Quasielastic neutron scattering experiment on water using TOFLAR \(Time Of Flight and LARmor precession\) technique at SNS

We present the results of the first TOFLAR (Time Of Flight and LARmor precession) experiment with a Larmor modulation device positioned after the sample. Using a setup based on the configuration of the SNS-NSE spectrometer (Spallation Neutron Source, USA), we did TOFLAR measurements on water at 5 and 20 degree Celsius, analyzed the spectra, and obtained diffusion coefficient values that are in a reasonable agreement with literature. Thus, our results demonstrate that TOFLAR may become a standard technique to study quasielastic neutron scattering in future.


Introduction
In an inelastic neutron scattering experiment two wavelengths, before and after scattering, have to be determined. Such an experiment can be done using the time-of-flight (TOF) method in two different modes. In a "direct geometry" mode a pulsed incident beam has a fixed incoming wavelength (λ 1 ) and TOF is used to measure the final wavelength (λ 2 ). In an "inverted geometry" mode λ 2 is fixed while TOF is used to measure λ 1 . For both modes, only a narrow wavelength band of neutrons (for incoming or scattered neutrons, respectively) is used, resulting in a reduction of available neutron intensity by at least two orders of magnitude, and in limitations for the range in momentum transfers Q and energy transfers ω.
In the TOFLAR technique [1] neither the incoming nor the scattered beam is required to be monochromatic. Larmor precession of polarized neutrons labels λ 1 or λ 2 (depending on whether the Larmor modulator is placed before or after the sample), and TOF yields a combination of λ 1 and λ 2 . A schematic setup is shown in Fig. 1.
The results of the first test of the TOFLAR technique were difficult to analyze mainly because of a poor counting statistics [2]. In this paper we present the results of a TOFLAR experiment on water done at SNS with a setup based on the configuration of the SNS-NSE spectrometer.

The TOFLAR technique
When a Larmor modulator (LM) device is placed after the sample, measured intensity is a function of scattering angle (θ ), TOF channel (t C H ), and the field line integral (Bl) a e-mail: kusmin@tudelft.nl b e-mail: m.ohl@fz-juelich.de c e-mail: a.a.vanwell@tudelft.nl produced by the LM device: where λ min is the interval of incident wavelengths, I 0 is proportional to the integrated neutron flux, p(λ 1 ) is the relative intensity of particular λ 1 , P N SE is the polarization of the sample scattering, R is the resolution function (which accounts for any depolarization not related to the sample scattering), c = mγ / h (m and γ are neutron mass and gyromagnetic ratio), k = 2π/λ, S(Q, ω) is the scattering function of the sample, and Q = (4π/λ) sin(θ/2). For simplicity, θ is assumed to be constant.
Using equations t C H = (m/ h)(L 1 λ 1 + L 2 λ 2 ) and ω = (h 2 /2m)(1/λ 2 2 − 1/λ 2 1 ), and the quasielastic approximation (λ 1 ≈ λ 2 ≈ λ = ht C H /(m(L 1 + L 2 ))), Eq. (1) can be written as (more details will be published elsewhere [4]): EPJ Web of Conferences and TOFLAR time is where An experiment consists of a scan of the field line integral Bl and yields for each TOF channel a modulated signal such as shown in Fig. 2, where an effect of a finite wavelength resolution (for rectangular resolution function) can be seen as well.
The aim of a quasielastic scattering experiment is to extract F(Q, t) and to test various dynamical models against it. Raw data (described by Eq. (3)) contain a few variables that can be removed by a suitable normalization. Just like in conventional NSE, I + -and I − -intensities (measured with π/2 flippers turned off, and π -flipper being off, and on, respectively) can be used for this purpose. Using scattered beam polarization P(λ) = I + (λ)−I − (λ) I + (λ)+I − (λ) , and differential scattering cross-section S(Q), we define I DI F F (λ) = I + (λ) − I − (λ) and I SU M (λ) = I + (λ) + I − (λ) which are (4b) where the last approximate identity can be a reasonable approximation (e.g. for small scattering angles/well 1 By contrast to [2], here L 1 , and not L 2 stands in the numerator because the LM device is after the sample. collimated incident beam, when π -flipper efficiency ( f ) is ≥ 0.98). P P and P A are polarizer and analyzer efficiency. Then, two possible normalizations are To extract F(Q, t) of the sample, R(λ, Bl) must be obtained in advance in a so-called resolution measurement using a sample that exhibits no dynamics (F(Q, τ T ) = 1) and has no spin-flip scattering (i.e. P N SE (λ, Bl) = 1). If Eq. (5a) is used for the normalization, fitting of the following equation to a normalized signal from the resolution measurement yields b R E S = R(λ, Bl)/P R E S (λ); P R E S accounts for any instrument-related depolarization of scattered beam when π/2-flippers are off. By fitting Eq.
can be obtained; here P S is polarization of neutrons scattered by the sample (when π/2 flippers are off) corrected for the instrument-related depolarization.  . The scheme of the instrument layout. P, B, S, A, and D stand for polarizer, a guide field to maintain neutron polarization, sample, analyzer, and detector. The first π/2-flipper turn the polarization of longitudinally polarized neutrons into the plane perpendicular to the plane of paper, the second π/2-flipper does the opposite. Due to the π flipper, the net field integral is For a fine adjustment of the net Bl-integral, either phase coil 1 (PC1) or 2 (PC2) could be used, leading to the net Bl-integral of F(Q, τ T ) can be readily obtained since P S can be calculated using I + − and I − − intensities from resolution and sample measurements.

Experimental
The measurements were done at the SNS-NSE spectrometer at SNS [5,6], modified to have a setup shown in Fig. 3. Distances between the source and the sample, and the sample and detector were L 1 = 23.361 m and L 2 = 3.939 m, respectively. Larmor modulation was set up in the second arm of the spectrometer to have a better TOF resolution (because L 1 L 2 ). The wavelength range, 3.6Å ≤ λ ≤ 6Å, was split into 1001 TOF channels. Measurements were done at θ = 25 • (we tried to measure at θ = 70 • but the polarization was not sufficiently high). Water was measured at 20 • C and 5 • C. The sample container was an aluminium rectangular flat cell (one of the standard containers used at BASIS, SNS); we used aluminium spacers to achieve water layer thickness of 0.25 mm. Resolution measurements were done with a zero alloy TiZr instead of Grafoil because of the targeted Q-region Q > 0.5Å −1 . This alloy has only isotopic incoherent (non-spin flip) scattering which is almost isotropic [7]. We did not measure scattering by either empty cell or empty instrument. Counting times per one Bl-point were 750 s, 900 s, and 600 s for resolution, and water at 20 • C, and 5 • C, respectively. For analysis, counts over the whole position sensitive detector were summed up; to further improve counting statistics 1001 TOF channels were grouped into 101 channels. It is obvious from Fig. 2 that to measure TOFLAR signal at closely spaced Bl-values (to see signal oscillations) in a wide Bl-interval a very long measurement time is required. Therefore, we selected six narrow Bl-intervals centered at different Bl = B 1 l 1 − B 2 l 2 , and slightly varied the net Bl by a fine adjustment of a phase coil current (see Fig. 3) to measure 20 Bl-points per interval. We intended to use the phase coil 2 but due to a misconfiguration used the phase coil 1 instead. To take that into account, the net Bl values were recalculated by the NSE control program. For every Bl-interval, I + and I − were measured at one Bl in the center of an interval.

Raw data
Intensities (in the form of I SU M (λ) = I + (λ) + I − (λ)) and polarizations of scattered neutrons, flipping ratios, and normalized TOFLAR signals are shown in Fig. 4 the dips in polarization. Note that after correction for the instrument-related beam depolarization, polarization of water scattering is ≈ −0.2/0.6 = −0.33 close to theoretical P = −1/3 for spin-incoherent scattering. This is an indication that multiple scattering and the absence of background corrections have no large effect. The flipping ratio generally increases with increasing λ but is lower than expected for small Bls such as here. Finally, the TOFLAR signal for a constant Bl as a function of λ demonstrates oscillations according to cos(cλBl). For water, oscillations have a smaller amplitude than for the resolution sample (as expected for a smaller (by modulus) polarization), and a phase shift by 180 • (because of spin-incoherent scattering). Note that the normalized TOFLAR signal does not oscillate around unity (as expected after normalization using Eq. (5b)). This may indicate that the phase coil 1 (actually used for a fine adjustment of the net Bl-integral) interfered with a function of the first π/2 flipper.
The TOFLAR signal shown in Fig. 4 allows an independent determination of Bl by fitting cos(cλBl) to it. When we first tried to do this, we found that fitted Blvalues were by ≈ 4 × 10 −4 Tm larger than the nominal Bls calculated from the magnetic field configuration. Another demonstration that nominal Bl should be corrected is given in Fig. 5. We determined Bl-offset (Bl 0 ) by fitting cos(cλBl) to all spectra (I (Bl, λ)) measured at 120 Blvalues (six Bl-intervals, 20 values per an interval), it was almost constant: Bl 0 = 3.9 ± 0.1 × 10 −4 Tm. The offset is known to exist (it is due to the flux pinning in superconducting solenoids) and was determined earlier [8] to be τ N SE = 5.0 ± 0.9 ps for 4 ≤ λ ≤ 7Å (which translates to a Bl 0 ≈ 4 × 10 −4 Tm).
TOFLAR signals (unnormalized in order to show counting statistics) are plotted as a function of corrected Bl-values for every Bl-interval in Fig. 6 where, again, a 180 • shift for water signal can be seen.

QENS analysis of TOFLAR signal from water
From the above results it is apparent that error bars for water measurements are rather large, and that grouping of a large number of TOF channels together would lead to smaller error bars at the expense of damping the modulated signal due to an increase in dλ/λ (see Fig. 2). There are two major concerns regarding the use of Eq. (6) to extract F(Q, τ T ) of water. First, normalizations using Eqs. (5a), (5b) lead to an increase of error bars, especially in case of normalization to (I + − I − )/2.  Second, since τ T is proportional to Bl, by using Eq. (6) we obtain an average of F(Q, τ T ) over the τ T -interval that corresponds to a given Bl-interval. Thus, depending on the τ T -dependence of F(Q, τ T ), and on the width of the Bl-interval, extracted F(Q, τ T )-values may deviate significantly from the true F(Q, τ T ).

EPJ Web of Conferences
Therefore, we chose to analyze TOFLAR signal that was only normalized to the integrated neutron flux, and we first fitted to the resolution signal (where F(Q, τ T ) ≡ 1); free fitting parameters were a R E S , b R E S and Bl 0 . Then we fitted to the sample signal the equation where b R E S was fixed, and a, b, and Bl 0 were free fitting parameters (we did not fix Bl 0 to respective values obtained from fitting of Eq. (7) to the resolution signal because Bl 0 is generally slightly different in every measurement). The expression for F(Q, τ T ) contains additional free fitting parameters that depend on the model used to describe water dynamics. To model F(Q, τ T ) in our Q-region long-range translational diffusion coefficient D T R , mean residence time (τ 0 ) between jumps, and rotational diffusion coefficient (D r ) are required [9] 2 but to reliably determine all three parameters a wide Q-region extending to larger Q-values is needed. Therefore, we used To keep the Q-range and τ T -range as wide as possible, TOFLAR signals measured in all TOF channels and in all Bl-intervals should be used. But, as seen in Fig. 4A, some TOF channels have much smaller intensities than others; we chose to use only channels with particularly high intensities (4.8 ≤ λ ≤ 5Å and 5.2 ≤ λ ≤ 5.7Å) to decrease uncertainties in fitted parameter values (at the cost of reducing Q-region to 0.48 ≤ Q ≤ 0.57Å −1 ). Furthermore, we chose to use only the first two Bl-intervals because a finite wavelength resolution (due to a pulse width and due to summing of several TOF channels together) destroys the signal at larger Bls, see Figs. 2, 6. At first we fitted Eqs. (7), (8) to TOFLAR signals measured in several Bl-intervals for one single TOF channel; we did it channel by channel. a R E S , b R E S , a, b, and Bl 0 were interval-dependent fit parameters, D T R was an interval-independent fit parameter. Thus, we obtained a range of D T R (λ)-values, but all those D T R 's had large uncertainties (because of experimental error bars and because the number of fitting parameters per TOF channel (for both resolution and sample) was N × 3 + 1, where N is the number of fitted Bl-intervals, and the number of data points was N × 20).
To reduce the number of free fitting parameters, we used an extra constraint, namely that D T R is λindependent, and simultaneously fitted Eqs. (7), (8) to all TOF channels. D T R was a single global fit parameter, Bl 0 was Bl-interval-dependent but λ-independent fit parameter, while a and b were Bl-interval-and λdependent fit parameters. Thus, when M TOF channels were used, the number of free fitting parameters per one TOF channel was (2 × N × M + M + 1)/M ≈ 2 × N + 1. However, even in this approach uncertainties in fitted D T R -values were too large. To further decrease them, we made a and b to be Bl-interval independent fit parameters and obtained D T R of 1.35 ± 0.3 and 2.37 ± 0.39 [10 −5 cm 2 /s] for water at 5 • C and 20 • C, respectively (literature value are 1.5 and 1.97 [10 • −5 cm 2 /s]). The agreement is reasonable taking into account i) poor counting statistics, ii) an uncertainty due to a Bl-offset, and iii) a limited Q-region where both τ 0 and D r must be used in addition to D T R to accurately model the intermediate scattering function of water.
In principle, instead of analyzing TOFLAR data as a set of spectra measured at fixed wavelengths, we could analyze I (Bl, λ) as a set of spectra measured at fixed Bls, but then a correction for p(λ) would be necessary,  and, consequently a normalization (cf. Eqs. (5a), (5b)) would be needed. In any case, to better assess different data reduction and analysis routes, experimental data of better statistical accuracy are required. Such data can in principle be obtained by optimizing the instrument setup to have higher flipping ratios, higher polarization of scattered beam, and/or by choosing a non-spin-flip scattering sample. The reader might ask: "If TOFLAR brings a large intensity gain due to the absence of monochromatization, why better statistical accuracy could not be achieved for water?" Apart from various reasons already mentioned above, it is also because an important idea of the TOFLAR technique was not implemented, which is to have both polarizer and analyzer either before or after the sample [1]. In our case, by having polarizer after the sample, the polarization of scattered neutron would be ≈ 1 instead of −1/3, leading to a significant improvement of measured signals.

Conclusion
We performed the first TOFLAR experiment with a Larmor modulation device positioned after the sample using a slightly modified SNS-NSE spectrometer setup at SNS, USA. After testing various data reduction and data analysis routes we successfully analyzed TOFLAR spectra of water recorded at two temperatures and extracted diffusion coefficients that are in a reasonable agreement with literature values. Thus, the results of our proof-of-principle experiment call for further experiments in order to test practical applicability of TOFLAR technique, and its performance compared to TOF-QENS 03009-p.5 EPJ Web of Conferences and classical NSE techniques, and in order to develop optimal measurement and data analysis strategies.