Upgrading of Resolution Elastic Neutron Scattering (RENS)

Abstract

An update of the Resolution Elastic Neutron Scattering (RENS) approach consisting in measuring the elastically scattered neutron intensity versus the instrumental energy resolution is presented. In particular it is shown that the measured elastic scattering law as a function of the logarithm of the instrumental energy of resolution gives rise to an increasing sigmoid trend whose inflection point can be connected with the system relaxation time. The validity of the RENS approach is supported by a numerical simulation, taking into account a Gaussian resolution function and a Lorentzian scattering law, and experimentally by integrated EINS and QENS measurements performed as a function of temperature on three homologous disaccharide/water mixtures showing different relaxation times. Furthermore, the most important advantages of the RENS approach are discussed; in particular, in comparison with QENS, the RENS approach requires a smaller amount of sample, which is an important point in dealing with biological and exotic systems, is not affected by the use of model functions for fitting spectra, and furnishes a direct access to the system relaxation time.

1. Introduction

The characterization of the molecular processes involved in condensed matter systems can be performed by Elastic Incoherent Neutron Scattering (EINS) by means of the so-called “fixed-windows” method [1], where the scattered intensity is collected at 𝜔=0 with a fixed “energy windows” corresponding to the instrumental energy resolution. In the case of purely elastic scattering, all the scattered neutrons fall within this energy window and the measured intensity remains constant at all temperatures, a part from the small decrease, due to the Debye-Waller factor, while changes in the elastically scattered intensity are registered when the system relaxation time becomes shorter than the instrumental resolution time. This technique takes advantage from the fact that, besides a relatively low number of fitting parameters, with respect to the quasielastic contribution at low energy transfer the elastic one is often a factor 100–1000 higher, and then for obtaining good quality data in reasonable times, due to the usually limited neutron fluxes, one can cope with a relatively small amount samples, with small sized samples, or with strongly absorbing samples. In this framework, the analysis of the EINS intensity, collected by varying some internal or external parameters, can provide useful information. For example, a wide wave-vector range (e.g., such as that of the back-scattering spectrometer IN13 at the ILL in Grenoble, with a 𝑄 range extending up to 4.28 Å⁻¹) can allow achieving an assignment based on spatial characteristics; in other words, the elastic intensity versus momentum exchange can allow us to characterize molecular motion according to their spatial extent and amplitude. Furthermore, a varying spectral resolution and energy window increase the capability of deciphering the nature of the recorded relaxations on the basis of characteristic time. Moreover, the use of hydrogen-deuterium labeling, highlighting contributions from specific scatterer groups or, separately, from protein and solvent, adds a dimension that can be crucial for spectral separation. Finally the temperature dependence of the measured elastic intensity can provide important clues on the activation energies and, thus, on the local potentials; in fact, for example, the guiding picture for a water soluble protein is that increasing temperature from the lowest values where only vibrations mainly contribute, one reveals some more or less marked variations in intensity or in some intensity derived physical quantities, such as the Mean Square Displacement (MSD), that may correspond to the “activation” of some internal motions (e.g., methyl groups rotations), to motions induced by solvent fluctuations, to the internal protein H-bonds, to the kinetic melting of solvent, to the melting of bulk solvent, and so forth [2–6].

It is well known that the time dependent pair correlation function 𝐺(𝑟,𝑡) represents the probability to find a particle at distance 𝑟 after a time 𝑡 when a particle was at 𝑡=0 in 𝑟=0. The scattering law 𝑆(𝑄,𝜔) and the intermediate scattering function 𝐼(𝑄,𝑡) are directly connected, in Planck’s units, through time and space Fourier transforms. As a matter of fact, the previously introduced functions make reference to the system properties and not to measured quantities; in fact, it should be taken into account that, for example, when the experimental technique gives access to the scattering law one should take into account the less straightforward connection with the measured scattering function, which is the convolution of the scattering law with the instrumental resolution function and that can be expressed as the time Fourier transform of the product between the Intermediate Scattering Function and the Resolution function in the time domain.

The aim of this work is to present an updated approach, that is, RENS, which is based on the acquisition of the measured elastic scattering law as a function of the instrumental energy resolution. In particular, the effect of the coupling between the system and the employed measuring instrument is considered and it is shown that the measured elastic scattering law as a function of the logarithm of the instrumental energy resolution behaves like an increasing sigmoid curve whose inflection point occurs when this instrumental resolution time matches the system relaxation time.

In order to test the validity of the RENS approach we shall consider three systems constituted by water mixtures of three homologous disaccharides, that is, trehalose, maltose, and sucrose; these systems are shown, by QENS measurements, to exhibit different dynamics as a function of temperature; as a consequence when tested by EINS measurements at a given instrumental energy resolution as a function of temperatures, these systems are supposed to show, following the RENS approach, different inflection point temperatures. The collection of the measured elastic scattering intensity versus temperature, showing different inflection points for the three investigated systems, confirms the validity of the approach.

2. Materials and Methods

Ultrapure powdered trehalose, maltose and sucrose, D₂O and H₂O, purchased by Aldrich-Chemie, were used for the experiment.

EINS data were collected in the 20–310 K temperature range on the three homologous disaccharides trehalose, maltose, and sucrose in H₂O at weight fraction values corresponding to 19 water molecules for each disaccharide molecule by the IN13 spectrometer at ILL (Grenoble, France). The used IN13 configuration is the following; incident wavelength: 2.23 Å, 𝑄-range: 0.28–4.27 Å⁻¹. Raw data were corrected for cell scattering and detector response and normalized to unity at 𝑄=0.00Å⁻¹.

Quasielastic neutron scattering (QENS) measurements were performed in a temperature range of 283–320 K on hydrogenated trehalose, maltose, and sucrose (C₁₂H₂₂O₁₁) in H₂O and on partially deuterated trehalose, maltose, and sucrose (C₁₂H₁₄D₈O₁₁) in D₂O at weight fraction values corresponding to 19 water (H₂O and D₂O) molecules for each disaccharide molecule. In order to obtain partially deuterated samples, the disaccharides were first dissolved in pure D₂O at a concentration of ~40% by weight, to exchange the eight labile hydrogen atoms of the disaccharides and the solutions were, subsequently, lyophilized. The procedure was repeated in order to allow the exchange of all the exchangeable hydrogens. The solution samples were obtained by dissolution of the D₂O-exchanged lyophilized disaccharides in pure D₂O. The QENS experiment was carried out by the IRIS at ISIS (RAL, UK). We used the high resolution configuration of IRIS (graphite 002 and mica 006 analyser reflections) to measure sets of QENS spectra covering a 𝑄,𝜔-domain extending from ℏ𝜔=−0.3 to 0.6 meV (energy transfer) and 𝑄=0.3 to 1.8 Å⁻¹ (momentum transfer). The detectors used give a mean energy resolution of Γ=8 μeV of Half Width at Half Maximum (HWHM) as determined by reference to a standard vanadium plate.

3. Results and Discussion

It is well known that the scattering law 𝑆(𝑄,𝜔) and the intermediate scattering function 𝐼(𝑄,𝑡) are connected by a time Fourier transform [7–12]: 𝑆()=1𝑄,𝜔2𝜋^+∞_−∞𝐼(𝑄,𝑡)𝑒^{−𝑖𝜔𝑡}𝐼()=1𝑑𝑡,𝑄,𝑡2𝜋^+∞_−∞𝑆(𝑄,𝜔)𝑒^𝑖𝜔𝑡𝑑𝜔. (1)

The frequency instrumental resolution function 𝑅(𝑄,𝜔) in the 𝜔 space is connected to the corresponding time instrumental resolution function by the following Fourier transform: 𝑅()=1𝑄,𝑡2𝜋^+∞_−∞𝑅(𝑄,𝜔)𝑒^𝑖𝜔𝑡𝑑𝜔. (2)

The resolution scattering law corresponds to the convolution of the scattering law: 𝑆_𝑅(𝑄,𝜔)=𝑆(𝑄,𝜔)⊗𝑅(𝑄,𝜔). (3)

Therefore, from a formal point of view, the convolution of the two functions 𝑆(𝑄,𝜔) and 𝑅(𝑄,𝜔) produces a third function 𝑆_𝑅(𝑄,𝜔) which can be viewed as a modified version of the original function 𝑆(𝑄,𝜔), given by the overlap area between the two functions, that is, the 𝑆(𝑄,𝜔) and the 𝜔-inverted 𝑅(𝑄,𝜔) functions when this latter is translated. As a consequence for instrumental resolution functions 𝑅(𝑄,𝜔) symmetric in 𝜔, the convolution coincides with the cross-correlation between 𝑆(𝑄,𝜔) and 𝑅(𝑄,𝜔).

By the convolution theorem (3) can be written as 𝑆_𝑅()()()=1𝑄,𝜔=𝑆𝑄,𝜔⊗𝑅𝑄,𝜔√2𝜋^+∞−∞𝐼(𝑄,𝑡)⋅𝑅(𝑄,𝑡)𝑒^{−𝑖𝜔𝑡}𝑑𝑡. (4)

In the ideal case of purely elastic scattering in which the resolution function is a delta in the 𝜔-space and, hence, a constant in the 𝑡-space, we obtain that having: 𝑆_𝑅()=1𝑄,𝜔=0√2𝜋^+∞−∞𝐼(𝑄,𝑡)⋅𝑅(𝑄,𝑡)𝑑𝑡 (5) the measured scattering law 𝑆_𝑅(𝑄,𝜔) reduces to a time average of the intermediate scattering function 𝐼(𝑄,𝑡).

A part from this ideal case, in the more general condition in which the resolution function in the ω-space has a nonnegligible width, the experimentally measured elastic scattering law, due to the finite energy instrumental resolution Δ𝜔, 𝑆^𝑀_𝑅(𝑄) is 𝑆^𝑀_𝑅(𝑄)=^+Δ𝜔/2_−Δ𝜔/2𝑆_𝑅(𝑄,𝜔)𝑑𝜔, (6) and hence, 𝑆^𝑀_𝑅(𝑄)=^+Δ𝜔/2_−Δ𝜔/2𝑆()()=𝑄,𝜔⊗𝑅𝑄,𝜔𝑑𝜔_{+Δ𝜔/2−Δ𝜔/2}1√2𝜋^+∞−∞(𝐼(𝑄,𝑡)⋅𝑅(𝑄,𝑡))𝑒^{−𝑖𝜔𝑡}𝑑𝑡𝑑𝜔. (7)

In order to show the operation approach of RENS as well as the expected experimental output we numerically test the behaviour of the measured elastic scattering function versus the instrumental energy resolution. On this purpose we operate in the frequency space and evaluate 𝑆^𝑀_𝑅(𝑄) as a function of the energy instrumental resolution Δ𝜔, in the case in which the scattering law 𝑆(𝑄,𝜔), is a Lorentzian curve with a fixed linewidth (as, for example, it occurs for a simple Brownian translational diffusion process) and the resolution function 𝑅(𝑄,𝜔) is a Gaussian curve. In particular, in order to numerically get the RENS output we consider an increasing instrumental resolution function linewidth Δ𝜔 and apply (7) for determining the value of the definite integral as a function of Δ𝜔. Figure 1 shows the result of such a calculation, that is, 𝑆^𝑀_𝑅 as a function of the logarithm of the instrumental resolution. As it can be seen it shows an increasing sigmoid trend whose inflection point occurs when the linewidth of the resolution function approaches the linewidth of the system scattering law; in other words, in such a semilogarithmic plot the inflection point occurs when the instrumental resolution time crosses the system relaxation time. Such a result shows the operating way of the RENS approach: from the inflection point of EINS profiles versus the logarithm of the energy resolution one can extract the system relaxation time. It should be noticed that, in a complementary way, for a given fixed instrumental energy resolution function from EINS profiles versus temperature one is able to obtain from the inflection point the temperature value for which the system relaxation time equals the resolution time. In the following we will apply such a procedure to demonstrate the validity of the RENS approach.

Figure 1 

𝑆𝑀𝑅 as a function of the logarithm of the instrumental resolution; the inflection point of the increasing sigmoid trend can be connected with the system relaxation time.

On the other hand, as far as the system cross-section is concerned, it is well known that the neutron scattering double differential cross-section can be written as 𝑑²𝜎=𝑘𝑑Ω𝑑𝜔𝑘¹₀12𝜋^+∞_−∞𝑑𝑡𝑒^{−𝑖𝜔𝑡}×1𝑁^𝑁𝑁_𝛼=1𝛽=1𝑏^∗_𝛼𝑏_𝛽𝑒^{−𝑖𝐐⋅𝐑𝛼(0)}𝑒^{−𝑖𝐐⋅𝐑𝛽(𝑡)}=𝑘₁𝑘₀12𝜋^+∞_−∞𝑑𝑡𝑒^{−𝑖𝜔𝑡}1𝑁×𝑏²_{𝛼𝛼≠𝛽}𝑒^{−𝑖𝐐⋅𝐑𝛼(0)}𝑒^{−𝑖𝐐⋅𝐑𝛽(𝑡)}+|𝑏|²_𝛼𝑒^{−𝑖𝐐⋅𝐑𝛼(0)}𝑒^{−𝑖𝐐⋅𝐑𝛼(𝑡)}=𝑘₁𝑘₀𝑏²𝑆_coh()+𝑘𝑄,𝜔¹𝑘₀|𝑏|²𝑆_inc()+𝑘𝑄,𝜔₁𝑘₀𝑏²−|𝑏|²𝑆_inc()=𝑘𝑄,𝜔₁𝑘₀𝑏²𝑆()+𝑘𝑄,𝜔¹𝑘₀𝑏²−|𝑏|²𝑆_inc()=𝑘𝑄,𝜔₁𝑘₀𝜎_coh𝑆()+𝑘4𝜋𝑄,𝜔¹𝑘₂𝜎_inc𝑆4𝜋^inc()=𝑘𝑄,𝜔₁𝑘₀𝜎_coh𝑆()+𝜎4𝜋𝑄,𝜔^inc𝑆4𝜋^inc(𝑄,𝜔), (8) where 𝑘₀ and 𝑘₁ are the incoming and outcoming neutron wavevectors, 𝑏_𝛼 and 𝑏_𝛽 are the scattering lengths of atom 𝛼 and 𝛽, respectively, and it is |𝑏|²=⟨𝑏^∗_𝛼𝑏_𝛽⟩, |𝑏|²=⟨𝑏^∗_𝛼𝑏_𝛼⟩, 𝜎_coh=4𝜋|𝑏|², and 𝜎_inc=4𝜋(|𝑏|²−|𝑏|²). 𝜎_coh takes into account interference effects among waves produced by the scattering of a single neutron from all the nuclei, whereas 𝜎_inc refers to single particle properties.

In our case, due to the high percentage of hydrogen atoms in the investigated systems (H: 𝜎_coh=1,76; 𝜎_inc=80,26) only this latter contribution is relevant: 𝑑²𝜎≅𝑘𝑑Ω𝑑𝜔𝑘¹₀𝜎_inc𝑆4𝜋^inc(𝑄,𝜔). (9)

In the following, the QENS and EINS data on disaccharides water mixtures are presented and discussed in the framework of the above-described RENS approach.

The analysis performed on the QENS data of disaccharide/D₂O and disaccharide/H₂O mixtures [13–16] allows us to characterise the diffusive dynamics of both solute and solvent. By determining the translational contribution, its linewidth Γ for disaccharides and water as a function of 𝑄² follows a typical Random Jump Diffusion (RJD) model [9]: Γ₁(𝑄)=𝐷_𝑠𝑄²1+𝐷_𝑠𝑄²𝜏, (10) where 𝐷_𝑠 is the self-diffusion coefficient of the molecule and 𝜏 is the residence time. The RJD model furnishes the diffusion coefficient 𝐷_𝑠 value from the extrapolation to 𝑄→0 and the residence time τ from the inverse of the asymptotic value at 𝑄→∞.

Such a model furnishes for trehalose the diffusion coefficient 𝐷_𝑠 and the residence time 𝜏 of 𝐷_𝑠=2.83×10⁻⁷ cm²/s and 𝜏=24.7 ps, 𝐷_𝑠=3.82×10⁻⁷ cm²/s and 𝜏=20.6 ps, 𝐷_𝑠=5.35×10⁻⁷ cm²/s and 𝜏=19.1 ps, and 𝐷_𝑠=8.50×10⁻⁷ cm²/s and 𝜏=18.3 ps for 𝑇=283, 295, 308, and 320 K, respectively.

The RJD model furnishes for the diffusion coefficient 𝐷_𝑠 and the residence time 𝜏 the values of 𝐷_𝑠=8.50×10⁻⁷ cm²/s and 𝜏=18.3 ps for trehalose, 𝐷_𝑠=1.00×10⁻⁶ cm²/s and 𝜏=14.3 ps for maltose, and 𝐷_𝑠=1.23×10⁻⁶ cm²/s and 𝜏=13.8 ps for sucrose at 𝑇=320 K.

The best fit according to the RJD model for the translational linewidth Γ₃ of water in trehalose solutions at different temperature values and of water in trehalose, maltose, and sucrose aqueous solutions at 𝑇=320 K as a function of 𝑄² is reported in Figure 2. The diffusion coefficient of water in the three disaccharide aqueous solutions at 𝑇=320 K are: 𝐷_𝑤=8.31×10⁻⁶ cm²/s for trehalose solution, 𝐷_𝑤=8.46×10⁻⁶ cm²/s for maltose solution, and 𝐷_𝑤=8.60×10⁻⁶ cm²/s for sucrose solution, with the values of residence times of 𝜏=3.7 ps, 3.4 ps, and 3.0 ps for trehalose, maltose and sucrose solutions, respectively. It should be noticed that the diffusion coefficient of pure water at the same temperature is 𝐷_𝑤=3.94×10⁻⁵ cm²/s.

(a)

(b)

Figure 2 

Translational linewidth Γ3 of (a) water in trehalose solutions for different temperature values and of (b) water in trehalose, maltose, and sucrose aqueous solutions at 𝑇=320 K as a function of 𝑄2.

For the investigated systems the water dynamics resembles that of water at ~268 K in the case of trehalose solution, at ~271 K in the case of maltose solution, and at ~277 K in the case of sucrose solution, while the whole water dynamics in trehalose solutions for 𝑇=283, 295, 308, and 320 K resembles that of water at ~256 K, ~261 K, ~263 K, and ~268 K, indicating that the water has a diffusive behaviour strongly triggered by the trehalose molecules and suffers from a noticeable frozen effect. Analogously to the trehalose aqueous solutions, all the disaccharides show a slowing down effect of the water dynamics, which is stronger for trehalose than the other disaccharides.

The higher slowing down effect of the diffusive dynamics observed for trehalose is evidently linked to its extraordinary capability to “switch off” the metabolic functions.

The EINS data on the same systems, that is, disaccharide/H₂O mixtures [17–21], give an experimental confirmation of the RENS approach. In particular, the experimental findings have been analyzed by means of a wavelet analysis that allows us to characterize the scattered elastic intensity behavior on different scales highlighting the correlation between the signal and the set of the scaled and translated mother functions [22]. From a formal point of view, a continuous wavelet transform can be written as 𝑊_𝐼(𝜏,𝑠)=^∞_−∞𝐼(𝑡)𝜓^∗_𝜏,𝑠(𝑡)𝑑𝑡, (11) where 𝜏∈ℝ is the translation parameter, whereas 𝑠∈ℝ⁺  (𝑠≠0) is the scale parameter, 𝐼(𝑡) is the function to be analyzed, the asterisks denote the complex conjugate, and 𝜓_𝜏,𝑠 is the mother wavelet: 𝜓_𝜏,𝑠=1√𝑠𝜓𝑡−𝜏𝑠 (12) the factor 𝑠^−1/2 is an adimensional normalization constant.

More precisely, the translated-version allow the wavelets to be localized, whereas the scaled-version wavelets allows us to analyze the signal in different scales through the so-called scalogram, that is, the square of the modulus of the wavelet transform: 𝑃_𝑊𝐼(𝑄,𝑠)=𝑊_𝐼(𝑄,𝑠)². (13)

More in details, the result of a wavelet analysis of the EINS data is shown in Figure 3, which reports the 3D scalograms at 𝑇=264 K. Such an analysis allows us to compare both the spatial properties of the three systems in the wavevector range of 𝑄=0.27Å⁻¹÷4.27Å⁻¹.

(a)

(b)

(c)

(d)

Figure 3 

3D scalograms of a wavelet analysis for sucrose/water mixtures, at three different temperatures, (a) 𝑇=19 K, (b) 𝑇=264 K, and (c) 𝑇=284 K, and (d) 3D scalogram of a wavelet analysis for trehalose/water mixtures at 𝑇=264.

More precisely, starting from the lowest investigated temperature, 19 K, the presence of only one spectral contribution extending in the whole investigated wavevector range is revealed. Such a wide and flat contribution spans the whole wavevector range and can be attributed to the vibrational motions of the scatterer particles, that is, the system’s protons. As it can be seen in Figure 3, by increasing temperature a different contribution, at low wavevector values, clearly emerges. As pointed out by the adopted chromatic scales for the wavelet coefficients, already at this temperature value such a contribution is much less marked in the case of trehalose. Finally the 3D scalograms of sucrose, maltose, and trehalose water mixtures at 𝑇=284 K reveal that the weight of this latter contribution tends to increase with temperature showing a different increasing rate for the three investigated systems. Therefore, the EINS results confirm that the dynamics of the trehalose/water system is slower with respect to that of the maltose/water and of the sucrose/water systems.

In order to test the validity of the RENS approach one should take into account that, since the three systems exhibit different relaxation times at all the investigated temperatures, when tested by EINS at a given instrumental energy resolution as a function of temperatures, they should show different inflection point temperatures.

In order to extract quantitative information from the 𝐼(𝑇) behaviors we performed a fit by using the following expression: 𝐼(𝑇)1=𝐴1−1+𝑒^{−𝐵(𝑇−𝑇0)}+(𝐶−𝐷𝑇). (14)

Here 𝐴 represents the amplitude factor connected with the global thermal intensity behavior, whose inverse we call “thermal restrain”; it should be noticed that this quantity is not connected with a local behavior but represents a global property of the system. 𝐵 is connected with the transition steepness, passing from smaller to higher values when the transition becomes more and more abrupt. As it can be inferred by following the RENS approach, 𝑇₀ is the inflection temperature, corresponding to the temperature value for which the system relaxation time equals the instrumental resolution time [4]. On this concern such a temperature value should occur at different values for the different disaccharides. Finally, 𝐶−𝐷𝑇 represents the low temperature vibrational contribution.

In Figure 4 the case of maltose intensity versus temperature is shown as an example while in the insert the behavior for the three disaccharides is reported. The results of the fitting procedure of EINS intensity as a function of temperatures for the three disaccharides performed by applying (14) are reported in Table 1. As it can be seen, the analysis indicates a higher thermal restrain together with a higher transition temperature for trehalose with respect to the other two homologous disaccharides. Such a result confirms the validity of the RENS approach. In fact, taking into account the meaning of the transition temperature, the disaccharides-water relaxation times cross the resolution time at different temperature values, that is, 264 K for trehalose, 259 K for maltose, and 257 K for sucrose.

Table 1 

Parameters of the fitting procedure according to (14) of the EINS intensity as a function of temperature for the three disaccharides.

Figure 4 

Intensity profile, obtained as sum for all the wavevector values, for maltose/H2O versus temperature with the fitting result according to (14).

In conclusion, this contribution shows the operating approach of RENS from an analysis performed on the frequency space: from the inflection point of EINS profiles versus the logarithm of the energy resolution one can extract the system relaxation time. In a complementary way, for a given fixed instrumental energy resolution function from EINS profiles versus temperature one is able to obtain from the inflection point the temperature value for which the system relaxation time equals the resolution time.