Optimising neutron polarizers—measuring the flipping ratio and related quantities

Nuclear Instruments and Methods in Physics Research A 481 (2002) 475–492

Optimising neutron polarizersFmeasuring the flipping ratio and related quantities D.J. Goossensa,1, L.D. Cussenb,* b

a Research School of Chemistry, Australian National University, Australia School of Communications and Informatics, Victoria University of Technology and ANSTO, PMB 1, Menai, NSW 2234, Australia

Received 16 May 2001

Abstract The continuing development of gaseous spin polarized 3He transmission filters for use as neutron polarizers makes the choice of optimum thickness for these filters an important consideration. The ‘‘quality factors’’ derived for the optimisation of transmission filters for particular measurements are general to all neutron polarizers. In this work optimisation conditions for neutron polarizers are derived and discussed for the family of studies related to measuring the flipping ratio from samples. The application of the optimisation conditions to 3He transmission filters and other types of neutron polarizers is discussed. Absolute comparisons are made between the effectiveness of different types of polarizers for this sort of work. r 2002 Elsevier Science B.V. All rights reserved. PACS: 61.12 Keywords: Neutron scattering instrumentation; Neutron polarizers; Spin flip fraction

1. Introduction Polarized neutron scattering is a powerful technique for the investigation of condensed matter systems on the microscopic scale. It is particularly useful in the exploration of magnetic properties. A polarization analysis experiment can determine scattered intensity not only as a function of momentum transfer and energy transfer but also as a function of neutron spin state, thus extracting unique information about the sample. It is desirable to use the neutron polarizing technique which best suits the experiment, to obtain the best possible data. In previous work the current authors have derived ‘‘quality factors’’ for measurements using polarized neutrons [1,2] and used these quality factors to quantitatively compare several different polarizing methods for measurements of two separate scattering cross-sections [3]. The present work considers

*Corresponding author. Tel.: +61-2-9717-3133; fax: +61-2-9717-3606. E-mail address: [email protected] (L.D. Cussen). 1 Now at ANSTO, PMB 1, Menai, NSW, 2234, Australia. 0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 3 4 8 - 1

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optimisation conditions and compares polarizer types for measurement of quantities related to the flipping ratio, F: The instrument used for the measurements is assumed to have an incident neutron beam with polarization p1 induced by the polarizer where the beam polarization is defined by Nm  Nk ; ð1Þ Nm þ Nk where N m (N k ) represents the number of spin up (down) neutrons in the beam. For a perfect polarizer one of these numbers is zero and p is 71. The sample scatters some fraction of the beam with spin unchanged (SNSF ) and some with spin flipped (SSF ). SNSF and SSF can be regarded as the total non-spin flip and spin flip sample scattering cross-sections, respectively. The scattered beam passes through an analyser which transmits neutrons preferentially on the basis of spin state. Its ability to do this is defined by the polarization, p2 ; it would induce in an initially unpolarized beam. A spin flipper, assumed to be perfect, is installed between polarizer and sample and can reverse the polarization of the incident beam. The measurement generally consists of separately measuring the scattered intensity with flipper off (I FOFF ) and with flipper on (I FON ) which can be expressed p¼

I FOFF ¼ I0 St1 t2 ð1 þ Sp1 p2 Þ;

I FON ¼ I0 St1 t2 ð1  Sp1 p2 Þ:

ð2Þ

Here, ti is the transmission of the ith polarizing element, I0 is the intensity incident on the polarizer, S ¼ SNSF þ SSF is the total sample cross-section and S is defined in Eq. (3), below. It should be noted that where the flipping ratio is measured, sometimes the scattered beam is not analysed. For two polarizers of equal performance we can set p1 ¼ p2 ¼ p: Four different measured quantities are considered in this work. These are the flipping ratio (F), the ratio of scattering cross-sections, (SF), the spin flip fraction (SFF) and the ‘‘Cross-section polarization’’ (S) defined by F¼

I FON ; I FOFF

SF ¼

SSF SNSF

SSF SNSF  SSF ; S ¼ : ð3Þ SNSF þ SSF SNSF þ SSF The value of F for a sample where one of SNSF and SSF is zero is commonly used to measure the polarization of the neutron beam through F ¼ ð1 þ p2 Þ=ð1  p2 Þ and thus to test the performance of an instrument or evaluate the polarizing efficiency of neutron polarizers [4]. Other than this use, F has no scientific meaning and should not be used to optimise an instrument as has already been concluded [1,5–7] and as discussed below. Measurements of F are sometimes used to derive SF [8]. Hence, while F is measured, the quantity of real interest is SF or, equivalently, SFF: In fact, the quantity of physical interest in much of this type of work is the absolute value of the magnetic scattering cross-section. However, given the practical complexity and difficulty in measuring absolute neutron scattering cross-sections, measuring the ratio of this quantity to some known quantity simplifies matters. SF is used to measure the polarization of Bragg reflections from magnetic crystals. This information can be used to assist in the solution of complex magnetic structures [9,10]. SF can also be used to measure magnetic form factors [8]. SFF has been used to discuss the optimisation of neutron polarization analysis instruments [1,5–7]. However, it is S that is a natural variable in the discussion of instrument optimisation as becomes clear when one examines the equations. S has a further physical significance relating to the neutron depolarization matrix, D: Neutron depolarization coefficients, derived from xyz polarization analysis, can reveal details of magnetic domain structure, spin glass phases and magnetic phase transitions, flux distributions in superconductors and grain sizes in thin films [11]. In a depolarization experiment, the SFF ¼

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depolarization coefficient is related to the flipping ratio through [12] F¼

1  p2 Dij p1 ; 1 þ ð2Z  1Þp2 Dij p1

ð4Þ

where Z is the flipper efficiency and Dij is the depolarization coefficient with the incident polarization along Cartesian direction i and the analyser selecting for polarization along j: In a typical 3-d depolarization experiment, the Dij are determined for all three orthogonal polarization directions and give rise to the 3  3 depolarization matrix, D: The rotation of the polarization direction is made using spin turning coils before and after the sample. If the flipper is considered perfect (Z ¼ 1) as is usual and sensible then combining Eqs. (2) and (3) to get an expression for F shows that S is equivalent to the diagonal elements of D: Thus the optimum measurement of S relates to the optimum measurement of the depolarization coefficient. All methods for polarizing neutron beams rely on having a material with different scattering or absorption properties for different neutron spin states. At present, the two most commonly used methods for polarizing neutron beams are supermirrors [13] which have been and remain the subject of intense development and Heusler (Cu2AlMn) alloy monochromators [14] also under continual development. Other ferromagnetic polarizing monochromating crystals have been tried. A third technique is the neutron transmission filter. Many variations have been used but at present gaseous 3He filters show great promise and are under intensive development at a number of locations around the world [15–17]. Transmission filters have the advantage that they allow large beam angular divergences and work effectively even at short neutron wavelengths but at present they suffer from relatively poor spin polarization of the filter medium, resulting in low transmission for a given polarization. Transmission filters decouple polarization from neutron optics, unlike supermirrors and Heusler alloy monochromators. Supermirrors give high polarization and transmission but limited acceptance angle, while Heusler alloy monochromators have suffered from relatively low reflectivity and small mosaic spreads resulting in a low ‘transmission’ by comparison with pyrolytic graphite (PG) monochromators. There has been some discussion of quality factors for transmission filter polarizers. Williams, as part of the study of a 149Sm polarizing filter at the Rutherford Appleton laboratory, [5] showed that the filter should be chosen to maximise p2 t when measuring SFF when S ¼ 0: Williams also concluded that when S was non-zero, thicker filters may be needed. A further study [7] derived a quality factor for measurements of g; the ratio of magnetic to nuclear scattering. This work showed that for jgjo0:4; the optimisation condition suggested by Williams was correct, while for higher g; thicker filters (higher beam polarization) were preferable. Other measurement types have also been discussed [1] and in particular quality factors have been developed for measurements of two cross-sections (spin flip and non-spin flip) to equal precision [2,3]. These quality factors were used to make an absolute comparison of the merits of supermirrors, Heusler alloy monochromators and 3He polarizing filters. The discussion began by defining a quality factor, Q over an instrument and then compared the optimum Q values, QOpt ; attainable using the different types of polarizers as a function of neutron wavelength (l) and beam angular divergence (a). Hence it differed from previous discussions of optimisation in that it compared transmission filters quantitatively with other techniques, and also in that it allowed quantitatively for l and a: A particular challenge in this sort of discussion is the sheer number of parameters. The optimisation conditions depend on the 3He spin polarization, the sample used (through the value of S), the neutron wavelength used (this does not affect transmission filters which can in principle be adjusted in thickness to give the same optimisation at all wavelengths), the desired beam angular width and the instrumental background. Because it is difficult to visualise many dimensional spaces and the number of parameters here is greater than two, detailed discussion must either ignore some of the variables or include a very large number of graphs showing many combinations.

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The 3He spin polarization is an important parameter in these studies. Previous work has used at least three different symbols. The current authors have used p in the past while Tasset and his coworkers have used the less ambiguous PHe ; a notation adopted here.

2. Formulae for quality factors The optimisation process used here is fully described elsewhere [1,3] and is briefly summarised below. The instrument used for the measurement is assumed to consist of some monochromator, a polarizer followed by a spin flipper, a sample, an analyser and a detector. A ‘‘Quality Factor’’ or ‘‘Figure of Merit’’, Q; is defined such that for the whole instrument the quality factor is Q2 while for a ‘half instrument’ (polarizer and associated collimators and monochromator) the quality factor is Q: This is a slight change from [3]. To derive Q the quantity to be measured is expressed in terms of the total sample scattering cross-sections, SNSF and SSF ; I0 ; the counting time T (separated into time with the flipper on T FON and with the flipper off T FOFF ¼ T  T FON ) and the effective beam polarization and transmission (pi and ti ) produced by the ith polarizer for an initially unpolarized beam. The error in the measured quantity, D; is assumed to depend only on the counting statistics on the instrument. The optimum division of counting time is deduced by choosing T FOFF to minimise D: This value of T FOFF is then substituted into the expression for D and polarizer parameters (pi and ti ) are sought to minimise D: Q can be taken to be the inverse of D for the measurement concerned and maximising Q produces the optimum measurement. general Q is a function of SNSF ; SSF ; p1 ; t1 ; p2 and t2 : In all cases Q is found to be proportional to ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pIn TI0 ðSSF þ SNSF Þ: This term scales Q for the magnitude of the measured counts and, being irrelevant to the filter optimisation, it is omitted from the expressions below along with all numerical constants. The polarizer and analyser are assumed to have identical performance, which is to say p1 ¼ p2 ¼ p and t1 ¼ t2 ¼ t: This has been rigorously shown [2] to be the optimum situation when measuring two cross-sections to equal precision. Numerical tests in the range of interest of S and PHe suggest that it is also optimal here but no analytical proof is presented. S appears to be a natural variable in the calculations and its use greatly simplifies the expressions for Q: A calculation of the optimum division of counting time for measurements of F; SF; SFF and S yields the same result for all cases, that is pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T I FON T 1  Sp2 FOFF T ¼ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð5Þ 1 þ Sp2 þ 1  Sp2 I FON þ I FOFF This is a (corrected) representation of Eq. (9) from [1]. As was pointed out there, this division of counting time requires spending most time measuring the smaller signal and in the limit could lead to spending all one’s time counting nothing. The instrumental quality factor for measuring F can be written Q2F ¼

t2 ð1 þ Sp2 Þ3 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii: ½1  Sp2 1 þ 1  S2 p4

ð6Þ

When S ¼ 0; the optimisation condition is to maximise t i.e. omit the polarizer and analyser. When S ¼ 1; the optimisation condition is to maximise p if PHe > 0:5 (i.e. infinite filter thickness) and to maximise t if PHe o0:5: These trends are disturbing of themselves but, in any case, measurements of F depend on the instrument and are physically meaningless as has already been noted [1,7]. Thus this potential criterion is ignored henceforth.

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The quality factor for measuring SFF has been previously determined (Eq. (11) in [1]) to be Q2SFF ¼

16I0 t2 p4 TðSNSF þ SSF Þ4 pffiffiffiffi 2A½ðSNSF þ SSF Þ þ A

A ¼ ð½SNSF 2 þ ½SSF 2 Þð1  p4 Þ þ 2ð1 þ p4 ÞSNSF SSF :

ð7Þ

On introducing the quantity S as a variable the quality factors for measuring SF; SFF and S can be written Q2SF ¼

t2 p4 ð1 þ SÞ4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1  S 2 p4 ½1 þ 1  S2 p4

Q2SFF ¼ Q2S ¼

ð8aÞ

t 2 p4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ½1  S 2 p4 ½1 þ 1  S2 p4

ð8bÞ

Since the ð1 þ SÞ4 term in Q2SF is entirely sample dependent, it has no effect on instrument optimisation and thus these three measurement types result in identical optimisation. Note that when S ¼ 0; these expressions for Q reduce to the widely used Q ¼ p2 t: Whatever the polarizer used, the optimisation of the instrument for a measurement consists of adjusting or choosing the polarizer to give p and t to maximise Q for the measurement.

3. Optimisation of 3He filters For all types of polarizing transmission filters, p and t depend on the physical thickness of the filter and are thus related variables. Strictly, they depend on the ‘‘effective filter thickness’’, x (denoted m in [7]) x ¼ rst;

ð9aÞ

where r is the density of absorbing/scattering centres, s is the centre cross-section and t is the physical thickness of the filter. For 3He filters, by far the most promising prospect for neutron polarizers at present, the cross-section is wavelength dependent and x ¼ rs0 lt;

ð9bÞ 3

where s0 is the cross-section presented by a He nucleus of antiparallel spin to a neutron of unit wavelength, so that ls0 gives the cross-section presented by the nucleus to a neutron of wavelength l: The cross-section presented by a 3He nucleus of parallel spin is assumed to be zero. For 3He filters, p and t depend on the spin polarization of the 3He nuclei within the filter, PHe : Assuming uniform polarization of the polarizing medium, p and t for an initially unpolarized neutron beam after passage through a spin polarized 3He filter of effective thickness x and spin polarization PHe are given by   PHe x pðPHe ; xÞ ¼ tanh 2 tðPHe ; xÞ ¼ ex=2 cosh



 PHe x : 2

A useful expression for xðp; PHe Þ can be derived from Eq. (10) x¼

2atanh ð pÞ : PHe

ð10Þ . ð10aÞ

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We wish to find an expression for xOpt ðPHe ; SÞ; the effective filter thickness which maximises Q: The value of Q corresponding to a filter of thickness xOpt ðPHe ; SÞ is denoted QOpt and the polarization and transmission of this filter pOpt and tOpt : Substituting the expressions for p and t from Eq. (10) into the expression for Qðp; t; SÞ yields QðPHe ; x; SÞ: Differentiating QðPHe ; x; SÞ with respect to x and finding the value of x which makes the result zero gives an expression for xOpt ðPHe ; SÞ: This particular approach, while clearest, proves to be relatively intractable in general for this type of problem and seems to be analytically impossible for nearly all cases investigated. To find a more useful approach it is helpful to develop expressions for pðt; PHe Þ and tðp; PHe Þ: tðp; PHe Þ can be shown to be  1=2p   ð1 þ pÞ1PHe 1 atanh ðpÞ ffiffiffiffiffiffiffiffiffiffiffiffiffi p tðp; PHe Þ ¼ ¼ : ð11Þ exp PHe ð1  pÞ1þPHe 1  p2 We have been unable to find an equally simple expression for pðt; PHe Þ: In the process of maximising Q it is useful to employ the relation  2   2    2   dQ qQ dp qQ dt þ ; ¼ dx dx dx qp qt where 

dp dx



PHe ð1  p2 Þ and ¼ 2



dt dx



t ¼ ðPHe p  1Þ 2

ð12Þ

rather than trying to use a full substitution. Thus, the derivative of Q2 (Eq. (8b)) with respect to x is found using Eq. (12) and the resulting expression dQ2 t2 p 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ dx ð1  S2 p4 Þ2 ð1 þ 1  S 2 p4 Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fPHe ð½2 þ ð2 þ S 2 p4 Þ 1  S 2 p4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  p2 ½1 þ S2 p4 þ 1  S2 p4 þ 2S2 p4 1  S 2 p4 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 p 1  S 2 p4 ½1 þ 1  S 2 p4 g

ð13Þ

is set equal to zero. This yields, PHe

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1  S 2 p4 Þð1 þ 1  S2 p4 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ½2 þ ð2 þ S2 p4 Þ 1  S2 p4  p2 ½ð1 þ S2 p4 Þ þ ð1 þ 2S 2 p4 Þ 1  S 2 p4

ð14Þ

While finding a general expression for pðPHe ; SÞ seems unlikely, Eq. (14) reduces to PHe ¼ p=ð2  p2 Þ when S ¼ 0 which can be solved to give qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8P2He  1 p¼ 2PHe and

0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ 8P2He  1 2 A: x¼ atanh @ PHe 2PHe

ð15Þ

This is an exact expression for xOpt when Q ¼ p2 t: When Sa0; a value is chosen for pOpt and PHe ðpOpt Þ is calculated analytically using Eq. (14). This value for PHe is that for which the chosen value of p is pOpt i.e. it corresponds to an optimised filter. Eq. (10a) can

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Fig. 1. (a) Optimum effective filter thickness, xOpt ; as a function of PHe for various S: Calculated points are shown. The lines are spline fits to these points. (b) Contour plot of xOpt ðPHe ; SÞ:

Fig. 2. (a) Polarization of an optimised filter, pOpt ; as a function of PHe for various S: (b) Detail of (a) with a magnified PHe scale.

then be used to find the optimum effective filter thickness, xOpt ; corresponding to the chosen value for p: Thus there is an analytic path to find the relation between xOpt and PHe : This has been done for a range of values of S and the results are shown as plots of xOpt ðPHe Þ in Fig. 1a and a contour plot of xOpt as a function of S and PHe in Fig. 1b. Note that the behaviour is quite pathological when S becomes large. Some understanding can be gained by examining the variation of pOpt and tOpt as PHe increases (Figs. 2 and 3). Once PHe is greater than 0.5 the optimisation condition snaps into a high p mode. Fig. 2b shows pOpt ðPHe Þ with an expanded p scale to illustrate this more clearly. It is of interest to know the tolerance of measurement precision to non-optimal filter thicknesses. The variation of Q with x has been calculated analytically and the variation in x needed to reduce Q from QOpt by 5% and 10% has been interpolated from the calculated values. The allowed variation in x is satisfactorily large as shown in Fig. 4 for the two extreme cases S ¼ 0 and S ¼ 0:99:

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Fig. 3. Transmission of an optimised filter, tOpt ; as a function of PHe for various S:

Fig. 4. Tolerances on xOpt for S ¼ 0 and S ¼ 0:99: The dashed lines show x for Q ¼ 0:95QOpt and the dotted lines show x for Q ¼ 0:90QOpt :

4. Background effects It has been shown numerically for some measurement types [1] that instrumental background can have a large effect on xOpt especially when xOpt is large in the absence of background. This was shown for measurements of two cross-sections to equal precision and measurements of a single cross-section. The demonstration was purely numerical. Given that the present work shows dramatic variations in xOpt under some conditions, it is important to test the effect of instrument background here. A complete investigation of the effect of background would be extremely time consuming and difficult. The equations are complex and there are many cases to considerFall measurement types discussed in this work, the three possible types of background (independent of analyser and polarizer, independent of polarizer and dependent on both) and variations in S: To establish the importance or otherwise of background, the case of measuring S with a background which is unaffected by the polarizer, the analyser and the flipper state is examined.

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Three measurements are needed to conduct such a measurement: sample with flipper on, sample with flipper off and background with the sample replaced by some matched attenuator. Thus I BG ¼ I0 SBG   I FOFF ¼ I0 St2 1 þ Sp2 þ I0 SBG   I FON ¼ I0 St2 1  Sp2 þ I0 SBG ;

ð16Þ

where SBG is the sample cross-section which would give the same count rate in the absence of background as the measured background intensity I BG : In this situation the variance in the measured value of S becomes (  2  2 I FOFF I FON  I BG I FON I FOFF  I BG 4 2 ðDS Þ ¼ þ T FOFF T FON p4 ½I FOFF þ I FON  2I BG 4  2 ) I BG I FOFF  I FON þ : T BG The optimum division of counting time for the three separate measurements needed is that where pffiffiffiffiffiffiffiffiffiffiffiffi I FOFF ½I FON  I BG FOFF pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¼ T pffiffiffiffiffiffiffiffiffiffiffiffi T I FOFF ½I FON  I BG þ I FON ½I FOFF  I BG þ I BG ½I FOFF  I FON pffiffiffiffiffiffiffiffiffiffiffi I FON ½I FOFF  I BG FON pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¼ T pffiffiffiffiffiffiffiffiffiffiffiffi T I FOFF ½I FON  I BG þ I FON ½I FOFF  I BG þ I BG ½I FOFF  I FON

T

BG

ð17Þ

ð18Þ

pffiffiffiffiffiffiffiffi I BG ½I FOFF  I FON pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ¼ T pffiffiffiffiffiffiffiffiffiffiffiffi : I FOFF ½I FON  I BG þ I FON ½I FOFF  I BG þ I BG ½I FOFF  I FON

Replacing SBG by RBG ¼ SBG =S and ignoring TI0 S; QSBG ¼ 

2t2 p2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffi : ½1  Sp2 t2 ð1 þ Sp2 Þ þ RBG þ ½1 þ Sp2 t2 ð1  Sp2 Þ þ RBG þ 2Sp2 RBG

ð19Þ

When RBG ¼ 0; this reduces to Eq. (8b). Adopting the analytic techniques described above becomes too complex for this expression and thus QSBG has been evaluated numerically. Values for xOpt have been extracted as a function of S and PHe for many values of RBG and those for RBG ¼ 0:005; 0.01, 0.02 and 0.05 are shown as Figs. 5a–d. These figures are analogous to Fig. 1a and show the dramatic effect that background has on xOpt : Further increases in RBG decrease xOpt even more and lessen its dependence on S and PHe : Even a background as small as 1% of the sample scattering removes the extreme variation of xOpt with PHe and S: A 1% instrumental background is a good performance on any instrument. Even when measuring a Bragg peak with 1000 counts this is a background of less than 10 counts. In general such backgrounds are only achievable with special samples in single crystal form. The implication is that the pathological optimisation result applies only to a situationFlarge S and PHe with a minute backgroundFwhich is rarely if ever seen in practice. The large values of xOpt only occur when S is very large. It is also noted that achieving large values of PHe is unlikely, at least in the medium term, and values higher than 0.75 may prove very difficult to achieve [18].

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Fig. 5. Contour plot of xOpt ðPHe ; SÞ with background effects included for RBG ¼ 0:005 (a), RBG ¼ 0:01 (b), RBG ¼ 0:02 (c) and RBG ¼ 0:05 (d).

Given the results in Fig. 4 which demonstrate that the exact value chosen for x is not very critical and these results on the effect of background, it is only in extremely unusual cases that x should be changed for an individual measurement. Thus it appears that optimising filter thickness is something which can generally be done once when the instrument is commissioned and then left with little performance loss for particular measurements. It also appears that a relatively modest value of x is all that is needed (xD4 ( ). corresponding to a physical thickness of about 4 cm for a filter at 3 atm 3He pressure at l ¼ 2:35 A It is worthwhile at this point discussing the relative merits of analytic and numerical modelling of optimisation. Extensive numerical modelling of this sort of problem has been conducted by Kulda [19] and Tasset and their associates. They have, however, almost universally assumed that Q ¼ tp2 : This was the condition derived by Williams and applies to the case of measuring the spin flip fraction (or as has been shown here, the related quantities SF and S) when S ¼ 0: As has been extensively demonstrated this expression for Q is incorrect for most measurements. Furthermore, the effect of even very small background has been shown here and elsewhere to be important. One can surmise that any higher order wavelength contamination of the beamFeven in small amountsFwould also be likely to significantly affect the optimisation. The chances of doing an experiment with truly negligibleFand the present work suggests this means much less than 1%Fhigher order contamination and background are minute. Fortunately the condition itself is not very sensitiveFthere is a broad range of thickness which gives close to optimal

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results. Our experience shows that the numerical modelling done for the backgroundFa more complex problem than the calculation without backgroundFis immeasurably quicker and easier than the analytic approach. Thus if the only thing wanted is a numberFfor example a filter thicknessFthe numerical approach is more sensible. However, this work is concerned with the general optimisation of measurements. This is far easier to do when an analytic calculation has been done. The formulae derived in the context of this problem are complicated but the fact that they enable the extension of the analysis beyond calculating xOpt to include the effect of a number of parameters and experimental conditions is evidence that analysis has substantial advantages over numerical modelling in exploring the parameter space.

5. Quality factors Fig. 6 plots the value of Q achievable with a 3He filter as a function of PHe for various values of S: It shows that it is easier to perform good measurements of S if S is itself large. This is reasonable given that S can be considered as the difference between quantities which are most similar when SD0: Note that the ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ptwo value for Q used in this work omits the TI0 ðSSF þ SNSF Þ term which places a limit on the precision of measurements. Knowing the optimum effective thickness for 3He polarizing transmission filters enables direct comparisons of the performance of such filters with that of other types of polarizers which can be calculated using the measurement quality factor (Eq. (8)). A useful form of presentation of this information is the value of PHe above which 3He filters are superior to other polarizers. This comparison gives development targets for 3He filters. Such a comparison has already been made [3] for the case of measuring two cross-sections to equal precision. That work suffered from shortcomings in assessing the effect of restricted beam angular width, a: These shortcomings arose from difficulties in describing the resolution of neutron instruments in qualitative ways. Recent work [20,21] has simplified this description. Essentially for a primary spectrometer consisting of a collimator–monochromator–collimator, the initial collimator and the monochromator mosaic combine to restrict l variation while the second collimator defines a (except for odd choices of components). Both supermirrors and Heusler alloy polarizing monochromators are the subject of intense development effort and there have been significant advances in the performance of both in recent years. Many of these

Fig. 6. Plot of QOpt against PHe for the p case of measuring ffiS or SFF for various values of S: QOpt increases dramatically with S: Note ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that the values here do not include the TI0 ðSSF þ SNSF Þ term which affects the absolute precision of a measurement.

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developments are not yet described in the literature as is normal in a field of rapid change. Thus absolute comparisons between polarizing techniques are comparisons between moving targets. The ability to make absolute comparisons and hence informed choices for instruments is novel and deserves regular updates. In the following sections the absolute performance of 3He filter, supermirror and Heusler alloy monochromator neutron beam polarizers are compared for measurements of S (and consequently, also for SF and SFF).

5.1. 3He filters While other devices have limitations in a and l; for transmission filters x can be adjusted to give a QOpt determined solely by PHe and S regardless of a and l: The comparison of devices needs to examine diffferent regions in an a  l space and should show where in this space individual devices are superior.

5.2. Supermirrors The performance of an individual supermirror is essentially independent of outside effects and the device produces a reflectivity and polarization which can be measured. However, the critical angle for reflection yC (measured as a ratio, m; to the critical reflection angle for nickel) is proportional to l; and thus a is limited at short l: At present, m ¼ 2 is routinely achieved and m ¼ 3 is becoming a reasonable expectation for polarizing supermirrors. At short l only neutrons with paths at very small glancing angles to the supermirror surface can be reflected. These neutrons then follow paths very close to the supermirror surface making many ‘‘garland’’ reflections. When individual supermirror blades are assembled to make a polarizer for a beam of real dimensions, unless the channel width of the device is very small, only a small fraction of the frontal area of the device is available for transmission of short l neutrons. This effect results in transmission which is large and constant for wavelengths above some critical value and decreases below that wavelength approximately as l2 : Ref. [3] treated the case of an ideal supermirror deviceFone which did not suffer from this effectFand thus overstated the usefulness of supermirror devices at short wavelengths. The wavelength and a of the transmitted beam become important considerations in assessing the quality factors for supermirror polarizing devices. The supermirror device considered here is assumed to be an assembly of m ¼ 3 supermirrors (7.5 cm long and 0.2 mm apart) with an overall device transmission of 35% of the incident neutron beam i.e. 70% of the desired spin state. The monochromator used with the supermirror is assumed to have 70% reflectivity and a mosaic spread and incident collimator chosen to give the desired l spread. Supermirrors give a rectangular transmission profile rather than the triangular profile offered by conventional collimators. This provides intensity gains for a given a but since it is now possible to build conventional collimators which mimic this beam profile no consideration is taken of this effect. The absolute value of Q depends on S and diverges as p and S approach 1. For convenience in viewing the results all values of Q have been divided by the value of Q (Qperfect ) when t ¼ 0:5 and p ¼ 1 (perfect polarization and transmission). QSM ða; lÞ is shown in Fig. 7a (S ¼ 0), 7b (S ¼ 0:6) and 7c (S ¼ 0:99). The relatively low Q=Qperfect values for S ¼ 0:99 are indicative of the rapid increase in Qperfect as S approaches unity and do not imply that the measurement quality decreases. Fig. 7 shows the strong l dependence of QSM arising from the change in acceptance angle. All plots show a plateau region where the acceptance angle of the supermirror is greater than the experimentally desired a: The a width of this plateau increases with l as expected. As desired width increases the acceptance angle becomes a limitation and QSM falls accordingly.

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Fig. 7. Contour plot of fractional Q for a supermirror (QSM ) as a function of neutron wavelength, l; and desired beam angular width, a; at (a) S ¼ 0; (b) S ¼ 0:6; (c) S ¼ 0:99: Background is ignored.

5.3. Heusler alloy monochromators There have been significant improvements in polarizing Heusler monochromators in recent years. Control of mosaic has improved, resulting in crystals of up to 0.51 mosaic. This improved mosaic overcomes most of the deficiencies in these polarizers pointed out in [3] and Heusler alloy monochromators are expected to be more competitive than was previously estimated. The effect of twisting of the lattice vectors in large crystals, which reduces average reflectivity over large crystals, has been appreciated and allowed for by using smaller crystal segments. Magnet assemblies have also improved. At present, the measured reflectivity of the crystals for the desired spin state varies from about 30% at short wavelengths to 70% or even more at long wavelengths although this variation does not seem to be fully understood [19]. tHeusler is this reflectivity divided by two to allow for the reflection of only one spin state. pHeusler is taken to be 0.96. As for supermirrors, fractional Q values for the polarizer are calculated and denoted QHeusler : Since it is the post-monochromator collimator which controls a at the sample and the Heusler alloy does not have the same angular width dependence on l as the supermirror, there is no a dependence in these results. Wavelength enters the calculation through the wavelength dependence of tHeusler which is assumed to vary linearly with wavelength. Fig. 8 shows QHeusler as a function of wavelength for S ¼ 0; S ¼ 0:6 and S ¼ 0:99:

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Fig. 8. Fractional Q for a Heusler alloy polarizing monochromator (QHeusler ) as a function of neutron wavelength, l; at S ¼ 0; S ¼ 0:6 and S ¼ 0:99: Background is ignored.

QHeusler varies little for S less than 0.6. QHeusler increases linearly with wavelength because tHeusler is assumed to do so. Naturally, this assumption and its resultant behaviour cannot hold for all wavelengths.

6. Comparisons This discussion is complicated by the large number of parametersFl; a; PHe and S: Contour plots are presented in a l  a space for various values of S: The contours represent the value of PHe needed for a 3He filter to match the performance of the other device for comparisons involving 3He filters. 6.1. 3He filters and Supermirrors Fig. 9 shows PHe ða; lÞ needed to match a supermirror device and demonstrates that while QSM is strongly S dependent, the required PHe is remarkably consistent, with the most notable effect being the slower fall off of required PHe with l when S is larger. In all cases the plateau region requires PHe values of greater than 0.75, suggesting that at small a and large l supermirrors will be superior for some timeFnot a surprising conclusion. More significantly, a large fraction of the space lies on the low-PHe side of the PHe ¼ 0:5 ( Fwhere 3He contour, suggesting that there are many experimentsFincluding all those requiring lo1 A filters are already superior to supermirrors. The fact that PHe required is only weakly dependent on S is a useful result, as it indicates that once a given level of 3He spin polarization is achieved, filters will perform similarly relative to a supermirror for samples with a wide range of S: 6.2.

3

He filters and Heusler alloy monochromators

As in Section 6.1, the value of PHe required to give equal performance for transmission filters and Heusler alloy monochromators is calculated. When calculating PHe ; the reflectivity of PG is assumed to be

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Fig. 9. Contour plot of PHe required for an optimised 3He filter to equal a supermirror in performance at (a) S ¼ 0; (b) S ¼ 0:6; (c) S ¼ 0:99: Background fraction is zero.

independent of l. This reduces the required PHe at short wavelengths because of the better reflectivity of the PG monochromator. Fig. 10 shows PHe required to equal the Heusler performance. The PHe required for S ¼ 0:99 is interesting and shows the effect of the quite different shapes of QOpt ðPHe Þ for the three values of S combined with the different wavelength dependencies of the two monochromators. Some insight into this can be gained from Fig. 6. For short wavelengths 3He filters can easily match Heusler alloys. In fact, the range over which the filter ( . This suggests that for all experiments at short wavelength would be superior is extended to around 2.5–3 A 3 He filters should be used, while Heuslers and supermirrors should be compared to determine which is better at larger wavelengths. 6.3. Supermirrors and Heusler alloy monochromators In order to compare the relative merits of Heusler alloy and supermirror polarizers, QSM =QHeusler is calculated. Because of the substantial functional differences between the polarizers we continue to use Q for a ‘half instrument’ as noted earlier. The results are plotted in Fig. 11. While the Heusler alloy ( olo3 A ( monochromator performs better over much of the space, the supermirror is superior for 1:5 A when a is smallFa common situation for real measurements. Further, as S increases, the wavelength range over which the supermirror is superior increases due to its marginally superior polarization and the extreme sensitivity of the optimisation to p for large S:

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Fig. 10. PHe required for an optimised 3He filter to equal a Heusler alloy polarizing monochromator in performance at S ¼ 0; S ¼ 0:6 and S ¼ 0:99: Background fraction is zero.

The true usefulness of a Heusler alloy monochromator is that it is a single component. Its polarization is comparable to that of the supermirror, while recent advances in control of crystal mosaic have resulted in increases in integrated reflectivity. With a supermirror, total transmission is that of the supermirror ( , compared to about 0.35 for the multiplied by that of the PG monochromator, that is about 0.28 at 4 A Heusler at the same wavelength. Given the same S and p; the Heusler tends to be superior in any regime where its reflectivity is comparable to that of the PG monochromator.

7. Conclusion There is some question whether measurements of the flipping ratio and related quantities have real physical significance. Many scientists routinely perform such measurements and produce useful and interesting results. Usually, these measurements are a simple means of measuring the quantity of interest, say the magnetic cross-section, with some internal calibration in the measurement. This removes the need to actually measure absolute scattering cross-sectionsFmeasurements which are notoriously difficult and involved. However, measuring flipping ratios may not the optimum way to obtain the quantities desired. Since such measurements are often done, they should be optimised and the analysis presented may help in this. Useful results on the optimisation of instruments for flipping ratio and associated measurements have been obtained through a strongly analytic approach. The generality of this approach and the understanding which it gives allows insight which may be lacking in purely numerical calculations. It has been recognised that sample polarization, S; has considerable impact on the optimisation of an experiment. With this in mind, the performance of different polarizer types have been compared for various values of S: This analysis shows that the range of wavelengths over which supermirrors are superior to Heusler alloy monochromators depends on S; as does the magnitude of the gain. It is also clear that 3He filters require much higher spin polarization to match supermirrors for large S:

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Fig. 11. Comparison of performance of supermirror and Heusler alloy polarizing monochromator; QSM =QHeusler ; at (a) S ¼ 0; (b) S ¼ 0:6; (c) S ¼ 0:99: Background fraction is zero.

Instrumental background tends to mitigate and effectively eliminate the effects of non-zero S if it is more than 1% of the sample scattering. Background tends to result in otherwise pathological optimisation conditions becoming less extremeFmaking it simpler and less necessary to optimise any individual realworld experiment, regardless of whether the S value of the sample is large or not. In general, an effective filter thickness of x close to 4 seems appropriate. There will always be special measurements where the particular benefits of 3He filters become very important, regardless of the value of PHe ; just as there are applications such as reflectometry where supermirrors have natural advantages. For the routine use of conventional polarization analysis techniques, it seems that 3He filters must reach a value of PHe > 0:7 before they can be truly competitive at the usual wavelengths used. Further works may be presented soon to discuss other types of measurements. In particular, measurements of magnetic cross-sections using the techniques developed by Moon et al. [22], xyz polarization analysis [23] and the most interesting recent development in polarized neutron work, neutron polarimetry e.g. [24], deserve consideration.

Acknowledgements LDC thanks Drs. F. Tasset and J. Kulda for useful discussions. DJG thanks Dr. T.R. Welberry.

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