# Concepts and engineering aspects of a neutron resonance spin-echo spectrometer for the National Institute of Standards and Technology Center for Neutron Research.

5. 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:

[[phi].sub.1] = 2N[[gamma].sub.n] [B.sub.0][L.sub.0]/[v.sub.i].[[??].sub.0] (106)

and

[[phi].sub.2] = 2N[[gamma].sub.n] [B.sub.1][L.sub.1]/[v.sub.f].[[??].sub.1], (107)

where [[gamma].sub.n] is the neutron gyromagnetic ratio, [v.sub.i,f] is the initial/final neutron velocity vector and [[??].sub.0,1] 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.

5.1 Small Divergence Approximation

For small beam divergences and coil axes that are perpendicular to vi and vf, we can approximate Eqs. (106) and (107) by

[[phi].sub.0] = 2N[[gamma].sub.n] [B.sub.0] [L.sub.0]/[v.sub.i] (108)

and

[[phi].sub.1] = 2N[[gamma].sub.n] [B.sub.1] [L.sub.1]/[v.sub.f] (109)

respectively, where [v.sub.i] and [v.sub.f] are scalars, so that the net spin turn at the analyzer is given by Eq. (69). For quasielastic non-spin flip scattering that is sufficiently low energy transfer ([delta]v [much less than] [v.sub.i]), we can write

[v.sub.f] [approximately equal to] [v.sub.i] + [delta]v (110)

so that Eq. (69) is approximately

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (111)

where

[delta](BL)= [B.sub.0][L.sub.0] - [B.sub.1][L.sub.1] (112)

is often called the spectrometer asymmetry. It is conventional to perform NRSE asymmetric scans by fixing the static field and scanning [delta](BL)= [L.sub.0]-[L.sub.1]. Also we will assume that [B.sub.0] = [B.sub.1] and replace [delta](BL) by the slightly less general expression [delta] (BL) = [B.sub.0]([L.sub.0] -[L.sub.1]) = [B.sub.0] [delta]L 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 [[phi].sub.NRSE], averaged over all the scattered neutron trajectories i.e.,

[P.sub.x] = <cos[[phi].sub.NRSE]>, (113)

where <> implies a statistical average over a large sample of scattered neutrons. From Eq. (111) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (114)

This expression must be averaged over all possible values of [v.sub.i] (the incident spectrum) and all possible values of [delta]v (= [v.sub.i]-[v.sub.f]) determined by the scattering. Noting that Q is approximately independent of [omega], i.e.,

Q [approximately equal to] 2[m.sub.n] [v.sub.i]/h sin[theta], (115)

where 2[theta] is the scattering angle, we can write

P ([v.sub.i], [delta]v) d ([delta]v) [equivalent] S (Q,[omega]) d[omega]. (116)

For small energy transfers (small [omega] and [v.sub.i] [approximately equal to] [v.sub.f]) we have from the definition of kinetic energy:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (117)

If S(Q, [omega]) is symmetric in a (usually a good approximation for quasielastic scattering) and substituting [delta]v [approximately equal to] [??][omega]/[m.sub.n][v.sub.i] from Eq. (117), the average over the [delta]v distribution characteristic of the scattering sample for a given [v.sub.i] becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (118)

where the "sine" part of the expansion in Eq. (114) disappears in the integral for symmetric S(Q, [omega]) and the denominator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for a normalized scattering function is implicit. Note that the quantity preceding [omega] in the second cosine argument in Eq. (118) when [delta](BL) = 0 (i.e., [B.sub.0][L.sub.0] = [B.sub.1] [L.sub.1]) is often referred to as the spin-echo time, [[tau].sub.NRSE], i.e.,

[[tau].sub.NRSE] = 2[??][[gamma].sub.n]N[B.sub.0][L.sub.0]/[m.sub.n][v.sup.3.sub.i] = [[gamma].sub.n]/[pi] [([m.sub.n]/h).sup.2] N[B.sub.0][L.sub.0][[lambda].sup.3.sub.i], (119)

where

[[tau].sub.NRSE] [ns] = 0.37271 N[B.sub.0] [T][L.sub.0] [m][([[lambda].sub.i] [[Angstrom]]).sup.3] [equivalent to] 1.27794 x [10.sup.-2] N[v.sub.0] [MHz][L.sub.0] [m] [([[lambda].sub.i] [[Angstrom]]).sup.3]. (120)

Equation (118) must be averaged over the normalized incident velocity distribution, F([v.sub.i]), so the final polarization is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (121)

where the denominator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is implied. Expressed in terms of the normalized incident wavelength distribution, I([[lambda].sub.i]), Eq. (121) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (122)

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 [lambda], 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.sub.0] and [L.sub.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.

5.2 Special Cases for No Sample, Isotope Incoherent Elastic Scattering, or Small [omega] ("Resolution Function")

Isotope incoherence implies scattering that is both Q-independent (so the [lambda]-dependence of the scattering function can be ignored). Elastic scattering, or small [omega], 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 [omega][[tau].sub.NRSE] [less than or equal to] 0.045[pi] Under these conditions Eq. (122) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (123)

where for brevity we define

A = 2N[[gamma].sub.n] [m.sub.n] [B.sub.0]/h [delta]L (124)

and the integral over [omega] evaluates to S(Q) since there is no other [omega]-dependence. The integral over [[lambda].sub.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.

5.2.1 Purely Monochromatic Beam

For a purely monochromatic incident beam, I([[lambda].sub.i]) = [delta]([[lambda].sub.0]), and we have simply:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (125)

Therefore, in the absence of instrumental imperfections (see Sec. 6), the resolution function has a pure cosinusoidal form of constant amplitude for any [delta](BL) with a periodicity given by

[delta][(BL).sub.2[pi]] = h/[m.sub.n] [pi]/[[gamma].sub.n]N 1/[[lambda].sub.0] = [pi][v.sub.0]/[[gamma].sub.n]N. (126)

5.2.2 Rectangular Incident Wavelength Spectrum

For a rectangular incident wavelength spectrum of full width [DELTA] [[lambda].sub.FW], centered about [[lambda].sub.i]= ([[lambda].sub.i]), the wavelength-dependent integral in Eq. (123) becomes: 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided that the lower wavelength limit of integration is greater than zero, so that

[P.sub.x] ([omega] [right arrow] 0) [approximately equal to] 2S(Q)/A[DELTA] [[lambda].sub.FW] sin (A [DELTA] [[lambda].sub.FW]/2) cos (A<[[lambda].sub.i]>), Rectangular incident spectrum, (127)

apart from depolarizations resulting from instrumental imperfections and flipper coil dispersion.

5.2.3 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 = [DELTA][[lambda].sub.FWHM] and mean wavelength <[[lambda].sub.i]>, the wavelength-dependent integral in Eq. (123) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided that the lower wavelength limit of integration is greater than zero, so that

[P.sub.x] ([omega][right arrow]0) [approximately equal to] S(Q)/[DELTA] [[lambda].sup.2.sub.FWHM][A.sup.2] {2cos (A <[[lambda].sub.i]>)[1 - cos (A[DELTA][[lambda].sub.FWHM])]} Triangular incident spectrum, (128)

apart from depolarizations resulting from instrumental imperfections and flipper coil dispersion.

5.3 Special Cases for Quasielastic Neutron Scattering (QENS) Symmetric Scans

In this case, [delta]L [right arrow] 0 in Eq. (122) and we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (129)

For quasielastic scattering, the scattering function is represented by a Lorentzian

S (Q,[omega]) = 1/[pi] [gamma](Q)/[gamma][(Q).sup.2] + [[omega].sup.2] (l30)

where [gamma] (Q) = [GAMMA] (Q)/[??], where [GAMMA](Q) is the energy half-width at half maximum. Performing the integral over co, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (131)

We now consider the possible [lambda]-dependence of [GAMMA](Q). Following Hayter and Penfold [17], we consider a common case of self-diffusion at low Q at fixed scattering angle, [theta]. For simplicity, we ignore the very small change in energy of the neutrons on scattering, so we may make the approximation

[GAMMA](Q) = [??]D[Q.sup.2] [approximately equal to] 16[??][[pi].sup.2] D[sin.sup.2][theta]/ [[lambda].sup.2.sub.i] QENS, self-diffusion low Q (Q [approximately equal to] [Q.sub.el]), (132)

where [theta]is the scattering angle, so that Eq. (131) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (133)

where we have set

K = 16[pi][[gamma].sub.n] [([m.sub.n]/h).sup.2] N[B.sub.0][L.sub.0] D [sin.sup.2] [theta] = 16[[pi].sup.2]D [sin.sup.2] [theta]/[[lambda].sup.3.sub.i] [[tau].sub.NRSE]. (134)

5.3.1 Purely Monochromatic Beam

For a purely monochromatic incident beam, I([[lambda].sub.i]) = [delta]([[lambda].sub.0]), and Eq. (133) becomes simply:

[P.sub.x] ([delta]L [right arrow] 0)[approximately equal to]exp(-K[[lambda].sub.0]) [equivalent to] exp([GAMMA](Q)[[tau].sub.NRSE]/[??]) I([[lambda].sub.i]) = [delta]([[lambda].sub.0]). (135)

5.3.2 Rectangular Incident Wavelength Spectrum

For a rectangular incident wavelength spectrum of full width [DELTA][[lambda].sub.FW], centered about [[lambda].sub.i]= <[[lambda].sub.i]>, Eq. (133) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (136)

5.3.3 Triangular Incident Wavelength Spectrum

For a triangular incident spectrum of FWHM = [DELTA] [[lambda].sub.FWHM] and mean wavelength <[[lambda].sub.i]>, Eq. (133) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (137)

5.3.4 Gaussian Incident Wavelength Spectrum

For a Gaussian incident spectrum of FWHM = [DELTA] [[lambda].sub.FWHM] (standard deviation [sigma]) and mean wavelength <[[lambda].sub.i]>, Eq. (133) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (138)

5.4 Detected Signal

5.4.1 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

[I.sub.+] = 1/2(1 + (<cos[[phi].sub.NRSE]>)=1/2(1 + <cos [omega][[tau].sub.NRSE]>)=1/2(1 + [P.sub.x]), (139)

where [P.sub.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.

Monte Carlo simulations illustrating the behavior of Eq. (139) in various situations are shown in Fig. 11. In these examples there are no sample size effects and [DELTA] [(B).sub.0] = [DLETA][l.sub.B0] = 0 (zero field inhomogeneity and perfect dimensions of the flipper coils), but the effects of beam divergence, incident neutron bandwidth, and non-elastic scattering are illustrated. The effects of spectrometer imperfections are analyzed further in Sec. 6 and additional simulation examples are given in Sec. 8. Columns 1 to 3 of Fig. 11 are for elastic, non-spin-flip (isotope incoherent elastic) scattering (or no sample). Column 4 is for quasielastic, isotope incoherent (non-spin-flip) scattering. Additionally for columns 1 and 4 zero beam divergence is assumed. These particular simulations were performed for a 4-N=2 bootstrap coil NRSE with [L.sub.l] = 2.0 m, [l.sub.B0] = 3.0 cm, [l.sub.g] = 0.0 cm, and [[lambda].sub.0] = 8 [Angstrom], for [10.sup.-3] [less than or equal to] [B.sub.0](T) [less than or equal to] 0.025. The asymmetric scan is performed with [absolute value of [B.sub.0]] = [absolute value of [B.sub.1]], with ([L.sub.0] - [L.sub.1]) varied ten minimum periods each side of the symmetric position (i.e., between [+ or -]10 [pi][v.sub.0]/[[gamma].sub.n]N[B.sup.max.sub.0], where [B.sub.0.sup.max] is the maximum applied static field [0.025 T], corresponding to [[tau].sub.NRSE] = 19.1 ns). [delta](BL) was varied by changing [L.sub.1] with respect to [L.sub.0]. (i.e., [delta](BL) = B[delta]L). The incoming and outgoing beam divergence, if any, is equal and uniform up to a maximum [DELTA][[theta].sub.i,max] = [DELTA][[theta].sub.f,max] = [DELTA][[theta].sub.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.sub.0] = [L.sub.1]. The left hand column of Fig. 11 illustrates the effect of broadening [DELTA][[lambda].sub.i], for elastic, non-spin flip scattering (or no sample). In the extreme, purely monochromatic case of [DELTA][[lambda].sub.i] = 0, the signal is cosinusoidal with respect to [delta]L (as predicted by Eq. (125) with [delta](BL) = B[delta]L for one signal period given by Eq. (126). For [DELTA][[lambda].sub.i] > 0, the maximum signal is achieved at the symmetrical spectrometer setting and, as [DELTA][[lambda].sub.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.sub.0] (= [B.sub.1]) (and hence [[tau].sub.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 [DELTA][[lambda].sub.i]/<[[lambda].sub.i]> = 10 % (column 3). The fourth column demonstrates the increasingly rapid decay of the echo point signal with respect to [[tau].sub.NRSE] as the quasielastic width is increased (as predicted by Eq. (135) for a purely monochromatic incident beam ([DELTA][[lambda].sub.i] = 0)).

5.4.2 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 "+" 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

[P.sub.P] = [I.sup.+.sub.P] - [I.sup.-.sub.P]/ [I.sup.+.sub.P] + [I.sup.-.sub.P] (140)

where [I.sup.+.sub.P] and [I.sup.-.sub.P] are the intensities of + and - neutrons in the beam after the polarizer P. Note that [P.sub.P] can vary between +1 and -1. The incoming unpolarized beam of total intensity [I.sub.0] is described by equal + and components, i.e.,

[I.sup.+.sub.0] = [I.sup.-.sub.0] = [I.sub.0]/2. (141)

The total intensity after the polarizer

[I.sup.tot.sub.P] = [I.sup.+.sub.P] + [I.sup.-.sub.P] = [T.sub.P] [I.sub.0]/2 = [T.sub.P] [I.sup.+.sub.0] (142)

where we have used the boundary condition that for perfect + polarization efficiency ([P.sub.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.sub.P] is the spinindependent transmission factor of the device with 0 < [T.sub.P] < 1 due to effects such as absorption or scattering. From Eqs. (140) and (142) it is easy to show that after the polarizer:

[I.sup.+.sub.P] = [T.sub.P] [I.sub.0]/4 (1 + [P.sub.P]) = [T.sub.P] [I.sup.+.sub.0]/2 (1 + [P.sub.P]) (143)

and

[I.sup.-.sub.P] = [T.sub.p] [I.sub.0]/4 (1 - [P.sub.P]) = [T.sub.P] [I.sup.-.sub.0]/2 (1 - [P.sub.P]). (144)

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 I

[I.sup.+.sub.PA] = [T.sub.A] [I.sup.+.sub.P]/2 (1 + [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sup.+.sub.0]/4 (1 + [P.sub.P])(1 + [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sub.0]/8 (1 + [P.sub.P])(1 + [P.sub.A]). (145)

Likewise

[I.sup.-.sub.PA] = [T.sub.A] [I.sup.-.sub.P]/2 (1 - [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sup.-.sub.0]/4 (1 - [P.sub.P])(1 - [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sub.0]/8 (1 - [P.sub.P])(1 - [P.sub.A]). (146)

with the total beam intensity after the analyzer

[I.sup.tot.sub.PA] = [I.sup.+.sub.PA] + [I.sup.-.sub.PA] = [T.sub.P][T.sub.A] [I.sub.0]/4 (1 + [P.sub.P] [P.sup.A]) . (147)

Therefore, the final polarization for non-spin flip scattering is

[P.sub.PA] = [I.sup.+.sub.PA] - [I.sup.-.sub.PA]/[I.sup.+.sub.PA] + [I.sup.-.sub.PA] = [P.sub.P] + [P.sub.A]/1+[P.sub.P][P.sub.A]. (148)

If two [pi]-flippers of efficiency [f.sub.1] and [f.sub.2] and spin-independent transmission factor [T.sub.f1] and [T.sub.f2] are placed between the polarizer and the analyzer, and remembering that the effect of a [pi]-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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (149)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (150)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] only flipper 2 on, (151)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] both flippers 1 and 2 on. (152)

As pointed out by Hayter [18], the ratio of the detector count rates, [I.sub.PA.sup.tot], measured with both [pi]-flippers switched off to the count rates with the two flippers switched "on-off', "off-on" and "on-on" provides three "flipping ratios", [R.sub.1], [R.sub.2], and [R.sub.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.sub.1], [f.sub.2], and the product of the polarizer and analyzer efficiencies, [P.sub.P][P.sub.A], which can be solved to obtain

[P.sub.P][P.sub.A] = [R.sub.12] ([R.sub.1]- 1)([R.sub.2] - 1)/([R.sub.1] [R.sub.2] - [R.sub.12]) (153)

[f.sub.i] = ([R.sub.i] - 1)/2[R.sub.i] (1 + [P.sub.P][P.sub.A]/[P.sub.P][P.sub.A]. (154)

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.sub.corr], is related to the measured signal, [P.sub.meas], by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](155)

5.4.3 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 [pi]-flipper (Sec. 2.4) and Eq. (55), the polarization after the sample, [P.sub.S], is related to the polarization before the sample, [P.sub.i], by

[P.sub.S] =(1 - 2q) [P.sub.i] . (156)

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 [pi]. Thus the sample flipping efficiency is given by q = 2/3, consequently

[P.sub.S]/ [P.sub.i] = -1/3 non-magnetic, pure isotope incoherent scatterer. (157)

This means that the spin-echo signal amplitude is reduced to 1/3 and the minus sign means that the echo signal is inverted. For this case, Eq. (155) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (158)

For a more general non-magnetic case where both spin-incoherent and coherent scattering are present, we might have

q [approximately equal to] 2[S.sub.inc] (Q)/3([S.sub.coh] (Q) + [S.sub.inc] (Q)). (159)

where we have assumed that the relative probabilities of coherent and spin-incoherent scattering are given by [S.sub.coh](Q) and [S.sub.inc](Q) respectively, therefore

[P.sub.S]/[P.sub.i] = 3[S.sub.coh](Q)-[S.sub.inc](Q)/3([S.sub.coh] (Q) + [S.sub.inc] (Q)) (160)

This represents an upper limit on the size of the spin-echo signal. For this case Eq. (155) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (161)

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.

6. 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.,

[[phi].sub.NRSE] = [[phi].sub.0] - [[phi].sub.1] = 2N[m.sub.n][[gamma].sub.n]/h [[B.sub.0][L.sub.0][[lambda].sub.i] - [B.sub.1][L.sub.1] [[lambda].sub.f]].

At the echo point, ([[phi].sub.NRSE]) = 0, however, even for [[lambda].sub.i] = [[lambda].sub.f] (elastic scattering or no sample), [[phi].sub.NRSE] has a distribution of values about the mean, ([[phi].sub.NRSE]), of characteristic width [DELTA][[phi].sub.NRSE]. This is because the terms [B.sub.0][L.sub.0] and [B.sub.1][L.sub.1] have non-zero spread, [DELTA]([B.sub.0][L.sub.0]) and [DELTA]([B.sub.1][L.sub.1]) respectively (2), arising from instrumental imperfections. Consequently, [DELTA][[phi].sub.0] and [DELTA][[phi].sub.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, [DELTA](BL), determines the elastic instrumental resolution function. In order to obtain a broad dynamic range, [DELTA][[phi].sub.NRSE] must be dominated by the distribution of [[lambda].sub.i] - [[lambda].sub.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 [[phi].sub.NRSE] also to have a Gaussian distribution, g([[phi].sub.NRSE]). At the echo point (([B.sub.0][L.sub.0]) = ([B.sub.1][L.sub.1]), g([[phi].sub.NRSE]) is symmetrically distributed about zero (polarization realigned along the original direction--the x axis in these examples). For the Gaussian distribution g([[phi].sub.NRSE]), the elastic scattering polarization along x, [P.sub.x.sup.0], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (162)

or the inverse relation

[DELTA][[phi].sup.FWHM.sub.NRSE] = 4 [square root of (ln 2 ln[1/[P.sup.0.sub.x])] [approximately equal to] [square root of 11.1 ln [1/[P.sup.0.sub.x]]. (163)

We use this convenient form when estimating spectrometer tolerances in the following sections. One notes that if the distribution g([[phi].sub.NRSE]) was uniform between the limits [+ or -] [DELTA][[phi].sub.NRSE.sup.max], rather than Gaussian, the analogue of Eq. (162) is a sinc function of [DELTA][[phi].sub.NRSE.sup.max]:

[P.sup.0.sub.x] ([DELTA][[phi].sup.max.sub.NRSE]) = sin[DELTA] [[phi].sup.max.sub.NRSE]/ [DELTA][[phi].sup.max.sub.NRSE]. (164)

The elastic and quasi-elastic signal count rate cannot exceed a maximum proportional to [P.sub.x.sup.0] (for pure coherent scatterers) and sometimes considerably less for incoherent scatterers (see Sec. 5.4.3), therefore [P.sub.x.sup.0]([[tau].sub.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.sub.x.sup.0] > 0.2 at the maximum required [[tau].sub.NRSE] in a quasielastic measurement. Purely coherent, elastic scatterers, such as Grafoil[R], 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 [[phi].sub.0] and [[phi].sub.1], which imposes [DELTA][[phi].sub.0] = [DELTA][[phi].sub.1], and that the spectrometer is operated at the echo point (i.e., ([p.sub.0]) = [[phi].sub.1]). If [[phi].sub.0] and [[phi].sub.1] are distributed normally, we can write

[DELTA][[phi].sub.NRSE] = [square root of [DELTA][[phi].sup.2.sub.0] + [DELTA][[phi].sup.2.sub.1]] [approximately equal to] [square root 2 [DELTA][[phi].sub.0]]. (165)

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.

6.1 Static Magnetic Field Inhomogeneities

We may consider the effect of static field inhomogeneity as creating a distribution of values of [[omega].sub.0] - [[omega].sub.rf]. In order to simplify the argument we consider [[omega].sub.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 [[omega].sub.0] - [[omega].sub.rf] to the distribution of [[omega].sub.0] caused by the field inhomogeneity, [DELTA][B.sub.0], and comparing the polarization with the equivalent system in which [[omega].sub.0] = [[omega].sub.rf] for all trajectories ([DELTA][B.sub.0] = 0). Further, we assume that the spectrometer is optimally tuned such that <[[omega].sub.0]> = [[omega].sub.rf] and that the field inhomogeneity gives rise to a normal distribution of [[omega].sub.0] with respect to <[[omega].sub.0]>. In this approximation, the effect of static field inhomogeneity is analogous to the effect of dispersion discussed in Sec. 2.2. The effect of [[omega].sub.0] [not equal to] [[omega].sub.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

[absolute value of [B.sup.eff.sub.rf]] = [square root of ([[omega].sub.0] - [[omega].sub.rf]).sup.2] + [[omega].sup.2.sub.p]]/[[gamma].sub.n] (166)

The effective field lies at an angle [[alpha].sub.eff] to the x-y plane given by

[[alpha].sub.eff] [tan.sup.-1] [[omega].sub.0] - [[omega].sub.rf]/[[gamma].sub.n][absolute value of [B.sub.rf]] = [tan.sup.-1] [[omega].sub.0] - [[omega.sub.rf]/[[omega].sub.p]. (167)

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 ([[omega].sub.0] = [[omega].sub.rf]) is given by Eq. (86) and in the general off-resonance case by Eq. (83). If [DELTA][[phi].sub.[pi]] 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 (cos [epsilon] cos[DELTA][[phi].sub.[pi]]), where [epsilon] here refers to the angle of the spin vector out of the x-y plane, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (168)

If [[lambda].sub.i] is the median wavelength and that the flipper is optimally tuned for [pi] flips at this wavelength, i.e., [[omega].sub.p] = [pi]h/[m.sub.n]l[B.sub.0] [[lambda].sub.i] (see Eq. (13)) and setting

[xi] = [absolute value of [B.sup.eff.sub.rf]]/[absolute value of [B.sub.rf]] = [square root of [[omega].sup.2.sub.p] + ([[lambda].sub.n][DELTA][B.sub.0]).sup.2]]/[[omega].sub.p], (169)

Eq. (168) simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (170)

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 [DELTA][B.sub.0] is sufficiently small that cos[epsilon] [approximately equal to] 1 and (cos[DELTA][[phi].sub.[pi]]) [approximately equal to] 1-[([DELTA][phi].sup.2.sub.[pi]])/2 = cos([DELTA][[phi].sub.[pi]] (rms)), we have after one [pi]coil due to the effect of [DELTA][B.sub.0]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (171)

For an M-coil unit spectrometer, we assume that [DELTA][[phi].sub.[pi]] 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)):

[DELTA][[phi].sup.FWHM.sub.NRSE] [approximately equal to][square root of M] [cos.sup.-1] (1/[[xi].sup.2][sin.sup.2]([pi]/2 [xi]])) = 4[square root of (ln 2 ln (1/[P.sup.0.sub.x])] (172)

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (173)

Specifically for a 4-N coil instrument we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (174)

In the range of operation of spectrometer configurations considered in this document, it can be shown that the value of [xi] is typically no greater than about 1.3. In this range, it turns out that the awkward term

[[cos.sup.-1]{1/[[xi].sup.2][sin.sup.2]([pi]/2 [xi])}].sup.2] may be replaced very successfully by 4 ([xi]-1) so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (175)

or specifically for a 4-N coil instrument:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (176)

It turns out, somewhat fortuitously, that Eqs. (175) and (176) produce a better approximation to [P.sub.x.sup.0] when [DELTA][B.sub.0] is too large to assume cos[epsilon] [approximately equal to] 1 (implicit in Eqs. (173) and (174)) or when the accumulated dephasing in the x-y plane approaches 2 [pi] The approximations in Eqs. (175) and (176) make the inverse problem significantly more tractable (i.e., what tolerance on [DELTA][B.sub.0] is required to obtain a given value of [P.sub.x.sup.0] under a given set of conditions [[lambda].sub.0], [B.sub.0] etc.?). The inverse expression, which is more useful in instrument design than the forward expression (Eq. (173)), then finally reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (177)

for M coils. Specifically for a 4-N coil instrument we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (178)

where [kappa] = ln 2 ln(l/[P.sup.0.sub.x]). 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.sub.0] = 0.0393 T, [l.sub.B0] = 0.03 mm, [L.sub.0] = 2 m, [[lambda].sub.0] = 8 [[Angstrom]], which gives [[tau].sub.NRSE] = 15 ns and 30 ns for N = 1 and N = 2 respectively.

6.2 Coil Flatness

In order to estimate tolerances on the flight path lengths, we return to the expanded equations representing [[phi].sub.0] (and [[phi].sub.1]) which contain the individual contributions (the flatness model used is that described in Sec. 3.6), and now we assume [DELTA][B.sub.0] = 0. (a) For a 4 (N = 1)-coil NRSE, we have from Eq. (72) for a given neutron trajectory,

[[phi].sub.0] = [[omega].sub.rf]/[v.sub.n] [2[L.sub.0] + [DELTA][f.sub.R](B) + [DELTA][f.sub.L](B)- [DELTA][f.sub.L](A) [DELTA][f.sub.R](A)]

where we have set [[phi].sub.in] = 0 (perfectly polarized incoming beam) and the terms [DELTA][f.sub.L] and [DELTA][f.sub.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, [DELTA][f.sub.L] and [DELTA][f.sub.R] have Gaussian distributions of equal FWHM =[DELTA][f.sub.FWHM], we can write

[DELTA][[phi].sub.0] = [square root of 4N[DELTA][f.sup.FWHM]/2N[L.sub.0] = [DELTA][f.sup.FWHM/[L.sub.0]] N = 1 (179)

whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (180)

We can also write the FWHM fluctuation in the coil length, [DELTA][l.sub.B0], in terms of [DELTA][f.sub.FWHM], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (181)

(b) For N=2 bootstrap coils, for a given neutron trajectory, we have from Eq. (73):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we have set [[phi].sub.in] = O (perfectly polarized incoming beam) and the terms [DELTA][f.sub.L] and [DELTA][f.sub.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 [DELTA][f.sub.L] and [DELTA][f.sub.R] have Gaussian distributions of equal FWHM = [DELTA][f.sub.FWHM], we can write

[DELTA][[phi].sub.0]/[[phi].sub.0] = [square root of 4N][DELTA][f.sub.FWHM]/2[NL.sub.0] = [DELTA][f.sub.FWHM]/[square root of N][L.sub.0], N = 2, (182)

whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (183)

We can also write the FWHM fluctuation in the coil length, (length of the [B.sub.0] field) in terms of [DELTA][f.sup.FWHM], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (184)

Inverting Eq. (184) we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (185)

Equation (184) seems to be generally valid, the "[square root] N" not being apparent in the N = 1 case (Eq. (181)). The success of Eq. (184) is demonstrated in Fig. 14 and Fig. 15 for N = 1 and N = 2 respectively.

6.3 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (186)

where [[??].sup.surf.sub.max] is the maximum tolerable angle between the entrance and exit surfaces of the static coil windings and a and [l.sub.axial] are the coil dimensions defined in Fig. 23.

6.4 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, [DELTA][[theta].sub.i]], or [DELTA][theta].sub.f], up to specified maxima [DELTA][[theta].sub.i,max] and [DELTA][[theta].sub.f,max] respectively, where all [DELTA][theta] are defined with respect to any axis parallel to the nominal beam axis. [DELTA][[theta].sub.i], and [DELTA][[theta].sub.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([DELTA][[theta].sub.if]).

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.sub.1][L.sub.1] = [B.sub.0][L.sub.0], <[[phi].sub.0][[phi].sub.0]), etc. The elastic resolution is still given by Eq. (113), i.e., [P.sup.0.sub.x] = (cos [[phi].sub.NRSE]) = (cos([[phi].sub.0] - [[phi].sub.1])>. We also assume small divergence, which allows one to write [[phi].sub.0] = 2N[m.sub.n][[gamma].sub.n] [B.sub.0][L.sub.0][[lambda].sub.i]/h etc. (see Sec. 5.1). With these assumptions the expression for [P.sub.x.sup.0] simplifies to

[P.sup.0.sub.x] = <cos([phi].sub.0] - [phi].sub.1])> = <cos([[phi].sub.0] + [DELTA][phi].sub.0]) + -[<[[phi].sub.1] + [DELTA][[phi].sub.1]]>] = <cos[DELTA][[phi].sub.0] - [DELTA][[phi].sub.1])>. (187)

For a trajectory in the incident arm of the spectrometer, we have

[DELTA][[phi].sub.0]/<[[phi].sub.0]> [approximately equal to] [DELTA][L.sub.0]/[L.sub.0] = [1/cos [DELTA][[theta].sub.i]]-1]. (188)

The distribution of [DELTA][[phi].sub.0] for random [DELTA][theta] is by no means Gaussian or uniform. Because we assume small divergence (i.e., [DELTA][[theta].sub.i,max] and [DELTA][[theta].sub.f,max] are small--certainly within the range of angles encountered in the NRSE), we write for all incident arm trajectories:

[DELTA][[phi].sub.0] [approximately equal to] <[[phi].sub.0]> (1-cos [DELTA][[theta].sub.i]) [approximately equal to] <[[psi].sub.0]> [DELTA][[theta].sup.2.sub.i]/2 (189)

and likewise at the echo point:

[DELTA][[psi].sub.1] [approximately equal to] <[[phi].sub.1]> (1-cos [DELTA][[psi].sub.f]) [approximately equal to]<[[phi].sub.1]> [DELTA][[theta].sup.2.sub.f]/2 = <[[phi].sub.0]> [DELTA][[theta].sup.2.sub.f]/2. (190)

(Note [DELTA][[phi].sub.0] and [DELTA][[phi].sub.1] are not necessarily small numbers because ([[phi].sub.0]) can be very large). Therefore, finally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (small divergence, at echo point, only angular uncertainties). (191)

The average in Eq. (191) can be expressed in terms of the double integral over the range of [DELTA][[theta].sub.i] and [DELTA][[theta].sub.f] which are both assumed to be uniform in probability in the range (o, [DELTA][[theta].sub.i,max]), (0, [DELTA][[theta].sub.f,max]), permitting the average to be written simply as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (192)

It can be shown that Eq. (192) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (193)

where [C.sub.1] and [S.sub.1] are the Fresnel cosine and sine integrals respectively, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (194)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (195)

Certain approximations for evaluating [C.sub.1] and [S.sub.1] have been discussed by Mielenz [19] (note that the [pi]/6 term in Eq. 3b of this reference should be multiplied by [x.sup.3]) and Heald [20]. The integrals can also be evaluated numerically. For the particular case of [absolute value of [DELTA][[theta].sub.i,max]] = [absolute value of [DELTA][[theta].sub.f,max]] = [absolute value of [DELTA][[theta].sub.max]], Eq. (193) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (196)

or in terms of the instrument parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (197)

The success of Eq. (193) in describing the relationship between [DELTA][[theta].sub.max] and [P.sub.x.sup.0] is demonstrated in Fig. 17 and Fig. 18 for realistic examples. The examples with [[tau].sub.NRSE] = 15 ns, N = 1, and [[tau].sub.NRSE] = 30ns, N = 2 have sufficiently large values of ([[phi].sub.0]) that the arguments of [C.sub.1] and [S.sub.1] exceed unity in the plotted range (the values shown on the right hand side y-axes). They also have [absolute value of [DELTA][[theta].sub.i,max]] = [absolute value of [DELTA][[theta].sub.f,max]] = [absolute value of [DELTA][[theta].sub.max]] (so that Eq. (196) is used).

In the present context it is useful to have [P.sub.x.sup.0] as the dependent variable and ask "what is the maximum permissible value of [absolute value of [DELTA][[theta].sub.max]] to achieve a given value of [P.sub.x.sup.0]?" Unfortunately, inversion of Eq. (196) is not trivial. The traditional approximations for [C.sub.1] and [S.sub.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.sub.1.sup-.2](x)+[S.sub.1.sup.2](x) involves terms in [x.sup.4n+2], n = 0, 1, 2,..., [infinity] alternating signs for the first few terms. Another function that has the the same powers and signs as these first terms would be [x.sup.2] exp(-a[x.sup.4]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (198)

The expansion of [C.sub.1.sup.2](x)+[S.sub.1.sup.2](x) for the first few terms is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (199)

therefore we try setting the parameter a in Eq. (198) to the magnitude of the second term coefficient in Eq. (199) = [[pi].sup.2]/45 [approximately equal to] 0.21932 which makes the two leading terms in Eqs. (198) and (199) identical. This should certainly work well for x < 1 since the higher order terms decrease rapidly. With this substitution Eq. (198) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (200)

for which the first few terms are quite similar to those of Eq. (199). It turns out that this approximation can be applied with about 1 % accuracy up to x ~ [x.sub.1%] ~ 1.15, where the fan-out of the spins due to the divergence (= [DELTA][[phi].sub.0] (see Eq. (189)) [approximately equal to][eta][x.sub.1%.sup.2]/2 ~ 0.7[pi]) is still below 2[pi]radians, i.e.,

[C.sup.2.sub.1] (x) + [S.sup.2.sub.1] (x) [approximately equal to] [x.sup.2] exp (-[[pi].sup.2][x.sup.4]/45] x < ~ 1.15. (201)

In fact the approximation is within 15 % for x up to about 1.8, at which point [DELTA][[phi].sub.0] ~ 1.6[pi](as is seen from

Fig. 17 and Fig. 18. Now identifying x with [square root of 2N[m.sub.n][[gamma].sub.n] [B.sub.0][L.sub.0[[lambda].sub.i]]/[pi]h [DELTA][[theta].sub.max]] Eq. (197) can be inverted using the approximation in Eq. (201) yielding:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (202)

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.

6.5 Approximation for Equal Contributions to Depolarization from [DELTA][B.sub.0], [DELTA][l.sub.B0], and [DELTA][[theta].sub.max]

In the preceding sections, the contributions of [DELTA][B.sub.0], [DELTA][l.sub.B0], or [DELTA][theta] to the elastic polarization [P.sub.x.sup.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 obtained by setting the [DELTA][B.sub.0], [DELTA][l.sub.B0], and [DELTA][theta] contributions to the depolarization approximately equal. For equal contributions, we assume that the tolerances will be approximately 1/[square root]3 times the values given by Eqs. (178), (184), and (202) respectively (for a 4-N coil instrument), i.e.,

[DELTA][B.sup.FWHM.sub.0] [approximately equal to] [absolute value of [B.sub.rf]][square root of K/3N (K/N)+ 2] (4-N instrument, equal contribs to [P.sub.x.sup.0]), (203)

where k = ln 2 ln (1/[P.sup.0.sub.x]), as before. Note that [DELTA][B.sub.0] is defined by N, [l.sub.B0], and [[lambda].sub.i] only and is independent of [B.sub.0] or zero field region parameters.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] meters (4-N instrument, equal contribs to [P.sub.x.sup.0]). (204)

Note that [DELTA][l.sub.B0] is defined by N, [l.sub.B0] and [lambda] only and is independent of [l.sub.B0] or zero field region parameters. Finally,

(4-N instrument, equal contribs to [P.sub.x.sup.0]), (205)

for [absolute value of [DELTA][[theta].sub.max]]<~ 6.7 x [10.sup.-3]/[square root of N[B.sub.0][T][L.sub.0][m][[lambda].sub.i][[Angstrom]]] [rad].

Note that [DELTA][[theta].sub.max] depends on [lambda] and on both the flipper coil and zero-field parameters (N, [B.sub.0], [L.sub.0] (i.e., [L.sub.AB], [l.sub.B0], and [l.sub.g])). Even though these parameters also appear in the expression for [[tau].sub.NRSE], the [[lambda].sup.3]-dependence of the latter means that [DELTA][[theta].sub.max] is not uniquely determined by the quantity [[tau].sub.NRSE] (i.e., the same value of [[tau].sub.NRSE] may require different values of [DELTA][[theta].sub.max] depending on the values of N, [B.sub.0], [L.sub.0] and [lambda]).

6.6 Some Examples (Equal Contributions to Depolarization)

Consider requiring the elastic (resolution) polarization [P.sub.x.sup.0] to be greater than some specified minimum value at a reference point with equal contributions coming from AB0, [DELTA][l.sub.B0], and [DELTA][[theta].sub.max]. We consider the point [[tau].sub.NRSE] [approximately equal to] 30 ns at [lambda] = 8 [Angstrom] with N = 2, for M = 8 [pi] coils ([l.sub.B0]=0.03 m), with [B.sub.0] = 0.0393 T, [L.sub.0] = 2 m. Using Eqs. (203-205), several results are shown in Table 7.

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.

7. NIST NRSE Project Goals

7.1 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 [[tau].sub.NRSE] = 30 ns at [lambda] = 8[Angstrom] 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 [Angstrom] for high-Q capability.

5. Must have a short Fourier time measurement capability.

7.2 Spectrometer Dimensions and Field Magnitudes Required to Access [[tau].sub.NRSE] = 30 ns at [lambda] = 8 [Angstrom]

From Eq. (120) we have

[[tau].sub.NRSE] [ns] = 0.37271 N [B.sub.0][T][L.sub.0][m][[lambda].sub.i][[Angstrom]].sup.3],

where we assume that [B.sub.0] = [B.sub.1] so that [L.sub.0] = [L.sub.1] at the QENS echo point. In order to access [[tau].sub.NRSE] = 30 ns at [lambda] = 8 [Angstrom], we must satisfy the condition

N[([B.sub.0][T][L.sub.0][m]).sub.max] [less than or equal to] 0.157 criterion for accessing [[tau].sub.NRSE] = 30 ns at [lambda]= 8 [Angstrom], (206)

where [([B.sub.0][L.sub.0]).sub.max] implies the maximum attainable value of the product [B.sub.0][L.sub.0]. If we chose N = 2 as the most likely bootstrap factor, noting the advantages and disadvantages outlined in Sec. 3.4, this condition amounts to fulfilling:

[([B.sub.0][T][L.sub.0][m]).sub.max] [less than or equal to] 0.079 Tm criterion for accessing [[tau].sub.NRSE] = 30 ns at [lambda] = 8[Angstrom] with N = 2. (207)

Obvious limitations on the maximum value of [B.sub.0] are imposed by the maximum current x winding density of the static field coils. This depends on the length, cross-section, material, winding temperature, and the ability to remove heat. Increasing the zero-field drift path lengths increases proportionately the maximum achievable value of [[tau].sub.NRSE], however disadvantages include the rapid reduction in solid angle ([varies] 1/[L.sup.2]) and possibly limitations imposed by available space. Owing to these constraints and the linear dependence of [[tau].sub.NRSE] on [B.sub.0], it seems reasonable to attempt to maximize the static magnetic field [B.sub.0] as far as possible. Evaluating [B.sub.0] and [L.sub.0] for [[tau].sub.NRSE] =30 ns at [[lambda].sub.i] = {[[lambda].sub.i]) = 8[Angstrom], we have, for example,

[B.sub.0] [approximately equal to] 0.08 T, [L.sub.0] = 1 m, N = 2

[B.sub.0] [approximately equal to] 0.04 T, [L.sub.0] = 2 m, N = 2.

To date, the largest static fields produced in water-cooled NRSE coils using pure aluminum windings are about [B.sub.0] [approximately equal to] 0.025 T. With this field we require [L.sub.0] = 3.14 m for N = 2 (which is a little long for available floor space) or else [L.sub.0] = 1.57 m for N=4. Apart from the increased restrictions on the maximum incoming bandwidth, [DELTA][lambda]/[lambda], when using N = 4, doubling the number of [pi]-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.sub.0] [less than or equal to] 2 m and pursuing the goal of increasing [B.sub.0] towards 0.04 T presents itself as one of the more attractive options. Some consequences are explored in the following sections.

7.3 Bootstrap NRSE Coil Components and Specifications

7.3.1 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. [mu]-metal plates capping each end of the static field coils conduct magnetic flux lines between the two coils. An outer [mu]-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.

7.3.2 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).

Very approximately, the macroscopic neutron cross-section of aluminum at all temperatures of interest is about 0.11 [cm.sup.-1] for [lambda] < 4.7[Angstrom]. Therefore, we have

[T.sub.Al] ~ exp(-0.11t[cm]), [lambda] 4.7 [Angstrom]. (208)

Estimating the equilibrium temperature and temperature gradients of the windings depends on the detailed coil design. In order to partially account for elevated winding temperatures at high-field operation of the coils, we approximate the macroscopic cross-section for [lambda] > 4.7[Angstrom] using the average of available data for pure aluminum [21] at T = 300 K and at T = 800 K. At 300 K data we have approximately

[[SIGMA].sub.Al] (300K)[[cm.sup.-1]][approximately equal to]6.4 + 8.94[lambda][[Angstrom]])x [10.sup.-3], [lambda] [greater than or equal to] 4.7[[Angstrom]] (209)

and for the 800 K data we have approximately

[[SIGMA].sub.Al] (800K)[[cm.sup.-1]][approximately equal to](1.91 + 1.175[lambda][[Angstrom]])x [10.sup.-2], [lambda] [greater than or equal to] 4.7[[Angstrom]] (210)

Therefore, we use an effective aluminum macroscopic cross-section of

[[SIGMA].sup.eff.sub.Al] [[cm.sup.-1]][approximately equal to](1.28 + 1.03[[Angstrom]])x [10.sup.-2], [lambda] [greater than or equal to] 4.7[Angstrom]. (211)

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.sub.Al][lambda] = 8 [Angstrom]) [greater than or equal to] 80 %, then Eq. (211) requires that t must not exceed a maximum value, [t.sub.max], of about 1.0 mm (i.e., the static field coil windings have thickness of about 1 mm, the r.f. windings have thickness of about 0.5 mm). For the r.f. coils the skin effect at ~1 MHz frequencies likely restricts the r.f. winding thickness to a smaller value (see Sec. 7.3.4.7).

Coils constructed at the Institut Laue-Langevin (ILL), Grenoble, France, Laboratoire Leon Brillouin (LLB), Saclay, France, and the Forschungs-Reaktor Munchen-II (FRM-II), Munich, Germany, have used 0.4 mm-thick anodized aluminum band windings, with anodization depth of about 3 [micro]m for insulation. The anodization layer can contain incorporated water, which gives rise to strong, anisotropic, small angle scattering. This small angle scattering is greatly reduced by boiling the wire in [D.sub.2]O under pressure at about 200[degrees]C [11].

7.3.3 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.

7.3.3.1 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

B = [[mu].sub.0]nI, (212)

where [[mu].sub.0] is the permeability of free space with [[mu].sub.0] =4[pi] x [10.sup.-7] [NA.sup.-2]. In SI units we have

[B.sub.0] [T] [approximately equal to] 4[pi] x [10.sup.-7]n [[m.sup.-1]]I[A][equivalent]1.26 x [10.sup.-6]n [[m.sup.-1]]I[A] long solenoid approximation, (213)

where [B.sub.0] is the static field in Tesla, n is the winding density in [m.sup.-1], and I is the current in Amps. Equivalently, the current in the coil at field [B.sub.0] is

I [A] = 2.5 x [10.sup.6]/[pi] [B.sub.0][T]/n[[m.sup.-1]] [approximately equal to] 8 x [10.sup.5] [B.sub.0] T]/n[[m.sup.-1]. (214)

Thus the required current is inversely proportional to the winding density and is directly proportional to the required field [B.sub.0].

7.3.3.2 Resistance of the static field coil windings

The resistance of the static field coil winding is

R = [rho](T)[l.sub.w]/[A.sub.w], (215)

where [l.sub.w] is the total length of the coil winding, [A.sub.w] is the wire cross-sectional area, and [rho](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.sub.B0]), assuming the winding thickness is negligible compared with a and [l.sub.B0]. For the particular case of single-layer windings, the total number of turns, [N.sub.B0], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (216)

so that the total length of any single-layer winding around the rectangular coil form shown in Fig. 23 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (217)

The outer surface area of the rectangular coil form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (218)

so Eq. (217) may be rewritten as

[l.sub.w] = [A.sub.surf] n any thin single-layer winding around rectangular coil form. (219)

7.3.3.3 Single-layer rectangular cross-section wire

The cross-sectional area, [A.sub.w], of rectangular cross-section wire (see Fig. 23) is

[A.sub.w] = th rectangular cross-section wire, width h, thickness t, (220)

so that, using Eqs. (219) and (220), and noting that for a single winding h [less than or equal to] 1/n, with the equality representing the tightly-wound limit, Eq. (215) becomes

R = [rho](T)n[A.sub.surf]/th any single-layer rectangular cross-section wire, (221)

with

R = [rho](T)[n.sup.2] [A.sub.surf]/t tightly-wound rectangular cross-section wire, thickness t (222)

representing the tightly-wound limit with h = 1/n. Therefore, for a given [A.sub.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.sub.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.sub.0] is, from Eqs. (214) and (221)

V = IR 2.5 X[10.sup.6]/[pi] [rho](T)[[OMEGA]m][A.sub.surf] [[m.sup.2]]/t[m]h[m][B.sub.0][T] any single-layer rectang cross-section wire winding (223)

with

V[V] = 2.5 x [10.sup.6][pi] [rho](T)[[OMEGA]m]n[[m.sup.-1]][A.sub.surf][[m.sup.2]]/ t[m] [B.sub.0] [T] tightly-wound, single-layer, rectang cross-section

wire windings (224)

representing the tightly-wound limit. Thus, for the tightly-wound case, the voltage required to maintain a field [B.sub.0] is proportional to [B.sub.0], proportional to the winding density, and inversely proportional to the winding thickness in the beam direction for a given coil surface area. For a given [B.sub.0], the voltage is minimized by tightly-winding the coil within the available surface area.

The power dissipated in the coil with single-layer, rectangular cross-section wire is (from Eqs. (214) and (221) or (223))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] any single-layer rectang cross-section wire winding. (225)

Specifically, for the tightly-wound, rectangular cross-section wire winding it is (from Eqs. (214) and (222) or (224))

P[W] [approximately equal to] 6.25 X[10.sup.12] [rho](T)[[OMEGA]m][A.sub.surf][m.sup.2]/[[eta].sup.2]/ [[eta].sup.2] t[m][([B.sub.0][T]).sup.2] tightly-woundrectang cross-section wire windings. (226)

Thus, for a given [A.sub.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.sub.0]. Like the voltage, the power dissipated is minimized for a given [B.sub.0] by tightly-winding the coil within the available surface area, since h [less than or equal to] 1/n.

7.3.3.4 Single-layer circular cross-section wire windings

The cross-sectional area of the circular cross-section wire, [A.sub.w], is

[A.sub.w] = [pi][r.sup.2.sub.w] circular cross-section wire of radius [r.sub.w]. (227)

Using Eq. (219) and noting that for a single-layer circular winding we have the constraint n [less than or equal to] l/2[r.sub.w], with the equality representing the tightly-wound case, Eq. (215) becomes

R = [rho](T)n[A.sub.surf]/[pi][r.sup.2.sub.w] any circular cross-section wire winding, (228)

with

R = 4[rho](T)[n.sup.3][A.sub.surf]/[pi] tightly-wound circular cross-section wire, (229)

representing the tightly-wound limit. Thus, for a given [A.sub.surf], the tight-winding resistance increases as the cube of n (as opposed to [n.sup.2] in the tightly-wound rectangular wire case with fixed t).

The D.C. voltage required to maintain a static field [B.sup.0] in the circular cross-section wire case is (from Eqs. (214) and (228))

V[V] = 2.5 x[10.sup.-6] [rho][T][[OMEGA]m][A.sub.surf][[m.sup.2]]/([pi][r.sub.w][m].sup.2])[B.sub.0] [T] any single circular cross-section wire winding. (230)

Specifically, for the tightly-wound case it is (from Eqs. (214) and (229))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] tightly-wound, circular cross-section wire winding. (231)

Thus, the voltage required to achieve a given [B.sub.0] in the circular cross-section wire case is independent of the winding density, other than n cannot exceed a value of 1/(2[r.sub.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.sub.0].

The power dissipated in the coil with single-layer, circular cross-section wire windings is (from Eqs. (214) and (228) or (230))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] any single circular cross-section wire winding, (232)

where the tightly-wound case with n = 1/(2[r.sub.w]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] tightly-wound circular cross-section wire windings. (233)

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.

7.3.3.5 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.sub.surf] becomes small with respect to the beam area it is increasingly difficult to maintain adequate field homogeneity within this region at high [[tau].sub.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. 7.3.2). Finally, there are very limited choices of winding material that have both good cold neutron transmission combined with low electrical resistivity. Although maximum fields of a few 10's of mT do not appear dauntingly high, the heat production from the coil is potentially quite large. This is illustrated by the following examples:

The coils produced for the neutron research laboratories Laboratoire Leon Brillouin (LLB), Institut LaueLangevin (ILL) (France), Forschungs-Neutronenquelle Heinz Maier-Leibnitz (FRM-II), and Helmholtz-Zentrum Berlin (HZB) (Germany) use tightly-wound 4 mm wide x 0.4 mm thick anodized aluminum band supplied by Wesselmann Umwelttechnik (3), with n [approximately equal to] 250 [m.sup.-1], [l.sub.axial] [approximately equal to] 0.2 m, a + [l.sub.B0] [approximately equal to] 0.25 m for a beam size of about 2.5 cm x 2.5 cm, so that [A.sub.surf] (see Eq. (218)) [approximately equal to] 0.1 [m.sup.2]. For these coils at maximum field ([B.sub.0] [approximately equal to] 0.025 T), we have (from Eq. (214)) I [approximately equal to] 80 A. For pure Al down to about liquid nitrogen temperature, we have

[[rho].sub.Al](T)[[OMEGA]m] [approximately equal to] 1.14 x [10.sup.-10]T (K) - 6.9 x [10.sup.-9]. (234)

Therefore, specifically for aluminum, we have (from Eq. (226))

[P.sub.Al][W] [approximately equal to] ([B.sub.0][T]).sup.2] [A.sub.surf][[m.sup.2]]/t[m](72.2T(K) - 4.37 x [10.sup.3]) tightly-wound rectangular wire windings. (235)

For T [approximately equal to] 300 K, [P.sub.Al] (0.025 T) [approximately equal to] 2.7 kW. For T [approximately equal to] 350 K, [P.sub.Al] (0.025 T) [approximately equal to] 3.3 kW. If similar coils are to achieve 0.04 Tesla, the current increases to I [approximately equal to] 128A with an increased power dissipation factor of approximately ([0.04.sup.2] /[0.025.sup.2]). The room-temperature power dissipation then increases to approximately 6.9 kW. If the coils are cooled to liquid nitrogen temperature [approximately equal to] 80 K, [P.sup.Al] is more than an order of magnitude smaller ([approximately equal to] 220 W at [B.sub.0] = 0.025 T, [approximately equal to] 560 W at [B.sub.0] = 0.04 T). This is discussed by Gahler, Golub, and Keller in Ref. [8]. One technical challenge is avoiding liquid coolant (water or liquid [N.sub.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.sub.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 [approximately equal to] 198 A at [B.sub.0] = 0.04 T with a corresponding current density of about 64 A [mm.sup.-2]. The winding would be supported by and cooled by four hollow tubes for the passage of liquid [N.sub.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.sub.2]-filled coil form, thus they should remain close to 80 K. In general, the liquid [N.sub.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.

7.3.3.6 Required static field coil current stability

The values in Table 7 imply that [DELTA][B.sub.0]/[B.sub.0] must be around 0.1 % in order to achieve [P.sub.x.sup.0] (8 [Angstrom], 30 ns) [greater than or equal to] 0.5 for typical spectrometer dimensions. Even for perfect static field coil homogeneity (AB0 = 0), this imposes a coil current stability of order of 0.1 % ([DELTA]I/I <~ [10.sup.-3]). The current stability should certainly not become the limiting factor on AB0. Preferably it should be at least an order of magnitude better ([DELTA]/I < [10.sup.-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.sup.-5] range are commercially-available, so this is not expected to impose any technical limitation.

7.3.3.7 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (236)

This can be re-expressed in terms of the dimensionless quantities

[mu] = 2x/[l.sub.axial] (237)

which is the axial distance from the solenoid center expressed as a fraction of the half-length of the solenoid and

[eta] = 2r/[l.sub.axial], (238)

which is the ratio of the diameter, d, of the coil to its axial length, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (239)

In the "long" solenoid limit ([l.sub.axial] [much greater than] r), the field at the coil center is maximized (=[[mu].sub.0]nI), whereas at its ends it is half this value (=[[mu].sub.0]nI/2--limit of Eq. (239) with [mu] = 1 and [[eta].sup.2] [much less than] [(1+[mu]).sup.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 (=[[mu].sub.0]nI) for solenoids with various ratios [eta] = d/[l.sub.axial], calculated according to Eq. (236). Figure 26 reveals that as [eta] 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 [eta] 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 [DELTA][B.sub.0]/[B.sub.0] is about 0.1 %. With reference to Fig. 26, this requires [B.sub.0]([mu] = 0.1) > 0.999 [B.sub.0]([mu] = 0). This occurs for [eta] <~ 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.

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:

[[phi].sub.1] = 2N[[gamma].sub.n] [B.sub.0][L.sub.0]/[v.sub.i].[[??].sub.0] (106)

and

[[phi].sub.2] = 2N[[gamma].sub.n] [B.sub.1][L.sub.1]/[v.sub.f].[[??].sub.1], (107)

where [[gamma].sub.n] is the neutron gyromagnetic ratio, [v.sub.i,f] is the initial/final neutron velocity vector and [[??].sub.0,1] 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.

5.1 Small Divergence Approximation

For small beam divergences and coil axes that are perpendicular to vi and vf, we can approximate Eqs. (106) and (107) by

[[phi].sub.0] = 2N[[gamma].sub.n] [B.sub.0] [L.sub.0]/[v.sub.i] (108)

and

[[phi].sub.1] = 2N[[gamma].sub.n] [B.sub.1] [L.sub.1]/[v.sub.f] (109)

respectively, where [v.sub.i] and [v.sub.f] are scalars, so that the net spin turn at the analyzer is given by Eq. (69). For quasielastic non-spin flip scattering that is sufficiently low energy transfer ([delta]v [much less than] [v.sub.i]), we can write

[v.sub.f] [approximately equal to] [v.sub.i] + [delta]v (110)

so that Eq. (69) is approximately

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (111)

where

[delta](BL)= [B.sub.0][L.sub.0] - [B.sub.1][L.sub.1] (112)

is often called the spectrometer asymmetry. It is conventional to perform NRSE asymmetric scans by fixing the static field and scanning [delta](BL)= [L.sub.0]-[L.sub.1]. Also we will assume that [B.sub.0] = [B.sub.1] and replace [delta](BL) by the slightly less general expression [delta] (BL) = [B.sub.0]([L.sub.0] -[L.sub.1]) = [B.sub.0] [delta]L 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 [[phi].sub.NRSE], averaged over all the scattered neutron trajectories i.e.,

[P.sub.x] = <cos[[phi].sub.NRSE]>, (113)

where <> implies a statistical average over a large sample of scattered neutrons. From Eq. (111) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (114)

This expression must be averaged over all possible values of [v.sub.i] (the incident spectrum) and all possible values of [delta]v (= [v.sub.i]-[v.sub.f]) determined by the scattering. Noting that Q is approximately independent of [omega], i.e.,

Q [approximately equal to] 2[m.sub.n] [v.sub.i]/h sin[theta], (115)

where 2[theta] is the scattering angle, we can write

P ([v.sub.i], [delta]v) d ([delta]v) [equivalent] S (Q,[omega]) d[omega]. (116)

For small energy transfers (small [omega] and [v.sub.i] [approximately equal to] [v.sub.f]) we have from the definition of kinetic energy:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (117)

If S(Q, [omega]) is symmetric in a (usually a good approximation for quasielastic scattering) and substituting [delta]v [approximately equal to] [??][omega]/[m.sub.n][v.sub.i] from Eq. (117), the average over the [delta]v distribution characteristic of the scattering sample for a given [v.sub.i] becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (118)

where the "sine" part of the expansion in Eq. (114) disappears in the integral for symmetric S(Q, [omega]) and the denominator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for a normalized scattering function is implicit. Note that the quantity preceding [omega] in the second cosine argument in Eq. (118) when [delta](BL) = 0 (i.e., [B.sub.0][L.sub.0] = [B.sub.1] [L.sub.1]) is often referred to as the spin-echo time, [[tau].sub.NRSE], i.e.,

[[tau].sub.NRSE] = 2[??][[gamma].sub.n]N[B.sub.0][L.sub.0]/[m.sub.n][v.sup.3.sub.i] = [[gamma].sub.n]/[pi] [([m.sub.n]/h).sup.2] N[B.sub.0][L.sub.0][[lambda].sup.3.sub.i], (119)

where

[[tau].sub.NRSE] [ns] = 0.37271 N[B.sub.0] [T][L.sub.0] [m][([[lambda].sub.i] [[Angstrom]]).sup.3] [equivalent to] 1.27794 x [10.sup.-2] N[v.sub.0] [MHz][L.sub.0] [m] [([[lambda].sub.i] [[Angstrom]]).sup.3]. (120)

Equation (118) must be averaged over the normalized incident velocity distribution, F([v.sub.i]), so the final polarization is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (121)

where the denominator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is implied. Expressed in terms of the normalized incident wavelength distribution, I([[lambda].sub.i]), Eq. (121) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (122)

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 [lambda], 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.sub.0] and [L.sub.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.

5.2 Special Cases for No Sample, Isotope Incoherent Elastic Scattering, or Small [omega] ("Resolution Function")

Isotope incoherence implies scattering that is both Q-independent (so the [lambda]-dependence of the scattering function can be ignored). Elastic scattering, or small [omega], 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 [omega][[tau].sub.NRSE] [less than or equal to] 0.045[pi] Under these conditions Eq. (122) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (123)

where for brevity we define

A = 2N[[gamma].sub.n] [m.sub.n] [B.sub.0]/h [delta]L (124)

and the integral over [omega] evaluates to S(Q) since there is no other [omega]-dependence. The integral over [[lambda].sub.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.

5.2.1 Purely Monochromatic Beam

For a purely monochromatic incident beam, I([[lambda].sub.i]) = [delta]([[lambda].sub.0]), and we have simply:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (125)

Therefore, in the absence of instrumental imperfections (see Sec. 6), the resolution function has a pure cosinusoidal form of constant amplitude for any [delta](BL) with a periodicity given by

[delta][(BL).sub.2[pi]] = h/[m.sub.n] [pi]/[[gamma].sub.n]N 1/[[lambda].sub.0] = [pi][v.sub.0]/[[gamma].sub.n]N. (126)

5.2.2 Rectangular Incident Wavelength Spectrum

For a rectangular incident wavelength spectrum of full width [DELTA] [[lambda].sub.FW], centered about [[lambda].sub.i]= ([[lambda].sub.i]), the wavelength-dependent integral in Eq. (123) becomes: 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided that the lower wavelength limit of integration is greater than zero, so that

[P.sub.x] ([omega] [right arrow] 0) [approximately equal to] 2S(Q)/A[DELTA] [[lambda].sub.FW] sin (A [DELTA] [[lambda].sub.FW]/2) cos (A<[[lambda].sub.i]>), Rectangular incident spectrum, (127)

apart from depolarizations resulting from instrumental imperfections and flipper coil dispersion.

5.2.3 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 = [DELTA][[lambda].sub.FWHM] and mean wavelength <[[lambda].sub.i]>, the wavelength-dependent integral in Eq. (123) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

provided that the lower wavelength limit of integration is greater than zero, so that

[P.sub.x] ([omega][right arrow]0) [approximately equal to] S(Q)/[DELTA] [[lambda].sup.2.sub.FWHM][A.sup.2] {2cos (A <[[lambda].sub.i]>)[1 - cos (A[DELTA][[lambda].sub.FWHM])]} Triangular incident spectrum, (128)

apart from depolarizations resulting from instrumental imperfections and flipper coil dispersion.

5.3 Special Cases for Quasielastic Neutron Scattering (QENS) Symmetric Scans

In this case, [delta]L [right arrow] 0 in Eq. (122) and we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (129)

For quasielastic scattering, the scattering function is represented by a Lorentzian

S (Q,[omega]) = 1/[pi] [gamma](Q)/[gamma][(Q).sup.2] + [[omega].sup.2] (l30)

where [gamma] (Q) = [GAMMA] (Q)/[??], where [GAMMA](Q) is the energy half-width at half maximum. Performing the integral over co, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (131)

We now consider the possible [lambda]-dependence of [GAMMA](Q). Following Hayter and Penfold [17], we consider a common case of self-diffusion at low Q at fixed scattering angle, [theta]. For simplicity, we ignore the very small change in energy of the neutrons on scattering, so we may make the approximation

[GAMMA](Q) = [??]D[Q.sup.2] [approximately equal to] 16[??][[pi].sup.2] D[sin.sup.2][theta]/ [[lambda].sup.2.sub.i] QENS, self-diffusion low Q (Q [approximately equal to] [Q.sub.el]), (132)

where [theta]is the scattering angle, so that Eq. (131) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (133)

where we have set

K = 16[pi][[gamma].sub.n] [([m.sub.n]/h).sup.2] N[B.sub.0][L.sub.0] D [sin.sup.2] [theta] = 16[[pi].sup.2]D [sin.sup.2] [theta]/[[lambda].sup.3.sub.i] [[tau].sub.NRSE]. (134)

5.3.1 Purely Monochromatic Beam

For a purely monochromatic incident beam, I([[lambda].sub.i]) = [delta]([[lambda].sub.0]), and Eq. (133) becomes simply:

[P.sub.x] ([delta]L [right arrow] 0)[approximately equal to]exp(-K[[lambda].sub.0]) [equivalent to] exp([GAMMA](Q)[[tau].sub.NRSE]/[??]) I([[lambda].sub.i]) = [delta]([[lambda].sub.0]). (135)

5.3.2 Rectangular Incident Wavelength Spectrum

For a rectangular incident wavelength spectrum of full width [DELTA][[lambda].sub.FW], centered about [[lambda].sub.i]= <[[lambda].sub.i]>, Eq. (133) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (136)

5.3.3 Triangular Incident Wavelength Spectrum

For a triangular incident spectrum of FWHM = [DELTA] [[lambda].sub.FWHM] and mean wavelength <[[lambda].sub.i]>, Eq. (133) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (137)

5.3.4 Gaussian Incident Wavelength Spectrum

For a Gaussian incident spectrum of FWHM = [DELTA] [[lambda].sub.FWHM] (standard deviation [sigma]) and mean wavelength <[[lambda].sub.i]>, Eq. (133) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (138)

5.4 Detected Signal

5.4.1 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

[I.sub.+] = 1/2(1 + (<cos[[phi].sub.NRSE]>)=1/2(1 + <cos [omega][[tau].sub.NRSE]>)=1/2(1 + [P.sub.x]), (139)

where [P.sub.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.

Monte Carlo simulations illustrating the behavior of Eq. (139) in various situations are shown in Fig. 11. In these examples there are no sample size effects and [DELTA] [(B).sub.0] = [DLETA][l.sub.B0] = 0 (zero field inhomogeneity and perfect dimensions of the flipper coils), but the effects of beam divergence, incident neutron bandwidth, and non-elastic scattering are illustrated. The effects of spectrometer imperfections are analyzed further in Sec. 6 and additional simulation examples are given in Sec. 8. Columns 1 to 3 of Fig. 11 are for elastic, non-spin-flip (isotope incoherent elastic) scattering (or no sample). Column 4 is for quasielastic, isotope incoherent (non-spin-flip) scattering. Additionally for columns 1 and 4 zero beam divergence is assumed. These particular simulations were performed for a 4-N=2 bootstrap coil NRSE with [L.sub.l] = 2.0 m, [l.sub.B0] = 3.0 cm, [l.sub.g] = 0.0 cm, and [[lambda].sub.0] = 8 [Angstrom], for [10.sup.-3] [less than or equal to] [B.sub.0](T) [less than or equal to] 0.025. The asymmetric scan is performed with [absolute value of [B.sub.0]] = [absolute value of [B.sub.1]], with ([L.sub.0] - [L.sub.1]) varied ten minimum periods each side of the symmetric position (i.e., between [+ or -]10 [pi][v.sub.0]/[[gamma].sub.n]N[B.sup.max.sub.0], where [B.sub.0.sup.max] is the maximum applied static field [0.025 T], corresponding to [[tau].sub.NRSE] = 19.1 ns). [delta](BL) was varied by changing [L.sub.1] with respect to [L.sub.0]. (i.e., [delta](BL) = B[delta]L). The incoming and outgoing beam divergence, if any, is equal and uniform up to a maximum [DELTA][[theta].sub.i,max] = [DELTA][[theta].sub.f,max] = [DELTA][[theta].sub.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.sub.0] = [L.sub.1]. The left hand column of Fig. 11 illustrates the effect of broadening [DELTA][[lambda].sub.i], for elastic, non-spin flip scattering (or no sample). In the extreme, purely monochromatic case of [DELTA][[lambda].sub.i] = 0, the signal is cosinusoidal with respect to [delta]L (as predicted by Eq. (125) with [delta](BL) = B[delta]L for one signal period given by Eq. (126). For [DELTA][[lambda].sub.i] > 0, the maximum signal is achieved at the symmetrical spectrometer setting and, as [DELTA][[lambda].sub.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.sub.0] (= [B.sub.1]) (and hence [[tau].sub.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 [DELTA][[lambda].sub.i]/<[[lambda].sub.i]> = 10 % (column 3). The fourth column demonstrates the increasingly rapid decay of the echo point signal with respect to [[tau].sub.NRSE] as the quasielastic width is increased (as predicted by Eq. (135) for a purely monochromatic incident beam ([DELTA][[lambda].sub.i] = 0)).

5.4.2 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 "+" 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

[P.sub.P] = [I.sup.+.sub.P] - [I.sup.-.sub.P]/ [I.sup.+.sub.P] + [I.sup.-.sub.P] (140)

where [I.sup.+.sub.P] and [I.sup.-.sub.P] are the intensities of + and - neutrons in the beam after the polarizer P. Note that [P.sub.P] can vary between +1 and -1. The incoming unpolarized beam of total intensity [I.sub.0] is described by equal + and components, i.e.,

[I.sup.+.sub.0] = [I.sup.-.sub.0] = [I.sub.0]/2. (141)

The total intensity after the polarizer

[I.sup.tot.sub.P] = [I.sup.+.sub.P] + [I.sup.-.sub.P] = [T.sub.P] [I.sub.0]/2 = [T.sub.P] [I.sup.+.sub.0] (142)

where we have used the boundary condition that for perfect + polarization efficiency ([P.sub.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.sub.P] is the spinindependent transmission factor of the device with 0 < [T.sub.P] < 1 due to effects such as absorption or scattering. From Eqs. (140) and (142) it is easy to show that after the polarizer:

[I.sup.+.sub.P] = [T.sub.P] [I.sub.0]/4 (1 + [P.sub.P]) = [T.sub.P] [I.sup.+.sub.0]/2 (1 + [P.sub.P]) (143)

and

[I.sup.-.sub.P] = [T.sub.p] [I.sub.0]/4 (1 - [P.sub.P]) = [T.sub.P] [I.sup.-.sub.0]/2 (1 - [P.sub.P]). (144)

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 I

[I.sup.+.sub.PA] = [T.sub.A] [I.sup.+.sub.P]/2 (1 + [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sup.+.sub.0]/4 (1 + [P.sub.P])(1 + [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sub.0]/8 (1 + [P.sub.P])(1 + [P.sub.A]). (145)

Likewise

[I.sup.-.sub.PA] = [T.sub.A] [I.sup.-.sub.P]/2 (1 - [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sup.-.sub.0]/4 (1 - [P.sub.P])(1 - [P.sub.A]) = [T.sub.P] [T.sub.A] [I.sub.0]/8 (1 - [P.sub.P])(1 - [P.sub.A]). (146)

with the total beam intensity after the analyzer

[I.sup.tot.sub.PA] = [I.sup.+.sub.PA] + [I.sup.-.sub.PA] = [T.sub.P][T.sub.A] [I.sub.0]/4 (1 + [P.sub.P] [P.sup.A]) . (147)

Therefore, the final polarization for non-spin flip scattering is

[P.sub.PA] = [I.sup.+.sub.PA] - [I.sup.-.sub.PA]/[I.sup.+.sub.PA] + [I.sup.-.sub.PA] = [P.sub.P] + [P.sub.A]/1+[P.sub.P][P.sub.A]. (148)

If two [pi]-flippers of efficiency [f.sub.1] and [f.sub.2] and spin-independent transmission factor [T.sub.f1] and [T.sub.f2] are placed between the polarizer and the analyzer, and remembering that the effect of a [pi]-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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (149)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (150)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] only flipper 2 on, (151)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] both flippers 1 and 2 on. (152)

As pointed out by Hayter [18], the ratio of the detector count rates, [I.sub.PA.sup.tot], measured with both [pi]-flippers switched off to the count rates with the two flippers switched "on-off', "off-on" and "on-on" provides three "flipping ratios", [R.sub.1], [R.sub.2], and [R.sub.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.sub.1], [f.sub.2], and the product of the polarizer and analyzer efficiencies, [P.sub.P][P.sub.A], which can be solved to obtain

[P.sub.P][P.sub.A] = [R.sub.12] ([R.sub.1]- 1)([R.sub.2] - 1)/([R.sub.1] [R.sub.2] - [R.sub.12]) (153)

[f.sub.i] = ([R.sub.i] - 1)/2[R.sub.i] (1 + [P.sub.P][P.sub.A]/[P.sub.P][P.sub.A]. (154)

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.sub.corr], is related to the measured signal, [P.sub.meas], by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](155)

5.4.3 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 [pi]-flipper (Sec. 2.4) and Eq. (55), the polarization after the sample, [P.sub.S], is related to the polarization before the sample, [P.sub.i], by

[P.sub.S] =(1 - 2q) [P.sub.i] . (156)

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 [pi]. Thus the sample flipping efficiency is given by q = 2/3, consequently

[P.sub.S]/ [P.sub.i] = -1/3 non-magnetic, pure isotope incoherent scatterer. (157)

This means that the spin-echo signal amplitude is reduced to 1/3 and the minus sign means that the echo signal is inverted. For this case, Eq. (155) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (158)

For a more general non-magnetic case where both spin-incoherent and coherent scattering are present, we might have

q [approximately equal to] 2[S.sub.inc] (Q)/3([S.sub.coh] (Q) + [S.sub.inc] (Q)). (159)

where we have assumed that the relative probabilities of coherent and spin-incoherent scattering are given by [S.sub.coh](Q) and [S.sub.inc](Q) respectively, therefore

[P.sub.S]/[P.sub.i] = 3[S.sub.coh](Q)-[S.sub.inc](Q)/3([S.sub.coh] (Q) + [S.sub.inc] (Q)) (160)

This represents an upper limit on the size of the spin-echo signal. For this case Eq. (155) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (161)

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.

6. 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.,

[[phi].sub.NRSE] = [[phi].sub.0] - [[phi].sub.1] = 2N[m.sub.n][[gamma].sub.n]/h [[B.sub.0][L.sub.0][[lambda].sub.i] - [B.sub.1][L.sub.1] [[lambda].sub.f]].

At the echo point, ([[phi].sub.NRSE]) = 0, however, even for [[lambda].sub.i] = [[lambda].sub.f] (elastic scattering or no sample), [[phi].sub.NRSE] has a distribution of values about the mean, ([[phi].sub.NRSE]), of characteristic width [DELTA][[phi].sub.NRSE]. This is because the terms [B.sub.0][L.sub.0] and [B.sub.1][L.sub.1] have non-zero spread, [DELTA]([B.sub.0][L.sub.0]) and [DELTA]([B.sub.1][L.sub.1]) respectively (2), arising from instrumental imperfections. Consequently, [DELTA][[phi].sub.0] and [DELTA][[phi].sub.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, [DELTA](BL), determines the elastic instrumental resolution function. In order to obtain a broad dynamic range, [DELTA][[phi].sub.NRSE] must be dominated by the distribution of [[lambda].sub.i] - [[lambda].sub.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 [[phi].sub.NRSE] also to have a Gaussian distribution, g([[phi].sub.NRSE]). At the echo point (([B.sub.0][L.sub.0]) = ([B.sub.1][L.sub.1]), g([[phi].sub.NRSE]) is symmetrically distributed about zero (polarization realigned along the original direction--the x axis in these examples). For the Gaussian distribution g([[phi].sub.NRSE]), the elastic scattering polarization along x, [P.sub.x.sup.0], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (162)

or the inverse relation

[DELTA][[phi].sup.FWHM.sub.NRSE] = 4 [square root of (ln 2 ln[1/[P.sup.0.sub.x])] [approximately equal to] [square root of 11.1 ln [1/[P.sup.0.sub.x]]. (163)

We use this convenient form when estimating spectrometer tolerances in the following sections. One notes that if the distribution g([[phi].sub.NRSE]) was uniform between the limits [+ or -] [DELTA][[phi].sub.NRSE.sup.max], rather than Gaussian, the analogue of Eq. (162) is a sinc function of [DELTA][[phi].sub.NRSE.sup.max]:

[P.sup.0.sub.x] ([DELTA][[phi].sup.max.sub.NRSE]) = sin[DELTA] [[phi].sup.max.sub.NRSE]/ [DELTA][[phi].sup.max.sub.NRSE]. (164)

The elastic and quasi-elastic signal count rate cannot exceed a maximum proportional to [P.sub.x.sup.0] (for pure coherent scatterers) and sometimes considerably less for incoherent scatterers (see Sec. 5.4.3), therefore [P.sub.x.sup.0]([[tau].sub.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.sub.x.sup.0] > 0.2 at the maximum required [[tau].sub.NRSE] in a quasielastic measurement. Purely coherent, elastic scatterers, such as Grafoil[R], 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 [[phi].sub.0] and [[phi].sub.1], which imposes [DELTA][[phi].sub.0] = [DELTA][[phi].sub.1], and that the spectrometer is operated at the echo point (i.e., ([p.sub.0]) = [[phi].sub.1]). If [[phi].sub.0] and [[phi].sub.1] are distributed normally, we can write

[DELTA][[phi].sub.NRSE] = [square root of [DELTA][[phi].sup.2.sub.0] + [DELTA][[phi].sup.2.sub.1]] [approximately equal to] [square root 2 [DELTA][[phi].sub.0]]. (165)

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.

6.1 Static Magnetic Field Inhomogeneities

We may consider the effect of static field inhomogeneity as creating a distribution of values of [[omega].sub.0] - [[omega].sub.rf]. In order to simplify the argument we consider [[omega].sub.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 [[omega].sub.0] - [[omega].sub.rf] to the distribution of [[omega].sub.0] caused by the field inhomogeneity, [DELTA][B.sub.0], and comparing the polarization with the equivalent system in which [[omega].sub.0] = [[omega].sub.rf] for all trajectories ([DELTA][B.sub.0] = 0). Further, we assume that the spectrometer is optimally tuned such that <[[omega].sub.0]> = [[omega].sub.rf] and that the field inhomogeneity gives rise to a normal distribution of [[omega].sub.0] with respect to <[[omega].sub.0]>. In this approximation, the effect of static field inhomogeneity is analogous to the effect of dispersion discussed in Sec. 2.2. The effect of [[omega].sub.0] [not equal to] [[omega].sub.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

[absolute value of [B.sup.eff.sub.rf]] = [square root of ([[omega].sub.0] - [[omega].sub.rf]).sup.2] + [[omega].sup.2.sub.p]]/[[gamma].sub.n] (166)

The effective field lies at an angle [[alpha].sub.eff] to the x-y plane given by

[[alpha].sub.eff] [tan.sup.-1] [[omega].sub.0] - [[omega].sub.rf]/[[gamma].sub.n][absolute value of [B.sub.rf]] = [tan.sup.-1] [[omega].sub.0] - [[omega.sub.rf]/[[omega].sub.p]. (167)

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 ([[omega].sub.0] = [[omega].sub.rf]) is given by Eq. (86) and in the general off-resonance case by Eq. (83). If [DELTA][[phi].sub.[pi]] 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 (cos [epsilon] cos[DELTA][[phi].sub.[pi]]), where [epsilon] here refers to the angle of the spin vector out of the x-y plane, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (168)

If [[lambda].sub.i] is the median wavelength and that the flipper is optimally tuned for [pi] flips at this wavelength, i.e., [[omega].sub.p] = [pi]h/[m.sub.n]l[B.sub.0] [[lambda].sub.i] (see Eq. (13)) and setting

[xi] = [absolute value of [B.sup.eff.sub.rf]]/[absolute value of [B.sub.rf]] = [square root of [[omega].sup.2.sub.p] + ([[lambda].sub.n][DELTA][B.sub.0]).sup.2]]/[[omega].sub.p], (169)

Eq. (168) simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (170)

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 [DELTA][B.sub.0] is sufficiently small that cos[epsilon] [approximately equal to] 1 and (cos[DELTA][[phi].sub.[pi]]) [approximately equal to] 1-[([DELTA][phi].sup.2.sub.[pi]])/2 = cos([DELTA][[phi].sub.[pi]] (rms)), we have after one [pi]coil due to the effect of [DELTA][B.sub.0]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (171)

For an M-coil unit spectrometer, we assume that [DELTA][[phi].sub.[pi]] 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)):

[DELTA][[phi].sup.FWHM.sub.NRSE] [approximately equal to][square root of M] [cos.sup.-1] (1/[[xi].sup.2][sin.sup.2]([pi]/2 [xi]])) = 4[square root of (ln 2 ln (1/[P.sup.0.sub.x])] (172)

and hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (173)

Specifically for a 4-N coil instrument we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (174)

In the range of operation of spectrometer configurations considered in this document, it can be shown that the value of [xi] is typically no greater than about 1.3. In this range, it turns out that the awkward term

[[cos.sup.-1]{1/[[xi].sup.2][sin.sup.2]([pi]/2 [xi])}].sup.2] may be replaced very successfully by 4 ([xi]-1) so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (175)

or specifically for a 4-N coil instrument:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (176)

It turns out, somewhat fortuitously, that Eqs. (175) and (176) produce a better approximation to [P.sub.x.sup.0] when [DELTA][B.sub.0] is too large to assume cos[epsilon] [approximately equal to] 1 (implicit in Eqs. (173) and (174)) or when the accumulated dephasing in the x-y plane approaches 2 [pi] The approximations in Eqs. (175) and (176) make the inverse problem significantly more tractable (i.e., what tolerance on [DELTA][B.sub.0] is required to obtain a given value of [P.sub.x.sup.0] under a given set of conditions [[lambda].sub.0], [B.sub.0] etc.?). The inverse expression, which is more useful in instrument design than the forward expression (Eq. (173)), then finally reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (177)

for M coils. Specifically for a 4-N coil instrument we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (178)

where [kappa] = ln 2 ln(l/[P.sup.0.sub.x]). 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.sub.0] = 0.0393 T, [l.sub.B0] = 0.03 mm, [L.sub.0] = 2 m, [[lambda].sub.0] = 8 [[Angstrom]], which gives [[tau].sub.NRSE] = 15 ns and 30 ns for N = 1 and N = 2 respectively.

6.2 Coil Flatness

In order to estimate tolerances on the flight path lengths, we return to the expanded equations representing [[phi].sub.0] (and [[phi].sub.1]) which contain the individual contributions (the flatness model used is that described in Sec. 3.6), and now we assume [DELTA][B.sub.0] = 0. (a) For a 4 (N = 1)-coil NRSE, we have from Eq. (72) for a given neutron trajectory,

[[phi].sub.0] = [[omega].sub.rf]/[v.sub.n] [2[L.sub.0] + [DELTA][f.sub.R](B) + [DELTA][f.sub.L](B)- [DELTA][f.sub.L](A) [DELTA][f.sub.R](A)]

where we have set [[phi].sub.in] = 0 (perfectly polarized incoming beam) and the terms [DELTA][f.sub.L] and [DELTA][f.sub.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, [DELTA][f.sub.L] and [DELTA][f.sub.R] have Gaussian distributions of equal FWHM =[DELTA][f.sub.FWHM], we can write

[DELTA][[phi].sub.0] = [square root of 4N[DELTA][f.sup.FWHM]/2N[L.sub.0] = [DELTA][f.sup.FWHM/[L.sub.0]] N = 1 (179)

whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (180)

We can also write the FWHM fluctuation in the coil length, [DELTA][l.sub.B0], in terms of [DELTA][f.sub.FWHM], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (181)

(b) For N=2 bootstrap coils, for a given neutron trajectory, we have from Eq. (73):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we have set [[phi].sub.in] = O (perfectly polarized incoming beam) and the terms [DELTA][f.sub.L] and [DELTA][f.sub.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 [DELTA][f.sub.L] and [DELTA][f.sub.R] have Gaussian distributions of equal FWHM = [DELTA][f.sub.FWHM], we can write

[DELTA][[phi].sub.0]/[[phi].sub.0] = [square root of 4N][DELTA][f.sub.FWHM]/2[NL.sub.0] = [DELTA][f.sub.FWHM]/[square root of N][L.sub.0], N = 2, (182)

whence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (183)

We can also write the FWHM fluctuation in the coil length, (length of the [B.sub.0] field) in terms of [DELTA][f.sup.FWHM], where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (184)

Inverting Eq. (184) we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (185)

Equation (184) seems to be generally valid, the "[square root] N" not being apparent in the N = 1 case (Eq. (181)). The success of Eq. (184) is demonstrated in Fig. 14 and Fig. 15 for N = 1 and N = 2 respectively.

6.3 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (186)

where [[??].sup.surf.sub.max] is the maximum tolerable angle between the entrance and exit surfaces of the static coil windings and a and [l.sub.axial] are the coil dimensions defined in Fig. 23.

6.4 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, [DELTA][[theta].sub.i]], or [DELTA][theta].sub.f], up to specified maxima [DELTA][[theta].sub.i,max] and [DELTA][[theta].sub.f,max] respectively, where all [DELTA][theta] are defined with respect to any axis parallel to the nominal beam axis. [DELTA][[theta].sub.i], and [DELTA][[theta].sub.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([DELTA][[theta].sub.if]).

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.sub.1][L.sub.1] = [B.sub.0][L.sub.0], <[[phi].sub.0][[phi].sub.0]), etc. The elastic resolution is still given by Eq. (113), i.e., [P.sup.0.sub.x] = (cos [[phi].sub.NRSE]) = (cos([[phi].sub.0] - [[phi].sub.1])>. We also assume small divergence, which allows one to write [[phi].sub.0] = 2N[m.sub.n][[gamma].sub.n] [B.sub.0][L.sub.0][[lambda].sub.i]/h etc. (see Sec. 5.1). With these assumptions the expression for [P.sub.x.sup.0] simplifies to

[P.sup.0.sub.x] = <cos([phi].sub.0] - [phi].sub.1])> = <cos([[phi].sub.0] + [DELTA][phi].sub.0]) + -[<[[phi].sub.1] + [DELTA][[phi].sub.1]]>] = <cos[DELTA][[phi].sub.0] - [DELTA][[phi].sub.1])>. (187)

For a trajectory in the incident arm of the spectrometer, we have

[DELTA][[phi].sub.0]/<[[phi].sub.0]> [approximately equal to] [DELTA][L.sub.0]/[L.sub.0] = [1/cos [DELTA][[theta].sub.i]]-1]. (188)

The distribution of [DELTA][[phi].sub.0] for random [DELTA][theta] is by no means Gaussian or uniform. Because we assume small divergence (i.e., [DELTA][[theta].sub.i,max] and [DELTA][[theta].sub.f,max] are small--certainly within the range of angles encountered in the NRSE), we write for all incident arm trajectories:

[DELTA][[phi].sub.0] [approximately equal to] <[[phi].sub.0]> (1-cos [DELTA][[theta].sub.i]) [approximately equal to] <[[psi].sub.0]> [DELTA][[theta].sup.2.sub.i]/2 (189)

and likewise at the echo point:

[DELTA][[psi].sub.1] [approximately equal to] <[[phi].sub.1]> (1-cos [DELTA][[psi].sub.f]) [approximately equal to]<[[phi].sub.1]> [DELTA][[theta].sup.2.sub.f]/2 = <[[phi].sub.0]> [DELTA][[theta].sup.2.sub.f]/2. (190)

(Note [DELTA][[phi].sub.0] and [DELTA][[phi].sub.1] are not necessarily small numbers because ([[phi].sub.0]) can be very large). Therefore, finally

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (small divergence, at echo point, only angular uncertainties). (191)

The average in Eq. (191) can be expressed in terms of the double integral over the range of [DELTA][[theta].sub.i] and [DELTA][[theta].sub.f] which are both assumed to be uniform in probability in the range (o, [DELTA][[theta].sub.i,max]), (0, [DELTA][[theta].sub.f,max]), permitting the average to be written simply as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (192)

It can be shown that Eq. (192) reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (193)

where [C.sub.1] and [S.sub.1] are the Fresnel cosine and sine integrals respectively, defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (194)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (195)

Certain approximations for evaluating [C.sub.1] and [S.sub.1] have been discussed by Mielenz [19] (note that the [pi]/6 term in Eq. 3b of this reference should be multiplied by [x.sup.3]) and Heald [20]. The integrals can also be evaluated numerically. For the particular case of [absolute value of [DELTA][[theta].sub.i,max]] = [absolute value of [DELTA][[theta].sub.f,max]] = [absolute value of [DELTA][[theta].sub.max]], Eq. (193) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (196)

or in terms of the instrument parameters:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (197)

The success of Eq. (193) in describing the relationship between [DELTA][[theta].sub.max] and [P.sub.x.sup.0] is demonstrated in Fig. 17 and Fig. 18 for realistic examples. The examples with [[tau].sub.NRSE] = 15 ns, N = 1, and [[tau].sub.NRSE] = 30ns, N = 2 have sufficiently large values of ([[phi].sub.0]) that the arguments of [C.sub.1] and [S.sub.1] exceed unity in the plotted range (the values shown on the right hand side y-axes). They also have [absolute value of [DELTA][[theta].sub.i,max]] = [absolute value of [DELTA][[theta].sub.f,max]] = [absolute value of [DELTA][[theta].sub.max]] (so that Eq. (196) is used).

In the present context it is useful to have [P.sub.x.sup.0] as the dependent variable and ask "what is the maximum permissible value of [absolute value of [DELTA][[theta].sub.max]] to achieve a given value of [P.sub.x.sup.0]?" Unfortunately, inversion of Eq. (196) is not trivial. The traditional approximations for [C.sub.1] and [S.sub.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.sub.1.sup-.2](x)+[S.sub.1.sup.2](x) involves terms in [x.sup.4n+2], n = 0, 1, 2,..., [infinity] alternating signs for the first few terms. Another function that has the the same powers and signs as these first terms would be [x.sup.2] exp(-a[x.sup.4]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (198)

The expansion of [C.sub.1.sup.2](x)+[S.sub.1.sup.2](x) for the first few terms is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (199)

therefore we try setting the parameter a in Eq. (198) to the magnitude of the second term coefficient in Eq. (199) = [[pi].sup.2]/45 [approximately equal to] 0.21932 which makes the two leading terms in Eqs. (198) and (199) identical. This should certainly work well for x < 1 since the higher order terms decrease rapidly. With this substitution Eq. (198) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (200)

for which the first few terms are quite similar to those of Eq. (199). It turns out that this approximation can be applied with about 1 % accuracy up to x ~ [x.sub.1%] ~ 1.15, where the fan-out of the spins due to the divergence (= [DELTA][[phi].sub.0] (see Eq. (189)) [approximately equal to][eta][x.sub.1%.sup.2]/2 ~ 0.7[pi]) is still below 2[pi]radians, i.e.,

[C.sup.2.sub.1] (x) + [S.sup.2.sub.1] (x) [approximately equal to] [x.sup.2] exp (-[[pi].sup.2][x.sup.4]/45] x < ~ 1.15. (201)

In fact the approximation is within 15 % for x up to about 1.8, at which point [DELTA][[phi].sub.0] ~ 1.6[pi](as is seen from

Fig. 17 and Fig. 18. Now identifying x with [square root of 2N[m.sub.n][[gamma].sub.n] [B.sub.0][L.sub.0[[lambda].sub.i]]/[pi]h [DELTA][[theta].sub.max]] Eq. (197) can be inverted using the approximation in Eq. (201) yielding:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (202)

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.

6.5 Approximation for Equal Contributions to Depolarization from [DELTA][B.sub.0], [DELTA][l.sub.B0], and [DELTA][[theta].sub.max]

In the preceding sections, the contributions of [DELTA][B.sub.0], [DELTA][l.sub.B0], or [DELTA][theta] to the elastic polarization [P.sub.x.sup.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 obtained by setting the [DELTA][B.sub.0], [DELTA][l.sub.B0], and [DELTA][theta] contributions to the depolarization approximately equal. For equal contributions, we assume that the tolerances will be approximately 1/[square root]3 times the values given by Eqs. (178), (184), and (202) respectively (for a 4-N coil instrument), i.e.,

[DELTA][B.sup.FWHM.sub.0] [approximately equal to] [absolute value of [B.sub.rf]][square root of K/3N (K/N)+ 2] (4-N instrument, equal contribs to [P.sub.x.sup.0]), (203)

where k = ln 2 ln (1/[P.sup.0.sub.x]), as before. Note that [DELTA][B.sub.0] is defined by N, [l.sub.B0], and [[lambda].sub.i] only and is independent of [B.sub.0] or zero field region parameters.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] meters (4-N instrument, equal contribs to [P.sub.x.sup.0]). (204)

Note that [DELTA][l.sub.B0] is defined by N, [l.sub.B0] and [lambda] only and is independent of [l.sub.B0] or zero field region parameters. Finally,

(4-N instrument, equal contribs to [P.sub.x.sup.0]), (205)

for [absolute value of [DELTA][[theta].sub.max]]<~ 6.7 x [10.sup.-3]/[square root of N[B.sub.0][T][L.sub.0][m][[lambda].sub.i][[Angstrom]]] [rad].

Note that [DELTA][[theta].sub.max] depends on [lambda] and on both the flipper coil and zero-field parameters (N, [B.sub.0], [L.sub.0] (i.e., [L.sub.AB], [l.sub.B0], and [l.sub.g])). Even though these parameters also appear in the expression for [[tau].sub.NRSE], the [[lambda].sup.3]-dependence of the latter means that [DELTA][[theta].sub.max] is not uniquely determined by the quantity [[tau].sub.NRSE] (i.e., the same value of [[tau].sub.NRSE] may require different values of [DELTA][[theta].sub.max] depending on the values of N, [B.sub.0], [L.sub.0] and [lambda]).

6.6 Some Examples (Equal Contributions to Depolarization)

Consider requiring the elastic (resolution) polarization [P.sub.x.sup.0] to be greater than some specified minimum value at a reference point with equal contributions coming from AB0, [DELTA][l.sub.B0], and [DELTA][[theta].sub.max]. We consider the point [[tau].sub.NRSE] [approximately equal to] 30 ns at [lambda] = 8 [Angstrom] with N = 2, for M = 8 [pi] coils ([l.sub.B0]=0.03 m), with [B.sub.0] = 0.0393 T, [L.sub.0] = 2 m. Using Eqs. (203-205), several results are shown in Table 7.

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.

7. NIST NRSE Project Goals

7.1 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 [[tau].sub.NRSE] = 30 ns at [lambda] = 8[Angstrom] 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 [Angstrom] for high-Q capability.

5. Must have a short Fourier time measurement capability.

7.2 Spectrometer Dimensions and Field Magnitudes Required to Access [[tau].sub.NRSE] = 30 ns at [lambda] = 8 [Angstrom]

From Eq. (120) we have

[[tau].sub.NRSE] [ns] = 0.37271 N [B.sub.0][T][L.sub.0][m][[lambda].sub.i][[Angstrom]].sup.3],

where we assume that [B.sub.0] = [B.sub.1] so that [L.sub.0] = [L.sub.1] at the QENS echo point. In order to access [[tau].sub.NRSE] = 30 ns at [lambda] = 8 [Angstrom], we must satisfy the condition

N[([B.sub.0][T][L.sub.0][m]).sub.max] [less than or equal to] 0.157 criterion for accessing [[tau].sub.NRSE] = 30 ns at [lambda]= 8 [Angstrom], (206)

where [([B.sub.0][L.sub.0]).sub.max] implies the maximum attainable value of the product [B.sub.0][L.sub.0]. If we chose N = 2 as the most likely bootstrap factor, noting the advantages and disadvantages outlined in Sec. 3.4, this condition amounts to fulfilling:

[([B.sub.0][T][L.sub.0][m]).sub.max] [less than or equal to] 0.079 Tm criterion for accessing [[tau].sub.NRSE] = 30 ns at [lambda] = 8[Angstrom] with N = 2. (207)

Obvious limitations on the maximum value of [B.sub.0] are imposed by the maximum current x winding density of the static field coils. This depends on the length, cross-section, material, winding temperature, and the ability to remove heat. Increasing the zero-field drift path lengths increases proportionately the maximum achievable value of [[tau].sub.NRSE], however disadvantages include the rapid reduction in solid angle ([varies] 1/[L.sup.2]) and possibly limitations imposed by available space. Owing to these constraints and the linear dependence of [[tau].sub.NRSE] on [B.sub.0], it seems reasonable to attempt to maximize the static magnetic field [B.sub.0] as far as possible. Evaluating [B.sub.0] and [L.sub.0] for [[tau].sub.NRSE] =30 ns at [[lambda].sub.i] = {[[lambda].sub.i]) = 8[Angstrom], we have, for example,

[B.sub.0] [approximately equal to] 0.08 T, [L.sub.0] = 1 m, N = 2

[B.sub.0] [approximately equal to] 0.04 T, [L.sub.0] = 2 m, N = 2.

To date, the largest static fields produced in water-cooled NRSE coils using pure aluminum windings are about [B.sub.0] [approximately equal to] 0.025 T. With this field we require [L.sub.0] = 3.14 m for N = 2 (which is a little long for available floor space) or else [L.sub.0] = 1.57 m for N=4. Apart from the increased restrictions on the maximum incoming bandwidth, [DELTA][lambda]/[lambda], when using N = 4, doubling the number of [pi]-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.sub.0] [less than or equal to] 2 m and pursuing the goal of increasing [B.sub.0] towards 0.04 T presents itself as one of the more attractive options. Some consequences are explored in the following sections.

7.3 Bootstrap NRSE Coil Components and Specifications

7.3.1 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. [mu]-metal plates capping each end of the static field coils conduct magnetic flux lines between the two coils. An outer [mu]-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.

7.3.2 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).

Very approximately, the macroscopic neutron cross-section of aluminum at all temperatures of interest is about 0.11 [cm.sup.-1] for [lambda] < 4.7[Angstrom]. Therefore, we have

[T.sub.Al] ~ exp(-0.11t[cm]), [lambda] 4.7 [Angstrom]. (208)

Estimating the equilibrium temperature and temperature gradients of the windings depends on the detailed coil design. In order to partially account for elevated winding temperatures at high-field operation of the coils, we approximate the macroscopic cross-section for [lambda] > 4.7[Angstrom] using the average of available data for pure aluminum [21] at T = 300 K and at T = 800 K. At 300 K data we have approximately

[[SIGMA].sub.Al] (300K)[[cm.sup.-1]][approximately equal to]6.4 + 8.94[lambda][[Angstrom]])x [10.sup.-3], [lambda] [greater than or equal to] 4.7[[Angstrom]] (209)

and for the 800 K data we have approximately

[[SIGMA].sub.Al] (800K)[[cm.sup.-1]][approximately equal to](1.91 + 1.175[lambda][[Angstrom]])x [10.sup.-2], [lambda] [greater than or equal to] 4.7[[Angstrom]] (210)

Therefore, we use an effective aluminum macroscopic cross-section of

[[SIGMA].sup.eff.sub.Al] [[cm.sup.-1]][approximately equal to](1.28 + 1.03[[Angstrom]])x [10.sup.-2], [lambda] [greater than or equal to] 4.7[Angstrom]. (211)

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.sub.Al][lambda] = 8 [Angstrom]) [greater than or equal to] 80 %, then Eq. (211) requires that t must not exceed a maximum value, [t.sub.max], of about 1.0 mm (i.e., the static field coil windings have thickness of about 1 mm, the r.f. windings have thickness of about 0.5 mm). For the r.f. coils the skin effect at ~1 MHz frequencies likely restricts the r.f. winding thickness to a smaller value (see Sec. 7.3.4.7).

Coils constructed at the Institut Laue-Langevin (ILL), Grenoble, France, Laboratoire Leon Brillouin (LLB), Saclay, France, and the Forschungs-Reaktor Munchen-II (FRM-II), Munich, Germany, have used 0.4 mm-thick anodized aluminum band windings, with anodization depth of about 3 [micro]m for insulation. The anodization layer can contain incorporated water, which gives rise to strong, anisotropic, small angle scattering. This small angle scattering is greatly reduced by boiling the wire in [D.sub.2]O under pressure at about 200[degrees]C [11].

7.3.3 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.

7.3.3.1 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

B = [[mu].sub.0]nI, (212)

where [[mu].sub.0] is the permeability of free space with [[mu].sub.0] =4[pi] x [10.sup.-7] [NA.sup.-2]. In SI units we have

[B.sub.0] [T] [approximately equal to] 4[pi] x [10.sup.-7]n [[m.sup.-1]]I[A][equivalent]1.26 x [10.sup.-6]n [[m.sup.-1]]I[A] long solenoid approximation, (213)

where [B.sub.0] is the static field in Tesla, n is the winding density in [m.sup.-1], and I is the current in Amps. Equivalently, the current in the coil at field [B.sub.0] is

I [A] = 2.5 x [10.sup.6]/[pi] [B.sub.0][T]/n[[m.sup.-1]] [approximately equal to] 8 x [10.sup.5] [B.sub.0] T]/n[[m.sup.-1]. (214)

Thus the required current is inversely proportional to the winding density and is directly proportional to the required field [B.sub.0].

7.3.3.2 Resistance of the static field coil windings

The resistance of the static field coil winding is

R = [rho](T)[l.sub.w]/[A.sub.w], (215)

where [l.sub.w] is the total length of the coil winding, [A.sub.w] is the wire cross-sectional area, and [rho](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.sub.B0]), assuming the winding thickness is negligible compared with a and [l.sub.B0]. For the particular case of single-layer windings, the total number of turns, [N.sub.B0], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (216)

so that the total length of any single-layer winding around the rectangular coil form shown in Fig. 23 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (217)

The outer surface area of the rectangular coil form is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (218)

so Eq. (217) may be rewritten as

[l.sub.w] = [A.sub.surf] n any thin single-layer winding around rectangular coil form. (219)

7.3.3.3 Single-layer rectangular cross-section wire

The cross-sectional area, [A.sub.w], of rectangular cross-section wire (see Fig. 23) is

[A.sub.w] = th rectangular cross-section wire, width h, thickness t, (220)

so that, using Eqs. (219) and (220), and noting that for a single winding h [less than or equal to] 1/n, with the equality representing the tightly-wound limit, Eq. (215) becomes

R = [rho](T)n[A.sub.surf]/th any single-layer rectangular cross-section wire, (221)

with

R = [rho](T)[n.sup.2] [A.sub.surf]/t tightly-wound rectangular cross-section wire, thickness t (222)

representing the tightly-wound limit with h = 1/n. Therefore, for a given [A.sub.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.sub.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.sub.0] is, from Eqs. (214) and (221)

V = IR 2.5 X[10.sup.6]/[pi] [rho](T)[[OMEGA]m][A.sub.surf] [[m.sup.2]]/t[m]h[m][B.sub.0][T] any single-layer rectang cross-section wire winding (223)

with

V[V] = 2.5 x [10.sup.6][pi] [rho](T)[[OMEGA]m]n[[m.sup.-1]][A.sub.surf][[m.sup.2]]/ t[m] [B.sub.0] [T] tightly-wound, single-layer, rectang cross-section

wire windings (224)

representing the tightly-wound limit. Thus, for the tightly-wound case, the voltage required to maintain a field [B.sub.0] is proportional to [B.sub.0], proportional to the winding density, and inversely proportional to the winding thickness in the beam direction for a given coil surface area. For a given [B.sub.0], the voltage is minimized by tightly-winding the coil within the available surface area.

The power dissipated in the coil with single-layer, rectangular cross-section wire is (from Eqs. (214) and (221) or (223))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] any single-layer rectang cross-section wire winding. (225)

Specifically, for the tightly-wound, rectangular cross-section wire winding it is (from Eqs. (214) and (222) or (224))

P[W] [approximately equal to] 6.25 X[10.sup.12] [rho](T)[[OMEGA]m][A.sub.surf][m.sup.2]/[[eta].sup.2]/ [[eta].sup.2] t[m][([B.sub.0][T]).sup.2] tightly-woundrectang cross-section wire windings. (226)

Thus, for a given [A.sub.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.sub.0]. Like the voltage, the power dissipated is minimized for a given [B.sub.0] by tightly-winding the coil within the available surface area, since h [less than or equal to] 1/n.

7.3.3.4 Single-layer circular cross-section wire windings

The cross-sectional area of the circular cross-section wire, [A.sub.w], is

[A.sub.w] = [pi][r.sup.2.sub.w] circular cross-section wire of radius [r.sub.w]. (227)

Using Eq. (219) and noting that for a single-layer circular winding we have the constraint n [less than or equal to] l/2[r.sub.w], with the equality representing the tightly-wound case, Eq. (215) becomes

R = [rho](T)n[A.sub.surf]/[pi][r.sup.2.sub.w] any circular cross-section wire winding, (228)

with

R = 4[rho](T)[n.sup.3][A.sub.surf]/[pi] tightly-wound circular cross-section wire, (229)

representing the tightly-wound limit. Thus, for a given [A.sub.surf], the tight-winding resistance increases as the cube of n (as opposed to [n.sup.2] in the tightly-wound rectangular wire case with fixed t).

The D.C. voltage required to maintain a static field [B.sup.0] in the circular cross-section wire case is (from Eqs. (214) and (228))

V[V] = 2.5 x[10.sup.-6] [rho][T][[OMEGA]m][A.sub.surf][[m.sup.2]]/([pi][r.sub.w][m].sup.2])[B.sub.0] [T] any single circular cross-section wire winding. (230)

Specifically, for the tightly-wound case it is (from Eqs. (214) and (229))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] tightly-wound, circular cross-section wire winding. (231)

Thus, the voltage required to achieve a given [B.sub.0] in the circular cross-section wire case is independent of the winding density, other than n cannot exceed a value of 1/(2[r.sub.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.sub.0].

The power dissipated in the coil with single-layer, circular cross-section wire windings is (from Eqs. (214) and (228) or (230))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] any single circular cross-section wire winding, (232)

where the tightly-wound case with n = 1/(2[r.sub.w]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] tightly-wound circular cross-section wire windings. (233)

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.

7.3.3.5 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.sub.surf] becomes small with respect to the beam area it is increasingly difficult to maintain adequate field homogeneity within this region at high [[tau].sub.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. 7.3.2). Finally, there are very limited choices of winding material that have both good cold neutron transmission combined with low electrical resistivity. Although maximum fields of a few 10's of mT do not appear dauntingly high, the heat production from the coil is potentially quite large. This is illustrated by the following examples:

The coils produced for the neutron research laboratories Laboratoire Leon Brillouin (LLB), Institut LaueLangevin (ILL) (France), Forschungs-Neutronenquelle Heinz Maier-Leibnitz (FRM-II), and Helmholtz-Zentrum Berlin (HZB) (Germany) use tightly-wound 4 mm wide x 0.4 mm thick anodized aluminum band supplied by Wesselmann Umwelttechnik (3), with n [approximately equal to] 250 [m.sup.-1], [l.sub.axial] [approximately equal to] 0.2 m, a + [l.sub.B0] [approximately equal to] 0.25 m for a beam size of about 2.5 cm x 2.5 cm, so that [A.sub.surf] (see Eq. (218)) [approximately equal to] 0.1 [m.sup.2]. For these coils at maximum field ([B.sub.0] [approximately equal to] 0.025 T), we have (from Eq. (214)) I [approximately equal to] 80 A. For pure Al down to about liquid nitrogen temperature, we have

[[rho].sub.Al](T)[[OMEGA]m] [approximately equal to] 1.14 x [10.sup.-10]T (K) - 6.9 x [10.sup.-9]. (234)

Therefore, specifically for aluminum, we have (from Eq. (226))

[P.sub.Al][W] [approximately equal to] ([B.sub.0][T]).sup.2] [A.sub.surf][[m.sup.2]]/t[m](72.2T(K) - 4.37 x [10.sup.3]) tightly-wound rectangular wire windings. (235)

For T [approximately equal to] 300 K, [P.sub.Al] (0.025 T) [approximately equal to] 2.7 kW. For T [approximately equal to] 350 K, [P.sub.Al] (0.025 T) [approximately equal to] 3.3 kW. If similar coils are to achieve 0.04 Tesla, the current increases to I [approximately equal to] 128A with an increased power dissipation factor of approximately ([0.04.sup.2] /[0.025.sup.2]). The room-temperature power dissipation then increases to approximately 6.9 kW. If the coils are cooled to liquid nitrogen temperature [approximately equal to] 80 K, [P.sup.Al] is more than an order of magnitude smaller ([approximately equal to] 220 W at [B.sub.0] = 0.025 T, [approximately equal to] 560 W at [B.sub.0] = 0.04 T). This is discussed by Gahler, Golub, and Keller in Ref. [8]. One technical challenge is avoiding liquid coolant (water or liquid [N.sub.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.sub.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 [approximately equal to] 198 A at [B.sub.0] = 0.04 T with a corresponding current density of about 64 A [mm.sup.-2]. The winding would be supported by and cooled by four hollow tubes for the passage of liquid [N.sub.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.sub.2]-filled coil form, thus they should remain close to 80 K. In general, the liquid [N.sub.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.

7.3.3.6 Required static field coil current stability

The values in Table 7 imply that [DELTA][B.sub.0]/[B.sub.0] must be around 0.1 % in order to achieve [P.sub.x.sup.0] (8 [Angstrom], 30 ns) [greater than or equal to] 0.5 for typical spectrometer dimensions. Even for perfect static field coil homogeneity (AB0 = 0), this imposes a coil current stability of order of 0.1 % ([DELTA]I/I <~ [10.sup.-3]). The current stability should certainly not become the limiting factor on AB0. Preferably it should be at least an order of magnitude better ([DELTA]/I < [10.sup.-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.sup.-5] range are commercially-available, so this is not expected to impose any technical limitation.

7.3.3.7 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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (236)

This can be re-expressed in terms of the dimensionless quantities

[mu] = 2x/[l.sub.axial] (237)

which is the axial distance from the solenoid center expressed as a fraction of the half-length of the solenoid and

[eta] = 2r/[l.sub.axial], (238)

which is the ratio of the diameter, d, of the coil to its axial length, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (239)

In the "long" solenoid limit ([l.sub.axial] [much greater than] r), the field at the coil center is maximized (=[[mu].sub.0]nI), whereas at its ends it is half this value (=[[mu].sub.0]nI/2--limit of Eq. (239) with [mu] = 1 and [[eta].sup.2] [much less than] [(1+[mu]).sup.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 (=[[mu].sub.0]nI) for solenoids with various ratios [eta] = d/[l.sub.axial], calculated according to Eq. (236). Figure 26 reveals that as [eta] 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 [eta] 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 [DELTA][B.sub.0]/[B.sub.0] is about 0.1 %. With reference to Fig. 26, this requires [B.sub.0]([mu] = 0.1) > 0.999 [B.sub.0]([mu] = 0). This occurs for [eta] <~ 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.

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Title Annotation: | p. 94-132 |
---|---|

Author: | Cook, Jeremy C. |

Publication: | Journal of Research of the National Institute of Standards and Technology |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2014 |

Words: | 12563 |

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