Spin echo small angle neutron scattering using a continuously pumped ³He neutron polarisation analyser

Liouville’s theorem has always been a limiting factor in neutron scattering measurements. Any attempt to focus a neutron beam onto a sample will result in increased beam divergence and consequently decreased resolution of the momentum transfer. In traditional elastic neutron scattering experiments, measuring small scattering angles, and, hence, large length scales, requires limiting the angular divergence of the neutron beam and losing much of the neutron flux. To escape this limitation, the Spin Echo Scattering Angle Measurement (SESAME) technique encodes the scattering signal into the polarisation of a beam of neutrons.^1–3 This eliminates, to a large extent, the coupling between beam collimation and momentum resolution, allowing for measurements of large scattering objects (10-1000’s nm) without unacceptable losses of neutron flux. The SESAME technique for studying large scale structures has been implemented at a number of facilities with the development of several new instruments.^4–6 In this paper, we explore this option in terms of practical performance details and implementation along with experimental results.

We have developed a multipurpose SESAME instrument⁷ for both transmission and reflection measurements. In transmission, the technique is referred to as Spin Echo Small Angle Neutron Scattering (SESANS)^2,8 while in reflection it is called Spin Echo Reflection Grazing Incidence Scattering (SERGIS).^9,10 The SESAME beamline has been installed at the pulsed 13 MeV Low Energy Neutron Source (LENS) at Indiana University.^11,12 This paper focuses purely on the SESANS technique; however, with additional motor control this instrument can be used in reflection geometry for SERGIS or diffraction geometry for Larmor diffraction.

The SESANS technique uses a neutron beam initially polarised in the | + Y > direction (see Figure 1 for the coordinate system). The beam first passes through an abrupt, 90° change in the magnetic field direction, which splits the neutron beam into two in-phase components with opposite spins, i.e., | + Z > and | − Z >. These spin components then enter triangular magnetic prisms with magnetic fields in each of the triangular regions, alternately arranged along the +Z and −Z directions, as shown in Figure 1. Since the total energy of each neutron is conserved, the neutron velocity depends on the Zeeman interaction with the surrounding magnetic field, and the two spin states refract in different directions when they pass through the inclined boundary between each triangular region of the prism. A second triangular prism, with opposite fields to the first, returns the neutron waves to being parallel, yet spatially separated by a distance of the order of a few hundred nanometers, known as the spin echo length (z) (see Figure 1). A secondary spectrometer, located after the scattering sample, recombines the two neutron spin states, leading to a spin echo. Any scattering from a sample will result in a relative phase between the two spin states at the end of the instrument, and the initial (instrumental) polarisation will not be retrieved; i.e., the echo signal will be diminished. In this geometry, the Larmor phase of a scattered neutron is linearly proportional to its scattering angle. The Larmor phase is manifested in a change in polarisation which occurs due to the difference in phase for the two spin states. As a result, the polarisation of the neutron beam is the cosine Fourier transform of the entire scattering signal, allowing SESANS to determine the scattering via measurement of the final beam polarisation.¹³ 

The accessible spin-echo length for a neutron of wavelength λ for our setup utilising a series of magnetic prisms is given by

where $c= γ m π h =1.476×1 0 14 T − 1 m − 2$⁠, L is the separation between the prisms (as shown in Figure 1), and θ is the angle between the hypotenuse of the prism and the beam direction (X in Figure 1). Therefore, in any time of flight experiment in which multiple wavelengths are used, a range of spin echo length is probed simultaneously. In time of flight measurements, the extent of this range is usually chosen by selecting a particular static magnetic field strength (B).

For details of interpreting SESANS data, the reader is directed to reference.¹⁴ Certain omissions in this work are addressed in recent work from our group,¹⁵ which presents the theory in terms of conventional notation used for Small Angle Neutron Scattering (SANS). Briefly, the salient points are summarised below. For SESANS, the measured polarisation is given by

where P(z) is the echo polarisation due to the scattering obtained by measuring the polarisation with the sample in (P_s(z)) and corrected for the instrumental polarisation (P₀(z)), i.e., (P(z) = P_s(z)/P₀(z)) and Σ_t is the fraction of neutrons that are scattered once by a sample of thickness t. G(z) is related to the Debye autocorrelation function γ(r) and is the projection of the autocorrelation function along the encoding direction (Y in Figure 1).

To practically determine G(z), a series of measurements are made of the spin echo polarisation with and without the sample. It is also important to note that the SESANS technique allows the measurement of the total scattering in absolute terms, as shown by Eq. (2) since G(z) → 0 as z → ∞.

As shown in Ref. 15, the total coherent neutron scattering scales with the neutron wavelength squared and linearly with the sample thickness then a plot of 1/λ²ln(P_s(z)/P₀(z)) should reach a constant value at large z as G(z) tends to zero at large z. This quantity has been recently termed the “normalised SESANS signal.” Therefore from Eq. (2), both G(z) and Σ_t can be determined. The normalised SESANS signal also allows comparisons between different neutron scattering instruments and spin echo lengths as the dependence upon the neutron wavelength λ and magnetic field B can be corrected. This correction allows several different measurements at different wavelengths and magnetic fields to be collapsed into one dataset.

SESANS instruments have been constructed at TU Delft⁶ and ISIS.^4,5 A further instrument, Larmor,¹⁶ is currently in development at ISIS and one is planned at the High Flux Isotope Reactor (HFIR)¹⁷ at Oak Ridge National Laboratory, with a second incorporating both SESANS and SERGIS on the proposed second target station, optimised for longer wavelength neutrons. To date, and as far as we are aware, all of the currently constructed SESANS instruments have utilised supermirror analysers, although Larmor is also going to use ³He spin filters as an analyser. In this work, we report our experience of operating a ³He based device as a polarisation analyser. We report the basic instrument configuration, performance of the various components and some recent results on silica Stöber particles. Considerable focus is given to the ³He neutron spin filter as this is, to our knowledge, the first time that such a device has been used for SESANS. The Appendices describe the effect of the stability of the ³He polarisation on the measured G(z) and also the algorithm used to stabilise the ³He polarisation.

The SESAME beamline at LENS^11,12 is first and foremost a time-of-flight polarised neutron instrument with a usable flux of neutrons over a wavelength range of 2.5 Å to 11 Å. The instrument is used for SESANS and also the development of polarised neutron instrumentation.^18–21 The neutrons are produced at a repetition rate of 20 Hz, are thermalized within a 50 cm diameter water reflector, and cooled with a 10 mm thick methane moderator at T ≈ 6 K. The neutrons are transported from the moderator via supermirror guides. Direct line-of-sight from the SESAME flight-path to the source is prevented through the use of a polarizing bender. NiTi supermirror guides are then used to transport the polarised beam to the instrument.

The beam passes through a series of resistive coils, which provide the necessary spin manipulation to encode the scattering; π flippers, $π 2$ flippers, and prisms. The $π 2$ flippers stop and start the precession; a π flipper at the centre of the precession region flips the neutron spins; whilst four prism coils with inclined current sheets serve to encode the scattering angle.²² There are two prism coils placed before the sample and two after. The polarisation of the scattered neutrons is analysed using a ³He neutron spin-filter. This continuously operating ³He neutron spin filter uses alkali (Rb) metal Spin-Exchange Optical Pumping (SEOP).²³ Since a ³He spin filter is based on spin dependent absorption of neutrons, it decouples the spin state selection from the neutron optics and has a higher angular acceptance than typical supermirrors.²⁴ Turning off the SEOP laser during data collection results in a loss of ³He nuclear polarisation. The change of ³He polarisation in turn affects the performance of the beamline as a whole, as the device has a time dependent analysing power. Our analyser overcomes this by continuously pumping during measurements^25–27 to maintain a constant efficiency. Finally, the analysed beam is detected using an array of 12.7 mm diameter 1-D position sensitive ³He detector tubes.

A 40 mm long solid-state polarising bender is used to polarise the incident beam.²⁸ The bender serves a dual purpose: (1) it produces a polarised incident beam and (2) prevents fast neutrons from propagating through the guide system along any straight line path. The bender is made from a stack of 170 curved silicon wafers each 150 μm thick. On the concave side of each wafer is an m = 3 polarising supermirror composed of layers of Fe₈₉Co₁₁ and Si, followed by Gd/Si antireflecting layers and an absorbing layer of Gd. On the convex side is an m = 2 polarising supermirror composed of layers of Fe and Si and the same antireflecting and absorbing layers.

The supermirror guides, manufactured by Osmic, have internal dimensions of 25 × 75 mm². Guides upstream from the bender are angled 1.5° above the horizontal, whilst guides after the bender are horizontal. The NiTi coatings are m = 3 and m = 2 on the 25 mm and 75 mm faces, respectively. Tests were carried out on the POSY-I reflectometer²⁹ at Intense Pulsed Neutron Source (IPNS) on a short 185 mm guide section with magnetic fields of 16, 45, and 300 G applied using permanent ceramic magnets. The loss in neutron polarisation was acceptably low (less than 2% per bounce) with ceramic magnets producing a 300 G field perpendicular to the m = 3 faces.

The various spin manipulation devices throughout the SESAME beamline are all made of pairs of adjacent solenoids that are wound with aluminium wire, which has low neutron absorption and scattering cross section, all the coils have non-adiabatic transitions for the neutron polarisation at the current sheet interface. The aluminium wire (1.4 mm diameter) is coated in an insulating layer of low neutron scattering cross section. Measurements of this wire show a transmission of 97.5%/layer. To increase transmission and prevent neutron depolarisation due to uncontrolled field transitions,²² the wires of the upstream and downstream faces of each device are folded away from the beam. The various types of coils are shown along with the magnetic field orientations in Figure 2. Electromagnetic guide fields are installed between each pair of prisms to preserve the polarisation and extend the accessible spin echo length of the instrument by increasing L in Eq. (1). The last coil before the $π 2$ flip is used to correct for any imbalance in the total magnetic field integral. This is done by scanning the current and setting it to achieve the maximum echo polarisation.

The solenoids from which the magnetic prisms are constructed have been discussed at length in previous publications.^30,31 The prism coils are powered by Kepco BOP 20-20 M bipolar power supplies operating in constant current mode. The measured resistance of a prism coil is 1.1 Ω and the magnetic field at the hypotenuse is 8.89 G/A determined from magnetic field measurements using a Hall probe. Care was taken to limit the power supply fluctuations including drift and noise to a the level of 0.02% over a 24 h period. To prevent the solenoids from overheating, each triangular frame is water-cooled and the wires at the hypotenuse are air-cooled allowing up to 15 A to be applied which presents an upper limit on the maximum magnetic field of 130 G.

The two π flippers are constructed in a manner similar to the prisms, but with a different cross sectional shape (the flipping plane is perpendicular to the neutron beam) and without water cooling. The same aluminium wire is used.

All solenoids used in the π and $π 2$ coils have a height (parallel to the field direction) of 140 mm, a width of 70 mm, and extend 70 mm along the neutron beamline. They are yoked in 1.5 mm thick mumetal for flux return. All the flipping in these coils occurs at the current sheet interface and, since there is no attempt to rotate the neutrons’ spins using Larmor precession, these flippers work for any neutron wavelength and do not require ramping of the applied current.

At the exit of the final neutron guide is a low efficiency (0.5%) ³He neutron monitor which is used to normalise all measurements to the same neutron flux incident on the sample. The active area of the monitor is 75.0 mm × 25.4 mm and thus covers the entire guide exit. The primary detector used is an array of sixteen 12.7 mm diameter position sensitive ³He filled tubes, the spatial variation of the efficiency of the tubes is shown in Figure 3. The tubes are arranged in two rows of eight, one behind the other and offset from one another such that the detector efficiency is almost constant perpendicular to the axis of the tubes except for a series of narrow gaps, corresponding to the thin stainless steel walls of the detector tubes. The gas in the tubes is at 10 bar and each tube has an efficiency of neutron capture >90% calculated for 5 Å neutrons. Each tube has ≈2 mm resolution along the tube length.

The ³He cell (Mars) was fabricated at NIST using the procedure outlined in Ref. 32. The cell body is made from Corning 1720 with optically flat windows made from re-blown GE180 alumina silicate glass. The flat windows provide a uniform analysing path for the scattered beam because they provide a uniform neutron path length through the cell, compared to re-blown cells. The flat windows reduce any lensing and birefringent effects which can occur in cells from re-blown glass. This is critical in our application as the cell cannot be side pumped as in similar online pumping experiments.²⁶ The windows are 8 mm thick and the cell body is 80 mm long with an inner diameter of 60 mm. The cell pressure measured by neutron transmission at NIST is 0.9 Bar.^32,33 However even with optically flat windows, there are a range of path lengths through the cell with the shortest going through the center of the cell, that arise due to the beam divergence and scattering. In SESANS, this variation in path lengths results in a difference of <1 % in analysing power for neutronswavelengths >2.5 Å over the whole cell.³⁴ 

The ³He cell is pumped by a Oclaro comet laser system with a fibre-coupled ∼30 W laser. The spectrum of the laser is narrowed down to 0.24 nm by a Volume Bragg Grating (VBG). The peak wavelength can be tuned to the Rb D₁ transition line mainly by adjusting the laser drive current with the laser at a temperature of 19.89 °C. The spatial profile of the laser is homogenised using a fibre and the polarisation of the laser emission is separated by a polarising beam cube at the output of the fibre (see Figure 4(a)). This splits the beam into two components in approximately a 2:1 ratio due to the elliptical polarisation imposed by the fibre. Each beam passes through a quarter-wave plate (CVI laser optics, USA) to produce circularly polarised light. One beam is incident on the side of the cell where the neutrons enter whilst the second beam (with opposite helicity) is used to illuminate the other side of the cell. The laser light is brought parallel to the ³He holding magnetic field using mirrors fabricated from single crystal silicon, these mirrors are coated in a proprietary coating (Rocky Mountain Instrument Company, USA) which provides >99% optical reflectivity at 795 nm to minimise losses of laser light. Fused silica windows with total thickness of 4 mm are used to enclose the cell, calculations of the transmission estimate a neutron wavelength independent transmission of 90% for each window. The cell is heated using an 750 W in-line air process heater (Omega, USA) and contained in a teflon oven.

The ³He cell is held in a shielded solenoid which contains Nuclear Magnetic Resonance (NMR), Electron Paramagnetic Resonance (EPR), and Adiabatic Fast Passage (AFP) coils as shown in Figure 4(a). The guide field is coupled into the solenoid by a circular coil on the outside of the solenoid. During the course of the measurements, the absolute polarisation of the ³He is measured using the change in the EPR resonance of the Rb frequency.³⁵ This frequency is monitored via the peak of the fluorescence from the D₂ electron transition emission using photodiodes whilst the ³He polarisation is reversed using an in-built coil via a frequency swept AFP pulse similar to Ref. 36 with losses of typically of 0.1% per flip. This could be improved by implementing a Gaussian envelope on the sweep function.^37,38 The AFP coil is a sine coil close to the solenoid winding and is not shown on the figure. A Free Induction Decay (FID) NMR system similar to that described in Ref. 39 is also used to monitor the relative polarisation of the ³He. Using FID NMR, a cell lifetime (T₁) of 107 ± 2 h was measured off-line.

The flux was measured using a high efficiency (≈100%) ³He pencil detector, measuring 100 mm wide × 25 mm tall and mounted at the sample position with the polariser in position. A thin slit was used to define the beam area and to ensure that the beam was incident on the centre of the detector. The spectrum is shown in Figure 5, the dip observed in the spectrum at 5.06 Å is a result of diffraction from the {111} plane of Si in the bender. Similar dependence has been reported for some previous solid-state benders⁴⁰ and can be avoided by more careful design. The total neutron flux at the sample position integrated over the wavelength range from 3-12 Å was 810 neutrons cm⁻² s⁻¹ kW⁻¹. The typical proton beam power for the LENS neutron source is 3 kW.

To check the instrument performance and alignment of the bender, the polarisation was measured with only guide field between the bender and analyser. Flipping of the ³He allows the measurement of the polariser and analyser (P_A × P_B) efficiency.⁴¹ The efficiency (P_A) of the analyser is given by

where $O He ( λ ) =7.33×1 0 − 2 ×plλ$ and P_He is the ³He nuclear polarisation. Using EPR to measure the ³He polarisation and with pl = 7.2 Bar ⋅ cm, determined from neutron transmission, all quantities in Eq. (3) are known and the performance of the bender can be extracted (see Figure 6).

The bender efficiency peaks at 4-6 Å, falling off at higher wavelengths, agreeing with earlier measurements.²⁸ This is well matched to the flux profile as shown in Figure 5. It may be possible to extend this range by adding in a second polariser using either a combination of reflection and transmission polarisers or a single-bounce polariser followed by a ³He cell as shown by Dalgliesh et al.⁴² 

In order to maximise the instrumental polarisation (P₀), the losses due to the coils must be minimised. In earlier work, it was shown how symmetry across the various components serves to cancel out most of the magnetic field aberrations.^22,31,43 Despite this symmetry allowing most aberrations to be cancelled, the total polarisation can still be low due to low flipping efficiency at the various current sheet interfaces between field regions.²⁰ 

Simulations from our group²⁰ using finite boundary methods and a Bloch equation solver⁴⁴ show that the majority of this depolarisation comes from the use of round wire at the interface between the two triangular solenoids that form the prism. Interdigitation of the wires belonging to neighbouring current sheets should significantly improve the performance. Simulations also show that the degree of depolarisation increases with the coil current density. This effect is not strong in the π-flipper as this is operated at relatively low currents of 3-5 A. However in order to reach long spin-echo lengths, it is important to achieve a high magnetic field (B) in the prisms. Hence, depolarisation at interfaces becomes a non-trivial loss effect which must be minimised. This is shown clearly in Figure 7 which shows a comparison of the flipping efficiency of two prism coils. In these measurements, the devices were tested as a simple π flipper without precessing polarisation, each prism was energised at 15 A (130G) with antiparallel (i.e., flipping) fields. Physical inspection of the coils used in Figure 7 shows that one has close to perfect inter-digitation at the current sheet interfaces whilst the second has gaps. The prism with gaps has a lower overall efficiency at the high current. Further examination of the spatial dependence of the efficiency shows that the efficiency varies with position and that this spatial dependence corresponds to the regions where the interdigitation of the wires has failed. The prism with good interdigitation has no corresponding spatial dependence in its flipping efficiency.

Measurements of the neutron wavelength dependence of the prism efficiencies were also made at applied currents of 5 A, again using them in a π flipping mode without precessing polarisation. In this flipping mode, a flipping efficiency of >90% was measured for neutrons with wavelengths of 2-10 Å. Using these measurements for the four prisms and the measured efficiencies of the π and $π 2$ flippers, the expected echo polarisation is calculated. A comparison between the calculated echo, measured echo, and instrumental polarisation is shown in Figure 8. This shows that at short wavelengths the full instrumental polarisation is recovered in spin echo. However at longer wavelength, additional depolarisation is observed due to Larmor aberrations. Moreover, this suggests that improvements in the flipping efficiency would allow a significant increase in echo polarisation. Such improvements may be possible using a new superconducting Wollaston prism that is under development.²¹ Currently, 2-7 Å neutrons are used for SESANS on SESAME due to our flux profile.

The saturated ³He polarisation was monitored by EPR during the course of several cycles of experiments. During the first two cycles of operation (250-300 h), a slow variation of the ³He polarisation of 3.0% and 3.3% was observed. There was also a difference in average polarisation between the two cycles which was due to realignment of the laser to better illuminate the cell. Any variation in P_He will change the total analysing power of the instrument according to Eq. (3) and hence should be minimised. For a detailed discussion, see Appendix  A.

The main source of fluctuations in the device was the optical pumping laser since the laser wavelength drifts with time, over the course of several hours (Appendix Figure 12). To solve this, a feedback mechanism was implemented to change the laser current to tune the laser back onto resonance. The details of the feedback loop are discussed in the Appendix. Once this feedback was implemented, variations in the ³He polarisation of 2.3% and 1.8% were observed during two later 2 week cycles. The effect of the ³He polarisation drift at this level is to render measurements below 2.5 Å unreliable as the drift in the analyser efficiency at these wavelengths is large. However above 2.5 Å, fluctuations in the ³He polarisation of 2% represents a total possible contribution to the accuracy of <1% of the determined SESANS signal P_s/P₀. For most SESANS measurements, this is acceptable.

The Stöber method can be used to produce large volumes of monodisperse colloidal silica particles.⁴⁵ In order to evaluate the SESAME instrument for SESANS, we prepared two samples which were measured ≈6 months apart on the SESAME instrument and at Offspec at the ISIS pulsed neutron source.

Shown in Figure 9 are measurements of the normalised SESANS signal for two different concentrations of silica Stöber particles in H₂O and H₂O : D₂O (4:1) on both SESAME and Offspec. Two points are clear; the first is that at long spin-echo length (G(z) → 0), the values of the normalised SESANS signal, which are dependent upon the total (single) scattering cross section (Σ_t) agree within error. Also at short spin echo length, the agreement is good, especially considering the differences in spin-encoding methods, wavelength ranges, beam sizes, and differences in analyser technology. Slight differences are observed in the measured G(z), these may be changes in the sample between measurements; however, more statistics on the SESAME measurements are required to unambiguously answer these questions. The second point is the difference in statistical error. Data collection times for Offspec and SESAME were ≈6 and 24 h, respectively. SESAME used a beam size of 200 mm² and Offspec 15 mm². Errors in SESAME are a factor four larger than Offspec. Accounting for differences in flux and beam size, these differences agree within a factor of two with expectations based on the fluxes of the ISIS and LENS source.

The work described in this communication describes to our knowledge the first continuously pumped ³He analyser used for SESANS; in fact to our knowledge, this is the first use of continuous SEOP in any long term neutron scattering application, although nuclear physics experiments have been using these devices for a number of years (for example Refs. 46 and 47) and the Maria instrument at JCNS also has a similar option. The principle characteristics of the analyser are a larger acceptance angle than conventional super mirrors and also a uniform analysing power. For example, supermirror channel polarisers may have strong positional dependence on both the transmission and polarisation characteristics.²⁴ Such behaviour would make the normalisation from the instrumental polarisation (P₀) invalid as the sample polarisation (P_s) would have a different spatial dependence. ³He is one technology which does not have such a dependence. However, the downside is the requirement of achieving a stable helium polarisation on the beam. This work shows that despite the additional equipment needed such a device is feasible to operate with a minimum of intervention.

The small ≈2% variation in the ³He polarisation renders the very short wavelengths unusable in our measurements; however, it should be noted that by definition this is where G(z) approaches unity and hence can be neglected in many cases. Alternatively, measurements could be performed at short spin echo length with a lower field in the prisms.

The continuous pumping removes the need for time dependent corrections as the gas polarisation decays. It also keeps the polarisation high to make maximum use of the available neutron flux. Additional modifications could be performed to the ³He analyser such as using a vertical magnetic field,⁴⁸ which would have the added advantage of removing the need to have mirrors in the beam and would allow the device to be placed closer to the sample, further increasing the angular acceptance. Also the angular acceptance can be further scaled up by the use of larger cells and/or narrower, higher-power lasers.⁴⁹ Indeed the laser system used in Ref. 49 allows 85% polarisation to be routinely achieved and these narrower high power lasers would improve the performance in our system.

The coils used in this work have shown the importance of the interdigitation in resistive coils the efficacy of such interdigitation requiring careful fabrication and alignment of the coils. Any failure in the interdigitation results in a spatial variation in the flipping efficiency. Such a spatial variation renders the determination of the ratio P_s/P₀ inaccurate.

This paper has described the operation and performance of the first SESANS instrument with a continuously pumped ³He spin analyser. The instrument is able to encode the neutron spin precession angle over a broad range of wavelengths and the encoding prisms are well suited to a time-of-flight source.

The ³He analyser has a large acceptance angle and allows large beams and samples to be used. The efficiency is uniform for all scattering angles to within 1% for wavelengths above 2.5 Å and hence normalisation to the instrumental polarisation (P₀) is valid for all scattering, unlike in the case of some supermirror polarisation analysers.

The authors would like to thank Jeff Andersen of the NIST shop for fabrication of the GE180 cell and Dr. W. A. Hamilton from Oak Ridge National Laboratory for helpful comments in the preparation of this manuscript. Also Dr. R. Dalgliesh from the ISIS pulsed neutron and muon source for help with the comparison between Offspec and SESAME results. W. M. Snow and H. Yan (now Chinese Academy of Engineering Physics) acknowledge support from the Indiana University Center for Spacetime Symmetries. This project was supported by the U.S. Department of Energy, Office of Basic Energy Sciences Grant Nos. DE-FG02-09ER46279 and DE-FG02-03ER46093. Construction of LENS was supported by the National Science Foundation Grant Nos. DMR-0220560 and DMR-0320627, the 21st Century Science and Technology fund of Indiana, Indiana University, and the Department of Defence.

For any SESANS measurement, the quantity measured is given by

where P(z) is the polarisation from the sample (P_s) normalised to the instrumental polarisation P₀(z). Therefore in any SESANS experiment, variations in the ³He polarisation will be convolved into the measurement of both P₀(z) and P_s(z). Inaccuracies in the determination of P(z) which will propagate through to inaccuracies in G(z). This inaccuracy can be mitigated by measuring both P₀(z) and P_s simultaneously, oscillating the measurements between the two by the use of an automated sample changer, or by using knowledge of the time dependence of the ³He polarisation, and applying corrections using similar methods to those suggested by Wildes.⁵⁰ However, the simplest method is to minimise the fluctuations in the ³He polarisation.

The total contribution of the derived polarisation, P(z), for these time-dependent fluctuations is given by standard error calculation. The associated inaccuracies for various different fluctuations are shown in Figure 10 as a function of neutron wavelength. At longer wavelength neutrons, fluctuations in the ³He polarisation have a minimal effect on the accurate determination of P(z), and hence G(z). Moreover, the effect is only prevalent below 2.5 Å, which is a region where both the neutron polarisation and flux are low.

Under steady state conditions, the ³He polarisation (P_He) is given by

where Γ_He is the ³He relaxation rate and γ_SE is the spin exchange rate. P_Rb is the Rb polarisation, given by

where Γ_SD is the electron spin polarisation destruction rate, which is dependent upon the gas density and alkali metal used. γ_opt is the optical pumping rate given by

where I(ν) is the photon flux at frequency ν and σ(ν) is the photon absorption cross section for the Rb electron transition. In order to achieve a stable ³He polarisation, then the alkali polarisation must be constant as the spin-exchange and relaxation rates are constant in time. Hence, the laser spectral distribution (⁠$I ν$⁠) must be constant.

Figure 11 shows an off-line measurement of the laser spectrum. During the measurement, the laser spectrum suffered from a fluctuating line shape, which causes the polarisation of Rb to be unstable in time, and eventually affects the polarisation of ³He. To reduce the variation of ³He polarisation, a feedback algorithm is added into the system. The idea is to tune the two small peaks either side of the D₁ absorption in Figure 11 to have equal height by adjusting laser current, so the absorption dip sits right on the Rb D₁ transition frequency, and Rb can absorb the most power from laser light. Figure 12 shows a comparison of the spectral stability before and after applying the feedback algorithm. The Y axis is the ratio of the heights of the two peaks (tuned to be 1). It shows that the feedback changes the ratio of the peak more rapidly in time than without the feedback. The feedback sweeps the laser emission wavelength back and forth across the D₁ absorption and hence wastes some laser light. However as the ³He polarisation/depolarisation rates are slow in comparison to the sweep rate of the laser, this produces good stabilisation. More sophisticated feedback may be possible using a proportional integral differential (PID) type algorithm. A further improvement may be to maximise the D₂ fluorescence (780 nm⁵¹) to lock the laser wavelength.