Concepts and Engineering Aspects of a Neutron Resonance Spin-Echo Spectrometer for the National Institute of Standards and Technology Center for Neutron Research

Following a brief introduction, the Neutron Resonance Spin-Echo (NRSE) principle is discussed classically in Sec. 2. In Sec. 3, two idealized 4-coil NRSE spectrometers are discussed (one using single π-flipper coil units and one using paired “bootstrap” coils); some idealized (exact π-flip) expressions are given for the spin-echo signal and some theoretical limitations are discussed. A more quantum mechanical discussion of NRSE is presented in Sec. 4 and additional theory related to the spin-echo signal, including wavelength-dependence, is given is Sec. 5. Factors affecting the instrumental resolution are discussed in Sec. 6. In Sec. 7, a variety of engineering issues are assessed in the context of challenging performance goals for a NIST Center for Neutron Research (NCNR) NRSE spectrometer. In Sec. 8, some Monte Carlo simulations are presented that examine the combined influences of spectrometer imperfections on the NRSE signal. These are compared with analytical predictions developed in previous sections. In Sec. 9, possible alternatives for a NCNR NRSE spectrometer configuration are discussed together with a preliminary assessment of the spectrometer neutron guide requirements. A summary of some of the useful formulas is given in Appendix A.


Introduction
Neutron Resonance Spin-Echo (NRSE) [1,2] is an alternative to the conventional Neutron Spin-Echo (NSE) technique [3], whereby the long solenoids of the latter are replaced by r.f. spin-flippers separated by regions in which there is ideally no magnetic field. For this reason, NRSE is occasionally referred to as "Zero-Field Spin-Echo". Neutron Spin-Echo spectrometers have the distinguishing characteristic of being able to resolve neutron scattering energy exchanges that are much smaller than the energy bandwidth of the incident neutron beam. This contrasts with conventional time-of-flight spectrometers, where the minimum time uncertainty is limited by the incident pulse duration and the associated velocity spread of the incident beam. Some of the important issues for high-resolution NRSE spectrometer design are explored in the following sections.

Classical Principle of Operation
In a common NRSE configuration, four short resonant r.f. flipper coils replace the static field boundaries of the classical NSE spectrometer and the intervening space has zero magnetic field. The r.f. fields in the first and second coils must be phased-locked and in the third and fourth coils. The phase of the r.f. field at the times of neutron passage through the coils acts effectively as a neutron clock (as opposed to the number of Larmor precessions performed in the solenoids of a conventional NSE instrument). In the following illustration, we adopt the coordinate system used by Gähler and Golub in Ref. [2], which differs from the one used in Ref. [1].
With reference to Fig. 1 (a), consider an incident monochromatic beam of neutrons of velocity v n , traveling along the y-axis, initially polarized parallel to the x-axis. In the flipper coil, a static magnetic field of magnitude B 0 is applied in the z direction and an oscillating r.f. field is applied in a plane perpendicular to B 0 (i.e., in the xy plane). We use the notation l π to define the length (in the beam direction) of the region of intersection of the static field region (length l B0 ) and the r.f. field region (length l rf ), i.e., , B rf l l l π = ∩ (1) since l B0 and l rf cannot be identical. Therefore, if the static field region completely encloses the r.f. region, l π = l rf and if the r.f. region completely encloses the static field region l π = l B0 . Note that l π is distinct from the length of the coil, which is a useful parameter in some instances and is defined by If one field region completely encloses the other, Eq. (2) may be restated as In r.f. flipper coils, the static field usually completely encloses the r.f. field region. Therefore, when discussing flipper coils we assume, by default, that l π ≡ l rf and l coil ≡ l B0 , as indicated in Fig. 1, and that the r.f. field is always in a region in which there is a perpendicular static field, with the possibility of a short static-only field region either side of the r.f. coil. The following provides a classical illustration of the NRSE principle. It agrees with the quantum mechanical result, provided that the following approximations are valid: 1. |B 0 | >> |B rf | (see Refs. [4,5]). 2. The interacting component of the oscillating field is considered as a purely rotating field. 3. The Zeeman splitting due to B 0 (2µ n B 0 ) << the kinetic energy of the neutron, m n v 2 /2. Therefore, decomposing the oscillating field into two counter-rotating components, as shown in Fig. 2, the resonant component is the one that rotates in the same direction as the Larmor precession induced by the static field, B 0 ; it is described by where the sign of ω rf is chosen appropriately. The present approximation ignores the much weaker interaction of the counter-rotating (non-resonant) component of the r.f. field. We consider first the neutron spin with respect to the static field B 0 . If B 0 >> B rf , the neutron spin may be assumed to precess in the xy plane with Larmor angular frequency 0 0 , n B ω γ = (5) where γ n is the gyromagnetic ratio of the neutron, defined as the ratio of the magnitude of the neutron magnetic moment to the magnitude of its angular momentum, where If the r.f. angular frequency, ω rf , is tuned exactly to the value ω 0 , one component of the r.f. field rotates about the z-axis synchronously with the precessing neutron spin (i.e., is "on resonance" - Fig. 1 (b)). In this case, we can write where t is the time of flight of the neutron across the region l rf . Note that the approximation in Eq. (12) implies that the beam divergence is sufficiently small that the substitution t ≈ l rf /v n can be made. In order to create a π flip of the neutron spin around B rf (such that the neutron spin returns to the xy plane), the magnitude of B rf must be tuned in order to satisfy n rf n rf n n rf n v h B l m l π π γ γ λ ≈ = for small beam divergence (13) Typically, the time of flight through such coils (of several cm in length) is around 50µs (within a factor of a few, depending on the neutron wavelength), therefore B rf is typically a few tenths of a mT. By contrast B 0 may range up to about 25 mT or more, so clearly for the larger values of B 0 the assumption B 0 >> B rf is valid.
At exact resonance, where π-flips return the spin initially in the xy plane back into the xy plane after passage through the combined field region, it is relatively straightforward to visualize the operation. Following Ref. [1], we denote the phase of the neutron spin in the xy plane relative to some fixed origin by ϕ and the phase of the resonant component of B rf with respect to the same origin by ψ. If the initial phase angle of the neutron spin lags the resonant component of B rf by α on entry into the r.f. field (see Fig. 1 (c)), and Eq. (13) is satisfied, then the neutron spin will lead B rf by α in the rotating frame on exiting the field region. Transforming back into the laboratory frame at the coil exit, we add on the phase change of the r.f. field during the neutron flight time through the coil (≡ω 0 l π /v n ) plus the (usually small) additional Larmor precession angle in the xy plane due to the B 0 field-only regions either side of the r.f. coil (≈ω 0 (l B0 -l π )/v n ). Consequently, we find that the neutron spin has changed its xy-plane phase angle in the coil by an amount ≈ 2α +ω 0 l B0 /v n . Thus for an ideal π-flipper, the neutron spin phase change in the coil is governed by the operator , ϕ ψ ψ ϕ where "unprimed" and "primed" refer to the "entrance" and "exit" of the coil respectively. When the coil is tuned for resonance We have chosen the initial polarization direction (along the x-axis) to define ϕ A =0, therefore Eq. (19) On exiting coil B, the total spin phase change is now independent of the initial r.f. phase angle, ψ Α , at the entrance to coil A (which is random for a continuous neutron beam). Similar arguments hold for the third and fourth coils (C and D), other than the signs of the spin phase changes are reversed by applying the static fields in the opposite direction to those in coils A and B, i.e.,  (23) This means that the phase locking of the r.f. fields between coils A and B may be performed independently of that in coils C and D, provided that the frequencies are equal. The spin turn in each arm of the spectrometer is proportional to the neutron time of flight in each arm irrespective of the time of entry. The net spin phase change in the whole spectrometer is therefore where we have substituted v i for v n in Eq. (22) (incident beam) and v f for v n in Eq. (23) (scattered beam) to account for the possibility of neutron speed changes in scattering events. This expression is entirely analogous with that for a conventional NSE spectrometer if the lengths of the precession solenoids, L, of the latter (operating with axial fields of magnitude B 0 ) are associated with the quantities 2( L AB +l Β0 ) or 2(L CD +l Β0 ). In the NRSE configuration L AB +l Β0 and L CD +l Β0 correspond to the separation of the midpoints of the coils in each arm of the spectrometer, with L AB and L CD typically >> l Β0 . The factor "2" implies a factor 2 advantage in NRSE resolution when comparing equivalent B 0 L in both techniques. However, large values of B 0 are usually more easily achieved in the long solenoids of a NSE spectrometer than in the compact r.f. flippers of the NRSE instrument (see for example Sec. 7.3.3.1). A generalization of Eq. (24) for different coil lengths is given in Sec. 3.3 and in Sec. 3.4, where it is shown how a "bootstrap" coil configuration [2] further increases the resolution factor from 2 to 2N, where N is http://dx.doi.org/10.6028/jres.119.005 commonly a small even integer (N ≥ 2). Henceforth, we concentrate on elastic or quasielastic applications of the NRSE technique.

Single Flipper
For polychromatic beams, the coil is tuned for π spin flips for the mean incident neutron velocity, 〈v i 〉 (or mean incident wavelength, 〈λ i 〉) such that, according to Eq. (12), we require It is clear that exact π turns about B rf occur for a unique neutron velocity or wavelength if |B rf | is kept constant 1 . The wavelength-dependence of the precession angle around B rf is referred to henceforth as dispersion. For a general wavelength, λ i , corresponding to a deviation from the mean ∆λ i = λ i -〈λ i 〉, we have: Equation (26) neglects the distribution in neutron flight times through the coil caused by beam divergence, which is typically narrow compared with that caused by the wavelength distribution. The particular case for λ i < 〈λ i 〉 is illustrated in Fig. 3, resulting in a less-than-π precession of the neutron magnetic moment around B rf in the rotating frame. Similarly, an over-rotation occurs for λ i > 〈λ i 〉. Thus, we see that some depolarization occurs due to the velocity spread and the maximum component in the xy plane no longer attains unity. For moderate ∆λ, the depolarization is largely determined by the component of the magnetic moment out of the xy plane, (i.e., the angle ε), however, a (usually) smaller shift in the spin direction within the xy plane also occurs (i.e., the angle χ). For symmetric distributions of λ with respect to the mean, the angle ε is uniformly distributed above and below the xy plane and its magnitude depends on the angle α. It is zero for α = 0 and maximum for α = π/2. In fact, 2 2 2 cos cos sin cos ε α α β = + (27) and 2 2 cos sin cos cos . cos α α β χ ε − = (28) We see from Fig. 3 that for ∆λ → 0, β → π, ε → 0, and therefore χ → 0, as expected.
For a continuous neutron beam with symmetrically-distributed wavelengths the ratio of the polarization "with dispersion" to that ignoring dispersion (or for a purely monochromatic beam, I(λ i ) = δ (λ i )) after passage through the device is therefore 1 It has been proposed to ramp |Brf | for situations where the neutron energy is strongly correlated with its time of emission, for example at pulsed neutron sources. However for continuous sources this is not feasible. http://dx.doi.org/10.6028/jres.119.005 Qualitatively this is the component (or dot product) of the actual unit spin unit vector projected onto the "perfect" spin direction, averaged over all α and over the neutron wavelength spectrum, I(λ i ). We note that α is random over 2π radians for a continuous beam and we may set 〈sin 2 α〉 = 1/2 in Eq. (29). Also using the identity cosβ = 1-2sin 2

Approximation for M Flippers
The above equations apply to a single flipper coil. If there are M identical flippers (e.g. M = 8 for a 4-coil N = 2 bootstrap instrument), and we assume that the neutron spectrum is unmodified through the spectrometer (elastic scattering, negligible absorption etc.), we can make the following approximation, provided that the cumulative spin rotation out of the xy plane remains small: We rewrite Eq. (30) approximately as the product of the M flipper coil efficiencies, prior to averaging over the spectrum, I(λ i ), i.e., The overall average flipping efficiency for the spectrometer is therefore described for the rectangular, triangular, and Gaussian incident spectra by expressions similar to Eqs. (32), (35), and (39), but with the sin 2 replaced by sin 2M , i.e., for the rectangular spectrum: The accuracy of the approximation for M > 1 is demonstrated by some special case Monte Carlo calculations in Sec. 8.2. Table 1 and Table 2 show predicted values of disp ideal M coils

Special Note
Gähler and Golub [2] ignore the (usually small) angle χ and give an expression for 〈cos ε〉 in terms of the root mean square (rms) value of ε, εrms, which is valid for small ε. For these reasons, their expression does not predict exactly the quantum mechanical flipping efficiency. The reader should beware of substituting, for example, FWHM values for δv into Eq. (33) of Ref. [2]. The latter equation is valid for discrete ±δv with respect to the mean, therefore their expression must be averaged over the appropriate velocity distribution, F(vn). For sufficiently small ε, the approximation is 2 2 with the combined effect of M similar π coils approximated by summing the ε's in quadrature [ii], whence

Coil Resonance Width
The coil flipping efficiencies given in the previous section are for optimally-tuned coils (exact resonance ω rf = ω 0 , and exact π-flips for the mean wavelength 〈λ i 〉). They account only for dispersion. For moderate ∆λ/λ, dispersion leads mainly to excursions of the spin vector out of the intended x-y plane accompanied by a usually small rotation of the spin component within the x-y plane. An additional question concerns non-optimal tuning of the coils arising either from (i) systematic differences between the Larmor frequency (ω 0 ) and r.f. frequency (ω rf ) or (ii) that caused by static field inhomogeneity when ω rf = 〈ω 0 〉; i.e., to what tolerance must ω rf match ω 0 ? Alternatively, what is the resonance width? Alvarez and Bloch [6] provided a quantum mechanical result for the flipping efficiency (valid for static field magnitudes that are much larger than the oscillating field magnitude), which (almost) in their original notation is ( ) ( ) where t is the time spent in the oscillating field, µH 1 t equates to β/2 (Eq. (12)), H 1 is the amplitude of the oscillating field ≡ 2B rf = pk rf B , and is the difference between the actual value of the static field and the value required for exact resonance (i.e., when ω 0 =ω rf ). Thus, Eq. (46) can be re-expressed as where it is understood that l π = l rf for the typical flipper coil. For the special case that B rf is tuned to produce exact π flips for the mean wavelength 〈λ i 〉, Eq. (48) becomes (see also Eq. (11)) 2 and the counter-rotating component increasingly plays a role. One manifestation of this is a shift in the resonance frequency as discussed in Ref. [5] (see Sec. 7.3.6.1). Eventually the r.f. flipper cannot function when B 0 /B rf falls below a certain threshold. The full width at half maximum of these resonance curves for a general value of l π is very well fitted by Thus for longer wavelengths and longer coils the resonance sharpens, requiring increased tuning accuracy.
Consequently, the tolerable field inhomogeneity also decreases with increasing l π and λ n .

Influence of π-Flipper Efficiency on Polarization
The term "flipper efficiency" usually excludes spin-independent effects such as scattering or absorption. Thus for a π-flipper of efficiency f, a fraction f of the spin-down component of a beam is converted to spin-up and vice-versa. Conversely, fractions (1-f ) of the spin-down and spin-up components are transmitted with no change of their spin directions. For an incoming beam with spin-up and spin-down intensities 0 I + and 0 I − respectively, the corresponding intensities in the outgoing beam are ( ) Thus, the outgoing beam polarization is just the incoming beam polarization multiplied by the factor (1-2f).

Illustrations of Idealized 4-Coil NRSE Instruments
In the following examples, we illustrate the performance of a 4-coil unit NRSE spectrometer by assuming "perfect" π-flipper coils (Sec. 3.1). In Secs. 6 and 7, we discuss departures from the idealized performance due to the non-ideal nature of the components.

The Perfect π-Flipper Coil
We define the "perfect" π-flipper coil as having the following properties: 1. "Dispersionless" -the exact π flip operation is assumed to be independent of wavelength (i.e., all neutron spins start and finish in the xy plane -see Sec. 2.2). 2. Perfectly uniform and stable applied static field B 0 within the beam passage. 3. Perfectly stable (frequency and magnitude) and sinusoidal r.f. field B rf . 4. Perfect perpendicularity of the static, r.f. fields, and beam direction (⇒ zero divergence beam). 5. Perfectly-defined field boundaries along the beam direction. 6. Zero stray fields or leakage fields in the "zero-field" regions. 7. Perfectly transmitting for neutrons.  Fig. 5. where the origin of the y-axis is chosen to coincide with the entrance to the first π flipper (A). We will assume that the static field magnitude, B 0 , in coils A and B is equal and that the static field magnitude, B 1 , in coils C and D is the same, i.e., and 0 0 When the π-flipper r.f. frequency is on-resonance, we can write 0 0 For elastic and small energy transfer quasielastic scattering (where the detailed balance factor is essentially 1 and the scattering function is symmetrical around zero energy transfer), we have 〈v i 〉 = 〈v f 〉. Therefore, the magnitude of the r.f. field is tuned to create π-flips for the mean incident velocity 〈v i 〉 for all coils. This For elastic and small energy transfer quasielastic scattering (where the detailed balance factor is essentially 1 and the scattering function is symmetrical around zero energy transfer), we have 〈v i 〉 = 〈v f 〉. Therefore, the magnitude of the r.f. field is tuned to create π-flips for the mean incident velocity 〈v i 〉 for all coils. This

A Note About Signs
In Sec. 3.3 and especially in Sec. 3.5 we must account for reversals of the directions of the static fields B 0 from one coil to the next. This is important because the reversed direction of B 0 reverses the direction of the Larmor precession and consequently switches the resonant r.f. field component to the counter-rotating component (that has a different absolute phase angle). This latter situation is simplified mathematically (with no loss of generality) if we assume that the r.f. field oscillates along the x-axis, since the shift of r.f. phase angle that accompanies the change of sign of B 0 amounts only to a flip of the sign of ψ. In the following sections, expressions for the phase changes throughout the spectrometer are written in tabular form, initially with signs that account for general static field directions in the coils.

Beams
The 4-single π flipper arrangement is illustrated in Fig. 5. We use the operator (Eq. (16)) for the neutron spin phase in the coil regions and assume truly zero field in the gaps between the coils (allowing Eq. (19) to be used). Phase locking of the r.f. frequency between coils A and B and between C and D is assumed (allowing expressions of the type (21) to be used), but no phase locking of the r.f. between the two arms of the spectrometer is required (hence ψ C is unrelated to ψ A ). By assuming well-collimated beams (cosθ ≈ 1), we have replaced neutron flight times with expressions of the type l/v n or L/v n where l or L is a dimension along the beam (y) axis. We now construct a table of phases through the spectrometer, applying the above assumptions and signs to account for general static field directions. For example, sgn(B 0 ) = "+" if the static field lies along +z and sgn(B 0 ) = "-" if the static field lies along -z. The result is shown in Table 3. http://dx.doi.org/10.6028/jres.119.005 Table 3. Phases for "perfect" single π coils with zero stray fields between coils showing how the field direction signs for the resonant component of the r.f. field apply (Brf is chosen to oscillate along the x axis).

Location y
Neutron spin phase angle ϕ r.f. field phase ψ Entrance D lA+LAB+lB+LBS+LSC+lC+LCD

Observations
1. The lack of a relation between the r.f. phases in each arm of the spectrometer is immaterial. This is because the r.f. phases, ψ A , and ψ C , on entry to coils A and C cancel on leaving coils B and D respectively. 2. The final spin phase angle of the neutron exiting coil D is independent of the distances between the second coil B and the sample (L BS ) and the sample and the third coil C (L SC ). 3. The final neutron spin phase from Table 4 is: If all coils are identical in length (i.e., l A = l B = l C = l D = l Β0 = l coil = l), then the phase angle of the neutron spin at the exit of coil D reduces to Equation (62)

Bootstrap Configurations
Gähler and Golub [2] appreciated that spin-echo configurations of resonant π-flippers are not limited to single π-flipper unit arrangements. Multiple flippers placed back-to-back with alternating static field directions can replace the single π-flippers at the zero-field region boundaries. Several 4-coil unit spin-echo arrangements are shown in Fig. 6. When more than one π-flipper (N > 1) comprises one unit, the combination is referred to as a "bootstrap coil". The technique was first demonstrated experimentally in Ref. [7]. Note that the static field directions in the second arm mirror-image those in the first arm. Because closed magnetic field loops are produced within the coil unit for even-N, leakage fields outside the coil regions are strongly reduced with respect to odd-N combinations and it has been demonstrated [8] that the field homogeneity within the beam area is improved for N = 2 with respect to N = 1. Furthermore, the (small) leakage fields each side of the even-N bootstrap coil cancel to first order because the Larmor precession that they induce is approximately equal in magnitude but of opposite sign.

Practical Limits to the Value of N
Bootstrap coils with N flippers effectively multiply the spin turn by a factor of N, thereby increasing the resolution of the spectrometer by the same factor N. This is illustrated for N = 2 in Sec. 3.5. However, instrumental non-ideality ultimately limits the maximum practical value of N.
1. The most obvious limitation is that N multiplies the number of coil windings traversed by the neutron beam, multiplying the absorption and scattering by the same factor. 2. The total power dissipation is proportional to N, negatively impacting the already challenging task of heat removal from the coils units.
3. The dispersion of the π-flippers means that increasing non-zero z-components of the spin vectors result as the neutron traverses additional coils. Gähler and Golub show [2] that the expectation values of 〈σ x 〉 and 〈σ z 〉 each contain 2 N terms in sin m (µB rf l/v n ) and/or cos m (µB rf l/v n ) where m runs up to N and v n is the neutron velocity. Because these rapidly-varying functions of velocity lead to depolarization of the beam, Gähler and Golub also show that ∆v n /v n must become increasingly small as N increases to compensate. In view of the compromises imposed by 1, 2, and 3, and the advantages of even-N for stray field suppression, N = 2 is almost universally used in existing NRSE spectrometers.

A 4 "Perfect" Dispersionless N = 2 Bootstrap Coil NRSE with Zero Stray Fields and Well-Collimated Beams
In the bootstrap pair, the main consequence of the field direction reversal mid-way across the coil unit is that the resonant component of the r.f. field switches to the counter-rotating component in the second πflipper. This reverses the sign of ψ and of ω 0 . In order to illustrate features that are likely present in a real bootstrap coil, it is assumed that the transition from one π-flipper of the pair to the other takes place across a small gap l g , which is equal for all coils. This gap is also assumed to be "zero field" (or a region where the stray fields of the adjacent coils exactly cancel). By adopting the procedure outlined in Sec. 3.3 and applying the specific field direction signs indicated in Fig. 6 for N = 2, we obtain the values given in Table  5.

Observations
1. The lack of a relation between the r.f. phases in each arm of the spectrometer is immaterial. 2. The final spin phase angle of the neutron exiting coil D 2 is independent of the distances between the coil B 2 and the sample (L BS ) and between the sample and the coil C 1 (L SC ). 3. The final neutron spin phase from Table 5 is: If the individual π-flippers are identical in length (i.e., l A1 = l A2 = l B1 = l B2 = l C1 = l C2 = l D1 = l D2 = l B0 ) and we consistently use the symbol l to define the total length of the bootstrap unit, i.e., l = 2l B0 + l g (including the gap in the middle), then the phase angle of the neutron spin at the exit of coil D 2 (Eq. (66)) reduces to  Table 5. Phase angles for a "perfect" 4-N = 2 bootstrap coil NRSE applying the static field signs as indicated in Fig. 6 for N = 2.
Location y Neutron spin phase angle ϕ r.f. field phase ψ Entrance A2 lA1+lg By comparing Eq. (67), with the equivalent equation for the single (N = 1) π-flipper case (Eq. (62)), we see that there is an additional doubling of the spin phase angle change by using bootstrap pairs. It can be shown [2] that this additional factor of 2 actually corresponds to N, the number of coils in the bootstrap coil unit, therefore we can rewrite Eq. (67) quite generally as Finally, Eq. (68) is represented more neatly by introducing L 0 and L 1 , the distances between the mid-points of the bootstrap coil units for the first arm and second arm respectively, so that Eq. (68) becomes With regard to differences that may exist in the flippers of a real spectrometer, it is interesting to note from Table 5 that the inner π-flippers in each arm contribute three times the spin turn of the outer coils whereas the zero-field gaps between the coils of a pair (designated by l g ) contribute at the same rate per unit length as the inter-coil zero-field gaps (see also Eq. (66)).

Coils with Dimensional Uncertainties
In order to analyze the effect of dimensional uncertainties, we assume that the center lines of each π coil are fixed and that the coil length (winding flatness on each side of the coil) fluctuates by ∆l B0 , according to equal, but independent, Gaussian distributions of width ∆f FWHM = ∆l B0 FWHM /√2. The zero-field flight paths between the coils are assumed truly zero field so that the neutron spin direction does not change in them.
We define the coil length deviation on the left and right hand sides of the coil as ∆ f L and ∆ f R respectively, where ∆ f is negative if the coil surface is on the -y side of the nominal position (neutron arrives earlier) and positive if on the +y side of the nominal position (neutron arrives later). The r.f. phase at the entrance to the first coil varies with respect to the nominal value due to fluctuations in the coil length where ( ) where ψ in is the instantaneous phase of the resonant component of the r.f. field with respect to the neutron spin at the coil entrance in the perfect situation. At the coil exit the r.f. phase is At the entrance to a third coil C, we have C B ϕ ϕ′ = . Assuming that the r.f. in coil C is phase locked to coil B (we do this so that this 4 π-flipper coil argument can be extended to an N = 2, 4 π-flipper per spectrometer arm arrangement in Sec. 3.6.2) At the entrance to the 4 th coil D D C ϕ ϕ′ so that the neutron phase at the exit of the 4th coil is We now consider two cases.

First Arm of a 4-N=1 π-Coil NRSE
In this case, the net spin turn in the first arm of the spectrometer in the absence of stray fields is given by Eq.

First arm of a 4-N=2 π-Coil NRSE
In this case, the net spin turn in the first arm of the spectrometer in the absence of stray fields is given by Eq. (71) where, for consistency with previous notation used in Sec. 3.5, we interpret "L AB " as the nominal gap l g between the bootstrap coil pair, "L BC " as "L AB " and "L CD " as l g of the second coil pair. We also change the notation A→A 1 , B→A 2 etc. for consistency with notation in the previous discussion of bootstrap coils (Sec. 3.5). In this case we have sgn(B 0 A1 ) = sgn(B 0 B2 ) = "+" and sgn(B 0 A2 ) = sgn(B 0 B1 ) = "-" and Eq. (71) becomes

Quantum Mechanical Description of NRSE
Gähler, Golub, and Keller [8] have discussed particle beam magnetic resonance in quantum mechanical terms and derived formulas for spin ½ particles passing along the y-axis. They show that for neutron magnetic interaction energies that are very much smaller than the neutron kinetic energy entering the field region (µ B 0 ≪ ½m n v i 2 -where reflected matter waves at field boundaries may be neglected), the quantum mechanical treatment reproduces the classical results with wave interpretations of physical processes such as Larmor precession.

Polarized Beam Traversing a Static Field
A spin s = ½ particle such as the neutron has spin angular momentum of magnitude ( ) 1 3 2 s s + =   with a component measured along any given axis of magnitude m s , where m s = ± ½. Consider a beam initially polarized along the x direction traveling along y. The wave function is written as a plane wave (which may be considered as the superposition of equally probable spin-up and spin-down states with respect to the z-axis) When the neutron enters a static magnetic field applied along the z-direction, the field gradient at the boundary and the associated magnetic force causes the kinetic energy of the ±σ z spin states to split by an amount ±µ n B 0 with the opposite splitting of the orientational potential energy. Inside the field, where there is no field gradient, the kinetic energies of the ±σ z spin states differ. If the total splitting is expressed as where ω 0 is interpreted as the classical Larmor precession frequency. The expectation value of the polarization along x inside the field region is Here, the relative phase of the spin-up and down waves is a cosine function of the distance traveled through the field. This is exactly equivalent to Larmor precession in the classical case.
On exiting the field, the ±σ z spin states once again become degenerate and the neutron spin states retain the accumulated relative phase angle ω 0 l B0 /v i with which they exited the field region. This phase angle does not change in the subsequent zero-field region, which is classically equivalent to the termination of the Larmor precession in the zero field region. This situation is illustrated in Fig. 7. Further accounts of these energy changes are found in Refs. [9][10][11][12][13]. Equation (75) is sometimes expressed in terms of a wavevector magnitude splitting of the two states.
For NRSE applications it is always true that ∆E B0 ≪ ω i , the incident neutron energy, therefore we can and the accumulated phase difference is l B0 ∆k.  7. When a neutron, of initial energy Ei = ωi , enters a constant static magnetic field region, the field gradient at the boundary causes the spin states that are parallel and antiparallel to the field direction to be split symmetrically by ±µnB0 with respect to Ei. Inside the field, the total (kinetic + potential) energy remains fixed at ωi. The total splitting is ω0, where ω0 is the classical Larmor angular frequency. Usually ω0 « ωi, such that reflections at the field boundaries can be ignored. On exiting the coil, the degeneracy of the two states is re-established, but the relative phase of the matter waves associated with the + and -states is shifted by an amount ω0 lB0/vi, corresponding to classical Larmor precession of the spin around the field direction during its passage through the field.

Passage Through a Static Field with Superimposed Perpendicular Oscillatory Field
With a superimposed perpendicular oscillating field (see Krüger [14]), tuned such that ω 0 ≈ ω rf , transitions between the Zeeman split states (separated by ∆E = ω 0 ) are induced via exchange of quanta with the r.f. field. Golub, Gähler, and Keller [13] treat a general case consisting of three regions; two static field regions (I and III) sandwiching an intermediate region (II) were the static field coexists with a perpendicular oscillating field of length d along the beam direction. These authors use the properties of Eq. (74) to simplify the problem by treating the +z and -z components of the eigenvector separately. Further, they assume ω rf ≪ ω i (equivalent to the µB 0 ≪ ½m n v i 2 assumption above) where ½m n v i 2 = ω i (whereby reflected matter waves at the potential boundary can be ignored and various simplifying approximations e.g. δk B0 ≈ ω 0 /v i etc. can be made outside of the exponentials). Using the symbols 0,1 T ± for the transmission amplitude with subscript "0" for elastic (no exchange of quanta -i.e., no spin flip) and "1" for inelastic (exchange of quanta with spin flip) and "+" and "-" for spin-up and spin-down final spin states respectively with respect to z, it can be shown: where α + and αare the amplitudes of the incoming + and -spin states respectively and ( ) with the parameter ε proportional to the resonance detuning, defined by where ω p =γ n B rf is the classical Larmor precession frequency of the neutron spin around B rf (Eq. (11)).
Substituting the values from Eq. (79) into Eq. (318) of Ref. [13], the wave function in region III becomes  The probability of a spin-flip involving a photon exchange with the r.f. field is which is equivalent to the expression given by Rabi, Ramsey, and Schwinger for the spin flip probability (Eq. (17) of Ref. [15]). Correspondingly, the non-spin flip probability (where the energy does not change) is which is equal to (1-spin flip probability), as expected.

Exact resonance (ω 0 =ω rf ) -general case
This is the condition for which ω rf = ω 0 , therefore ε = 0, ω A = ω p /2, consequently and the probability of a spin-flip involving a photon exchange with the r.f. field is ( ) ( ) and the non-spin flip probability is We see that unless ω p d/v i = (2N+1)π, where N is an integer, an incomplete spin inversion occurs (corresponding to a flipper efficiency < 1); i.e., r.f. π-flippers only produce exact π-flips for a unique and ( ) ( ) respectively. Note that the quantum mechanical spin-flip probability for a neutron of wavelength λ i (Eq. (89)) is exactly equivalent to the quantity cos cos "No-dispersion" implies that the classical condition for exact π-flips around B rf is satisfied for all velocities, i.e., we replace ω p d/v i by π for any v i in Eq. (85), which becomes and the probability of a spin-flip involving a photon exchange with the r.f. field is ( ) ( ) Note that under these conditions exact spin inversion occurred in region II. Region II is defined from 0 ≤ y ≤ d so that shifting the coordinate system to the exit of region II, we have http://dx.doi.org/10.6028/jres.119.005 The expectation value of σ x at the exit of the coil is Thus, at a fixed position y, the spin precesses at angular frequency 2ω 0 . Because the superimposed r.f. field induces a spin state inversion in the resonant coil, the kinetic energy splitting of ±ω 0 /2 induced by the static field gradient at the entrance to the coil adds to the splitting produced by the opposite field gradient at the coil exit (rather than canceling as it would with a static field alone). Thus the two spin states emerge from the coil with frequencies split by ±ω 0 = 2ω 0 (≡ 2ω rf at resonance) with correspondingly different momenta. The beating due to this difference in ω and k of the two spin states after exiting the coil corresponds to a Larmor precession seen at a fixed position y. This precession in a zero field has been observed directly in a MIEZE ("Modulation of Intensity Emerging from Zero Effort!") setup [16]. The argument of Eq. (95) equates with the spin phase angle of a neutron exiting coil A in Table 4, where the neutron spin was initially polarized along x, and where exact resonance (ω 0 = ω rf ) and zero dispersion were also assumed, i.e., Table 4 (96) with the argument in Eq. (95) evaluated at y = 0 (the coil exit) By identifying ψ A with ω rf t and d with l A , we see that the two equations are equivalent.
Qualitatively, a kinetic energy splitting of the two spin states occurs on entry to the coil (∆E k = ω 0 ) due to the static field gradient, whilst the total energy (= kinetic + potential) remains constant initially. The total spin inversion that occurs in the flipper involving exchange of photons with the r.f. field (see Eq. (92)) causes the total energy of the two states to become split by 2ω 0 , but leaves their kinetic energy unchanged (still split by ∆E k = ω 0 ). At the coil exit, when crossing the static field boundary, the kinetic energy splitting does not disappear (as it did in the static-only field case (Sec. 4.1)). If 100 % spin inversion occurs due to the r.f. field, an additional splitting in kinetic energy (by ω 0 ) due to the static field gradient at the exit boundary takes place (i.e., the kinetic energy splitting doubles at the exit of the first coil (∆E k = 2ω 0 )). This explains the 2ω 0 precession frequency in region III (see Eq. (95)). This situation is shown (for perfect spin state inversion) in Fig. 8. Further accounts of these energy changes are found in Refs. [9][10][11][12][13].
http://dx.doi.org/10.6028/jres.119.005 The r.f. region is deliberately shown shorter than the static field region (as is the case for an r.f. coil placed inside a static field coil). The flipper is tuned for exact π-flips within the r.f. field region so that γnBrf lrf /vi =π. In this case, by the time the neutrons exit the r.f. field region a complete inversion of the spin states has occurred via exchange of photons with the r.f. field (of angular frequency ωrf = ω0). The absorption and emission of r.f. photons means that, in contrast to the case shown in Fig. 7, the total energy of individual spin states is not conserved, as shown. The splitting of the total energy reaches a maximum of 2ω0 at the exit boundary of the r.f. field, whilst the additional splitting in kinetic energy experienced due to the spin inversion does not manifest itself until the neutron crosses the static field boundary on the exit side of the coil. Because the kinetic energies of the two states differ by 2ω0 at the exit of the coil, the relative phase of the spin-up and down states go in and out of phase corresponding to a Larmor precession in the subsequent zero field region of angular frequency 2ω0. This precession in zero field has been referred to as anomalous or "wrong" Larmor precession by Mezei [10]. http://dx.doi.org/10.6028/jres.119.005

Off-resonance, exact π-flips for all velocities (i.e., no dispersion or monochromatic)
"No-dispersion" implies that the classical condition for exact π-flips around B rf is satisfied for all velocities, i.e., ω r d/v i = π, therefore from Eq. (80) so there is no convenient simplification of the wave function for ε ≠ 0.

Passage through a second similar coil a distance L downstream (on-resonance, no dispersion or monochromatic)
Golub, Gähler, and Keller [13] extrapolate the wave function exiting the first coil through a zero-field path length, L, to the entrance of a second similar coil downstream (equivalent to the first arm of a N = 1 NRSE) and show that the wave function exiting the second coil is given by so that the expectation value of the polarization with respect to x at the exit of the second coil is The presence of the second similar flipper coil with the same static field orientation "mirror images" the neutron energy history shown in Fig. 8 so that the kinetic and total energies of the two spin states revert to their starting value E i . Therefore, the precession occurring downstream of the first coil is not observed downstream of the second coil. The final spin phase angle, 2ω 0 (L+d)/v i , agrees exactly with the classical result for the phase change in the first arm of the N = 1 spectrometer, as expected (see Eq. (62)).

What is the Effect of l rf ≠ l B0 ? -Coil Tuning
For any spin inversion to take place, the r.f. field must be applied in a region where the static field is also present (i.e., there is a Zeeman splitting to induce transitions between states). Secondly, the r.f. photons must have a frequency close to that of the splitting (resonance) so that transitions occur with high probability (Sec. 2.3). Finally, for optimum flipping probability, Eq. (13) must be satisfied in the overlap region between the static and r.f. fields, of length l π , applying the definition in Eq. (1).

Monochromatic Beam with Static Field Region
Enclosing the r.f. Region (l coil = l B0 , l π = l rf ) A monochromatic beam eliminates the complication of dispersion and for a tuned coil, therefore the spins remain in the x-y plane after passage through the coil. This is exactly the situation depicted in Fig. 8. shift in the x-y plane. However, because the effect of the spin inversion in the "r.f. + B 0 " region (length l π = l rf ) is not manifested until the neutron reaches the exit of the static field region, the effective Larmor precession continues at a rate ω 0 over the entire coil length (l coil = l B0 ) and at a rate 2ω 0 in the zero field region after the coil. Thus for an initial spin direction along x, the phase of the spin a distance L downstream of the static field region is given by with l π = l rf so that the specific π-flip condition is This case time-averaged is depicted in Fig. 9. Note that in the "r.f.-field-only" regions each side of the static field, the time-averaged field direction is random with respect to the neutron spin and so the average kinetic energy and total energy of the neutron spin states remain unchanged. Spin inversion and photon exchange occur only in the region where the static and r.f. fields are coincident, where all the change in total energy takes place. As before, the effect of the spin inversion on the kinetic energy is only felt at the static field boundary after which the time-averaged r.f. field does not change the total or kinetic energy. Thus for an initial spin aligned along x, the phase of the spin a distance L downstream of the static field region is given by However, using the definition of coil length in Eq. (3), l coil = l rf , no longer allows us to write an expression for σ x in terms of the coil length l coil as in Eq. (102). Therefore, if we always define L as the distance downstream of the static field region, Eq. (101) may be used for both geometrical cases. For this case, we have l π = l B0 so that the specific π-flip condition is

With Dispersion or When Detuned from Resonance (B 0 Region Encloses B rf )
When dispersion is present or when the coil is not resonant for all neutrons (e.g., due to static field inhomogeneities), a fraction of the spins do not flip in the superimposed "static field + r.f. field" region. The unflipped neutrons behave similarly to neutrons in a pure static field (see Sec

Analysis of the Spin-Echo Signal
The following applies to an idealized spectrometer (no uncertainty on B, L etc.). The effects of instrumental uncertainties on the spin-echo signal are discussed in the Sec. 6. The emphasis here is on quasielastic applications of the NRSE. The terms in Eq. (68) corresponding to the neutron spin phase gain in the first arm and loss in the second arm of the spectrometer, expressed vectorially, are respectively: where γ n is the neutron gyromagnetic ratio, i,f v is the initial/final neutron velocity vector and 0 1 , L is a unit vector parallel to the axes of the first and second arms of the spectrometer (and perpendicular to the coil axis). The "-" sign in Eq. (107) implies that the field directions in the second arm are such that they reverse the spin phase angle change with respect to the first arm.

Small Divergence Approximation
For small beam divergences and coil axes that are perpendicular to i v and f v , we can approximate Eqs.
the static field and scanning δL = L 0 -L 1 . Also we will assume that B 0 = B 1 and replace δ (BL) by the slightly less general expression δ (BL) = B 0 (L 0 -L 1 ) = B 0 δL in the following.
The measured quantity in neutron spin-echo is related to the polarization of the scattered beam. If the polarization is analyzed in the same direction as the polarization direction of the incident beam (assumed here to be the x axis), the polarization of the scattered beam is related (classically) to the cosine of ϕ NRSE , averaged over all the scattered neutron trajectories i.e., cos , where 〈〉 implies a statistical average over a large sample of scattered neutrons. From Eq. (111) we have This expression must be averaged over all possible values of v i (the incident spectrum) and all possible where 2θ is the scattering angle, we can write For small energy transfers (small ω and v i ≈ v f ) we have from the definition of kinetic energy: If S(Q,ω) is symmetric in ω (usually a good approximation for quasielastic scattering) and substituting δv ≈ ω/m n v i from Eq. (117), the average over the δv distribution characteristic of the scattering sample for a where the "sine" part of the expansion in Eq. (114) disappears in the integral for symmetric S(Q,ω) and the denominator ( ) for a normalized scattering function is implicit. Note that the quantity preceding ω in the second cosine argument in Eq. (118) when δ (BL) = 0 (i.e., B 0 L 0 = B 1 L 1 ) is often referred to as the spin-echo time, τ NRSE , i.e., http://dx.doi.org/10.6028/jres.119.005 Equation (118) must be averaged over the normalized incident velocity distribution, F(v i ), so the final polarization is where the denominator The extent to which the spectrometer asymmetry cosine term contributes to the wavelength-dependence of the integrand depends on the situation. Since Q is also a function of λ, Q-dependent scattering also contributes to the wavelength-dependence of the integrand. However, for now we assume this part of the wavelength-dependence is weak or else the scattering is Q-independent. It should be remembered that Eq. (122) is valid for an essentially "perfect" spectrometer, i.e., negligible beam divergence and uncertainty in the value of B 0 and L 0 , and negligible flipper coil dispersion. These effects must be included separately. The effect of flipper coil dispersion has already been dealt with in Sec. 2.2 and the other instrumental effects are considered in Sec. 6. Examples are compared with simulation results in Sec. 8.

Function")
Isotope incoherence implies scattering that is both Q-independent (so the λ-dependence of the scattering function can be ignored). Elastic scattering, or small ω, implies that there is negligible neutron wavelength change through the spectrometer and therefore the second cosine term in Eq. (122) is either unity or very close to unity. For the cosine term to exceed 0.99 requires ωτ NRSE ≤ 0.045π. Under these conditions Eq. (122) becomes where for brevity we define http://dx.doi.org/10.6028/jres.119.005 and the integral over ω evaluates to S(Q) since there is no other ω-dependence. The integral over λ i is readily performed for simple wavelength spectral functions, allowing analytical approximations for the "resolution" echo signal to be obtained in the absence of depolarizations resulting from instrumental imperfections. Expressions for purely monochromatic, rectangular, and triangular incident wavelength distributions are given in the following. The less trivial results for rectangular and triangular distributions are compared with simulations in Sec. 8.6.

Purely Monochromatic Beam
For a purely monochromatic incident beam, I(λ i ) = δ (λ 0 ), and we have simply: Therefore, in the absence of instrumental imperfections (see Sec. 6), the resolution function has a pure cosinusoidal form of constant amplitude for any δ (BL) with a periodicity given by ( )

Rectangular Incident Wavelength Spectrum
For a rectangular incident wavelength spectrum of full width ∆λ FW , centered about λ i = 〈λ i 〉, the wavelength-dependent integral in Eq. (123) becomes: provided that the lower wavelength limit of integration is greater than zero, so that apart from depolarizations resulting from instrumental imperfections and flipper coil dispersion.

Triangular Incident Wavelength Spectrum
The triangular incident spectrum is useful in many practical situations as it is approximately the shape delivered by neutron velocity selectors when the source spectrum varies slowly within the selected wavelength range. For a triangular incident spectrum of FWHM = ∆λ FWHM and mean wavelength 〈λ i 〉, the wavelength-dependent integral in Eq. (123) becomes http://dx.doi.org/10.6028/jres.119.005 provided that the lower wavelength limit of integration is greater than zero, so that apart from depolarizations resulting from instrumental imperfections and flipper coil dispersion.

Special Cases for Quasielastic Neutron Scattering (QENS) Symmetric Scans
In this case, δL → 0 in Eq. (122) and we have: For quasielastic scattering, the scattering function is represented by a Lorentzian where γ (Q) = Γ (Q)/ , where Γ(Q) is the energy half-width at half maximum. Performing the integral over ω, we have We now consider the possible λ-dependence of Γ(Q). Following Hayter and Penfold [17], we consider a common case of self-diffusion at low Q at fixed scattering angle, θ. For simplicity, we ignore the very small change in energy of the neutrons on scattering, so we may make the approximation ( ) where we have set

Rectangular Incident Wavelength Spectrum
For a rectangular incident wavelength spectrum of full width ∆λ FW , centered about

Triangular Incident Wavelength Spectrum
For a triangular incident spectrum of FWHM = ∆λ FWHM and mean wavelength 〈λ i 〉, Eq. (133) becomes

Gaussian Incident Wavelength Spectrum
For a Gaussian incident spectrum of FWHM = ∆λ FWHM (standard deviation σ) and mean wavelength

Perfect Polarizer, Analyzer and Non-Spin Flip Scattering
In an NRSE instrument, for a perfect polarizer and analyzer, the non-spin flip quasielastic signal in the detector is where P x has been derived for some specific cases in the preceding section and is proportional to the intermediate scattering function. Measuring scattering in the time domain rather than in energy has the significant advantage that the scattering function is obtained from the measured data by simple division by the instrumental resolution function, rather than by deconvolution. This feature allows for very sensitive line shape analysis. 19.1 ns). δ (BL) was varied by changing L 1 with respect to L 0 . (i.e., δ (BL) = BδL). The incoming and outgoing beam divergence, if any, is equal and uniform up to a maximum ∆θ i,max = ∆θ f,max = ∆θ max , and is symmetrical with respect to the nominal axes for both spectrometer arms (see also Sec. 6.4). Under these conditions, the echo maximum is found at L 0 = L 1 . The left hand column of Fig. 11 illustrates the effect of broadening ∆λ i for elastic, non-spin flip scattering (or no sample). In the extreme, purely monochromatic case of ∆λ i = 0, the signal is cosinusoidal with respect to δL (as predicted by Eq. (125) with δ (BL) = BδL for one signal period given by Eq. (126). For ∆λ i > 0, the maximum signal is achieved at the symmetrical spectrometer setting and, as ∆λ i increases, the primary envelope of the echo signal tightens around this point. Note that the period (in L) also decreases inversely proportional to B 0 (= B 1 ) (and hence τ NRSE ), as predicted by Eq. (126). The second and third columns show the effect of increasing the neutron flight path differences via increasing beam divergence for (i) a purely monochromatic incident beam (column 2), and (ii) a triangular wavelength distribution with ∆λ i /〈λ i 〉 = 10 % (column 3). The fourth column demonstrates the increasingly rapid decay of the echo point signal with respect to τ NRSE as the quasielastic width is increased (as predicted by Eq. (135) for a purely monochromatic incident beam (∆λ i = 0)).  The axis oriented into the paper represents the static field B0 from 10 -3 T at the front to 0.025 T (τNRSE ≈ 19 ns for these spectrometer parameters) at the rear for each plot. The horizontal axis more parallel to the plane of the paper is the spectrometer asymmetry (L0 -L1), in units of mm, between about ± 1.7 mm for each plot. Columns 1 to 3 are for elastic non spin-flip scattering (or no neutron energy change through the spectrometer). For the pure monochromatic elastic scattering examples there is no coil dispersion. Column 4 is for quasielastic non spin-flip scattering. For each of these simulations there are no sample size effects and ∆B0 = ∆lB0 = 0 (zero field inhomogeneity and perfect dimensions of the flipper coils). Additionally for columns 1 and 4 zero beam divergence is assumed.

Imperfect Polarizers with Non-Spin Flip Scattering or No Sample
Real polarizing devices transmit a fraction of the wrong spin state, which results in a reduction of the NRSE signal. It is important to correct data in such a way as to isolate depolarization due to sample dynamics from instrumental depolarization as far as it is possible. Considering the quantum mechanical description of the polarization in terms of spin-up and spin-down neutrons, the polarizing efficiency of a http://dx.doi.org/10.6028/jres.119.005 "+" polarizer is numerically equal to the polarization of an initially unpolarized beam obtained after action of the polarizer. Using the definition in Eq. (54), the polarization after the action of the initial polarizer is where P I + and P I − are the intensities of + and -neutrons in the beam after the polarizer P. Note that P P can vary between +1 and -1. The incoming unpolarized beam of total intensity I 0 is described by equal + andcomponents, i.e., where we have used the boundary condition that for perfect + polarization efficiency (P P =1), only the + state neutrons of the originally unpolarized beam are transmitted (i.e., one half of the neutrons of the incoming beam) and we assume that this total number is conserved for inefficient polarizers. T P is the spinindependent transmission factor of the device with 0 < T P < 1 due to effects such as absorption or scattering. From Eqs. (140) and (142) it is easy to show that after the polarizer: Therefore, the combined action of the polarizer (P) with the analyzer (A), both oriented to transmit + spin neutrons, for non-spin flip scattering is expected to give transmitted intensities with the total beam intensity after the analyzer If two π-flippers of efficiency f 1 and f 2 and spin-independent transmission factor T f1 and T f2 are placed between the polarizer and the analyzer, and remembering that the effect of a π-flipper of efficiency f is to multiply the incoming polarization by the factor (1-2f ) (see Sec. 2.4), we infer by analogy with Eqs. (145) and (146) that the + and -intensities downstream of the analyzer (i.e., at the detector) are and both flippers 1 and 2 on.
As pointed out by Hayter [18], the ratio of the detector count rates, I PA tot , measured with both π-flippers switched off to the count rates with the two flippers switched "on-off", "off-on" and "on-on" provides three "flipping ratios", R 1 , R 2 , and R 12 respectively, which no longer have the spin-independent pre-factors common to each measurement. We thus have three equations for the three unknowns: f 1 , f 2 , and the product of the polarizer and analyzer efficiencies, P P P A , which can be solved to obtain The flipping ratios are determined for multi-angle instruments by using a diffuse, non-spin flip scattering sample such as quartz.
In an M-coil NRSE instrument with non-spin flipping samples (e.g. pure nuclear coherently-scattering samples), the polarization (NRSE signal) is reduced from the ideal value by the product of these instrumental inefficiencies. Therefore, the corrected signal, P corr , is related to the measured signal, P meas , by ( )

Imperfect Polarizers with Spin Flip Scattering
When there is a sample that modifies the spin state of the incoming neutrons, the spin transfer function of the sample has to be taken into account just like the function (1 -2f) for the flipper. Table 6 shows relative spin-flip probabilities for various types of nuclear scattering for non-magnetic samples. Consider a non-magnetic sample that flips a fraction q of the neutron spins so that, in exact analogy with the π-flipper (Sec. 2.4) and Eq. (55), the polarization after the sample, P S , is related to the polarization before the sample, P i , by ( ) For the simple example of a single isotope, pure incoherent scatterer, 1/3 of the neutrons have their spins unchanged whilst 2/3 of the neutrons have their spins flipped by π. Thus the sample flipping efficiency is This means that the spin-echo signal amplitude is reduced to 1/3 and the minus sign means that the echo signal is inverted.
where we have assumed that the relative probabilities of coherent and spin-incoherent scattering are given by S coh (Q) and S inc (Q) respectively, therefore This represents an upper limit on the size of the spin-echo signal. For this case Eq. (155) becomes Other scattering cases including paramagnetic, ferromagnetic, and antiferromagnetic samples have been discussed by Mezei [3]. In order to determine the exact spin-flip/non-spin flip behavior of the sample, a conventional polarization analysis arrangement may be used with a polarizer and analyzer and only one flipper switched on or off.

Analysis of Contributions to the Elastic Instrumental Resolution Function: Allowable Flight Path Differences and Static Magnetic Field Inhomogeneity
The spin-echo phase is given by Eq. (69), i.e., At the echo point, 〈ϕ NRSE 〉 = 0, however, even for λ i = λ f (elastic scattering or no sample), ϕ NRSE has a distribution of values about the mean, 〈ϕ NRSE 〉, of characteristic width ∆ϕ NRSE . This is because the terms B 0 L 0 and B 1 L 1 have non-zero spread, ∆(B 0 L 0 ) and ∆(B 1 L 1 ) respectively 2 , arising from instrumental imperfections. Consequently, ∆ϕ 0 and ∆ϕ 1 are non-zero and the valued information, which is the depolarization due to the scattering energy transfer distribution, is modified by the instrumental depolarization. The instrumental uncertainty, ∆(BL), determines the elastic instrumental resolution function. In order to obtain a broad dynamic range, ∆ϕ NRSE must be dominated by the distribution of λ i -λ f from sample energy exchanges (rather than the uncertainties in the BL terms) to the largest field magnitudes possible.
If we assume Gaussian uncertainties on the values of B and L and that B and L are independent variables, we expect ϕ NRSE also to have a Gaussian distribution, g(ϕ NRSE ). At the echo point (〈B 0 L 0 〉 = or the inverse relation 0 0 1 1 4 ln 2 ln 11.1 ln . We use this convenient form when estimating spectrometer tolerances in the following sections. One notes that if the distribution g(ϕ NRSE ) was uniform between the limits ± ∆ϕ NRSE max , rather than Gaussian, the analogue of Eq. (162) is a sinc function of ∆ϕ NRSE max : The elastic and quasi-elastic signal count rate cannot exceed a maximum proportional to P x 0 (for pure coherent scatterers) and sometimes considerably less for incoherent scatterers (see Sec. 5.4.3), therefore P x 0 (τ NRSE ) must remain comfortably greater than zero. In order to avoid excessive counting times or poor signal-to-noise ratio we suggest a practical minimum P x 0 > 0.2 at the maximum required τ NRSE in a quasielastic measurement. Purely coherent, elastic scatterers, such as Grafoil®, Carbopack TM , and carbon black are all used for measuring the resolution function in spin-echo spectrometers.
In order to estimate the depolarization produced by static field inhomogeneities, dimensional uncertainties, and beam divergence, we use the convenience of Eq. (163). We further assume similar distributions of ϕ 0 and ϕ 1 , which imposes ∆ϕ 0 = ∆ϕ 1 , and that the spectrometer is operated at the echo point (i.e., 〈ϕ 0 〉 = 〈ϕ 1 〉). If ϕ 0 and ϕ 1 are distributed normally, we can write In order to isolate individual contributions, we analyze first the effect of static magnetic field inhomogeneities in the absence of flight path uncertainties, and secondly, the flight path uncertainties in the absence of field inhomogeneities. We also separate the flight path uncertainties due to spectrometer dimensional fluctuations from those due to beam divergence. For the beam divergence, we cannot assume Gaussian distributions, as explained in Sec. 6.4.

Static Magnetic Field Inhomogeneities
We may consider the effect of static field inhomogeneity as creating a distribution of values of ω 0 -ω rf . In order to simplify the argument we consider ω rf as being precisely fixed (a reasonable assumption for a high quality frequency generator). The effect of field inhomogeneity is isolated by attributing all the fluctuation in ω 0 -ω rf to the distribution of ω 0 caused by the field inhomogeneity, ∆B 0 , and comparing the polarization with the equivalent system in which ω 0 =ω rf for all trajectories (∆B 0 = 0). Further, we assume that the spectrometer is optimally tuned such that 〈ω 0 〉 = ω rf and that the field inhomogeneity gives rise to a normal distribution of ω 0 with respect to 〈ω 0 〉. In this approximation, the effect of static field http://dx.doi.org/10.6028/jres.119.005 inhomogeneity is analogous to the effect of dispersion discussed in Sec. 2.2. The effect of ω 0 ≠ ω rf is conveniently visualized in the rotating coordinate system, as proposed by Rabi, Ramsey, and Schwinger in Ref. [15], whereby the rotating field magnitude transforms to an effective field of magnitude ( ) The effective field lies at an angle α eff to the x-y plane given by This is implicit in the quantum mechanical treatment of Ref. [13] discussed in Sec. 4.2. We now find an approximate relation between the static field inhomogeneity and the consequent depolarization that works well within certain limits.
The spin-flip probability for exact resonance (ω 0 = ω rf ) is given by Eq. (86) and in the general offresonance case by Eq. (83). If ∆ϕ π represents the difference in the x-y spin turn in the off-resonance case with respect to exact resonance case, then, in analogy with Sec. 2.2, we equate the ratio of the spin-flip probabilities with the quantity 〈cosε cos∆ϕ π 〉, where ε here refers to the angle of the spin vector out of the If λ i is the median wavelength and that the flipper is optimally tuned for π flips at this wavelength, i.e., for the median wavelength, which we assume is approximately true for the entire incident wavelength band.
(Note that Eq. (170) is analogous to Eq. (49).) If we assume that ∆B 0 is sufficiently small that cosε ≈ 1 and 〈cos∆ϕ π 〉 ≈ 1-〈∆ϕ π 2 〉/2 = cos〈∆ϕ π (rms)〉, we have after one π coil due to the effect of ∆B 0 : For an M-coil unit spectrometer, we assume that ∆ϕ π is uncorrelated between coils and that the cumulative effect for M coils is obtained by summing in quadrature. Taking FWHM values, we have (by combining Eqs. (163) and (171)): and hence Specifically for a 4-N coil instrument we have: In the range of operation of spectrometer configurations considered in this document, it can be shown that the value of ξ is typically no greater than about 1.3. In this range, it turns out that the awkward term may be replaced very successfully by ( ) or specifically for a 4-N coil instrument: It turns out, somewhat fortuitously, that Eqs. (175) and (176) produce a better approximation to P x 0 when ∆B 0 is too large to assume cosε ≈ 1 (implicit in Eqs. (173) and (174) for M coils. Specifically for a 4-N coil instrument we have: The success of Eq. (178) is demonstrated in Fig. 12 and Fig. 13 for N = 1 and N = 2 respectively for a spectrometer setting with B 0 = 0.0393 T, l B0 = 0.03 mm, L 0 = 2 m, λ 0 = 8 Å, which gives τ NRSE = 15 ns and 30 ns for N = 1 and N = 2 respectively.

Coil Flatness
In order to estimate tolerances on the flight path lengths, we return to the expanded equations representing ϕ 0 (and ϕ 1 ) which contain the individual contributions (the flatness model used is that described in Sec. 3.6), and now we assume ∆B 0 = 0.
(a) For a 4 (N = 1)-coil NRSE, we have from Eq. (72) for a given neutron trajectory, ∆B0. The red curve represents the approximation obtained using 4(ξ-1) for the inverse cosine squared part of the exponential in Eq.
(174) which just so happens to give a better approximation at larger ∆B0 (Eqs. (176) and (178)).  where we have set ϕ in = 0 (perfectly polarized incoming beam) and the terms ∆ f L and ∆ f R are the deviations of the coil surface from perfect flatness on the left and right hand sides of the coil respectively.
Assuming, for similar coils, ∆ f L and ∆ f R have Gaussian distributions of equal FWHM =∆ f FWHM , we can where we have set ϕ in = 0 (perfectly polarized incoming beam) and the terms ∆ f L and ∆ f R are the deviations of the coil surface from perfect flatness on the left and right hand sides of the coil respectively.
Assuming, for similar coils, that We can also write the FWHM fluctuation in the coil length (length of the B 0 field) in terms of ∆ f FWHM , Inverting Eq. (184) we also have 0 0 Equation (184)

Coil Parallelism
Related to the coil flatness is the question of parallelism, which may actually impose the major engineering limitation. The tolerances on the coil length are the same as indicated in Sec. 6.2, however, a lack of parallelism leads to a predictable and continuous change of field paths over the beam area. If we assume that Eq. (184) defines approximately the maximum tolerance in the static field length, we can approximate the coil parallelism tolerance by where max surf ϑ is the maximum tolerable angle between the entrance and exit surfaces of the static coil windings and a and l axial are the coil dimensions defined in Fig. 23.

Beam Divergence (Simplified Model)
We use a simplified model in order to estimate analytically the effects of beam divergence on the elastic resolution (polarization). More realistic beam divergence models, which are treated numerically, are described in Sec. 8.5. The simplified model assumes that the spectrometer components (coil boundaries, samples, etc.) are described by thin planes perpendicular to a nominal beam direction. A divergent incident or scattered beam is represented by selecting random trajectory polar angles, ∆θ i or ∆θ f , up to specified maxima ∆θ i,max and ∆θ f,max respectively, where all ∆θ are defined with respect to any axis parallel to the nominal beam axis. ∆θ i and ∆θ f are assumed to affect all path lengths upstream and downstream of the sample plane respectively. This situation is illustrated in Fig. 16. Therefore, the effect of beam divergence is to increase all distances between planes normal to the nominal beam axis by the factor 1/cos(∆θ i,f ). In order to isolate the influence of the beam divergence on the elastic resolution one can consider a symmetrical spectrometer at the echo point with no field inhomogeneities such that B 1 L 1 =B 0 L 0 , 〈ϕ 0 〉=〈ϕ 1 〉, etc. The elastic resolution is still given by Eq. (113) cos cos cos .
For a trajectory in the incident arm of the spectrometer, we have The distribution of ∆ϕ 0 for random ∆θ is by no means Gaussian or uniform. Because we assume small divergence (i.e., ∆θ i,max and ∆θ f,max are small -certainly within the range of angles encountered in the NRSE), we write for all incident arm trajectories: and likewise at the echo point: (Note ∆ϕ 0 and ∆ϕ 1 are not necessarily small numbers because 〈ϕ 0 〉 can be very large). Therefore, finally where C 1 and S 1 are the Fresnel cosine and sine integrals respectively, defined by ( ) ( ) Certain approximations for evaluating C 1 and S 1 have been discussed by Mielenz [19] (note that the π/6 term in Eq. 3b of this reference should be multiplied by x 3 ) and Heald [20]. The integrals can also be evaluated numerically. For the particular case of |∆θ i,max | = |∆θ f,max | = |∆θ max |, Eq. (193) becomes: 196) or in terms of the instrument parameters: The success of Eq. (193) in describing the relationship between ∆θ max and P x 0 is demonstrated in Fig. 17 and Fig. 18  In the present context it is useful to have P x 0 as the dependent variable and ask "what is the maximum permissible value of |∆θ max | to achieve a given value of P x 0 ?" Unfortunately, inversion of Eq. (196) is not trivial. The traditional approximations for C 1 and S 1 discussed in Refs. [19,20] and others do not lend themselves to neat closed forms either, even for small arguments, since the numerator of Eq. (196) involves large powers of the argument for sufficient accuracy. However, we note that the expansion of C 1 2 (x)+S 1 2 (x) involves terms in x 4n+2 , n = 0, 1, 2,…, ∞ with alternating signs for the first few terms. Another function that has the same powers and signs as these first terms would be x 2 exp(-ax 4 ): ( ) 4 202)) that should be valid for |∆θmax| <~ 6.0×10 -3 rad to within 1 % is shown by the blue dashed curve.
therefore we try setting the parameter a in Eq. (198) to the magnitude of the second term coefficient in Eq.
(199) = π 2 /45 ≈ 0.21932 which makes the two leading terms in Eqs. (198) and (199)   . 4 In fact the approximation is within 15 % for x up to about 1.8, at which point ∆ϕ 0 ≈ 1.6π (as is seen from T m A 6.7 10 for~1. 15 rad . The results of this latter approximation are plotted as the blue curves in Fig. 17 and Fig. 18. Although the suggested limits of applicability implied by Eq. (202) (for 1 % accuracy of Eq. (201)) are 8.4 mrad and 6.0 mrad for the N = 1 and N = 2 cases respectively shown in the figures, the approximation works quite well also for larger angles.

Approximation for Equal Contributions to
Depolarization from ∆B 0 , ∆l B0 , and ∆θ max In the preceding sections, the contributions of ∆B 0 , ∆l B0 , or ∆θ to the elastic polarization P x 0 were taken in isolation. Because all three parameters will have some uncertainty, their individual tolerances must be correspondingly tighter to compensate for the depolarization created by the other two. It is difficult to assess which parameter tolerance is easiest to achieve but some idea of the spectrometer requirements is  T Note that ∆l B0 is defined by N, B 0 and λ only and is independent of l B0 or zero field region parameters.
Note that ∆θ max depends on λ and on both the flipper coil and zero-field parameters (N, B 0 , L 0 (i.e., L AB , l B0 , and l g )). Even though these parameters also appear in the expression for τ NRSE , the λ 3 -dependence of the latter means that ∆θ max is not uniquely determined by the quantity τ NRSE (i.e., the same value of τ NRSE may require different values of ∆θ max depending on the values of N, B 0 , L 0 and λ).

Some Examples (Equal Contributions to Depolarization)
Consider requiring the elastic (resolution) polarization P x 0 to be greater than some specified minimum value at a reference point with equal contributions coming from ∆B 0 , ∆l B0 , and ∆θ max . We consider the point τ NRSE ≈ 30 ns at λ = 8 Å with N = 2, for M = 8 π coils (l B0 =0.03 m), with B 0 = 0.0393 T, L 0 = 2 m. Using Eqs. (203-205), several results are shown in Table 7.
http://dx.doi.org/10.6028/jres.119.005 The results in Table 7 are summarized in Fig. 19. Note the particular sensitivity of the instrumental resolution on the beam divergence once a certain threshold angle is reached.

Desired Function
Desirable criteria for a NIST NRSE instrument are summarized as follows: 1. Emphasis on quasi-elastic scattering -coil tilting is not necessary. 2. Large solid angle coverage and multi-angle measurement capability.
3. If possible, the spectrometer should be able to access Fourier times of τ NRSE = 30 ns at λ = 8Å and be fabricated with sufficient precision to allow useful measurements to be performed at this measurement point. 4. Offer usable incident wavelengths at least down to 3 Å for high-Q capability. 5. Must have a short Fourier time measurement capability.  To date, the largest static fields produced in water-cooled NRSE coils using pure aluminum windings are about B 0 ≈ 0.025 T. With this field we require L 0 = 3.14 m for N = 2 (which is a little long for available floor space) or else L 0 = 1.57 m for N=4. Apart from the increased restrictions on the maximum incoming bandwidth, ∆λ/λ, when using N = 4, doubling the number of π-flipper coils has the obvious disadvantage of increasing the complexity and setup of the spectrometer and increasing the amount of material in the beam. Thus an N = 4 option is unattractive for a multi-angle instrument. Restricting N to 2 with L 0 ≤ 2 m and pursuing the goal of increasing B 0 towards 0.04 T presents itself as one of the more attractive options. Some consequences are explored in the following sections.

General Description
The N=2 bootstrap NRSE coil, a most recent example of which is shown in Fig. 20, is composed of back-to-back static field coils with equal but oppositely-opposed field directions. Each static field coil encloses an r.f. coil (whose coil axis is perpendicular to that of the static coil). The r.f. coil must be placed inside the static field coil in order to avoid significant r.f. attenuation that would otherwise occur in the metallic structures of the static field coil. µ-metal plates capping each end of the static field coils conduct magnetic flux lines between the two coils. An outer µ-metal shield enclosing the entire assembly, apart from the beam windows, helps reduce the stray field magnitude entering the zero field regions. For quasielastic applications, both the static and the r.f. coil axes are perpendicular to the beam direction. To profit from the advantages of the NRSE technique over conventional NSE, the NRSE coils must be moderately compact in the beam direction. Because the neutron beam traverses both the static and the r.f. coil windings, there are particular restrictions on the winding materials that may be used in the beam passage (see Sec. 7.3.2). High resolution requirements also impose restrictions on the shape of the windings themselves. These and other factors are discussed in the following sections.

Aluminum Windings: Transmission and Small Angle Scattering
Because the beam must traverse both the static field coil and the r.f. coil windings with this design, the neutronic properties of copper exclude it as a winding material within the beam region. For nonsuperconducting windings, the most obvious choice is aluminum. However, even pure aluminum has resistivity that is almost 60 % greater than pure copper at room temperature. For a 4-N = 2 coil NRSE instrument, the beam must traverse a total of 16N = 32 layers of static and r.f. coil windings. Assuming that each winding layer has the same thickness, t, we can estimate the anticipated maximum transmission of all the coils from the total cross-section of pure aluminum at room temperature. Some results for different winding thicknesses t are shown in Fig. 21. Note that the values in Fig. 21 are optimistic because (i) impurities (e.g. from anodization of the actual winding material) are not accounted for, and (ii) the transmission will be reduced by increased phonon scattering if the operational winding temperature exceeds 300 K (which it is likely to do significantly).
for the purposes of estimating the coil transmission. We now assume that the static field coil windings (which usually have to carry higher maximum currents than the r.f. windings) have thickness t and the r.f. windings have thickness t/2, such that the total thickness of windings traversed by the beam in the spectrometer is 12Nt = 24t for N = 2. If we choose a transmission criterion such that T Al (λ = 8 Å) ≥ 80 %, then Eq.

Static Field Coils
An early static field coil using circular section aluminum wire developed for the Zeta spectrometer at the ILL, Grenoble, is shown in Fig. 22.

Current in the static field coil
Sufficient static field homogeneity within the beam passage may be achieved by passing the beam through a suitably restricted area close to the axial center of a long solenoid. The field at the center of a long solenoid is where µ 0 is the permeability of free space with µ 0 =4π ×10 -7 NA -2 .  where B 0 is the static field in Tesla, n is the winding density in m -1 , and I is the current in Amps. Equivalently, the current in the coil at field B 0 is Thus the required current is inversely proportional to the winding density and is directly proportional to the required field B 0 .

Resistance of the static field coil windings
The resistance of the static field coil winding is where l w is the total length of the coil winding, A w is the wire cross-sectional area, and ρ (T) is the resistivity of the winding at its operating temperature, T. The winding length per turn (see Fig. 23) for the rectangular cross-section coil form is approximately 2(a+ l B0 ), assuming the winding thickness is negligible compared with a and l B0 . For the particular case of single-layer windings, the total number of turns, N B0 , is 0 B axial N l n = any single-layer winding, so that the total length of any single-layer winding around the rectangular coil form shown in Fig. 23 is The outer surface area of the rectangular coil form is

Single-layer rectangular cross-section wire
The cross-sectional area, A w , of rectangular cross-section wire (see Fig. 23) is representing the tightly-wound limit with h = 1/n. Therefore, for a given A surf , the resistance of the tightlywound coil increases as the square of the winding density and is inversely proportional to the winding thickness, t. Logically, the resistance is minimized for a given n, A surf , t, by ensuring that the windings are tightly-wound. For a single-layer rectangular cross-section wire winding, the D.C. voltage required to maintain a static field B 0 is, from Eqs. (214) Thus, for a given A surf , the power dissipated in the tightly-wound coil is inversely proportional to the winding thickness, t, and is independent of n or h (essentially a current sheet). We also note that the power increases as the square of the required field, B 0 . Like the voltage, the power dissipated is minimized for a given B 0 by tightly-winding the coil within the available surface area, since h ≤ 1/n.

Single-layer circular cross-section wire windings
The cross-sectional area of the circular cross-section wire, A w , is representing the tightly-wound limit. Thus, for a given A surf , the tight-winding resistance increases as the cube of n (as opposed to n 2 in the tightly-wound rectangular wire case with fixed t).
The D.C. voltage required to maintain a static field B 0 in the circular cross-section wire case is (from Eqs. (214) and (228) tightly-wound, circular cross-section wire winding.
Thus, the voltage required to achieve a given B 0 in the circular cross-section wire case is independent of the winding density, other than n cannot exceed a value of 1/(2r w ) for a single layer. Qualitatively, this is because decreasing n decreases R at the same rate that I (Eq. (214)) must increase to maintain B 0 . The power dissipated in the coil with single-layer, circular cross-section wire windings is (from Eqs.
Therefore, the tightly-wound coil represents the minimum power condition for circular cross-section wire.
Furthermore, the circular wire should be as thick as is tolerable to minimize the power.

Summary and static field coil power concerns
The coil flatness requirements for high resolution operation (see Sec. 6.2) favor rectangular crosssection wires for the static field coils. Two potential concerns are: (i) the magnitude of the currents supplied to the coils, (ii) excessive heat dissipation in the coils and the associated cooling difficulties. Item (i) is somewhat mitigated by choosing the largest value of n that is feasible. Item (ii) is mitigated by tightlywinding the coil as indicated in Sec.7.3.3.3. Beyond these measures Eq. (226) identifies the remaining constraints: Firstly, if A surf becomes small with respect to the beam area it is increasingly difficult to maintain adequate field homogeneity within this region at high τ NRSE (see e.g. Sec. 6.1 and Sec. 6.6).
Secondly, the winding thickness in the beam direction, t, must be limited so as to maintain high neutron transmission (see Sec Ωm 1.14 10 K 6.9 10 . For T ≈ 300 K, P Al (0.025 T) ≈ 2.7 kW. For T ≈ 350 K, P Al (0.025 T) ≈ 3.3 kW. If similar coils are to achieve 0.04 Tesla, the current increases to I ≈ 128A with an increased power dissipation factor of approximately (0.04 2 /0.025 2 ). The room-temperature power dissipation then increases to approximately 6.9 kW. If the coils are cooled to liquid nitrogen temperature ≈ 80 K, P Al is more than an order of magnitude smaller (≈ 220 W at B 0 = 0.025 T, ≈ 560 W at B 0 = 0.04 T). This is discussed by Gähler, Golub, and Keller in Ref. [8]. One technical challenge is avoiding liquid coolant (water or liquid N 2 ) in the beam passage since both scatter thermal neutrons strongly. A separate issue is the evidently undesirable increased beam divergence from small angle scattering that occurs in Aluminum. A concept for a liquid N 2 -cooled static field coil with the above requirements has been proposed by Carl Goodzeit of M.J.B. Consulting, De Soto, TX, USA (Fig. 24). The basic shape of this coil is a racetrack-shaped toroid ( Fig. 24 (a)). A section of one side of this hollow coil provides the beam passage ( Fig. 24 (b)) requiring high purity aluminum (99.999 %) conductor. The specific example shown has 0.5 mm thick and 6.2 mm wide conductor which implies I ≈ 198 A at B 0 = 0.04 T with a corresponding current density of about 64 A mm -2 . The winding would be supported by and cooled by four hollow tubes for the passage of liquid N 2 (Fig. 24 (c)) running the full height of the coil. On the sides which do not transmit the beam, additional thermal contact and support is provided by heat-conducting side plates. Because the effective resistance of each turn is combined with the resistance of the turns in the remainder of the toroid, all turns, except at beam transit, can be of a lower resistivity material and are in thermal contact with the N 2 -filled coil form, thus they should remain close to 80 K. In general, the liquid N 2 would be admitted at the bottom of the racetrack coil form and would vent from the top (these features and eventual feed-throughs for the r.f. coil are not shown). The coils would be contained in an environment that prevents condensation of water vapor on the windings.

Required static field coil current stability
The values in Table 7 imply that ∆B 0 /B 0 must be around 0.1 % in order to achieve P x 0 (8 Å, 30 ns) ≥ 0.5 for typical spectrometer dimensions. Even for perfect static field coil homogeneity (∆B 0 = 0), this imposes a coil current stability of order of 0.1 % (∆I/I <~ 10 -3 ). The current stability should certainly not become the limiting factor on ∆B 0 . Preferably it should be at least an order of magnitude better (∆I/I < 10 -4 ). Long-term current drift (e.g. in response to temperature changes) should also be in this range. Current supplies offering stabilities in the 10 -5 range are commercially-available, so this is not expected to impose any technical limitation.

Effect of coil dimensions on field homogeneity and field magnitude
With respect to geometry, field homogeneity, field strength, and winding resistance, it is preferable that the static field coils be short in the beam direction, given that the coil width must be somewhat wider than the beam. Reducing the coil thickness in the beam direction tends to allow the perpendicular axial length of the coil to be reduced without loss in field homogeneity in the beam passage. This principle is illustrated by considering the axial field of a cylindrical open-ended solenoid (Fig. 25), where instead of the coil thickness in the beam direction we refer to the coil radius. The field at axial position x is ( ) In the "long" solenoid limit (l axial ≫ r), the field at the coil center is maximized (=µ 0 nI), whereas at its ends it is half this value (=µ 0 nI /2 -limit of Eq. (239) with µ = 1 and η 2 ≪ (1+µ) 2 ). This alone implies that the axial length of the coil must be substantially greater than the height of the neutron beam. Figure 26 shows the variation of the axial field normalized to the maximum attainable field (=µ 0 nI) for solenoids with various ratios η = d/l axial , calculated according to Eq. (236). Figure 26 reveals that as η increases: (i) The axial range over which the field can be held close to B(x = 0) decreases.
(ii) The maximum achievable field (at the center) decreases. This reduction becomes quite significant once η increases above about 0.4.  This latter consideration is particularly important in the present application where the goal of achieving the highest fields is already hampered by large currents. Usually, detailed field calculations are required to optimize the coil windings and dimensions. Using the example of the cylindrical solenoid, suppose the maximum axial length of the static field coil is 0.3 m, the beam height is 0.03 m, and the required ∆B 0 /B 0 is about 0.1 %. With reference to Fig. 26, this requires B 0 (µ = 0.1) ≥ 0.999B 0 (µ = 0). This occurs for η <~ 0.045, i.e., for coil diameters of 0.0135 m or less. Although this example just considers the axial field variation for a cylindrical solenoid, it suggests that careful control of the coil dimensions perpendicular to the coil axis are required to achieve sufficient field homogeneity within the beam passage of the NRSE coils.

Coil flatness issues
As demonstrated in Sec. 6 the dimensional tolerances for high resolution coils are demanding. The winding support must be accurately machined and the windings themselves must be very flat. The use of anodized pure aluminum band not only creates a geometrically well-defined field region but also eliminates curved field lines that are generated in the vicinity of circular cross-section wires (Dubbers et al. [22]). The existence of a magnetic pressure (see Sec. 7.3.3.10) is also of concern for maintaining the shape of the windings. For high fields this usually requires clamping of the windings outside of the beam passage.

Winding methods
Commercial coil winding tools are available, however, good experience has been obtained using machinist's lathes. These machines offer the desirable combination of precise translational and rotational speeds, and adjustable torque settings.

Magnetic pressure on the coil windings and their mechanical constraint
Magnetic pressure in a coil refers to the radial force exerted on the coil windings due to the difference in magnetic flux density inside and outside the coil. The magnetic pressure, P mag , exerted on the windings at the center of a long circular solenoid is For B = 0.04 T, P mag ≈ 637 Nm -2 (≈ 0.0063 Atm).
For the approximately rectangular section coils used in NRSE, Ampere's law predicts that the magnitude of the field inside the coil is similar to that of a cylindrical coil carrying the same current, assuming that the field outside the coil is negligible with respect to that inside. Therefore, we assume that the magnetic pressure is also given by Eq. (241) near the center of a long rectangular section coil.
For a coil wound on a rectangular former with slight pre-tension, we can approximate the action of the magnetic pressure on the band-like windings by the mechanical problem of an evenly-loaded beam whose ends are constrained. According to Ref. [23], the maximum winding deflection at the center is where w is the load per unit length of the beam, l u is the unconstrained length of the beam, E is Young's modulus for the winding material, and I is the moment of inertia. For the band-like (rectangular) windings of width h and (small) thickness t, the moment of inertia is 3 . 12 For pure Al windings (E = 7.1×10 10 Nm -2 ) with t = 0.4 mm (as used in existing coils), B = 0.04 T, and a typical l u for the coil face traversed by the beam of about 0.25 m, we have y max ≈ 17 mm. This is clearly unacceptably large, therefore in order to maintain the coil dimensions within required tolerances, the windings must be clamped for high fields. Note that Eq. (245) contains the unconstrained length of the winding to the 4th power, therefore it is often feasible for the clamping plates to incorporate an open window allowing passage of the beam (see Fig. 20). For example, if this window is 0.03 m wide, then l u ≈ 0.03 m and Eq. (245) yields y max ≈ 3.5 µm, which is well within the acceptable range (Table 7).

Existing r.f. coil designs
The r.f. coils, two examples of which are shown in Fig. 27, are of similar overall design with the beam passing through the (gray) aluminum windings at the coil center. The fields are returned through the two arch-shaped coils which greatly reduce r.f. power loss from induced currents in (and consequent heating of) surrounding metallic structures, including the static field coils. This also prevents significant perturbations to the static field. The coils outside of the beam passage are wound with high-frequency (very thin stranded) copper wire to maximize electrical conductivity. The electrically-insulating return coil former material in the units shown is similar to the fiberglass/epoxy composite used in printed circuit boards. The method used at the ILL for maintaining tension on the aluminum windings is to stretch the windings over silicon-based rubber O-rings covered with Kapton tape (rubber containing carbon has been found to burn).

r.f. circuit and impedance matching
The NRSE spectrometer operates at a single frequency for each scan point (or τ NRSE ). Typically, an NRSE scan might consist of 10 or 20 points and therefore 10 or 20 different r.f. frequencies. Thus, even though the drive circuit is "narrow" band for each measurement point, it must be tunable through more than a decade of r.f. frequencies.
http://dx.doi.org/10.6028/jres.119.005 In this application at high frequencies, it is important to match the characteristic impedances of the transmission line with that of the load to prevent reflections of r.f. power from the load toward the source. Reactive elements in a circuit (inductance and capacitance) store and return energy to the source unless the circuit appears purely resistive (i.e., the voltage and the current are in phase). This is the condition for impedance matching. Equivalently-stated, the power factor (= cosθ), where θ is the phase angle between the current and the voltage must ideally equal 1or the net capacitative reactance of the circuit cancels the net inductive reactance. Impedance matching not only maximizes the efficiency of the circuit, but also prevents distortion of the r.f. signal caused by reflected, delayed signals. A lossless coaxial cable may be considered as an inductance in parallel with a capacitance as shown in Fig. 28. By "lossless", we mean a perfectly insulating coaxial dielectric with negligible wire resistance. In this case, the characteristic impedance, Z 0 , of an impedance-matched cable at any frequency appears purely resistive with magnitude where L′ and C′ are the characteristic inductance and capacitance per unit length of (uniform) cable. Typically, for coaxial r.f. cables, Z 0 is 50 Ω by design. The task is then to match the impedance of the rest of the circuit (including the r.f. coil) to emulate a resistive value of magnitude Z 0. Consider the r.f. filter circuit shown in Fig. 28. The power supply acts like a current source and the choke protects the source by giving it high output impedance at high frequency. The r.f. coil may be considered as the combination of the inductance and the series resistance, R. The parallel tunable capacitance C 2 allows maximization of the power factor by canceling the inductive reactance of the r.f. coil (which increases proportional to the frequency). In other words, the tunable capacitance C 2 is necessary to maintain the imaginary part of the impedance of the circuit at zero, because the capacitance C 1 must also change with frequency to maintain the real part of the circuit impedance at the value Z 0 ; the impedance would otherwise decrease with increasing frequency causing signal reflection towards the source, and most of the voltage drop would occur across the cable.  . 28. r.f. filter with lossless cables. R includes the resistance (likely mainly from the r.f. coil). where we have used X C = -j/ωC, X L = jωL, and the complex identity y = y y * /y * , etc. For exact impedance matching, we require Re(Z load ) = Z 0 with a zero voltage-current phase difference which means that Im(Z load ) (and consequently Im(1/Z load )) is zero. From Eq. (248), therefore, we have Note that we consider R, L, and Z 0 as fixed (neglecting a possible high frequency dependence of R(ω) due to the reduction of the conducting cross-sectional area of the wire caused by the skin effect (Sec. 7.3.4.7)). From the impedance matching conditions Eqs. (249) and (250), we obtain: and ( ) ( ) ( ) (for exact impedance matching).
For these particular values of C 1 and C 2 , we construct a simplified table of voltages, currents, and impedances remembering that the cable acts like a pure resistance of Z 0 (as does the C 1 , C 2 , L, R part of the circuit), and these two sub-circuits reduce to a 2:1 voltage divider (Table 8). Table 8. Circuit values at exact impedance matching (C1 and C2 are given by Eqs. (251) and (252)). More realistically, the transmission line insulator has some conductance (represented by G cable ) and nonzero resistance, represented by R cable (as shown in Fig. 29). Distortionless cables are fabricated such that where again the prime represents "per unit length". Note that if such a circuit is used to drive M coils in parallel (for example the four coils of one arm of a 4 -N = 2 coil NRSE), the current through C 1 will be M times greater than for the single coil. Therefore, the capacitors must be rated to handle these currents. Fig. 29. r.f. filter with losses in cables (cable resistance, represented by Rcable, and conductance of dielectric, represented by Gcable).

r.f. coil frequency, currents, and voltages
The r.f. coil dimensions used in this and subsequent sections are shown in Fig. 30. From Table 8 at exact impedance matching we have (substituting the value of C 2 (ω) from Eq. (252)): where V in is the supply voltage, with where n rf is the winding density of the r.
Applying Faraday's law to a coil of inductance L, From Eq. (266) we see that the maximum voltage occurs at maximum B 0 and minimum λ. We will assume that the minimum useful wavelength is 2 Å. From Eq. (14) for a typical r.f. coil thickness l rf (in the beam direction) of 2.5 cm, we have B rf pk (l rf = 0.025 m, 〈λ i 〉 = 2 Å) ≈ 2.71 mT. The peak current in the r.f. coil with winding density n rf = 250 m -1 is approximately I rf pk (n rf = 250 m -1 , l rf = 0.025 m, 〈λ i 〉 = 2 Å) ≈ 8.64 A. This is approximately one order of magnitude less than the maximum currents required in the static field coils with similar winding densities.
The voltage between adjacent windings in a tightly-wound coil, V pk ww , must also be maintained below breakdown. For a simple voltage divider, this is simply the total voltage difference across the coil multiplied by the fractional length of one turn with respect to the total winding length on the coil. For tightly-wound rectangular windings this fraction is h rf /l axial = 1/(n rf l axial ) = 1/N rf , where N rf is the total number of turns on the r.f. coil, i.e., . max max pk pk ww pk rf axial rf

r.f. power supply voltage at exact impedance matching
If V rms PS is the rms voltage of the power supply, we equate the magnitude of the rms current in the coil at exact impedance matching (from Table 8 For R ≈ 1 Ω, n rf [m -1 ]l rf [m] ≈ 10, the supply voltage is approximately 100 V/λ [Å]. Note that this is typically much smaller than the high frequency voltages generated across the r.f. coil itself.

Power dissipation in the r.f. circuit and in the r.f. coil
At exact impedance matching, Re(Z load ) = Z 0 , Im(Z load ) = 0 (see Sec. 7.3.4.2), and the resistance of the entire circuit is 2Z 0 , therefore the heat dissipated in the whole circuit is Therefore the power dissipated as heat in the coil at exact impedance matching is ( ) ( ) where we have used the value of C 2 from Eq. (252), i.e., one half of the total power dissipated in the circuit (c.f. Eq. (270)) is dissipated in the r.f. coil at exact impedance matching.

r.f. coil cooling
Because the maximum currents required in the r.f. coils are not so large (of order 10 A), experience has shown that compressed air cooling is usually adequate to maintain them below 100 °C for frequencies of up to about 750 kHz.

The skin effect and the resistance of the r.f. coil windings
An unfortunate consequence of induced eddy currents and Lenz's law at high frequencies is the concentration of current towards the outer surface of the conductor, commonly referred to as the "skin effect". The skin depth (or thickness within which the current falls to 1/e of its outer surface value for a thick conductor), δ, is inversely proportional to the square root of the frequency: 2 , where σ is the conductivity and µ is the permeability of the wire. Two undesirables result. Firstly the resistance of the wire increases rapidly when δ is comparable to, or less than, the wire diameter. depth for copper at the highest frequencies required (~1 MHz) is about 66 µm (0.065 mm). Secondly, the increase in resistance with increasing frequency induces a frequency-dependence of the signal velocity causing dispersion even in a 'distortionless' cable. However, the latter effect is negligible for the NRSE due to the extremely narrow r.f. bandwidth for each spectrometer setting.
One solution for r.f. coils operating at about 1 MHz is to use multiple small diameter (preferably < δ ), individually-insulated copper wires in parallel rather than fewer thicker conductors. This is the case for the coils shown in Fig. 27 Note that in the low-frequency limit δ ≫ t, therefore ( ) as expected, and in the high-frequency limit t ≫ δ, therefore i.e., the band winding of thickness t in the high frequency limit has approximately the D.C. resistance of a conductor of thickness 2δ with the central core volume behaving like a perfect insulator. Applying Eqs.
(275) and (276) to the t = 0.4 mm-thick aluminum band example from Sec. 7.3.2, we see that the resistance at 1 MHz is about 5 times greater than the D.C. resistance with a current at the center of the conductor of less than 9 % of the value near the surface. In this particular example, the 1 MHz conductor resistance would drop by less than 10 % even if the conductor were made arbitrarily thick. Consequently, increasing the aluminum winding thickness beyond a few 1/10ths of a millimeter (for MHz frequencies) results in only modest power reduction with unnecessary losses of neutron transmission.

Allowable ∆B rf /B rf and ∆l rf /l rf
In each π-flipper coil, the neutron spin ideally precesses through an angle π around B rf during neutron passage through the coil. The actual angle of precession of the spin about B rf is determined by the dispersion due to the spread of incident wavelengths, as discussed in Sec. 2.2. However, an additional loss of polarization/flipping efficiency by a similar mechanism results if there is a spread in the magnitude of B rf or of the length of the r.f. field, l rf . In order to maximize intensity, we wish to maximize the operational neutron wavelength bandwidth of the spectrometer. Therefore, it is reasonable to require that the effects of We choose a reasonable criterion whereby (∆(B rf l rf )/B rf l rf ) 2 is at most 10 % of (∆λ FWHM /〈λ i 〉) 2 , i.e., 1.05 .  Table 9.
The results demonstrate that the latter creates measurable but tolerably small reductions in the flipping efficiency with respect to the effect of dispersion alone. Using Eq. (282) for l rf = 0.025 m, we have ∆l rf <~ 0.6 mm (which is not excessively demanding), with ∆B rf /B rf = ∆B rf pk /B rf pk <~ 2 % (which is more than an order of magnitude more relaxed than the required static field homogeneities (see Table 7 For the same l rf (= 0.025 m) and a wavelength range from 2 Å to 12 Å, (B rf pk varies from about 2.7 mT to about 0.45 mT respectively), we require ∆B rf pk <~ 60 µT at 2 Å and ∆B rf pk <~ 10 µT at 12 Å.

Stray Fields in the "Zero Field" Regions
There are inevitably stray fields within the "zero-field" gaps that give rise to unwanted Larmor precession around the local stray field direction. In the worst cases, these can severely reduce or even destroy the echo signal. Sources of stray fields are leakage fields from the coils themselves, the Earth's magnetic field, and other externally-produced magnetic fields. Even-N bootstrap coils greatly reduce the coil contribution by providing compact, closed return paths due to the oppositely-opposed field directions. Furthermore, the leakage field has opposite sign each side of the bootstrap coil, resulting in a first order cancellation of the Larmor precession upstream and downstream of the coil. Tight conduction of field lines between the coil pairs (and away from the zero field regions) is greatly improved by using high permeability µ-metal caps linking the coil ends. Leakage fields into the zero-field flight paths are further reduced by encapsulating the coil in a µ-metal screen with the exception of the beam path. External sources of stray field (such as the Earth's field) are practically eliminated by surrounding the sensitive flight paths with multi-skinned µ-metal shielding [24] (see also Sec. 7.7). For a mean net stray field of magnitude B stray , integrated along the spectrometer arm of length L, the net additional precession angle is Keller [25] shows that the field integral of such coils could be reduced by an additional factor of about 30 by adding the µ-metal screen around the N = 2 coil. In this case we anticipate a typical stray field integral magnitude on one side of the coil unit of about ∆ 1 ≈ 6.3 × 10 -7 Tm at B 0 = 0.04 T, with a similar stray field integral magnitude ∆ 2 on the other side (of opposite sign). If the 4-N = 2 coils are arranged as shown in Fig.  6 For the λ = 8Å example, this requires 〈∆ 1 -∆ 2 〉 <~ 3.4×10 -7 Tm for 10° net stray field precession in each arm. For ∆ 1 ≈ 6.3 × 10 -7 Tm, this means that the typical stray field cancellation need only be about 50 % in this case. This appears entirely achievable.

The Bloch-Siegert shift
In the previous discussion, the resonant component of the r.f. field has been approximated as a pure rotating field and the influence of the counter-rotating component has been ignored. However, the applied r.f. field is an oscillating field, not a pure rotating field. Bloch and Siegert [5] treated the case that really exists in the resonance coil for a spin-1/2 particle traversing a static field with a superimposed, perpendicular oscillating field. This problem does not have an exact solution. However, they showed that for increasing B 0 /B rf , the solution increasingly approximates to that of a "static + circular" field with a similarly-shaped resonance curve, but with a resonance frequency that deviates from the classical Larmor frequency, ω 0 , by a fractional amount equal to

Solution using NSE mode operation of coils
When B 0 becomes comparable to B rf , the flippers do not perform well. Köppe et al. [11] provide some idea of when this is likely to occur. Their coils cease to operate satisfactorily for static fields B 0 < 2.7 mT for λ ≤ 6 Å. Assuming that these r.f. coils are no greater than about 0.025 m thick in the beam direction, we infer from Eq. (15) that the peak r.f. field at which problems occur is for B rf pk >~ 0.9 mT or B rf >~ 0.45 mT.
Thus we assume that the π-flippers must be operated under the following condition:  The situation is more complicated in a bootstrap coil configuration (for example N = 2). In this case, all r.f. fields are switched off (as in the N = 1 case), however the static field directions are inappropriate for operation in NSE mode. Several solutions to this problem are illustrated in Fig. 32. In case (a), all static fields remain on but a π flipper is placed between the two opposing coils of a bootstrap pair, essentially reversing the field direction of the second coil. Conceivably, the field direction in the second coil of the pair could be reversed by reversing the current direction in the coil, but the two coils are often constructed from a single winding rendering this option impractical. In case (b), if the coils can be switched out of the circuit independently, only the fields that have the correct direction are energized. (c) is like (b) but only one coil of the correct field direction is energized in each arm. Note that if the static fields are provided by permanent magnets (see Sec. 9.2), only option (a) is feasible unless magnets of the wrong field direction are physically removed from the beam. For permanent magnets the NSE scan must be performed by rotating the magnets to change the field integrals in a way that does not introduce unwanted Q-dependence (see Sec. 9.3). http://dx.doi.org/10.6028/jres.119.005

An example of a combined NSE-NRSE mode scan
For NRSE operation we have τ NRSE given by Eq. (120) where M is the total number of static field coils energized. If the maximum B 0 is 0.04 T, with M = 8 (i.e., the configuration shown in Fig. 32 (a)), we can use NSE mode up to τ NSE ~ 1.8 ns for λ = 8 Å, i.e., there is a possible overlap between the upper NSE mode and the lower NRSE mode if the scheme in Fig. 32 (a) is adopted. Some dummy data points are plotted in Fig. 33 showing the scan points that result from evenlyspaced values of B 0 in the range from about 1.7 mT (the minimum field for NRSE operation in this example) to 0.04 T for both NSE mode (red circles) and NRSE mode (blue squares).

Defining the Major Instrument Parameters for the NRSE Instrument Using Coils
We now explore some of the major constraints on the instrument parameters imposed by the proposed instrumental performance goals, when combined with some technical constraints for an NRSE instrument using resonance coils. This is by no means an exhaustive list and additional compromises may be necessary. Probably the major factors are as follows: 1. We wish to access τ NRSE = 30 ns at λ = 8Å. This has implications for the minimum achievable magnitude of the product B 0 L 0 expressed in Eq. (206).
2. Once 30 ns at λ = 8 Å is accessible, we wish to achieve a resolution function signal (polarization) greater than or equal to a stated minimum value, P x 0 . The static field homogeneity, coil flatness, and beam divergence required to achieve these conditions are given approximately by Eqs. (203-205) respectively.
3. The anticipated neutron transmission of the windings should be >~ 80 % at λ = 8 Å. This concerns the thickness of the windings in the beam direction, t. The transmission for aluminum windings may be estimated using the macroscopic cross-section given in Eq. (211). 4. Maximum limitations on the static field coil current/minimum coil winding density (see Eq. (214)). 5. Capacity to remove heat from the static field coils (see for example Eq. (235) for tightly-wound rectangular cross-section aluminum windings), given the estimated constraints on the coil surface area, the winding thickness, t, the maximum operating static field, B 0 max , (as constrained by condition 1 above), and the means available for cooling outside of the neutron beam passage (see Sec. 7.3.3.5). 6. Maximum limitations on the voltage across the r.f. coil at B 0 max (see Eq. (266)). These are dictated by cabling and insulation breakdown issues with a practical maximum of about 1.5 kV. We assume, for the reasons given in Sec. 3.4, that the number of coils in the bootstrap is universally N = 2 and that the windings are made of aluminum. http://dx.doi.org/10.6028/jres.119.005 We also assume that these conditions must be satisfied at least for λ = 8 Å, thereby being automatically satisfied for λ < 8 Å, but not for λ > 8 Å. Choosing Condition 3 amounts to having a total aluminum winding thickness traversed by the neutron beam of less than 24 mm. Using the 2:1 winding thickness ratio for the static: r.f. field coil windings used in the example in Sec. 7.3.2, we treat this condition as only influencing one of the critical parameters in the above list, namely the thickness t of the static field coil windings. The condition for the present purposes is therefore stated as t ≤ 1 mm on the understanding that the r.f. coil can work satisfactorily with winding thickness ≤ 0.5 mm.
We will assume an equilibrium winding temperature of T ≈ 400 K. Minimizing P(B 0 max ) in condition 5 is aided by choosing the maximum allowable value of t (i.e., 1 mm from condition 3), so that condition 5 becomes: We assume from previously-developed coils that l axial and a must be at least about 7 times larger than the beam dimensions to achieve sufficient static field homogeneity within the beam passage. For a 3 cm × 3 cm beam, this equates to a ≈ l axial ≈ 0.2 m. Also l B0 ≈ 0.03 m, therefore, substituting these values for a typical situation we have: This condition must be true for all operating conditions and consequently also for the minimum operating wavelength, which we choose as λ = 2 Å, where the voltage is maximized. Because the r.f. field homogeneity requirements are typically an order of magnitude more relaxed (see Sec. 7.3.4.8) than the required static field homogeneities (see Table 7) at high resolution, we assume that l rf axial and a rf need be only three times the beam size (i.e., we will make l rf axial = a rf = 0.09 m). We make one further simplification to express condition 1 in terms of the maximum B 0 (B 0 max ) only. The technical conditions 2(b), 4, 5, and 6 are all worst-case at maximum field (B 0 max ). We might reduce the maximum necessary field B 0 by reasonably maximizing the inter-coil separation L 0 in condition 1. A value of L 0 = 2 m is about the longest practical value in terms of available floor space for the instrument. However the disadvantage of further increasing L 0 is that the instrumental solid angle of acceptance reduces proportional to 1/L 0 2 . Fixing L 0 = 2 m, therefore, condition 1 (Eq. (292)) becomes: The parameter values 2(c), and 3 have already been determined in this example as ∆θ max ≤ 4 ×10 -3 rad and t = 10 -3 m respectively. The remaining parameter range to be determined is 2(a). This is somewhat driven by what is achievable in the coil design, but we have seen (Sec. 7.3.3.7) that small values of l B0 aid in achieving the required static field homogeneity. Given that the static field coil must enclose both the r.f. coil and the necessary structures for heat removal, we anticipate l B0 ≈ 0.03 m as imposing an approximate practical lower limit on the static field coil length (as has been assumed in many of the examples given above). Using this value, condition 2(a) amounts to designing a coil that can achieve

Coupling Coils
The author is grateful to Roland Gähler of the ILL, Grenoble, for providing information about these coils: The µ-metal shield surrounding the coils cannot be closed because a polarized neutron beam cannot be passed through µ-metal without significant depolarization. Thus the µ-metal tube must be open-ended. The open-ended tube by itself has field lines penetrating partially into the openings, thus in order to maintain control of the polarization direction at the entrance and exit of the µ-metal shield, coupling coils (CCs) are used. Gähler et al. use a µ-metal tube of about 0.1 m diameter into which is introduced a (0.15 to 0.2) m long (in the beam direction), rectangular cross-section CC. An example of a CC penetration into a µ-metal shield on the NRSE-TAS spectrometer at the FRM-II is shown in Fig. 34. The magnetic field axis of the CC is perpendicular to the beam. The residual field of the polarizer (and analyzer) at the entrance (exit) of these coils is usually a few hundred µT. The field magnitude in the CCs is also typically a few hundred µT.
The windings on the polarizer side are bent outwards (this is visible in Fig. 34) in order to ensure an adiabatic transition from the polarizer field to the CC guide field, whilst eliminating CC windings from the neutron beam path. If the adiabatic condition is met, the neutron spins follow the direction of the CC guide field. On the inner side of the CCs, the neutrons pass abruptly through the windings and a non-adiabatic transition results, whereby the polarization direction immediately prior to passing through the windings is preserved. In order to ensure this, the CC return fields are conducted sharply into an additional µ-metal shield that surrounds the inner ends of the coil, thus avoiding a gradual stray field gradient downstream that could affect the polarization direction. Finally, the CCs (and hence the polarization direction) can be rotated through 90° without loss of polarization. For the ILL "Zeta" instrument, both the polarizer field and the initial polarization direction are vertical (parallel to z). The CCs are used to rotate the polarization to lie along x for normal instrument operation, or along z for individual tests of the flipper coils. Fig. 34. A coupling coil at the exit of a µ-metal housing on the NRSE-TAS spectrometer at the FRM-II, Garching, Germany (photo kindly allowed by T. Keller). The windings are bent outwards at the exit to avoid contact with the beam and to ensure an adiabatic transfer from the polarizer/analyzer field to the CC guide field.

Conditions for Adiabatic and Non-Adiabatic Field Transitions
"Adiabatic" and "non-adiabatic" spin transitions in spatially and/or temporally-varying magnetic fields refer to two extremes: (i) Adiabatic: the spin direction follows the field direction at all times.
(ii) Non-adiabatic: the neutron passage is sufficiently fast that the spin cannot follow the change of field direction and preserves its original direction. a neutron of constant velocity v n sees a magnetic field rotating at frequency Ω, where This situation has been represented by Ramsey [26] and other authors in terms of an effective field in a coordinate frame fixed to the rotating field (see Fig. 35). In the adiabatic case, B eff ≈ B guide (i.e., θ → 0), therefore s remains approximately parallel to B guide in the rotating frame and consequently the spin follows the change of direction of the guide field in the lab frame. In the non-adiabatic case, B eff ≈ Ω /γ n (approximately independent of B guide , θ → π/2), and the spins precess at a rate γB eff ≈ Ω. In the lab frame, therefore, where B guide rotates with Ω, the spins stand still, i.e., they do not follow the change of direction of B guide . We note that the angle θ is given by An order of magnitude for the required guide fields is obtained by considering several examples of an increasing approach to pure adiabatic rotation of spins (decreasing θ) through an angle ψ = π/2. The angle ψ is brought about by a uniform rotation of a guide field (of constant magnitude |B guide |) over a flight path of 0.5 m (a typical spacing between the polarizer and the coupling coil). The results are shown in Table 10. Table 10. Estimated minimum (fixed) guide field magnitudes, |Bguide|, required to produce a π/2 rotation of a neutron spin that follows a guide field rotation of π/2 radians over a distance of 0.

Alignment of the B 0 Fields Using Coupling Coils
The coupling coils (Sec. 7.5) provide a convenient means of aligning the static fields of the coils in the spectrometer. This is performed by rotating the field axis of the coupling coil such that the neutron spins are aligned along the required B 0 field axis. The B 0 field of each coil is switched on one at a time and the static field coil is adjusted until the maximum signal is measured in a detector placed downstream of the analyzer.

Magnetic Shielding
It is essential to reduce net stray field integrals in the "zero-field" flight paths to the order of a few ×10 -7 Tm (see Sec. 7.3.5). At high static fields B 0 , this involves magnetic screening of the individual coils units outside of the beam area. Significant sources of external magnetic fields must also be excluded. For example, the action of the unshielded Earth's magnetic field may give rise to a precession of several turns over a typical 2 m drift path, which additionally is variable depending on the orientation of the spectrometer arm. Uncompensated neighboring magnetic environments may cause worse complications, especially if the field magnitude changes. Therefore, the neutron drift paths between the coils must also be magnetically shielded.
One of the best magnetic shielding materials is so-called "µ-metal". µ-metal is an alloy with typical composition 75 % Ni, 2 % Cr, 5 % Cu, 18 % Fe and density of about 8.75 gcm -3 . It has the property of being very soft magnetically, having a very small coercive field, and an extremely high permeability at low field strengths. With a single-skinned, 1 mm thick µ-metal tube, Dubbers et al. [22] were able to obtain a shielding factor for the Earth's magnetic field of about 20, from about 40µT to about 2µT. However, the resulting several µTm field integral over 2 m drift paths is insufficient by nearly an order of magnitude for achieving the goals discussed in Sec. 7.3.5. The magnetic shielding factor is significantly improved by using multiple-skinned shields with intervening air gaps [27]. The case of triple-skinned, concentric cylindrical and spherical shields was first treated in an elementary way by Wills [24]. Dubbers [28] further simplified the cylindrical geometry, multi-skinned µ-metal case in the thin-shell approximation that agrees with the rigorous calculations to about 1 percent in most cases. He reiterates that the shielding is most effective when the shell diameters, D i , grow in geometric progression, i.e., 1 , where κ is a constant. Using this approximation, the total shielding factor, S, for n concentric shells with a constant diameter ratio κ is given approximately by ( ) ( ) where µ 1 t 1 /D 1 is the shielding factor of the innermost shell of diameter D 1 , thickness t 1 , and permeability µ 1 . Equation (301) demonstrates the value of using high permeability with n > 1, given that t 1 cannot be large for practical purposes and D 1 cannot be smaller than is allowed by the enclosed instrumentation. However, minimizing D not only increases shielding performance but also reduces the cost and weight of the shield. Magnetic shields should also be closed wherever possible since magnetic field lines can penetrate into openings by up to about five times the opening diameter. Closure maintains the reluctance path continuity, increasing shielding performance. Shield closures should also be rounded where possible because flux lines negotiate gentle radii better than sharp angles. One disadvantage with the high permeability of µ-metal is its low saturation field (the saturation field is inversely proportional to the permeability). If necessary, the magnetic shielding layer closest to the high field is fabricated from a lower permeability material to avoid saturation and successive shielding layers may be fabricated from increasingly high permeability material, as the field magnitude at each layer reduces.
After fabrication µ-metal shielding structures must be annealed in a dry hydrogen atmosphere at about 1200 °C for several hours. The hydrogen atmosphere helps remove carbon and other trace impurities. The high temperature relieves stresses from fabrication and allows the nickel crystallite grain boundaries to expand. The annealing can increase the permeability of the alloy significantly -typically by a factor of 40.
However, careful handling of the µ-metal after annealing is required. Mechanical shocks readily disrupt the nickel grain structure, negating the permeability gain.

Beam Optics for High-Resolution Operation
In order to achieve the highest resolution goals of the instrument, the neutron flight path length distribution must be narrowed by corrective optics. Some evidence for this is presented in Sec. 8.5. A detailed study of the beam optics will be presented in a separately.

Monte Carlo Simulations of NRSE Instrument Performance
Some Monte Carlo simulations are presented that illustrate and validate some of the analytical models of the NRSE developed in the previous sections. Numerical techniques are invaluable for modeling complex cases where coupled variables are involved, whilst the analytical models are useful for making rapid predictions of the instrument parameters and performance.

General Description of the Monte Carlo Simulation Method
The time-dependence of τ NRSE is implicit in the simulations. The neutrons are treated as discrete particles, each having a particular spin vector and all spin coordinate transformations are performed exactly within the limitations of the following assumptions: 1. The r.f. field is rotating in a plane perpendicular to the static field B 0 . 2. The interaction of the r.f. field component that is rotating counter to the direction of Larmor precession in the static field can be ignored. 3. r.f. frequencies of successive coils are phase locked. 4. The magnitude of B 0 is assumed large with respect to B rf .
The neutron beam is assumed to be directed along the y-axis of a right-handed coordinate system with the neutron spin initially polarized along the x-axis. The applied static fields of the coils are applied parallel or antiparallel to the z-axis and the applied r.f. field is in the x-y plane. The simulation is built around a single r.f. flipper coil module which transforms the entry momentum-spin state of the neutron into an exit momentum-spin state with a main module which handles the spectrometer geometry, source distribution, sample setup, and gathers statistics. The signs of static field directions (and hence the sense of the resonant field rotations) are handled by explicitly applying sgn(B 0 ) to the frequencies and angles in each coil as described below. For successive coils of a 4-N coil NRSE we choose , where N is the bootstrap factor. The option to fix L AB = L CD for a "symmetric" scan is also available.
The incoming wavelength distribution may be selected from rectangular, triangular, or Gaussian distributions, or else a δ function (pure monochromatic) symmetrically with respect to a specified nominal (true mean) wavelength, λ 0 . Uncertainties in the coil lengths and the static field homogeneity are handled by randomly selecting values from Gaussian distributions which are centered on the nominal values for each coil. This means that for each Monte Carlo trajectory, the uncertainty is uncorrelated with the position in the coil; however, this allows for comparison with the simple "beam-average" formulations for the resolution contributions described in Sec.6. The beam divergence is considered to be uniform and symmetric up to specified limits of the incoming beam at the entrance to the first coil and may be specified independently for x and z for a beam traveling along y. Thereafter, the collimation is imposed by the dimensions of the coil windows or sample (if present) though which the beam is required to pass. An option to specify divergences according to the simplified model in Sec. 6.4 is also available.
T T − , is also calculated to transform from the rotating frame back to the lab frame. (Note that for exact resonance (ω rf = ω 0 ), T 2 is just the identity matrix and T 2 T 1 ≡ T 1 ).

The total Larmor precession by an angle
so that the compound transformation of the spin is 3 The coil dispersion is accounted for at this stage. i.e., for the selection of Q only, the very small change in wavelength due to the scattering is ignored. For example, for an incident wavelength λ i = 8 Å (E i = 1.278 meV) with a typical NRSE energy transfer of 0.025 µeV, ∆λ/λ i ≈ ½ ∆E/E i ≈ 10 -5 , i.e., the Q-value is accurate to about 10 -3 % which is very much smaller than the incident wavelength bandwidth, which is of order several %. Also the distribution of θ s in a real situation would broaden Q significantly more than 10 -3 %. With the chosen value of Γ(Q), a Lorentziandistributed energy transfer is randomly selected according to subject to the maximum sample energy gain restriction ω ≤ E i , where E i is the incident neutron energy. Finally, the scattered neutron wavelength (velocity) is calculated from the incident wavelength and the randomly selected value of ω. The resulting value is used for propagation of the neutron downstream of the sample position.

Numerical Verification of Analytical Approximations for Coil Dispersion
The approximations represented by Eqs.  Table 1). These are shown in Fig. 37 and Fig. 38 respectively. The results corresponding to Λ FW = 0.1 (black symbols and curves) show that the approximations made in Sec. 2.2.2 for extending the single coil case to the M-coil case agree with the simulations to within about 0.01 % for all spectral shapes. For the results corresponding to Λ FW = 0.2 (red symbols and curves), the agreement is at about the 0.1 % level, and for the Λ FW = 0.5 family (magenta symbols and curves), the approximations agree to about 2.3 % for all spectral shapes. For the perfect spectrometer, the loss of echo signal due to dispersion appears to be independent of B 0 (τ NRSE ) for all practical cases.

Effects of Field Inhomogeneity, Coil Length Uncertainty, and Beam Divergence (Simplified Divergence Model) in the Absence of Flipper Dispersion
For all calculations in Sec. 8.3 the simplified beam divergence model described in Sec. 6.4 is adopted and incident and scattered beams are assumed to have uniform divergence of the same magnitude. Furthermore, all effects of flipper dispersion are effectively switched off by choosing a purely monochromatic incident beam with no subsequent energy changes. This means that the simulated polarizations tend to unity as τ NRSE → 0. The spectrometer configuration in each case is 4-N=2 bootstrap http://dx.doi.org/10.6028/jres.119.005   Fig. 43 is equivalent to the blue curve in Fig. 42. Note that 0.658 mrad Å -1 is a little less than is characteristic of polished glass (similar to a neutron guide with no metallic coating). The curves are plotted for identical ranges of B 0 from 0.001 T to 0.0393 T, but because of the λ 3 -dependence of τ NRSE , the range of the abscissa is a sensitive function of wavelength. The echo signals corresponding to the cases shown in Fig. 43 at B 0 = 0.0393T (maximum τ NRSE ) are shown in Fig. 44. Therefore, the polarizations at symmetry (L 0 = L 1 ) are those of the curves in Fig. 43 at maximum τ NRSE . Note that there is no modulation of the peak magnitude when L 0 ≠ L 1 because a purely monochromatic incident beam is being simulated and that the periodicity is inversely proportional to the wavelength, as predicted by Eq. (126). Using the same reference spectrometer setup that reaches τ NRSE = 30 ns at λ = 8 Å, Fig. 45 demonstrates the significant suppression of the echo signal as the incident and http://dx.doi.org/10.6028/jres.119.005 scattered arm divergence approaches that of a natural Ni guide (≈ 1.73 mrad Å -1 ) at λ = 8Å. In this example ∆θ i,max = ∆θ f,max = ∆θ max (simplified divergence model) in an otherwise perfect spectrometer (∆B 0 = ∆l B0 =0). The simulation is for a purely monochromatic incident beam, I(λ) = δ (8 Å).

Simulations of Spectrometer Signal Revealing Flipper Coil Dispersion
All the simulations in this section adopt the same reference spectrometer configuration used previously,   effect of dispersion for a triangular spectral distribution (Eq. (43)), the simulation results are very well reproduced by the analytical approximation (solid black curve). Figure 47 is the exact analogue of Fig. 46 except that ∆λ i FWHM /〈λ i 〉 is increased from 10 % to 30 %, which exaggerates the effect of coil dispersion.

Simulations with an Improved Divergence Model and Sample/Beam Size Effects (No Corrective Optics)
In the preceding calculations (Sec. 8.3 and Sec. 8.4), the simplified beam divergence model (Sec. 6.4) was used to verify the validity of the analytical approximations given in Sec. 6. This model is useful for predicting order-of-magnitude divergence effects, however, the incident beam is usually provided by a neutron guide that gives rise to approximately random x and z components of the trajectory angle up to maxima of θ c x (λ i ) and θ c z (λ i ) respectively. Furthermore, the scattered beam divergence, in the absence of special optics, is usually defined by the sample size and the collimation between the sample and the detector. This more realistic situation is sketched in Fig. 52   The resulting function P(θ) is illustrated in Fig. 54. Therefore, clearly an improved model is necessary for more typical instrumental scenarios. In the following all coils are assumed to have equally-sized beam-defining windows at their entrances and exits. The entrance of the first coil is assumed to be uniformly-illuminated with a beam that has uniform x-y and z-y plane angular distributions with |θ |up to θ c x (λ i ) = κ x λ i and θ c z (λ i ) = κ z λ i respectively, where κ x and κ z are independently-specified constants. The sample is assumed to be a thin cylindrical shell of radius r with its axis parallel to the z-axis. The neutron trajectories arriving at the sample are those that join random points on the first coil entrance window and random points on the sample without obstruction, subject to the maximum divergence constraints |θ x | ≤ θ c x (λ i ) and |θ z | ≤ θ c z (λ i ). The sample is assumed to scatter isotropically without self-shielding so that all unobstructed trajectories between the scattering point and the exit window of the final coil are equally probable and 100 % detected. Figure 55 shows example resolution functions using this model for λ = 8 Å with ∆λ/λ = 10 % (triangular) for three coil window sizes (w win = h win = 1 cm, 2 cm, and 3 cm) assuming that a natural Ni guide (i.e., with κ x = κ z = 1.73 × 10 -3 rad Å -1 ) is placed very close to the first coil entrance. The sample diameter, D sam , and height, h sam , in each case are chosen so that the projected sample cross-sectional area is equal to the window size (i.e.,    Fig. 57. In this case, w win /L 0 ≈ 9θ c (1 Å), therefore we expect that the incoming beam divergence is determined by the guide characteristics rather than the coil window size. Indeed, from Fig. 57 we see that P(θ) for the incident neutrons resembles that of the neutron guide (c.f. Fig. 54), whereas the scattered divergence is determined more by w win (h win ) and resembles that of the maroon curve in Fig. 56, as expected. It is clear from Fig. 55 that, without corrective optics to narrow the flight path distribution, significant degradation of the resolution function is expected with typical neutron beam delivery systems and beam sizes, if high instrument resolution is required. Reducing w win (h win ) and the incident beam divergence could significantly compromise data collection rates. However, corrective optics requirements will be considered elsewhere.

Resolution Effects for Asymmetrical Configurations of the Spectrometer
This section deals specifically with instrumental resolution effects in the general asymmetrical spectrometer case (δ (BL) ≠ 0). In NRSE spectrometers, it is customary to fix B 0 and vary δL, hence results are plotted in terms of δL = L 0 -L 1 . The resolution as a function of asymmetry relates to the range of frequencies in the scattering function that can usefully contribute to the signal, which ultimately limits the incident neutron wavelength bandwidth.

Simulated Versus Theoretical Resolution Curves and Asymmetry-Dependence of Flipper Coil Dispersion
In   Fig. 36 and Fig. 37), the fitted theoretical functions also describe well the simulated echo functions (as evidenced by the relatively small oscillations in the residuals). Therefore, the fitted constants are quite close to the values provided by these equations with M = 8 total coils (see also Table 1). The theoretical echo functions do not account specifically for the cumulative dispersive spin excursions out of the r.f. field plane as the neutron passes through multiple coils and the increased structure in the residuals at larger ∆λ i /〈λ i 〉 is likely due to this shortcoming rather than a real asymmetry-dependence http://dx.doi.org/10.6028/jres.119.005 of the dispersion. Nonetheless, the indications are that dispersion is only weakly dependent on the spectrometer asymmetry, if at all, under typical conditions.

Asymmetry-Dependence of Static Field Inhomogeneity, Coil Length Uncertainty, and Beam Divergence in Typical Circumstances
In the previous section it was shown that the depolarization due to flipper coil dispersion is roughly asymmetry-independent for moderate ∆λ i /〈λ i 〉. The resolution function for the case of a rectangular incident spectrum with ∆λ i /〈λ i 〉 = 10 % in an otherwise perfect spectrometer (one in which the only source of instrumental imperfection is flipper coil dispersion) has already been shown in Fig. 58 for the reference spectrometer configuration. Figure 62, Fig. 63  simulations are compared directly (with no fit parameters) against the product of the theoretical "perfect (dispersionless) instrument" resolution function for the rectangular incident wavelength spectrum (Eq.   Fig. 49 with ∆λi FWHM /〈λi〉 = 30 % (triangular). The peak polarization at zero asymmetry should match the values at τNRSE = 30 ns for the resolution (red symbols) and quasielastic (black symbols) in Fig. 49 Table 1 and Fig. 36. The small residual fluctuations about zero (turquoise curve) imply that the effect of dispersion is approximately independent of the spectrometer asymmetry in this case.
http://dx.doi.org/10.6028/jres.119.005  Table 1 and Fig. 37. The small residual fluctuations about zero (turquoise curve) imply that the effect of dispersion is approximately independent of the spectrometer asymmetry in this case.
http://dx.doi.org/10.6028/jres.119.005  Table 1 and Fig. 36. The increased structure of the residuals (turquoise curve) is probably due to cumulative out-ofrotating plane excursions of the spin that is not accounted for in Eq. (127), nonetheless, there appears to be no strong asymmetrydependence of the dispersion, even at these large values of ∆λi/〈λi〉.   Table 1 and Fig. 37, although clearly the data is much less well represented with only a constant fitting parameter. The increased structure of the residuals is probably due to cumulative out-of-rotating plane excursions of the spin that is not accounted for by Eq. (128), nonetheless, there appears to be no strong asymmetry-dependence of the dispersion, even at these large values of ∆λi/〈λi〉.    in this example is revealed by the relatively small residuals (blue curve). There is apparently no strong asymmetry-dependence of the effects of ∆θmax in this case.

General Instrument Features
The general features of a so-called MIEZE-II spectrometer [29] are shown in Fig. 65. The configuration shown is equivalent to a multi-arm conventional arrangement of flipper coils, but with the fourth coil unit replaced by a "thin" detector at exactly the same location. The second and third bootstrap coils are created from the annular coil surrounding the sample area. In contrast to the similarly-named MIEZE spectrometer (see Ref. [16]), the r.f. frequency of all coil units is identical. The discussion in Sec. 4.2.1.3 illustrates the effect of eliminating the fourth coil: When the r.f. angular frequency is tuned to the Larmor frequency, ω 0 , the neutron spin-up and spin-down states retain their kinetic energy splitting after leaving the third coil unit, corresponding to a Larmor precession of angular frequency 2ω 0 (for N = 1 coils), or 4ω 0 (for N = 2 coils). Because the quasielastic echo point occurs at L 0 = L 1 for the 4 identical coil unit arrangement, the polarization at L 1 = L 0 (the detector plane in the MIEZE-II) is modulated at angular frequency 2ω 0 (N = 1), or 4ω 0 (N = 2), with maximum amplitude.

Toroidal r.f. Solenoid
An annular π-flipper illustrated in Fig. 65 would be a new development for NRSE. The design of the r.f. coil depends on maintaining voltages within reasonable limits.

Self-inductance of the toroidal r.f. solenoid
The self-inductance of a toroidal r.f. solenoid, radius r toroid , is approximately where N rf is the total number of turns, l rf axial is the axial length of the solenoid (in this case the mean circumference -l rf axial = 2π r toroid ), and A rf is the cross-sectional area of the r.f. solenoid (the area enclosed by a single r.f. winding -i.e., the equivalent of a rf × l rf in Fig. 30 We now consider the dimensions of the toroidal coil. The tolerable uncertainty on B rf × l rf is somewhat relaxed for the r.f. coils when compared with the static field coil requirements at the highest values of B 0 , because only a π rotation of the spin is required around B rf . The flipping efficiency is naturally limited by coil dispersion so that relaxing the tolerance on ∆(B rf l rf ) is usually accompanied by a restriction of the bandwidth, ∆λ (see Sec. 7 As an example, we assume that a coil height (side of the rectangle perpendicular to the beam direction) a rf ≈ 0.1 m provides sufficient r.f. field homogeneity within the beam area. The toroid radius, r toroid , must be sufficiently large to accommodate typical scattering sample environments. A reasonable value is r toroid ≈ 0.3 m. The choice of r toroid does not affect the instrumental resolution significantly, but it affects the usable solid angle. In order to estimate a worst case, we use the maximum value of B 0 and the minimum value of 〈λ i 〉 from the previous discussions (about 0.04 T and 2 Å, respectively). It follows, from Eq. (314), that satisfying n rf (B 0 = 0.04 T, 〈λ i 〉 = 2 Å, r toroid = 0.3 m, a rf = 0.1 m) ≤ 16 m -1 (1 turn every 6.3 cm) maintains the r.f. voltage below 1.5 kV in this case.

Resistance and inductive reactance of the toroidal r.f. solenoid
The resistance of the r.f. coil winding is where l w rf is the total length of the winding and A w rf is the cross-sectional area of the r.f. wire, i.e., On resonance, with B 0 = 0.04 T, we have ω rf ≈ 7.3×10 6 rad s -1 (ν rf = 1.17 MHz) (Eq. (9)). For n rf = 16 m -1 , r toroid = 0.3 m, and choosing a typical A rf = 0.1 × 0.025 = 2.5 × 10 -3 m 2 , we have (from Eq. (313)) L ≈ 1.5 µH, from which we obtain X L ≈ 11 Ω. Therefore, X L ≫ R toroid at the highest frequencies (in this example, by more than three orders of magnitude).

Current and power dissipated in r.f. coil
With a maximum required r.f. field magnitude of about 2.7 mT (see Sec. 7.3.4.3) and a winding density n rf = n rf (r toroid = 0.3 m) = 16 m -1 , Eq. (213) implies that a peak current I pk ≈ 2.7 × 10 -3 /(1.26 × 10 -6 × 16) = 134 A is required. Therefore, having eliminated the high voltage problem we appear to run into problems with peak current. This is mitigated by increasing the minimum operational wavelength. Nonetheless, because the r.f. coil load is almost entirely inductive at high frequencies (see previous example), the high frequency current in the coil lags the voltage by approximately 90°. The heat dissipated in the r.f. coil is only that due to the resistance. In the example given in Sec. 9

Requirements for the MIEZE Detection System
As demonstrated in Sec. 6.6, accessing high-resolution requires flight path length uncertainties in the several tens of microns range, with frequencies in the 1 MHz range. Therefore, the active part of the detector must be flat and capture neutrons within tens of micron thicknesses with reasonable efficiency. The most suitable detector type appears to be a scintillator-photomultiplier combination. The charged particles for activation of the scintillation originate from a nuclear reaction produced by the absorption of thermal neutrons. The very small absorption depth probably requires a 6 Li-containing compound such as 6 LiF, which produces negligible gamma radiation. Scintillator material such as ZnS:Ag, ZnS:Cu,Al,Au have the advantage of rapid decay times (no afterglow). The data acquisition response time should be preferably within the 1 ns to 10 ns range with signal handling up to about 4 MHz, if N = 2 bootstrap coils are used. (Note that the signal frequency is 4ω 0 for N = 2 -see Sec. 9.1.1).

Criteria for Permanent Magnet NRSE Options
An important limitation on the static field coil is the restriction on the winding thickness parallel to the beam, imposed by neutron absorption and scattering. In Sec. 7.3.3.1 we saw that this may lead to significant heat generation at high fields, unless the coils are cryogenically cooled. We now consider the feasibility of replacing the static field coils by a ferromagnetic or anti-ferromagnetic material that transmits neutrons. The static field magnitude is fixed in this scenario, therefore, a scan of τ NRSE might involve a scan of the r.f. unit separation in each arm of the spectrometer, such that that δ (BL) is maintained at zero (the echo point). This contrasts with varying the static field magnitude at fixed L in the coil case. A quasi-elastic NRSE spectrometer using permanent magnets to provide the static field (N = 1) is shown in schematically in Fig. 66.   Fig. 66. Schematic of a quasi-elastic NRSE instrument using electrically-insulating permanent magnets to provide the static field with a superimposed r.f. field.

Comparison of Static Field Coil and Permanent Magnet NRSE (a) Coil
• τ NRSE scan usually fixes L 0 , L 1 and varies B 0 .
• High resolution applications typically require very flat and parallel windings. • Production of (0.03 to 0.04) Tesla fields with thin (in the beam direction) Al windings is challenging. Heat dissipation is proportional to the coil surface area and can reach several kW with typical windings, unless cryogenically cooled. • Maintaining adequate coil cooling without interfering with the beam path is challenging. (b) Permanent magnet • τ NRSE scan has fixed B 0 , B 1 , vary both L 0 and L 1 .
• High resolution applications require similar dimensional tolerances to the coils (but probably easier to achieve). • Field homogeneity is very good inside the magnet and field boundaries are abrupt.
• Requires a neutron-transparent magnetic material.
• Magnet must reside inside the r.f. coil -requires magnetic material to be electrically-insulating.
• Magnet likely requires an externally-applied saturation field.
• Resonance width requires field magnitudes in each unit to be similar to within a few tens of µT.
• Significantly reduced heat removal problem.
• Compact, with no electrical coil circuitry.

Definition of the Required Instrument Parameters Using Permanent Magnets
In the following, we develop a set of inequalities defining the major parameters required to achieve the desired spectrometer performance using permanent magnets.

Coil unit geometry
The coil unit consists of a permanent magnetic material enclosed by an r.f. coil (with a perpendicular field axis) as shown in Fig. 67. Henceforth, we refer to the dimensions defined in Fig. 67.  (323)

Minimum wavelength (λ = 2 Å) r.f. voltage criterion
An approximate expression for the peak r.f. voltage in terms of the r.f. coil parameters and the neutron wavelength is given by Eq. (266), where a rf and l rf axial are shown in Fig. 67

Tolerance criterion for B rf -r.f. penetration of the permanent magnet and absorbed r.f. power
Variations of the magnitude of B rf within the static field region lead to reduced flipping efficiencies and, consequently, reduced signal magnitudes. One source of attenuation of B rf is absorption of the r.f. field by the magnetic material with the associated heating. In the medium-wave (MW) to short-wave (SW) band that is relevant to the NRSE (far from molecular vibrations that reside in the > 100 GHz microwave range), the average magnitude of the Poynting vector (which, for a plane wave, is the energy density × the phase velocity) in a material of conductivity, σ, permeability, µ, and permittivity, ε, may be expressed as which is the power per unit slab area, ac. From Eq. (335) we note that almost 90 % of the r.f. energy is absorbed in the initial thickness δ, as indicated in Fig. 68. In fact, after a thickness l B0 , the power absorbed per unit slab area is The power absorbed per unit area increases as the square root of the frequency. The required electromagnetic properties of the magnetic material are estimated by assuming that the r.f.
field attenuation corresponds to a value ∆B rf which produces a precession angle π-β around B rf (see Sec. 2.2), in which β must not exceed ± 2.5°. This ensures that ∆B rf does not significantly compromise the usable bandwidth ∆λ/λ, (± 2.5° corresponds to the equivalent effect produced by ∆λ/λ ≈ 3 % FWHM).
Thus, by setting 0.03, subject to the minimum B 0 criterion (Eq. (323)). If the criterion in Eq. (342) is satisfied, the r.f. power absorption is almost uniformly distributed over the slab thickness and the power absorbed in the slab volume is ( )    Ref. [30]). The left hand side in this case is about 1.5 × 10 -8 , which is much closer to the requirement.
Alternatively, the ∆B rf /B rf condition must be relaxed. Nonetheless, the materials issues for a permanent magnet option appear to be the principal challenge, especially in view of the neutron transmission constraints.

Potential Problems with the Multi-Angle Permanent Magnet Configuration
In a permanent magnet, multi-angle NRSE arrangement, a potential geometrical issue is either that of crowding of the fourth flipper coil units (or detectors in a MIEZE-II configuration) as τ NRSE (and hence L 0 ) is reduced, or that of mechanical interference of the coil units with the high-resolution optical elements (Fig. 69)  optimized for the highest resolution (i.e., for L 0 = L 0 max ) and may not be required when measuring shorter Fourier times. An NSE mode of operation could also take over at short Fourier times. Because the magnetic field magnitude is fixed, one cannot adopt the method described in Sec. 7.3.6.2 for the NSE mode. A possible solution consists of rotating the magnets to change the field integral (see following section).

An NSE Permanent Magnet Configuration?
One may compare the Fourier time ranges of a NRSE spectrometer with that of the NSE configuration using permanent magnets. As there is no oscillating field in the NSE, the precession field integral is varied by changing the magnet tilt, as opposed to the magnet separation. However, the spin-up and spin-down neutron k-vector components that are normal to the field boundaries are split in magnitude inside the field, whereas the parallel components are not. The result is that the locus of constant spin-echo phase is Q-dependent. Whilst this property is exploited for measuring widths of dispersive excitations [29], it is problematic for quasielastic scattering, where a given spin-echo phase is obtained for a range of energy transfer-Q magnitude combinations allowed by the broad incident wavelength band and the beam divergence. A possible mitigating solution uses opposing symmetric tilts (as opposed to tilts in the same sense), as shown schematically in Fig. 70  For any ϕ max this is unacceptably thick for thermal neutron transmission, therefore a permanent magnet NSE configuration is feasible only for measuring the lower Fourier time range (as was the case with the coils -see e.g. Fig. 33).

Neutron Guide Requirements
The static field homogeneity and corrective optics requirements demand a small area, low divergence, cold neutron beam. This may be provided by a curved or curved-straight polarizing neutron guide or a conventional neutron guide followed by a polarizer. If the beam monochromatization is provided by a velocity selector, the polarizing elements are placed downstream. A polarizing neutron guide at FRM-II is shown in Fig. 71.
A curved-straight neutron guide arrangement, designed according to the prescription for "Phase-Space Tailoring" (PST) [31,32], is particularly suitable for this application. Despite the curved section, a PST guide is capable of delivering a beam with optimal intensity and uniform spatial and angular distributions, for all wavelengths exceeding a threshold, λ′, determined by the guide geometry and reflective coatings.
From the estimates given in Table 7, it is likely that the incident beam divergence tolerances are stringent for high-resolution operation. Even if beam divergence dominates the instrumental depolarization, the tolerances in Table 7 are relaxed by only a factor of √3. This requires a critical angle of reflection of about 50 % of natural Ni -about that of polished glass -to obtain P x 0 = 0.5 at (τ NRSE = 30 ns, λ = 8 Å) in this example. This degree of beam collimation is not required for lower resolution measurements, therefore a likely design would introduce additional collimation, as necessary, into a more divergent beam.
To illustrate the implications of the high-resolution limits for a PST guide design, we choose a beam size W × H = 3 cm × 3 cm with a total length of the curved-straight guide combination, L tot = L c + L str = 50 m, where L c is the length of the curved section and L str is the length of the straight section. The short neutron wavelength filtering ability of the curved section (assuming no direct line of sight) is expressed in terms of the "characteristic wavelength", λ c . In the small angle approximation, λ c is given by http://dx.doi.org/10.6028/jres.119.005 asymmetries introduced by the curved section [33]. A curved guide is considered "long" if it has no direct line-of-sight [LOS]. For this to be true, L c must satisfy: 8 .
Particularly favorable gamma-ray filtering occurs when L c ≥ 2L LOS , since neither the direct nor the oncescattered gamma rays from the source are viewable from the guide exit. Thus, in the following examples we set a desirable (but not necessary) constraint L c = 2L LOS , i.e., L c = 4√(2Wρ). The lateral displacement of the curved-straight guide exit with respect to the projected axis at the guide entrance (a useful quantity when considering instrument placement) is given by Parameters for several guides that satisfy the PST guide conditions with the above constraints are summarized in Table 11. The guide systems described in Table 11 are illustrated schematically in Fig. 72. Their simulated performance (with no velocity selector or polarizer) at the NCNR Unit 2 liquid hydrogen cold source is shown in Fig. 73. The simulated intensity of the NG-5 guide (a 58 Ni-coated optical filter) at the NSE instrument is also shown under the same conditions of no velocity selector and no polarizing cavity. Figure   74 shows the horizontal angular distributions at λ = 8 Å (greater than λ′ for all models in Table 11). The horizontal angular distributions show the expected uniformity within the critical angle limits of the straight sections (indicated by the vertical lines). Figure 75 shows simulated integral fluxes that could be expected at the guide exits when using a typical Dornier-type velocity selector operating at 10 % ∆λ/λ (FWHM), with a polarizing cavity of wavelength-independent transmission 0.45. The predicted flux of the m str = 1 guide is comparable to that of the NG-5 under similar conditions, with a slightly reduced beam divergence (see Fig. 74).
http://dx.doi.org/10.6028/jres.119.005 Fig. 72. Modeled curved-straight guide geometries for simulation of the PST guide systems described in Table 11. Fig. 73. Simulated differential flux spectra (dϕ/dλ) at the exits of the PST guide systems described in Table 11 and for the NCNR guide NG-5 (a 58 Ni-coated optical filter), all assuming no velocity selector and no polarizer. The source model is the NCNR Unit 2 liquid hydrogen cold source.

Appendix A: Summary of Useful NRSE Formulas
(L 0 = length between coil centers, l=total coil length, l Β0 = length of one π-flipper coil assuming the static field region encloses the r.f. field region, l π =length of combined r.f. and static field region (usually=l rf for π flipper coils), N=number of π flipper coils in Bootstrap, M=total number of π-coils traversed by the beam in the instrument).

Effects of Dispersion: Flipping efficiencies
Flipping efficiency for a wavelength λ i for single flipper tuned for resonance and for exact π flips for the mean neutron wavelength 〈λ i 〉. Approximate instrumental tolerances to achieve resolution goal P x 0 (equal contributions from ∆B 0 , ∆l B0 , and ∆θ max ) -see Sec. 6 (i) Tolerance on B 0 field in each π-flipper for a 4-N coil instrument (Gaussian -equal contributions)   (ii) Rectangular incident wavelength spectrum ("perfect instrument")  (128)). (iv) Gaussian incident wavelength spectrum ("perfect instrument") (