Imaging of dynamic magnetic fields with spin-polarized neutron beams

Precession of neutron spin in a magnetic field can be used for mapping of a magnetic field distribution, as demonstrated previously for static magnetic fields at neutron beamline facilities. The fringing in the observed neutron images depends on both the magnetic field strength and the neutron energy. In this paper we demonstrate the feasibility of imaging periodic dynamic magnetic fields using a spin-polarized cold neutron beam. Our position-sensitive neutron counting detector, providing with high precision both the arrival time and position for each detected neutron, enables simultaneous imaging of multiple phases of a periodic dynamic process with microsecond timing resolution. The magnetic fields produced by 5- and 15-loop solenoid coils of 1 cm diameter, are imaged in our experiments with ∼100 μm resolution for both dc and 3 kHz ac currents. Our measurements agree well with theoretical predictions of fringe patterns formed by neutron spin precession. We also discuss the wavelength dependence and magnetic field quantification options using a pulsed neutron beamline. The ability to remotely map dynamic magnetic fields combined with the unique capability of neutrons to penetrate various materials (e.g., metals), enables studies of fast periodically changing magnetic processes, such as formation of magnetic domains within metals due to the presence of ac magnetic fields.


Introduction
The unique ability of neutrons to penetrate materials opaque to other types of radiation, combined with their ability to interact with a magnetic field through the neutron spin precession, makes neutrons an excellent probe for remotely studying magnetic field distributions, even internally within solid objects. Various techniques utilizing polarized neutrons have been developed for the spatially resolved investigation of magnetic fields [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. One of these techniques correlates the variation of neutron intensity with the magnetic field strength and direction, allowing effective magnetic field imaging [2,3,[5][6][7][8][9][10]. Also interferometric and diffraction methods have been available for a number of years, where the dependence of the real part of the refractive index on the magnetic field strength can be exploited [1,4,11,12] including methods utilizing spin echo [13,14]. However the latter techniques operate with neutron beams of low intensity caused by all the necessary beam conditioning, and further, some of them require spatial scanning through the sample, and therefore provide very limited imaging statistics, but with the tradeoff of providing very good spatial resolution. The alternative approach combines the conventional neutron imaging technique (whereby the two-dimensional neutron flux intensity distribution is measured after the sample) with the neutron spin analysis [5][6][7][8][9][10]. Magnetic fields within relatively large areas (tens of centimeters in linear dimension) can be studied by this technique without spatial scanning. Here the precession of the neutron spin in the presence of a magnetic field can be measured as a modulation of the intensity. Compared to a conventional neutron radiographic setup a spin polarizer/spin analyzer pair needs to be added, enabling studies of magnetic fields in combination with conventional absorption contrast imaging as well as other modalities such as neutron transmission tomography and Bragg edge imaging. Previous magnetic field studies concentrated on the imaging of static magnetic fields [5][6][7][8][9][10]. In this paper we demonstrate how similar methods can be implemented for studies of dynamic magnetic fields. However, the limited neutron flux available even at the present state-of-the-art neutron imaging facilities [15][16][17][18][19][20] necessarily limits these dynamic studies to either slowly changing magnetic fields (<1 Hz ) or to periodic or cyclical processes where measurements of multiple cycles can be performed, in order to acquire sufficient neutron statistics for adequate imaging. Importantly however, the frequency of the periodic field can be as high as ∼100 kHz, as the timing resolution of present neutron imaging detectors is in a sub-microsecond range [21]. This represents a significant advance in capability, compared with previous instrumentation.
In the presence of a magnetic field with non-zero component perpendicular to the neutron spin vector the spin undergoes precession around the magnetic field vector. That precession is described by the spin wave equation of an individual neutron spin ⃑ S t ( ) propagating in a magnetic field ⃑ B t ( ) [5,8] where γ is the gyromagnetic ratio of the neutron (−1.8 324 × 10 8 rad s −1 T −1 ). Thus the neutron spin precesses around the vector ⃑ B t ( ) and has the instantaneous angular velocity ω γ α = ⃑ t B t ( ) ( ) sin , where the angle α is the angle between the vectors ⃑ S t ( ) and ⃑ B t ( ), figure 1. The angle of rotation of spin component perpendicular to the magnetic field around the vector ⃑ B t ( ) can be found then from the equation: Substituting the time variable t to the neutron path variable l (t = l/v) equation (2) becomes where v and λ are the neutron velocity and wavelength, respectively, m is the mass of neutron, h is the Plank constant and dl is a short section of neutron path l. Integrating equation (3) over the neutron path it is possible to calculate the angle of spin precession around the vector ⃑ B l ( ): From equation (4) it is clear that the spin precession depends on the value and direction of the magnetic field and the neutron wavelength: for the same magnetic field the precession angle is larger for shorter wavelength neutrons. Therefore, to accurately probe the magnetic field by neutron spin precession at a continuous neutron source, a monochromatic beam of neutrons is required. At the same time, a polychromatic neutron beam can be used if the energy of each neutron is detected and therefore its wavelength λ is known, allowing for the reconstruction of value α ⃑ B l ( ) sin . The full three-dimensional reconstruction of the magnetic field vector ⃑ B l ( ) requires multiple measurements with all possible combinations of the polarization vectors of incident neutron beam and the polarization analyzer, as explained in [6] and in section 1.1 below.
Let's consider a simple case, when neutrons propagate along the Z-axis and the magnetic field is directed along the same axis: ).
x y z z In that case the neutron spin always precesses around the Z-axis and the angle of precession φ is proportional to the value of the magnetic field and the neutron wavelength. Therefore the value of the magnetic field can be recovered if the angle φ is measured and the two-dimensional map of magnetic field values B z (x, y) can be found from the map of angles φ(x, y). In our imaging experiments at a continuous neutron source we measured rather sin φ(x, y) than the angle φ. Therefore we measure the magnetic field accurate only with respect to an unknown offset (due to sin φ = sin (φ + 2nπ). The reconstruction of the absolute value of the magnetic field B z (x, y) requires a priori knowledge of the absolute value of the magnetic field at least in one point (e.g. the zero value of the field at a certain position x, y) or multiple measurements at different wavelengths [5,24]. In the general case, when the magnetic field is not aligned with the Z-axis, the precession of the vector ⃑ S l ( ) around the particular magnetic field vector ⃑ B l ( ) at a point l along the neutron path can be calculated for a small increment dl. That process repeated for the entire path of the neutron travel between the polarizer and the analyzer can be used to correlate the measured spin precession angle with the magnetic field vector and thus be used for the reconstruction of the magnetic field. Note, however, that a discrete numerical calculation requires very small time steps and pixel sizes in the adiabatic case when the neutron spin projection on the vector B remains constant, i.e. the component of the neutron polarization parallel to the field follows directional changes of the magnetic field adiabatically.
where Σ(x, y, z) is the linear attenuation coefficient of the sample at position (x, y, z), I 0 (x, y) is the intensity of the incident beam, and φ is the angle between the neutron spin at the analyzer position and the analyzer maximum transmission direction. For experiments in which the magnetic field components perpendicular to the initial spin orientation are both zero (the angle φ is always equal to 0 as no spin precession is present for α = 0) the measured intensity is determined only by the sample attenuation, as seen from the equation (5). Therefore if the magnetic field of the sample can be switched off (e.g. experiments with the magnetic field produced by a current) the influence of the sample attenuation can be measured first and used later for normalization to isolate only the magnetic field component from the equation (5). Another possibility is to switch the direction of polarizer by 180°and perform the measurements with spin up and spin down. From the measurement of spin up and down for both sample and open beam the degree of beam depolarization can be reconstructed accurately [8] and the attenuation due to the sample can be separated. As soon as at least one of the perpendicular field components is larger than 0 the neutron spin precesses and arrives at the analyzer with certain angle φ(x ,y) and therefore the measured intensity at that position is reduced by a factor of φ + x y [ 1 cos ( , ) ] . 1 2 Thus both perpendicular components of the magnetic field are probed in the experiment with a spin-polarized incident beam. Realignment of the spin polarizer and the analyzer along the other axes allows studies of the other pairs of the magnetic field components. In order to reconstruct the full three-dimensional distribution of the magnetic field (not intended in the present study) a set of nine measurements can be acquired with the polarizer P and analyzer A oriented along X or Y or Z axis in pairs as P x A x , P y A y , P z A z , P x A y , P x A z , P y A x , P y A z , P z A x , P z A y . In practice, instead of realignment of polarizer and analyzer it is easier to use a pair of spin flippers installed after the polarizer and before the analyzer [6].

Experimental setup
Our imaging experiments are conducted at the cold neutron beamline facility CONRAD [15] at the Helmholtz-Zentrum Berlin (HZB) providing a continuous flux of neutrons with the spectrum shown in figure 2. A spin polarizer with a magnetic field directed along the Y-axis was installed at the exit of cold neutron guide. The sample was placed in front of a spin-analyzer followed by a position sensitive detector, figure 3. Both the polarizer and the analyzer were bending-type spin polarization devices [25,26], which transmit only neutrons with a particular spin state (parallel to its own magnetization axis) and absorbs neutrons with the other spin state. The polarizer and the analyzer were co-aligned so that the intensity of the transmitted neutron beam with no external magnetic fields was maximized. The polarization axes of the polarizer and the analyzer were along the Y-axis.
A neutron counting detector with microchannel plates, providing both position and time of neutron arrival was used in the experiments [21,22,27]. The active area of the detector was limited to 14 × 14 mm 2 and the spatial and temporal resolution was 55 μm and ∼1 μs, respectively. High detection efficiency exceeding 50% for thermal and cold neutrons [28,29] and was essential for the present studies of the dynamic magnetic fields where acquisition of many frames corresponding to various phases of the magnetic field variation was required. The detector was synchronized with the ac current source for the coil generating magnetic field with 3 kHz frequency. A coil with 5 and another coil with 15 loops of ∼10 mm diameter were installed at a few cm distance from the analyzer, figure 3. A 1 mm aluminum wire coated by 0.1 mm insulation was used to form the coils shown in figure 4. The neutron attenuation of these coils was very low and thus the contrast in the images acquired in our experiments was determined almost exclusively by the spin precession in the magnetic field and not by the neutron attenuation by the sample. All the images reported in this paper were normalized by the 'open beam' images to eliminate the stripes from the structure of the polarizer and the analyzer, which consisted of multiple layers of the polarizing mirrors and Gd absorbers [25,26].
For the static magnetic fields a dc current of 1-10 A was supplied to the coils and the resulting transmission images were accumulated by the detector over a 100 s period in most cases. The measurements of the dynamic magnetic field was performed simultaneously for all phases of the ac cycle, unlike a stroboscopic mode often used in time resolved neutron imaging experiments, when most of the neutrons are excluded from images by electronic shutter and only a short time range within each cycle is measured. In our experiments the time of each detected neutron was measured with sub-μs accuracy relative to the trigger synchronized to the ac current source. The events were binned online into a stack of images, each corresponding to a 2 μs wide period of the ac cycle, so that 166 phases of 3 kHz cycle were accumulated simultaneously, allowing an efficient data collection for a dynamic process with 333 μs period.
We used the full spectrum of the beam in order to increase the counting statistics for both dynamic and static experiments. As mentioned earlier, the presence of neutrons of various energies leads to a certain blurring of the magnetic field imaging as the rate of spin precession is proportional to neutron energy and more accurate future experiments will have to be conducted with somewhat monochromatized neutron beams (e.g. by a velocity  selector with its ∼10% energy resolution or a double crystal monochromator with ∼3% energy resolution). A pulsed neutron beam can also be used as the energies of neutrons can be recovered from their time of flight and thus images corresponding to different energies can be acquired simultaneously. The advantage of the pulsed neutron beam for polarization measurements is in the fact that a wide spectrum of energies can be used at the same time, thus allowing efficient quantification. Indeed the angle φ can be recovered explicitly from equation (5) without the bias of an undefined offset due to the modulus of the cosine [6,24,30].

Theoretical predictions of polarized neutron radiography of magnetic fields
All our experimental results of the corresponding measurements were compared to the calculated images for the 15 loop coil operating in dc and ac modes. The utilized calculation procedure of the resulting images can be applied to any magnetic field and is not limited to the field produced by our coil sample. To calculate the expected intensity distribution for a given sample we first compute the three-dimensional magnetic field ⃑ = B x y z B x y z B x y z B x y z ( , , ) ( , , ), ( , , ), ( , , ) x y z (in our case the field in a 20 × 20 × 30 mm 3 volume around the coil, appendix A). Then for each position (x, y) the integrated angle of neutron precession as it propagates through the magnetic field is calculated numerically from equation (4) for the neutrons of various wavelengths λ. The details of integration of equation (4) are described in appendix B. Once the angle φ was found for the neutron arriving at the analyzer position the intensity of the neutron flux at position (x, y) is calculated from equation (5) assuming there is no attenuation of the neutron beam by absorption (low absorbing aluminum wire in our experiments). The image otherwise is modified by the attenuation of the sample if required, or results are correlated to attenuation corrected images through normalization. The calculation is repeated for each pixel (x, y) of the detector active area and the resulting transmission image I(x, y) is calculated for a given current I in the coil and given neutron wavelength λ. Finally, to simulate the experiment at CONRAD beamline with a wide neutron spectrum multiple images are calculated for various neutron wavelengths (for the same magnetic field value) and those images are combined with weighted coefficients corresponding to the beam spectrum shown in figure 2.

Dc magnetic fields
Initially we have measured the spin-polarized neutron images of static magnetic fields produced by 1-10 A dc currents in the 5 loop and 15 loop coils, figures 5 and 6. The observed fringing from the 5-loop coil indicates that even at 10 A the number of full 2π precessions of the spin in the XY-plane is small, while for the 15 loop coil 10 A create a magnetic field strong enough to produce multiple fringes in the resulting images. Some of those fringes cannot be resolved because of the use of the polychromatic neutron beam, image blurring by the beam divergence and the short period between the fringes at positions with maximum field values. The agreement between the calculated (figure 7) and measured polarized images confirms the validity of our modeling, which can be used for the pre-experimental analysis in future studies for the estimation of which magnetic fields can be resolved by our technique and what is the requirement for the beam monochromatization. The measured magnetic fields produced by opposite currents, figure 8, also agree quite well with the results of our corresponding calculations, figure 9, although the asymmetry of the coil lead wires results in some asymmetry of the polarized images.

Time-varying periodic magnetic fields
Imaging of dynamic magnetic fields in general is difficult as the polarized neutron fluxes are relatively low (10 5 -10 7 n cm −2 s −1 ) and a short exposure time (microseconds to milliseconds) is not sufficient to form an image with acceptable statistics. However, if the magnetic field varies periodically the neutron statistics can be improved by the integration of polarized neutron images at the same phase over multiple cycles. Stroboscopic imaging is one of the options where only a particular phase of the dynamic process is imaged by gating a fast imaging detector (e.g. scintillator screen coupled to a gated image intensifier followed by a CCD or CMOS readout). However, this approach has a serious limitation: the timing resolution of such experiments is fixed by the acquisition shutter and cannot be changed by the later analysis. Even worse, a large fraction of neutrons is not used for the imaging and multiple acquisitions have to be performed in order to cover the full cycle of the process, which is highly inefficient. Another approach is to use a neutron counting detector, which can provide both position and time resolution for each detected neutron. Synchronization of timing in such experiment to the dynamic process under study enables quasi-simultaneous detection of multiple phases of the cycle and flexible time binning. In that case only one measurement is required and the resulting time-resolved images are integrated over multiple cycles. In our dynamic experiments a neutron counting detector with Timepix readout was used to image a dynamic magnetic field produced by the 15-loop coil conducting a 3 kHz ac current. The detector and the coil were the same as in static measurements, only the current was changed from dc to ac. In the resulting data set the 333 μs wide cycle was split into 166 images each corresponding to a specific 2 μs wide phase of the process. The flexibility of this acquisition mode allowed subsequent grouping of the images into 10 μs wide slices (with the phase of each slice shown in figure 10). The measured neutron polarized images for the 2 A rms ac current are shown in figure 11 and for 10 A rms current in figure 12. A better visualization of the measured magnetic field dynamics can be found in the animation of the acquired images in the online supplemental materials 7 . The neutron statistics of dynamic images is lower than in dc current experiments as the process was split into 166 individual frames and a lower number of neutrons were acquired in each 55 μm pixel of our detector per acquisition frame compared to the static measurement. The simulated images corresponding to a 10 A rms current shown in figure 13 were calculated for the polychromatic beam spectrum of CONRAD beamline and Figure 9. Simulated spin-polarized neutron radiographs of a dc magnetic field produced by a 15-loop coil. Positive and negative 5 and 10 A currents are calculated for the CONRAD beamline spectrum shown in figure 2. 7 The dynamic magnetic fields measured for a 3 kHz AC current in a coil imaged at CONRAD beamline facility as well as simulated images for the same coils are presented in the online supplemental material to this article (stacks.iop.org/NJP/17/043047/mmedia). demonstrate the blurring of the fringes through the superposition of images corresponding to various neutron wavelengths. A detailed analysis of the wavelength dependence of such dynamic polarized neutron imaging is described in the next section.

Wavelength dependence of polarized neutron imaging
As seen from equation (4) the neutron spin precession angle depends both on the value of the magnetic field and the wavelength of the neutrons. In the experiments with a static magnetic field the contrast in the measured images is determined by the variation of the magnetic field across the field of view and by the monochromaticity of the beam. With a neutron beam of a relatively narrow energy range (e.g. formed by a double crystal monochromator at a continuous source) the magnetic field variation is represented by the oscillation of intensity across the image. Measurements with a pulsed neutron beam allow simultaneous detection of multiple energies and flexible energy binning improving the statistics and enabling more accurate quantification of the magnetic field values. However, in dynamic magnetic field measurements there is another factor affecting the  . Spin-polarized neutron radiographs measured for different phases of a dynamic magnetic field. The field is produced by a 15-loop coil with a 3 kHz ac current with 2 A peak-to-peak amplitude. Each image here represents a 10 μs wide time slice of the ac cycle. The corresponding phases of the 3 kHz cycle are shown in figure 10. Multiple cycles of the process were imaged and grouped together to provide adequate statistics. All phases of the dynamic magnetic field are imaged quasi-simultaneously, which is enabled by the time tagging of each detected neutron relative to the phase of the ac current.  contrast of the measured images and thus complicating the data analysis: neutrons of different wavelengths impinging on the detector at the very same time have experienced the magnetic field at different times and hence represent different phases of the dynamic magnetic field. Neutrons of various energies travel at different speeds through the gap between the sample and the detector. That leads to the fact that faster neutrons which interacted with a given phase of the magnetic field arrive at the detector earlier, simultaneously with colder neutrons which interacted with a different phase of the magnetic field. Thus two blurring effects on fringes occur in the dynamic imaging with polarized neutrons due to limited wavelength resolution: the first is due to the increase of the spin precession with neutron wavelength; the second is due to the fact that slower neutrons corresponding to a particular phase of the dynamic process arrive at the detector plane with some delay compared to the faster neutrons. Figure 14 demonstrates the time difference of the corresponding neutron arrival for various wavelengths. Here the time of arrival of neutrons corresponding to the same phase of the dynamic process is calculated for neutrons of various energies, assuming that neutrons of 3.45 Å arrive at time t = 0. The difference in arrival time between 1 and 6 Å neutrons is ∼200 μs for the 170 mm distance between the coil and the detector used in our experiment. That implies that there is about ∼1-2 μs un-sharpness for 1% wavelength resolution in that energy range. To demonstrate the extent of this dynamic blurring by the polychromatic neutron beam we have calculated the images corresponding to the same phases of ac cycle in the coil as shown in figure 13 for the neutrons of three different wavelengths, figures 15-17. For 2.025 Å neutrons the minimum number of fringes is observed at the phase corresponding to the phase marked as (a) in figure 10, while it is close to the phase (b) for 3.45 Å neutrons and phase (c) for neutrons of 4.875 Å wavelength. Therefore if a wide spectrum neutron beam is used for imaging of dynamic magnetic fields multiple phases of the dynamic field overlap in the resulting images. Consequently, at a continuous source a compromise needs to be found between the reduced neutron statistics and the degree of beam monochromatization. The possibility to image with intrinsic energy resolution over a broad wavelength band at a pulsed neutron source is probably the best option for imaging the dynamic magnetic fields as it also provides the basis for accurate data analysis. However, since the frequency of neutron sources is fixed to a certain value and the frequency of the dynamic process of interest can be different, for each neutron pulse the dynamic process will be probed by a random correlation of process phase and neutron energy. With multiple neutron pulses acquired, all phases can be imaged at all neutron energies available in a particular beam. However, care has to be taken to avoid some harmonics between the two frequencies involved. Analyses of the probed magnetic field can then be performed wavelength by wavelength. These results can subsequently be combined for the accurate quantification of the magnetic field values without the need for the absolute field calibration required for a single monochromatic measurement.
To demonstrate the potential of quantitative analysis of magnetic fields measured at a pulsed neutron beamline we simulated the data set which could be obtained in such experiment with the same 15-loop coil sample as in our continuous source experiment. A static magnetic field is simulated in that case, although similar methods will be applicable for the dynamic studies, where each phase is analyzed separately in the same way as in case of a corresponding static field. At a pulsed source a set of images, each measured with a certain neutron energy can be acquired. Thus the energy dependence of the measured intensity is obtained for each pixel of the resulting data set. In the simplest case, when the magnetic field distribution has only one component, which is perpendicular to the initial beam polarization vector, the spin precession has a sinusoidal dependence on the neutron wavelength as seen from equations (4) and (5). The period of that fluctuation can be used to calculate the value of the measured magnetic field. As mentioned earlier, this field quantification technique does not   Thus images corresponding to significantly different phases of the magnetic field overlap at the detector plane. Hence, a beam with relatively narrow spectrum needs to be used to measure dynamic magnetic fields accurately. More fringes are also seen for the longer wavelength neutrons due to the fact that these neutrons spend more time moving through the magnetic field. The numbers in image (b) indicate the areas where the neutron transmission spectra of figure 18 were obtained. require counting of fringes from the zero-field area and provides the field value for a given pixel of the image. However, in a general experiment with a three-dimensional magnetic field distribution the precession of neutron spin is more complicated and a more detailed analysis has to be performed with measurements of different orientation of spin analyzer and polarizer. Figure 18 demonstrates the calculated intensity variation with neutron wavelengths for several areas in and around the coil for 5 and 10 A dc currents. In the center of the coil (area 1 of figure 16(b)), where the magnetic field is predominantly directed along the X-axis, the spin precession occurs mostly in the XY-plane resulting in a nearly sinusoidal variation of the measured neutron intensity in that area. Same is valid in area 2, where the field is not as strong as in area 1, the variation of intensity is with a larger period with respect to the neutron wavelength. However, in areas 3 and 4, the presence of considerable magnetic field components in the YZ-and XZ-planes results in a complicated pattern. To reconstruct the three-dimensional magnetic field distribution quantitatively more measurements are required, as explained earlier. However, the field pattern can still be somewhat visualized in a single measurement, as seen in simulated images in figures 15-17.

Summary and conclusion
The remote imaging of static magnetic fields demonstrated in previous studies [5][6][7][8][9][10], has been extended in this paper to time-varying periodic fields, which provides a powerful new tool for dynamic applications where probing the magnetic field can otherwise be difficult if not impossible. The ability of neutrons to penetrate some materials (e.g., metals) and at the same time to provide quantified information on the distribution of the magnetic field within (or even behind) these materials, enables unique studies to be carried out-for example, the realignment of magnetic domains under the influence of external magnetic fields, measurement of the magnetic fields within complicated structures, and so on. It is shown that careful planning of the experimental setup is required for accurate quantification of the magnetic field distribution. In particular, a compromise between the image statistics (or integration time) and beam monochromatization-factors which determine the accuracy of the reconstructed distribution-must be found for any specific experiment at a continuous source. Selection of a particular neutron energy range is required for the measurement of a given range of magnetic field values, because the spatial resolution of polarized neutron imaging sets the limits on the distance between the fringes which are still resolvable. Thus stronger magnetic fields may be better imaged with neutrons having shorter wavelengths, while weaker magnetic fields require imaging with longer wavelengths. The limits of the quantification of spin precession are defined by the minimum distance between the fringes which can still be resolved by a particular experimental setup. In that respect, providing a more compact analyzer device will be very helpful, allowing reduction in the distance between the sample and the detector area, thus reducing image blurring due to the neutron beam divergence. In principle, spatial resolutions down to ∼10 μm can be achieved using the latest high resolution neutron imaging detectors [27,31]. Thus fringes as close as 20 μm can be resolved, provided adequate neutron statistics can be achieved and the image blurring remains below 10 μm.
The limit on the frequency of periodically changing magnetic fields in neutron imaging experiments is determined by the energy resolution of the beam (ΔE/E) and the timing resolution of current neutron imaging detectors-which can be of the order of sub-microsecond [21][22][23]. The limited energy resolution can cause overlap of adjacent phases of a dynamic process due to the fact that neutrons have different travel times between sample and detector, including the finite length of the spin analyzer. Faster neutrons representing a particular phase of the B field will arrive at the detector simultaneously with slower neutrons representing a preceding phase of the B field, and that will constrain the frequency of the dynamic process, which can be studied at a particular beam facility. At a reactor source, the energy resolution is usually defined by the mosaicity of the double crystal monochromator being used, leading typically to resolutions of ΔE/E ∼ 3%-a compromise between the neutron flux and energy resolution. At pulsed sources, energy resolution is limited by the width of the neutron pulse, typically 10-100 μs for thermal and cold neutrons at short pulse sources. Therefore, processes as fast as 100 kHz can still be studied with the present reactor-based and pulsed experimental facilities, using compact spin analyzers which minimize the distance between the sample and the detector.
The potential to quasi-simultaneously measure images at multiple energies makes pulsed sources very attractive for imaging of magnetic fields with an expanded dynamic range. However, accurate quantitative reconstruction algorithms still need to be established, especially for three-dimensional reconstruction problems. Current developments at pulsed neutron sources [24] are very promising, and represent a significant addition to methods already being used at continuous neutron sources.
Moreover, simulation and modeling of images for a specific experimental setup and magnetic field distribution should prove very beneficial for the optimization of a particular experiment, especially taking into account relatively long integration times required for accurate quantification with accompanying high spatial resolution. 10 A dc current. The nearly sinusoidal variation of measured neutron intensity is observed in the middle and above the coil (areas 1 and 2), where the field is predominantly perpendicular to the neutron spin polarization. There the spin precession is almost exclusively in the XY plane. However, in areas 3 and 4 the presence of the magnetic field fringes leads to a three-dimensional precession of the neutron spin, resulting in a deviation of transmission spectra from a purely sinusoidal curve.
Initially the vector ⃑ B l ( ) is aligned with axis Z″ by two rotations of the system of coordinates: first counterclockwise by the angle φ −   After that precession the system of coordinates X‴Y‴Z‴ is transferred back to XYZ by three inverse rotations by angles −η, θ −