium ðco ¼ 66%Þ the rotational transition J ¼ 1-2 appears as a peak at E14 meV, which intensity scales with the number density of para molecules. At E5 meV there is a clear signal of the acoustic phonons, but no clear signal of the optic phonons at E9 meV, as it was published in the paper of Yu et al. [15] and Nielsen [1]. Their analysis of neutron scattering data (measuring phonon dispersion curves and determining the elastic constants of solid deuterium) leads to a density of states, which shows two narrow peaks at 5 meV and at 9 meV. These peaks are attributed to acoustic and optical phonons in solid D2 (hcp structure). Our results for the GDOS indicate a smoothing-out of the higher energy optic phonons, which was also reported for solid hydrogen [16]. The data exhibit strong multiphon contributions to the GDOS above E410 meV and also a peak at E12 meV. In a first analysis (see Fig. 4) the GDOS divided by o2 was parameterized by a sum of Gaussian functions, in order to get a preliminary picture of the excitations which contributes to the density of states of solid D2 . ðGDOS=o2 Þ is proportional to the dynamical neutron cross-section ds=dE (ð1=nðEÞ þ 1ÞE for E-0; see Eq. (1)) in the low energy limit. This approximation is only valid for small energies, but it can be used for a first analysis of the major excitations. There are two narrow peaks at E5 and 7:4 meV, which are induced by the acoustic phonons and the J ¼ 0-1 transition. The broad peak at E5 meV is phonon background, which comes from the averaging over all crystal orientations. The other broad peaks are very likely a combination of multiphon and smeared out optic phonons, the peak at E14 meV is maybe a sign of the transition J ¼ 1-2. A detailed analysis of our data concerning the density of
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E. Gutsmiedl et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 256–258
Fig. 4. Generalized density of states (divided by o2 ) of solid D2 at T ¼ 4 K with 95% ortho concentration. Fit of GDOS with a sum of Gaussian functions ðw2red ¼ 0:99Þ. Data from IN4 measurements.
states using the theory of Turchin [17] is published in another paper [18]. 3.2. UCN production With the aid of ds=dE (see Fig. 2) it is possible to calculate the UCN production cross-section. Unfortunately the absolute value of our measured cross-section is not known in first order, but it can be used to calculate the temperature dependence of the UCN production as function of the effective temperature ðTn Þ of the incoming neutrons in the solid D2 . UCN’s are produced by downscattering of thermal or subthermal neutrons. The neutron has after the scattering almost zero (neV-region) energy (Ef ). Therefore the energy transfer E ¼ E0 Ef is approximately EE0. The UCN production rate ðUCN cm3 s1 Þ in sD2 is calculated by PðTn Þ ¼ ND2
Z
Emax U 0
Z
Emax 0 0
dFðTn Þ ds dEU dE0 : dE0 dE0
ð2Þ
P is determined by integrating the product of UCN production cross-section with the spectral flux (Maxwell spectrum) of the incoming neutrons over a finite UCN energy range (EU max maximum allowable UCN energy) and over the spectrum of the incoming neutrons (see Fig. 5). The UCN production has a maximum at a neutron temperature of Tn 45 K (cold neutrons), which is larger than the result published by Yu et al. [15] ðTn ¼ 30 KÞ. This result is not a surprise, because our data are showing at higher phonon energies ðE410 meVÞ still contributions to the neutron cross-section, and therefore the overlap between the Maxwell neutron spectrum and ds=dE0 will be shifted to a higher effective neutron temperature Tn . This result fits well to recently published data of UCN production crosssection measurements [19], where the contribution of phonons at E410 meV is also seen. Our data show that the rotational transition J ¼ 0-1 (see Fig. 2) has a significant contribution to this production cross-section.
Fig. 5. UCN production gain of solid D2 ðco ¼ 95%Þ-temperature dependence.
4. Conclusion The generalized density of states of solid D2 was determined with neutron scattering methods for different ortho concentrations. The data show excitations (phonons and multiphonons) beyond E410 meV. The rotational transition J ¼ 0-1 is clearly seen in the neutron cross-section, and contribute significantly to the UCN production in solid D2 . The optimal effective neutron temperature Tn of UCN production is higher compared to earlier published results [15].
Acknowledgments This work was supported by the Maier-Leibnitz-Laboratorium Garching and by the Cluster of Excellence EXC 153. We thank T. Deuschle and H. Ruhland for their help during the experiments. References [1] H. Nielsen, et al., Phys. Rev. B 7 (1973) 1626. [2] V.K. Ignatovich, The Physics of Ultracold Neutrons, Clarendor Press, Oxford, 1990. [3] R. Golub, et al., Ultra-Cold Neutrons, Adam Hilger, Bristol, Philadelphia, New York, 1991. [4] S. Arzumanov, et al., Phys. Lett. B 483 (2000) 15. [5] A. Serebrov, et al., Phys. Lett. B 605 (2005) 72. [6] C.A. Baker, et al., Phys. Rev. Lett. 97 (2006) 131801. [7] A. Steyerl, et al., Phys. Lett. A 116 (1986) 347. [8] S.K. Lamoreaux, arXiv:nucl-ex/0103005, 2007. [9] A. Frei, et al., EPJ A 34 (2007) 119. [10] T. Unruh, et al., Nucl. Instr. and Meth. A 580 (2007) 1414. [11] H. Mutka, Nucl. Instr. and Meth. A 338 (1994) 144. [12] C.Y. Liu, Phys. Rev. B 62 (2000) R3581. [13] W.A. Kamitakahara, et al., Phys. Rev. B 44 (1991) 94. [14] G.L. Squires, Theory of Neutron Scattering from Condensed Matter, vol. 1, Cambridge, 1978. [15] Z. Ch. Yu, et al., Z. Phys. B 62 (1985) 137. [16] A. Bickermann, et al., Z. Phys. B 31 (1978) 345. [17] V.F. Turchin, Slow Neutrons, Israel Program for Scientific Translations, Jerusalem, 1965. [18] A. Frei, et al., Phys. Rev. B 80 (2009) 064301. [19] C.F. Atchison, et al., Phys. Rev. Lett. 99 (2007) 262502-1.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Understanding UCN production in solid D2 : The generalized density of states measured via inelastic neutron scattering a,e ¨ E. Gutsmiedl a,, A. Frei a, A.R. Muller , S. Paul a, M. Urban a, H. Schober b, C. Morkel c, T. Unruh d a
Technische Universit¨ at M¨ unchen, Physik Department E18, D-85747 Garching, Germany Institute Laue-Langevin, F-3802 Grenoble, France c Technische Universit¨ at M¨ unchen, Physik Department E21, D-85747 Garching, Germany d Technische Universit¨ at M¨ unchen, Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), D-85747 Garching, Germany e Technische Universit¨ at M¨ unchen, Exzellenzcluster Universe, D-85748 Garching, Germany b
a r t i c l e in f o
a b s t r a c t
Available online 6 August 2009
The dynamics of solid deuterium [H. Nielsen, et al., Phys. Rev. B 7 (1973) 1626] ðsD2 Þ is studied by means of inelastic thermal neutron scattering (coherent and incoherent scattering) at different temperatures and para–ortho ratios. In this paper, the results for the generalized density of states (GDOS) are presented and discussed. The measurements were performed at the thermal time-of-flight (TOF) instrument IN4 at the ILL Grenoble, and at the cold neutron TOF instrument TofTof at the FRM II. The GDOS contains beside the hcp phonon excitations of the sD2 the rotational transition J ¼ 0-1 and 1-2. The measured and expected intensities of these rotational excitations are strongly depended on the ortho concentration in the sD2 . Above E ¼ 10 meV there are still strong excitations, which are very likely smeared out higher energy optic phonons and multiphonon contributions. An empirical method of separation of the one- and multiphonon contributions to the GDOS will be presented and discussed. The impact of these measurements on the UCN production in sD2 will be shown in detail. & 2009 Elsevier B.V. All rights reserved.
Keywords: Neutron Phonons Ultra-cold neutrons (UCN) UCN production
1. Introduction Ultra-cold neutrons (UCN) are slow enough to be confined [2,3] in traps, which can be formed by material with a high Fermi potential or by a magnetic field (60 neV T1 ). UCN can be observed for more than 1000 s in these traps, and are excellent tools for high precision measurements, concerning the life time [4,5] of the neutrons itself, and also for determining a possible small electric dipole moment of the neutron [6] (current upper limit 1026 e cm). Improving these high precision measurements makes it necessary to develop a new generation of strong UCN sources (UCN densities up to 104 UCN cm3 ). At the moment only sources [7,8] with UCN densities up to 50–300 UCN cm3 are available. Solid deuterium is one converter material (converting thermal or cold neutrons into UCN), which seems to be capable to deliver high UCN densities [9]. The downscattering of neutrons in solid D2 is mainly done by excitation of phonons in the deuterium crystals. A precise knowledge of the phonon system in the solid D2 is the base for a clear understanding of UCN production in the solid, and also a guideline for optimization of a high UCN density deuterium source. The aim of this paper is to present new neutron
Corresponding author.
E-mail address: [email protected] (E. Gutsmiedl). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.07.082
scattering measurements on solid D2 , concerning the phonons in this solid system.
2. Experiments The neutron scattering experiments have been performed at the time-of-flight machines TofTof at the FRM II [10] and at the IN4 [11] at the ILL Grenoble. The measurements at the IN4 have been carried out at two different wavelengths of the incoming ˚ whereas a wavelength of l2:0 A˚ at neutrons (l2:2 and 1:1 A), ˚ ˚ the TofTof was used. The solid D2 was frozen out in a double cylinder shell sample holder. Typically the thickness of the sample was 2–3 mm (see Fig. 1).
3. Results and discussion 3.1. Generalized density of states (GDOS) A typical dynamical neutron scattering cross-section (integrated over all scattering angles) for two different ortho concentrations of solid D2 is shown in Fig. 2. The phonon contribution to the cross-section is dominant between 0 and 18 meV on the neutron energy loss side, on the neutron energy
ARTICLE IN PRESS E. Gutsmiedl et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 256–258
257
Fig. 1. D2 sample cell.
Fig. 3. Generalized density of states of solid D2 at T ¼ 4 K. Comparison of two ortho concentrations (95%/66%). Data from IN4 measurements.
that the incoherent response of the sample should be large enough, or a large area of scattering angle should be covered by the experiment. In our scattering experiments at the IN4 or TofTof both conditions are fulfilled. Furthermore it should be pointed out that Eq. (1) is only valid if the scattering system is in thermal equilibrium. The GDOS enhances excitation modes with higher energy compared to the appearance in the dynamical neutron cross-section Z ymax " 2 # ds d s ¼ siny dy dE dO dE ymin
Fig. 2. An example of dynamical neutron cross-section of solid D2 at T ¼ 7 K. Comparison of two ortho concentrations. Data from TofTof measurements.
gain side close to the elastic line (E0). At E7:5 meV the rotational transition J ¼ 1-0 is visible. The influence of the ortho-concentration co is clearly seen in the spectrum. The J ¼ 1-0 transition (neutron energy gain side Eo0) increases with increasing concentration ðcp ¼ 1 co ) of the para molecules. On the neutron energy loss side one can see also an enhancement of the J ¼ 0-1 transition with increasing ortho concentration. The energy transfer of this transition is E7:5 meV [12]. These experimental observations are more pronounced in the plot of the generalized density of states GDOSðEÞ. The GDOS can be extracted from the data by [13] GDOSðEÞ
‘2 2M
E0 ðnðEÞ þ 1Þ
!2
Z
E 4 ðQmax
4 Þ Qmin
ymax ymin
"
2
d s dO dE
# siny dy:
ð1Þ
data
E0 is the energy of the incoming neutron in the scattering, while E is the energy transfer on the neutron. nðEÞ is the Bose statistic distribution. Qmax and Qmin are the limits of the scattering area in momentum space, while ymin and ymax are the limits for the scattering angles. This expression is only valid in the framework of the incoherent approximation [14], which means
due to the factor E in Eq. (1) (nðEÞ þ 11 for higher energies), and is therefore a good tool for inspecting these excitations. The enhancement of the J ¼ 0-1 at ðE7:5 meVÞ transition with increasing ortho concentration can be seen clearly in Fig. 3. In the case of natural deuterium ðco ¼ 66%Þ the rotational transition J ¼ 1-2 appears as a peak at E14 meV, which intensity scales with the number density of para molecules. At E5 meV there is a clear signal of the acoustic phonons, but no clear signal of the optic phonons at E9 meV, as it was published in the paper of Yu et al. [15] and Nielsen [1]. Their analysis of neutron scattering data (measuring phonon dispersion curves and determining the elastic constants of solid deuterium) leads to a density of states, which shows two narrow peaks at 5 meV and at 9 meV. These peaks are attributed to acoustic and optical phonons in solid D2 (hcp structure). Our results for the GDOS indicate a smoothing-out of the higher energy optic phonons, which was also reported for solid hydrogen [16]. The data exhibit strong multiphon contributions to the GDOS above E410 meV and also a peak at E12 meV. In a first analysis (see Fig. 4) the GDOS divided by o2 was parameterized by a sum of Gaussian functions, in order to get a preliminary picture of the excitations which contributes to the density of states of solid D2 . ðGDOS=o2 Þ is proportional to the dynamical neutron cross-section ds=dE (ð1=nðEÞ þ 1ÞE for E-0; see Eq. (1)) in the low energy limit. This approximation is only valid for small energies, but it can be used for a first analysis of the major excitations. There are two narrow peaks at E5 and 7:4 meV, which are induced by the acoustic phonons and the J ¼ 0-1 transition. The broad peak at E5 meV is phonon background, which comes from the averaging over all crystal orientations. The other broad peaks are very likely a combination of multiphon and smeared out optic phonons, the peak at E14 meV is maybe a sign of the transition J ¼ 1-2. A detailed analysis of our data concerning the density of
ARTICLE IN PRESS 258
E. Gutsmiedl et al. / Nuclear Instruments and Methods in Physics Research A 611 (2009) 256–258
Fig. 4. Generalized density of states (divided by o2 ) of solid D2 at T ¼ 4 K with 95% ortho concentration. Fit of GDOS with a sum of Gaussian functions ðw2red ¼ 0:99Þ. Data from IN4 measurements.
states using the theory of Turchin [17] is published in another paper [18]. 3.2. UCN production With the aid of ds=dE (see Fig. 2) it is possible to calculate the UCN production cross-section. Unfortunately the absolute value of our measured cross-section is not known in first order, but it can be used to calculate the temperature dependence of the UCN production as function of the effective temperature ðTn Þ of the incoming neutrons in the solid D2 . UCN’s are produced by downscattering of thermal or subthermal neutrons. The neutron has after the scattering almost zero (neV-region) energy (Ef ). Therefore the energy transfer E ¼ E0 Ef is approximately EE0. The UCN production rate ðUCN cm3 s1 Þ in sD2 is calculated by PðTn Þ ¼ ND2
Z
Emax U 0
Z
Emax 0 0
dFðTn Þ ds dEU dE0 : dE0 dE0
ð2Þ
P is determined by integrating the product of UCN production cross-section with the spectral flux (Maxwell spectrum) of the incoming neutrons over a finite UCN energy range (EU max maximum allowable UCN energy) and over the spectrum of the incoming neutrons (see Fig. 5). The UCN production has a maximum at a neutron temperature of Tn 45 K (cold neutrons), which is larger than the result published by Yu et al. [15] ðTn ¼ 30 KÞ. This result is not a surprise, because our data are showing at higher phonon energies ðE410 meVÞ still contributions to the neutron cross-section, and therefore the overlap between the Maxwell neutron spectrum and ds=dE0 will be shifted to a higher effective neutron temperature Tn . This result fits well to recently published data of UCN production crosssection measurements [19], where the contribution of phonons at E410 meV is also seen. Our data show that the rotational transition J ¼ 0-1 (see Fig. 2) has a significant contribution to this production cross-section.
Fig. 5. UCN production gain of solid D2 ðco ¼ 95%Þ-temperature dependence.
4. Conclusion The generalized density of states of solid D2 was determined with neutron scattering methods for different ortho concentrations. The data show excitations (phonons and multiphonons) beyond E410 meV. The rotational transition J ¼ 0-1 is clearly seen in the neutron cross-section, and contribute significantly to the UCN production in solid D2 . The optimal effective neutron temperature Tn of UCN production is higher compared to earlier published results [15].
Acknowledgments This work was supported by the Maier-Leibnitz-Laboratorium Garching and by the Cluster of Excellence EXC 153. We thank T. Deuschle and H. Ruhland for their help during the experiments. References [1] H. Nielsen, et al., Phys. Rev. B 7 (1973) 1626. [2] V.K. Ignatovich, The Physics of Ultracold Neutrons, Clarendor Press, Oxford, 1990. [3] R. Golub, et al., Ultra-Cold Neutrons, Adam Hilger, Bristol, Philadelphia, New York, 1991. [4] S. Arzumanov, et al., Phys. Lett. B 483 (2000) 15. [5] A. Serebrov, et al., Phys. Lett. B 605 (2005) 72. [6] C.A. Baker, et al., Phys. Rev. Lett. 97 (2006) 131801. [7] A. Steyerl, et al., Phys. Lett. A 116 (1986) 347. [8] S.K. Lamoreaux, arXiv:nucl-ex/0103005, 2007. [9] A. Frei, et al., EPJ A 34 (2007) 119. [10] T. Unruh, et al., Nucl. Instr. and Meth. A 580 (2007) 1414. [11] H. Mutka, Nucl. Instr. and Meth. A 338 (1994) 144. [12] C.Y. Liu, Phys. Rev. B 62 (2000) R3581. [13] W.A. Kamitakahara, et al., Phys. Rev. B 44 (1991) 94. [14] G.L. Squires, Theory of Neutron Scattering from Condensed Matter, vol. 1, Cambridge, 1978. [15] Z. Ch. Yu, et al., Z. Phys. B 62 (1985) 137. [16] A. Bickermann, et al., Z. Phys. B 31 (1978) 345. [17] V.F. Turchin, Slow Neutrons, Israel Program for Scientific Translations, Jerusalem, 1965. [18] A. Frei, et al., Phys. Rev. B 80 (2009) 064301. [19] C.F. Atchison, et al., Phys. Rev. Lett. 99 (2007) 262502-1.