Direct Measurements of Quantum Kinetic Energy. Tensor in Stable and Metastable Water near the. Triple Point: An Experimental

Transcription

1 Direct Measurements of Quantum Kinetic Energy Tensor in Stable and Metastable Water near the Triple Point: An Experimental Benchmark:Supporting Information Carla Andreani, Giovanni Romanelli, and Roberto Senesi, Università degli Studi di Roma "Tor Vergata", Dipartimento di Fisica e Centro NAST, Via della Ricerca Scientica 1, Roma, I, Consiglio Nazionale delle Ricerche, CNR-IPCF, Sezione di Messina, I, and ISIS Pulsed Neutron and Muon Source, Science Technology Facility Council, Chilton, Oxfordshire, OX11 0QX, UK roberto.senesi@uniroma2.it To whom correspondence should be addressed Università degli Studi di Roma "Tor Vergata", Dipartimento di Fisica e Centro NAST, Via della Ricerca Scientica 1, Roma, I Consiglio Nazionale delle Ricerche, CNR-IPCF, Sezione di Messina, I ISIS Pulsed Neutron and Muon Source, Science Technology Facility Council, Chilton, Oxfordshire, OX11 0QX, UK 1

2 Supporting Information DINS measurements Spectrometer and sample environment set up The DINS measurements of water near the triple point are performed using the VESUVIO spectrometer at the ISIS Pulsed Neutron and Muon Source (Rutherford Appleton Laboratory, Chilton, Didcot, UK). This is an inverse geometry spectrometer which uses a white beam of incident neutrons, with energies in the range 1 ev ev. Neutrons with initial energy E 0 travel a distance L 0 from the moderator to the sample; after scattering at an angle θ, neutrons with nal energy E 1 travel a distance L 1 to the detector. At each angle the nal energy of the scattered neutrons, E 1 = 4897 mev, is selected using an Au resonance lter that absorbs neutrons in a narrow range of energies centred around E 1. 1 The time of ight (t.o.f.) technique is used to reconstruct the kinematics of the scattering process, yielding the wave vector and energy transfers for each t.o.f. channel. The scattering signal is recorded by individual detectors: in the forward direction scattered neutrons are detected by 64 Yttrium Aluminum Perovskite (YAP) scintillators, 2 located at a distance L 1, ranging between 0.50 m and 0.75 m from sample position, in the angular range o to 72.5 o ; in the backward direction scattered neutrons are detected by Li scintillators, located at a distance ranging between 0.46 m and 0.67 m from sample position, in the angular range 130 to 163. The instrument operates using the Foil Cycling technique in forward scattering, 1,3 and the Double Dierence technique 4 in backward scattering. The range of the wave vector and energy transfer is 27 Å 1 q 230 Å 1 and 2.5 ev ω 800 ev respectively. 5,6 In previous experiments on SW 7,8 the water samples were contained in a at disk shaped Al can (5 cm diameter, 1 mm sample thickness) with inner Teon coating. This circular disk was embraced by a circular sector containing continuously owing Ethylene Glycol during the SW experiment. In order to eliminate any possible source of systematic spurious signal from the refrigerator bath, in this paper we have planned a completely dierent sample container and refrigerator set up, as described in the following. Samples are contained inside a circular at stainless steel can, 1.0 mm window thickness and 7 cm diameter, and positioned perpendicularly to the beam. The sample container is attached to a standard VESUVIO centre-stick connected to a Closed Circuit Refrigerator. The samples' thickness is 1 mm. A specic sample temperature control and sensing system is realised, with 4 heaters and 4 sensors, Rh/Fe thermocouples, placed in the top and bottom edges of the container close to the sample volume. This set up is designed in order to guarantee an homogeneous sample heating and to carefully monitor temperature in the supercooled and stable phases of the water samples during the whole duration of the DINS experiment. The temperature stability of the samples is ±0.1 K. The temperature readings are logged every 5 seconds, to directly monitor the phase of SW. In particular, in the event of freezing of the SW sample the temperature increase due to the release of the heat of solidication is measured in real time. A second procedure to monitor the phase of the metastable supercooled liquid is set up using the diraction capability of the VESUVIO back scattering detector banks (see below). A picture of the sample used in the experiment is shown in Figure 1. The temperature sensors, two of which are placed on the top of 2

3 the sample and two on the bottom, are labelled as Top1, Top2, Bottom1 and Bottom2, respectively. The sample is attached to a standard centre-stick placed in the VESUVIO Closed Circuit Refrigerator. Figure 1: (colour online) Water sample stainless steel container attached to a standard VESUVIO centre-stick. The sample is equipped with four heaters and four sensors (Top1, Top2, Bottom1, Bottom2) with temperature values recorded every 5 s. Sample preparation and data acquisition The stainless steel container is loaded with ultrapure distilled water. The container is sealed, attached to the centre-stick and placed in the VESUVIO sample position. The sample is slowly cooled down from room temperature to few degrees above the melting point. The sample temperature is then set to T=271 K and the temperature readings reach the set point value within 15 minutes. Figure 2 shows the temperature log for the four sensors, during a 24 hours DINS measurement on SW at T = 271 K. The inset (a) reports the nal approach from 2 degrees above the triple point to the set-point temperature at 271 K, which is reached within 15 minutes after few smooth oscillations. The DINS acquisition is then started 10 minutes after the temperature is stable within ±0.1 K. The set-point temperature is stable for about 25 hours. The inset (b) shows the temperature increase associated to the heat release from the transition to polycrystalline ice. DINS data acquisition is terminated within 5 minutes from the temperature increase, and the last 5 minutes of acquisition are discarded from the data analysis. The sample is then warmed up to room temperature in order to complete the melting of the ice inside the container. The sample cooling procedure is then repeated to perform a new DINS measurement on SW. Ice sample preparation is carried out by setting the temperature to 250 K to guarantee freezing and then heating to the set-point of the measurement. The room temperature water sample preparation is carried 3

4 out by setting the temperature at 300 K. DINS spectra are recorded for a total integrated proton current of about 5000 µah for SW sample at T=271 K, 4300 µah for ice samples at T = 271 K and 2000 µah at T = 270 K and 4000 µah for water sample at T=300 K. For each sample, the acquisition is split into 180 µah runs. Temperature [ C] (a) (a) Time [minutes] (b) Top1 Top2 Bottom1 Bottom Time [hours] (b) Time [minutes] Figure 2: (colour online) Temperature logs of the 4 sensors located on the sample container during the measurement of SW at T = 271 K. The insets highlight (a) the temperature stabilization at the beginning of the DINS measurement and (b) the heat release during the sample freezing at end of the measurement. In addition to the temperature log monitoring, the analysis of the diraction signal is carried out for each 180 µah DINS acquisition run. This allows an independent inspection of the supercooled sample phase. The t.o.f. spectra of the backscattering detectors are converted to d-spacing and the Bragg peak pattern is compared to that of ice at T = 271 K and of ice at T = 270 K measured under the same experimental conditions. Ice spectra at T=270 K are acquired in order to have of an additional temperature point to be used in comparison with the SW diraction signal. The d-spacing patterns are dominated by Bragg peaks from the steel container. Ice samples show additional Bragg peaks which are not present in the diraction data of the SW sample DINS acquisition. The d-spacing diraction patterns from the SW and ice samples are reported in Figure 3. The SW pattern does not show Bragg peaks associated to a polycrystalline phase, thus providing evidence on the liquid state of the SW sample at T=271 K. The analysis of temperature logs and diraction patterns provide independent monitoring of the samples' phase, with special regards to the metastable SW sample. Data reduction and analysis In DINS experiments the VESUVIO spectrometer operates in the Impulse Approximation (IA) regime. The latter holds for high energy, ω and q, i.e where the incident neutron wavelengths are much less than the inter-atomic spacing and thus atoms scatter incoherently, with the total scattering intensity being the sum of intensities from individual atoms in the 4

5 1.04 A 1.08 A 1.27 A d-spacing [ Å ] Figure 3: (colour online) Bragg peaks from all the VESUVIO backscattering detectors in the case of SW at T = 271 K (red line) and of ice at T = 271 K (blue line) and T = 270 K (black line). sample. 6,9,10 The DINS count rate as a function of t.o.f. at each l-th detector yields the following expression: ( ) 8E0 3 d 2 σ m C(t) = I(E m n L 2 0 )D(E 1 ) N m dω (1) 0 dωde 1 where I(E 0 )de 0 is the number of incident neutrons s 1 with energies between E 0 and E 0 + de 0, D(E 1 ) is the probability that a neutron of energy E 1 is detected, m n is the neutron mass, L 0 is the distance between moderator and sample, m is the mass of the particle being struck by the neutron, N m is the number of atoms of mass m in the sample and d2 σ m dωde 1 is the partial dierential cross-section for mass m. In the IA regime the d2 σ m dωde 1 and the dynamical structure factor, S IA (q, ω), are expressed in terms of the neutron NCP, J(y, ˆq), the West scaling variable, y and n(p) as follows: 6,10 d 2 σ m dωde 1 = b 2 m q ( E1 E 0 M ) 1/2 J IA(y, ˆq) (2) where b is the neutron scattering length of the mass m atom. and q m S IA(q, ω) = J IA (y, ˆq) = n(p)δ (y p ˆq) dp (3) 5

6 where the scaling variable y: y = m q ) (ω q2 2m is the projection of the particle momentum distribution n(p) along the ˆq direction and J IA (y, ˆq) is the NCP within the IA framework. 6,9 For isotropic samples, the particle momentum distribution depends on p only and the ˆq direction becomes immaterial. Thus the NCP is expressed by J IA (y) = 2π pn(p)dp. y Data reduction and analysis of the hydrogen DINS signal is carried out using the forward scattering detectors. Raw t.o.f. data of individual detectors for SW and ice are corrected by γ-background, multiple scattering, steel sample-container and water's oxygen contributions, using a standard procedure available on VESUVIO. 11,12 The t.o.f. data for SW at the angle θ = 35 degrees is reported in Figure 4 together with the simulated multiple scattering contribution. In this gure the peak of the hydrogen signal is located between µs and the sample-container signal is between µs. (4) normalised count rate t.o.f. [µs] Figure 4: Raw spectrum of SW at T = 271 K for a detector at the angular position θ = 35 degrees. Experimental data are reported as (blue error bars) and simulated multiple scattering as a red line. Due to the nite q values in the scattering process, the NCP at each detector retains the q dependence, expressed by the function F (y, q). The F (y, q) function is related to the count rate via the expression: F l (y, q) = BM E 0 I(E 0 ) q C l(t) (5) where B is a constant taking into account several contributions: the detector solid angle, its eciency at E = E 1, the time-energy Jacobian, the free-atom neutron cross section and the number of particles hit by the neutron beam. DINS data sets of all samples are y-scaled 6

7 according to Eq. (5). In a DINS experiment the asymptotic IA prole, strictly valid in the limit of innite-q (asymptotic regime), is broadened for each individual l-th detector by nite q corrections terms, J l (y, q), known as nal state eects (FSE) and by the the instrumental resolution function, R(y, q) (see Eq.10): F l (y, q) = [J IA (y) + J l (y, q)] R l (y, q). (6) where R l (y, q) is determined using standard Monte Carlo routines available on VESUVIO. This equation is used to describe the experimental NCP of Eq.5 for each individual l-th detector, F l (y, q). In order to derive the n(p), a line-shape analysis of F (y, q) has been performed using models M1 and M2. In the rst case the equations for the NCP, related to a Gauss-Laguerre momentum distribution, takes the form of a Gauss-Hermite expansion: J M1 (y) = e y 2 2σ [ πσ Ω n=2 ( ) ] c 2n y 2 2n n! H 2n (7) 2σ in the second case, the NCP related to a multivariate Gaussian momentum distribution takes the form: ] 1 dω J M2 (y) = [ 2πσx σ y σ z 4π exp y2 S 2 (θ, φ) (8) 2S 2 (θ, φ) with ( ) 1 cos 2 S 2 (θ, φ) = φ sin2 θ + sin2 φ + cos2 θ σx 2 σy 2 σz 2 In both models corrections to IA, due to nal state eects, take the form of the additive term J(y, q): ) J(y, q) = J IA (y) + J(y, q) = (1 A 3 (q) 3 J y 3 IA (y) (10) where A 3 (q) = σ4 9q. The model tting function is obtained by a numerical convolution of J(y, q) with the simulated experimental resolution R(y, q), yielding F th (y, q) = J(y, q) R(y, q). Analysis of the NCP spectra is carried out either by a global tting over each individual l-th detector, F l (y, q), or by tting spectra from individual groups of detectors positioned at similar scattering angles and distances. The two procedures yield the same sets of parameters within uncertainties. In the former case the tting parameters have been derived by minimization of the following chi-square: χ 2 = l i ( F th l (y i, q i ) F exp l (y i, q i ) ) 2 ɛ 2 l,i where l labels the detector and i labels the i-th bin in the y spectra. In the second case, for detectors positioned at the same scattering angle within 2 degrees (9) (11) 7

8 and distance from the sample within 4 cm, their NCP spectra are averaged to obtain a total number of nine groups out of the individual 64 detectors. Data for each group are tted via the F th (y, q) line-shape in order to derive, in the case of M1, values of σ and c 4, and, in the case of M2, values of σ x, σ y and σ z. The tting parameters have been derived by minimization of the following chi-square: χ 2 = i ( F th l (y i, q i ) F exp l (y i, q i ) ) 2 ɛ 2 l,i (12) where l labels the detector group and i labels the i-th bin in the y spectra. Each of the groups is tted separately in order to check the individual E K value and its θ dependence. Results of the individual ts are reported in gure 5. The simultaneous t performed on the nine groups of detectors provides a value of E K = ± 2.0 mev. The region within the two green lines in the gure allow to evaluate the mean value and standard deviation of E K determinations. In Figure 5 one can see that two groups, six and seven, are slightly outside the average value. If one excludes these two groups the t yields a value of E K = ± 2.0 mev, in agreement with the simultaneous t determination. E K [mev] detector group Figure 5: The values of E K from t on each group of detectors sharing the same scattering angle and distance from the sample. The green lines represent the mean value on the ensemble of nine groups plus and minus one standard deviation. Sensitivity analysis on the tting parameters of the M2 model In order to carry out an analysis of the sensitivity of the parameter derived from the M2 model, we report in Figure 6 the dierences between the raw F (y, q) data for SW and ice at T=271 K and the dierences between the ts using the M2 model. This gure shows that the modelling we obtain using M2, of the kinetic energy tensor components and their dierences in SW and ice, are well reected into the dierences between the raw experimental data of SW and ice. Furthermore, to test the sensitivity of the tting model to small variations of the E K α parameters, we report in Figure 7 the results of a t of the SW data with E K x (SW)= E K x (ice), 8

9 Difference [error units] y Å 1 Figure 6: Dierence (in unit of error bars) between NCP F (y, q) for SW and ice at T=271 K (red squares); for clarity a ve point smoothing has been applied. Dierence of M2 ts for SW and ice at T=271 K (green continuous line). and E K y (SW)= E K y (ice). This Figure shows that the best t is obtained in Figure 7 (a), corresponding to the E K x,y,z (SW) reported in Table I of the manuscript. 9

10 Å ] F (y, q) [ (a) Å ] F (y, q) [ (b) Å ] F (y, q) [ (c) y Å 1 Figure 7: Panel (a). Angle averaged hydrogen NCP F (y, q) for SW at T = 271 K (blue dots with error bars). The angle average of the best ts on the individual detectors, obtained using the M2 model, is plotted as red line. Both are the same as in Figure 1 of the manuscript. The t residual is reported in green dots with error bars. Panel (b). Angle averaged hydrogen NCP F (y, q) for SW at T = 271 K (blue dots with error bars). The best t of SW with E K x (SW)= E K x (ice), and E K y (SW)= E K y (ice) is reported as a red line. The t residual is reported in green dots with error bars. Panel (c). Angle averaged hydrogen NCP F (y, q) for SW at T = 271 K (blue dots with error bars). The best t of SW with E K x (SW)= E K x (ice), E K y (SW)= E K y (ice), and E K z (SW)= E K z (ice) is reported as a red line. The t residual is reported in green dots with error bars. 10

11 References (1) Senesi, R.; Andreani, C.; Bowden, Z.; Colognesi, D.; Degiorgi, E.; Fielding, A. L.; Mayers, J.; Nardone, M.; Norris, J.; Praitano, M. et al. VESUVIO: a novel instrument for performing spectroscopic studies in condensed matter with ev neutrons at the ISIS facility. Physica B Condensed Matter 2000, 276, (2) Tardocchi, M.; Pietropaolo, A.; Andreani, C.; Bracco, A.; D'Angelo, A.; Gorini, G.; Imberti, S.; Senesi, R.; Rhodes, N. J.; Schooneveld, E. M. Cadmium-Zinc-Telluride photon detector for epithermal neutron spectroscopy-pulse height response characterisation. Nuclear Instruments and Methods in Physics Research Section A 2004, 526, (3) Schooneveld, E. M.; Mayers, J.; Rhodes, N.J.; Pietropaolo, A.; Andreani, C.; Senesi, R.;Gorini, G.; Perelli Cippo, E.; Tardocchi, M.; Foil cycling technique for the VESUVIO spectrometer operating in the resonance detector conguration. Review of Scientic Instruments 2006, 77, (4) Andreani, C.;Colognesi, D.;Degiorgi, E.; Filabozzi, A.;Nardone, M.;Pace, E.;Pietropaolo, A.; Senesi, R. Double dierence method in deep inelastic neutron scattering on the VESU- VIO spectrometer. Nuclear Instruments and Methods in Physics Research Section A 2003, 497, (5) Reiter, G. F.; Mayers, J.; Noreland, J. Momentum-distribution spectroscopy using deep inelastic neutron scattering. Physical Review B 2002, 65, (6) Andreani, C.; Colognesi, D.; Mayers, J.; Reiter, G. F.; Senesi, R. Measurement of momentum distribution of light atoms and molecules in condensed matter systems using inelastic neutron scattering. Advances in Physics 2005, 54, (7) Pietropaolo, A.; Senesi, R.; Andreani, C.; Botti, A.; Ricci, M. A.; Bruni, F. Excess of Proton Mean Kinetic Energy in Supercooled Water. Physical Review Letters 2008, 100, (8) Pietropaolo, A.; Senesi, R.; Andreani, C.; Mayers, J. Quantum Eects in Water: Proton Kinetic Energy Maxima in Stable and Supercooled Liquid. Brazilian Journal of Physics 2009, 39, (9) Gunn, J. M. F.; Andreani, C.; Mayers, J. A new approach to impulsive neutron scattering. Journal of Physics C Solid State Physics 1986, 19, L835L840 (10) West, G. B. Electron scattering from atoms, nuclei and nucleons. Physics Reports 1975, 18, (11) Flammini, D.; Pietropaolo, A.; Senesi, R.; Andreani, C.; McBride, F.; Hodgson, A.; Adams, M. A.; Lin, L.; Car, R. Spherical momentum distribution of the protons in hexagonal ice from modeling of inelastic neutron scattering data. The Journal of Chemical Physics 2012, 136,

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