REVIEW OF SCIENTIFIC INSTRUMENTS 84, 105113 (2013)

Neutron spin evolution through broadband current sheet spin flippers P. Stonaha,1 J. Hendrie,2 W. T. Lee,3 and Roger Pynn1,4 1 Indiana University Center for the Exploration of Energy and Matter, Indiana University, Bloomington, Indiana 47405, USA 2 Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA 3 Australia Nuclear Science and Technology Organisation, Lucas Heights, NSW 2234, Australia 4 Neutron Science Directorate, Oak Ridge National Lab, Oak Ridge, Tennessee 37831, USA

(Received 18 July 2013; accepted 4 October 2013; published online 25 October 2013; publisher error corrected 30 October 2013) Controlled manipulation of neutron spin is a critical tool for many neutron scattering techniques. We have constructed current-sheet, neutron spin flippers for use in Spin Echo Scattering Angle Measurement (SESAME) that comprise pairs of open-faced solenoids which introduce an abrupt field reversal at a shared boundary. The magnetic fields generated by the coils have been mapped and compared with both an analytical approximation and a numerical boundary integral calculation. The agreement is generally good, allowing the former method to be used for rapid calculations of the Larmor phase acquired by a neutron passing through the flipper. The evolution of the neutron spin through the current sheets inside the flipper is calculated for various geometries of the current-carrying conductors, including different wire shapes, arrangements, and common imperfections. The flipping efficiency is found to be sensitive to gaps between wires and between current sheets. SESAME requires flippers with high fields and flipping planes inclined to the neutron beam. To avoid substantial neutron depolarization, such flippers require an interdigitated arrangement of wires. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4826086] INTRODUCTION

Elastic neutron scattering has been used for many decades to interrogate the structure of materials and to complement information obtained from other structural probes. Traditionally, elastic neutron scattering produces information over a range of length scales between ∼0.1 nm and a few tens of nanometers. The information is obtained in terms of a structure factor, S(Q), which is proportional to the fraction of neutrons that suffer a wavevector change equal to Q during the scattering process. Over the past decade, a new interferometric method has been introduced which allows neutron scattering to present information as a real-space correlation function rather than as the function S(Q) in wavevector space. This method, dubbed Spin Echo Scattering Angle Measurement (SESAME), has a number of additional features that make it useful in particular applications.1–8 For example, it allows mesoscopic length scales up to 10 μm or more to be probed and it automatically accounts for multiple scattering of neutrons, making it a useful technique for studying strongly scattering samples.9–11 SESAME operates by using suitably designed magnetic fields to encode the scattering angle of each polarized neutron into the Larmor phase accumulated by the neutron beam.12–14 The arrangement of magnetic fields before the scattering sample produces a Larmor phase for each neutron that depends only on its direction of travel. This is accomplished using devices that produce a relative inversion of the neutron spin and magnetic field along an interface that is inclined to the neutron beam.15 Magnetic field regions of identical shape and magnitude placed after the sample produce the same Larmor phase if no scattering occurs.

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Another device placed near the sample produces a similar inversion, but with an interface that is perpendicular to the neutron beam. This ensures that the Larmor phases on either half of the instrument have opposite signs and cancel one another, producing a spin echo—a neutron beam whose polarization is the same as the incident beam. When a scattering sample is present to deflect the neutrons, the cancellation of Larmor phase is not assured, and the echo polarization is a measurement of the beam deflection. The amplitude of the echo polarization turns out to be given by the cosine Fourier transform of the structure factor S(Q), which is simply a projection of the usual Debye density-density correlation function.1, 7, 10, 16–18 Each of these devices is called a π -flipper. The devices which ensure the Larmor coding are “inclined π -flippers,” while the latter, for the purposes of this paper, is referred to as a “non-inclined π -flipper.” The optimization of the π flippers for SESAME is the subject of this contribution. The π -flippers we consider in this paper comprise two adjacent solenoids, each wound with thin wire. Polarized neutrons are made to pass through the shared boundary; accordingly, the “current sheet” that creates the non-adiabatic field transition is actually made from two planar sheets of parallel wires that are in close proximity to one another. Since neutron spins are unable to follow the abrupt change in field direction, they are flipped with respect to the magnetic field. For applications to SESAME, it is important that π flippers do not cause excessive depolarization or introduce aberrations of the Larmor phase. The complete apparatus comprises 4 inclined π -flippers and one non-inclined π flipper, so depolarization of more than a few percent in each device makes the overall instrument less sensitive. We will

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show that a simple model provides an accurate estimate of the fields within the π -flippers and that this model agrees both with measurement and detailed finite-element simulations. This model allows us to easily calculate the Larmor phase generated by the flipper. To calculate the neutron beam depolarization caused by the flipper, we solve the Bloch equation for the evolution of a neutron spin in the neighborhood of the field reversal for both idealized and realistically shaped conductors. Based on these considerations, we present an optimal winding scheme for the solenoids that minimizes unwanted neutron beam depolarization.

spin precession is not able to follow the changing magnetic field. This can be used to select one component of a neutron spin by transmitting the neutron through an abrupt 90◦ rotation of the magnetic field. It also means that a fast reversal of the local magnetic fields can produce a relative “flip” of the neutron spin and magnetic field. When calculating the evolution of a neutron spin, the  greater the maximum value of κ(r ) throughout a magnetic field transition, the smaller the requisite step size in the RKA. In practice, a suitable maximum step size can be found heuristically by reducing the step until the calculated evolution of the spin stabilizes. Because of compounding errors in the

NEUTRON SPIN IN MAGNETIC FIELDS

RKA, we found our numerical solution to S(r ) in a constant magnetic field diverged from the analytic solution after ∼200 000 steps. The region of space over which the RKA can be used to predict a spin’s evolution must be limited to satisfy the constraints on both the maximum step size and this maximum number of steps. The measured polarization of a neutron beam is the pro-

 

 

The evolution of a classical neutron spin, S(r ), moving  

in a magnetic field, B(r ), follows the Bloch equation:  

  dS(r ) γ  = (S(r ) × B(r )), dr v

(1)

where v is the neutron velocity and γ ≈ 1.8325 × 108 s−1 T−1  is the neutron gyromagnetic ratio. The coordinate r is a generalized spatial coordinate describing the neutron’s trajectory. For a neutron moving at a speed v through a constant  

magnetic field B(r ) = Bz zˆ , Eq. (1) is satisfied by the solution,      |r | − η sin(χ )xˆ S(r ) = cos γ Bz v    |r | + sin γ Bz − η sin(χ )yˆ + cos(χ )ˆz, v ˆ y, ˆ zˆ } factors are unit vectors along the three where the {x, Cartesian axes, the constant χ is the opening angle between  

S(r ) and zˆ , and the phase factor η describes the spin’s initial orientation. As the neutron moves through the field, its spin will precess about zˆ at the Larmor precession frequency ωL = γ Bz .19 The Larmor phase ϕ L is simply the total phase acquired by the precessing neutron spin  as it passes through the region of magnetic field, i.e., ϕL = γv Bz dl where the integral is taken along the neutron path. Equation (1) is, in general, not analytically solvable. It can be solved numerically using a Runge-Kutta algorithm (RKA)20 or other numerical routine. To obtain accurate results using an RKA, the magnetic field must be sampled with an appropriate step size. We define B (r ) and S (r ) to represent the orientations of the magnetic field and spin moment, respectively, with respect to a fixed laboratory reference frame. For a neutron spin propagating through a field of varying direction, the correct step size can be quantified using the dimensionless adiabaticity parameter, |d B (r )/dr| 

κ(r ) = 

|d S (r )/dr| 

, 

ˆ r ) is the unit vector pointed along B(r ). Two cases where B(  can be considered here. When κ(r )  1, the rate of change of the orientation of the magnetic field is slow compared to the Larmor frequency, and the axis about which the neutron spin precesses is able to adiabatically follow the changing mag netic field. If it is not the case that κ(r )  1, then the axis of

 

jection of the classical vector S(r ) onto the field direction of the polarization analyzer. In SESAME, the spin is forced to adiabatically follow the magnetic field onto the polarization analyzer, and the average polarization at the end of the instrument can be written P = cos χ , with χ as defined above. Using the RKA, the expectation value of the polarization, P, can be calculated for any single path through a field transition  ˆ r ) · S(r ). as P  = B( CONSTRUCTION, MEASUREMENT, AND MODELING OF MAGNETIC FIELDS

The π -flippers we have constructed for SESAME comprise resistive wire solenoids yoked in high permeability μmetal. The wire is wound so that one face of each solenoid presents an opening for neutrons (c.f. Figure 1(a)). Pairs of solenoids are abutted on the sides opposite the openings. Current flows through the solenoids so that the magnetic fields inside them are in opposite directions, denoted +z and −z (c.f. Figure 1(b) for coordinate axes). Each π -flipper solenoid is tightly wound with 100 turns of 1.39 mm diameter aluminum wire with a thin electrically insulating coating. The opening in each solenoid is made by winding each coil with an extra length of wire for several turns near the coil’s midplane. The extra length of each turn is then folded back against the face of the coil above or below (in the z-direction) its original location (c.f. Figure 1(c)). The extent of the opening in the z-direction is G = 58.9 mm. The μ-metal box extends 140 mm in both the y- and z-directions and has length L = 140 mm. The wires are wrapped with no space between the first and last turns and the adjacent pieces of μ-metal. The folded wires extend ∼90 mm in each direction from the y = 0 plane. The magnetic fields from an open-faced solenoid can be easily approximated. Detailed finite element calculations show that a closed solenoid that is yoked in μ-metal produces a uniform magnetic field B0 at all points inside of the solenoid to within the noise of the calculation, with B0 {Tesla} = 4π nj/107 , where n is the turn density (m−1 ) and j is the

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FIG. 1. (a) Image of the conductors in a single coil of a π -flipper. The arrows indicate the direction of the current when the magnetic field is directed upward. (b) Schematic cartoon of the arrangement of conductors in the y = 0 plane. Dark gray regions indicate current flowing into the page; thatched regions have current flowing out of the page. Arrows indicate the direction and shape of the magnetic fields. The x = 0 plane is defined as the midplane between the dark gray current sheets. (c) Photograph of a constructed π -flipper.

current (A). Thus, a yoked closed solenoid mimics an infinite solenoid and can be modeled as such. The opening in each solenoid can be modeled as an infinitesimally thin current sheet of width G with current flowing in the opposite direction to that in the solenoid. The current sheets above and below the opening (that is, outside of the μ-metal) can be modeled similarly, albeit with width G/2. In this approximation, in which the current sheets have vanishing thickness, there is an abrupt field transition across each current sheet. With finite-thickness wires, we expect the transition to occur more or less linearly within the thickness of the current sheet. We refer to model depicted in Figure 1(b) as the Infinite Solenoid plus Current Sheet (ISCUS) model. To check the validity of the approximations made in the ISCUS model, the magnetic fields of an open-faced solenoid were mapped using a three-axis Hall probe. Measurements of the magnetic field were made by stepping the Hall probe into a coil at a fixed (y, z) coordinate. At each location, a background measurement was taken with zero current in the coil. The coil was then energized to 5 A and the measurement was repeated. By repeatedly measuring the magnetic field at a single position, it was found that this method gave stable and reproducible results. Measurements of the minor hysteresis loop from −10 → +10 A reveal that magnetization in the μ of 0. The magnetic field throughout the current sheets can be approximated as    B(r )x≈0 ≈ Bx (z)xˆ + B0 x/ x zˆ , (4) where we assume x  L so that the Bx field is independent of x throughout the field transition. Expanding each component of Eq. (1), we can obtain a system of three coupled first order differential equations, which is solved in Appendix B. The average polarization for a neutron beam of height h0 is sin(ω3 ) P3 ≈ , ω3  (5) γ mn Gh0 nj xλ, ω3 = c3 (L2 + G2 ) 5h





FIG. 6. A depiction of the changing relative orientations of S and B across 

in infinitesimally thin current sheet. Initially, the spin S is aligned with the 

Values for P1 are close to unity for reasonable dimensions of the π -flipper. For the design parameters discussed in “Construction, measurement, and modeling of magnetic fields” section, the polarization of a 20 mm tall neutron beam after passing through the transition is 99.93%.



field B i . The inversion of Bz causes the spin to begin precessing around −B f after passing through the current sheet.

where c3 = 250 A1/2 (T m)−1/2 . For the same operating parameters as in “Construction, measurement, and modeling of magnetic fields” section and with x = 1/n (i.e., a non-inclined π -flipper), the polarization after passing through this field transition is P3 ≈ 99.84% and 99.67% at 0.5 nm and 1 nm neutron wavelengths,

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FIG. 7. Final polarization of a neutron after passing through a π -flipper with finite-thickness current sheets as calculated using the RKA with RADIAcalculated fields (lines) and Eq. (5) (circles) along z = 10 mm. The full size plot is calculated with 5 A in the π -flipper. The inset is with 20 A, showing the oscillations of Eq. (5) about the RKA solution.

respectively. For an inclined π -flipper in which x → x/sin(30◦ ), the polarization is P3 ≈ 99.67% and 99.34% at 0.5 nm and 1 nm wavelengths, respectively. The result from Eq. (5) is plotted against the solution of the RKA using the RADIA-calculated fields in Figure 7. The oscillations of the solution to the RKA arise from the ex r ) at the end of act orientation of Sy (r ) with respect to B( the field transition, which is averaged out in the derivation of Eq. (5). It is apparent that π -flippers with these three types of interfaces do not produce any substantial depolarization of a transmitted neutron beam. In reality, the interfaces are not so ideal but are characterized by a pair of wire arrays, which introduce additional potential for neutron beam depolarization. WIRE ARRANGEMENT

Field profiles through shaped conductors were obtained using the RADIA model. We consider three scenarios; one in which the wires have a circular cross section and the neighboring wire sheets have wires positioned with mirror symmetry (SYM, c.f. Figure 8(a)), one in which the wires have a circular cross section and are arranged in an interdigitated

FIG. 8. A depiction the wire shapes and arrangements considered in this paper. (a) Round symmetric (SYM), (b) round interdigitated (INT), and (c) square (SQR). The horizontal lines indicate the 8 slices in the y = 0 plane through which the spin evolution was calculated and averaged in Tables I and II. The parameters d, x, and 1/n are discussed in the text.

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pattern (INT, Figure 8(b)), and one in which the wires have a square cross-section and are arranged with mirror symmetry (SQR, Figure 8(c)). Both the symmetric, round wires and the square wires have rectangular packing, while the interdigitated round wires have hexagonal packing. We use the RKA to calculate the depolarizing effect of the wire shape and to deduce the polarization P of a neutron spin after passing through the wire screen. Using the RADIA model, the fields from the opening in the coils are combined with those from the individual wires themselves, producing a comprehensive model of a SESAME π -flipper. In order to obtain a representation of the effect across the entire beam, we average a series of such calculations over one wire period (in z). In each calculation, the neutron is assumed to be traveling parallel to the x-axis, with S initially aligned with B. We consider the possibility of imperfections in the winding of the coils resulting in regular spacing between neighboring wires. Specifically, the diameters of round wires in Figure 8 were made progressively smaller while the wire period was kept the same. This is a common manufacturing error that can occur if the aluminum wire is stretched and made thinner as it is wound. We also consider the possibility of small space between the two coils. These two types of imperfections are shown in Figure 8. The wire period 1/n is kept constant in all calculations. The examined values of √ the intercoil-space d are d = d , 0, or −d , where d = x( 3/2 − 1) is the distance the interdigitated wires can be compressed to force a hexagonal closed packed arrangement. When x = 1/n = 1.39 mm, one finds d ≈ 187 μm. The results of these calculations near z = 0 are shown in Table I for a non-inclined π -flipper at 5 A and a 30◦ inclined π -flipper at 5 and 15 A. The polarization was calculated at neutron wavelengths of 0.5 nm and 1 nm (1 nm results in parenthesis). These calculations are repeated for a neutron beam near z = 10 mm in Table II. DISCUSSION

For a uniform finite thickness current sheet, which is a realistic approximation to an actual π -flipper, the phase of a spin after passing through the current sheet scales with the square roots of n, j, λ, and x. There are two striking results here. The first is that increasing the current in a π -flipper will always result in a decrease in polarization. The second is that because n and x appear as a product in ω3 , the polarization is insensitive to wire size for the non-inclined π -flippers. The consideration of shaped wires adds yet another layer of complexity. The source of all depolarization as the neutrons pass through the current screens in the π -flipper is the Bx field. Because both current sheets of the π -flipper have current flowing in the same direction near the central plane, the Bx field from the wires arranged in an interdigitated pattern tends to cancel. Typical operation of a SESAME apparatus requires one non-inclined π -flipper running at 5 A and four inclined π flippers, each running at anywhere from 3 to 20 A. The non-inclined π -flipper performs worst with mirror symmetric round wires, but deviation from unit flipping efficiency is no more than ∼4% for any neutron of wavelength 99%) in almost all cases we have investigated. On the other hand, far more care is needed for inclined π -flippers operated at high fields. Four of these devices are required for the implementation of SESAME, and they can only be made reasonably efficient at large fields if the wire screens within the device are arranged so that wires of neighboring screens are interdigitated. ACKNOWLEDGMENTS

This work was supported by the US Department of Energy through its Office of Basic Energy Sciences, Division

Rev. Sci. Instrum. 84, 105113 (2013)

of Material Science and Engineering (Grant No. DE-FG0209ER46279).

APPENDIX A: MAGNETIC FIELD IN THE ISCUS MODEL

The magnetic field Bx (x = 0, z) in the ISCUS model is Bx (z)|x=0

nj G ≈ c0 4 − 3





 1 −α z L2 + G2

  3L2 − G2 − β z3 , (L2 + G2 )3

(A1)

with c0 = 6.25 × 105 A(Tm)−1 . The above equation for Bx (x, z), truncated at third order, is an expansion of the sum of the six tanh −1 () functions that describe the magnetic field perpendicular to the six current sheets in the model of the π -flipper. Two current sheets are in the plane of the face of the “infinite” solenoid at x = ±L/2 with current flowing in the opposite direction to that in the solenoid, and the other four “outer sheets” are above and below the openings at x = ±( L2 + n1 ). The former sheets produce the leading terms in the z and z3 terms, while the latter sheets lead to the α and β terms, which are two orders of magnitude smaller than the leading terms. The magnitude of α and β decrease with increasing distance of the outer sheets from the z = 0 plane of the prism. Equation (A1) is obtained in the limit that the length of all of the current sheets approaches infinity.

APPENDIX B: SOLUTION TO THE BLOCH EQUATION

Equation (1) can be written as three coupled differential equations. Plugging Eq. (4) into this system and differentiating twice more, we come to the expression    g2      . Sy (r ) = −a 2 σ 2 3Sy (r )|r | + Sy (r ) |r |2 + 2 σ (B1) Here we have made the substitutions a = γ /v, σ = B0 / x, and g = Bx (x, z). The prime ( ) indicates derivation with respect to r. Like Eq. (1), Eq. (B1) is not generally solvable. In some special cases, the solution to (B1) can be reduced to a Parabolic Cylinder Function.23 To continue, we solve Eq. (B1) near the boundary where we can make the approximation x → x. In this region, we take the limit Bx /B0  x/ x. The last term in (B1) falls out, giving Sy (r ) = −a 2 σ 2 (3Sy (r )|r | + Sy (r ) |r |2 ) 









(B2)

The range over which the approximation used in (B2) fails is |x|  10 μm. The phase acquired by a neutron spin over this range is extremely small – less than 200 μrad for a 0.5 nm neutron – and so the approximation that Bx /B0  x/ x near x = 0 is justified.

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Using the exact initial conditions,   Sy (r )− x = 0,   Sy (r ) − x = 0,  ag  , Sy (r ) − x = − ρ x

(B3)

we come to the result g(1 + i)   Sy (r ) = exp(−2ib2 (|r |2 + x 2 )) 4 xρσ × [(1 + i)(exp(4ib2 |r |2 ) − exp(4ib2 x 2 )) √ + 2i xb π exp(2ib2 x 2 ) 

× (exp(4ib2 |r |2 )(Erf ((i + 1)b|r |) 



+ Erf ((i + 1)b x)) + Erf i((i + 1)b|r |) 

+ Erf i((i + 1)b x))].

(B4)

In Eqs. (B3) and (B4), we have used √ ρ = 1 + (Bx (x, z)/B0 )2 , b = |aσ |/2, and the imaginary Error function Erf i(ϒ) = √−iErf (iϒ). The second term in Eq. (B4), the one with the π factor, is the dominant term – especially at the endpoints, where the competing term in parentheses is zero. For typical operating parameters with √ 2 cold neutrons, the error functions evaluate to ∼ 2 . At the  

end of the field transition, S(r ) will be tracing a cone with opening angle √ 2gb 2π  −1 χ3 = sin (Sy (r ))| x ≈ ρσ

2π γ x ≈ Bx (z) . (B5) B0 v 1 R.

Andersson et al., “Analysis of spin-echo small-angle neutron scattering measurements,” J. Appl. Crystallogr. 41(5), 868–885 (2008). 2 R. Ashkar et al., “Spin-echo resolved grazing incidence scattering (SERGIS) at pulsed and CW neutron sources,” J. Phys.: Conf. Ser. 251(1), 012066 (2010). 3 W. Bouwman et al., “Spin-echo methods for SANS and neutron reflectometry,” Physica B 357(1–2), 66–72 (2005).

G. Bouwman et al., “Real-space neutron scattering methods,” Nucl. Instrum. Methods Phys. Res. A 586(1), 9–14 (2008). 5 R. Pynn et al., “Neutron spin echo scattering angle measurement (SESAME),” Rev. Sci. Instrum. 76(5), 053902 (2005). 6 M. Rekveldt, “Novel SANS instrument using Neutron Spin Echo,” Nucl. Instrum. Methods Phys. Res. B 114(3–4), 366–370 (1996). 7 M. Rekveldt et al., “Elastic neutron scattering measurements using Larmor precession of polarized neutrons,” in Neutron Spin Echo Spectroscopy, edited by F. Mezei, C. Pappas, and T. Gutberlet (Springer, Berlin, 2003), pp. 87–99. 8 M. T. Rekveldt, Spin-echo small-angle neutron scattering in reflectometry, J. Appl. Cryst. 36, 1301–1306 (2003). 9 W. G. Bouwman et al., “First quantitative test of spin-echo small-angle neutron scattering,” Appl. Phys. A 74, s115–s117 (2002). 10 T. Kruglov et al., “Structure of hard-sphere colloid observed in real space by spin-echo small-angle neutron scattering,” Physica B 357(3–4), 452– 455 (2005). 11 C. Rehm et al., “DCD USANS and SESANS: a comparison of two neutron scattering techniques applicable for the study of large-scale structures,” J. Appl. Crystallogr. 46(2), 354–364 (2013). 12 M. T. Rekveldt et al., “Larmor precession applications: magnetised foils as spin flippers in spin-echo SANS with varying wavelength,” Physica B 335(1–4), 164–168 (2003). 13 O. Uca et al., “An analysis of magnetic field inhomogeneities in a spin-echo small-angle neutron scattering instrument,” Physica B 276–278, 136–137 (2000). 14 O. Uca, W. G. Bouwman, and M. Theo Rekveldt, “Magnetic design of a spin-echo small-angle neutron-scattering instrument,” Nucl. Instrum. Methods Phys. Res. A 496(2–3), 437–445 (2003). 15 R. Pynn et al., “Birefringent neutron prisms for spin echo scattering angle measurement,” Physica B 404(17), 2582–2584 (2009). 16 T. Keller et al., “Neutron resonance spin echo — triple axis spectrometry (NRSE-TAS),” in Neutron Spin Echo Spectroscopy, edited by F. Mezei, C. Pappas, and T. Gutberlet (Springer, Berlin, 2003), pp. 74–86. 17 C.-Y. Shew and W.-R. Chen “Elucidation of spin echo small angle neutron scattering correlation functions through model studies,” J. Chem. Phys. 136(6), 064506 (2012). 18 L. Xin et al., “Contrast variation in spin-echo small angle neutron scattering,” J. Phys.: Condens. Matter. 24(6), 064115 (2012). 19 R. Golub, R. Gahler, and T. Keller, “A plane wave approach to particle beam magnetic resonance,” Am. J. Phys. 62(9), 779–788 (1994). 20 P. A. Seeger and L. L. Daemen, “Numerical solution of Bloch’s equation for neutron spin precession,” Nucl. Instrum. Methods Phys. Res. A 457(1– 2), 338–346.(2001). 21 O. Chubar, P. Elleaume, and J. Chavanne, “A three-dimensional magnetostatics computer code for insertion devices,” J. Synchrotron Radiat. 5(3), 481–484 (1998). 22 A. Serebrov et al., “New method for precise determination of neutron beam polarization,” Nucl. Instrum. Methods Phys. Res. A 357(2), 503–510 (1995). 23 R. T. Robiscoe, “A spin flip problem,” Am. J. Phys. 39(2), 146–150 (1971).

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Neutron spin evolution through broadband current sheet spin flippers.

Controlled manipulation of neutron spin is a critical tool for many neutron scattering techniques. We have constructed current-sheet, neutron spin fli...
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