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Neutron resonance spin echo using spin echo correction coils
Chemical Physics 292 (2003) 501–510 www.elsevier.com/locate/chemphys
Neutron resonance spin echo using spin echo correction coils Wolfgang H€ aussler a,*, Ulrich Schmidt b, Georg Ehlers a, Ferenc Mezei c b
a Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France Physikalisches Institut der Universit€at Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany c Hahn-Meitner-Institut, Glienicker Straße 100, 14109 Berlin, Germany
Received 31 October 2002; in final form 10 March 2003
Abstract We report on a recent experiment which was performed in order to examine a new experimental realization of the neutron resonance spin echo (NRSE) technique. In neutron spin echo (NSE) neutrons accumulate spin phase in long magnetic fields before and after the sample. The measured quantity is the final scattered beam polarization, which contains information on the sample dynamics (the intermediate scattering function SðQ; tÞ). This method is limited by the inhomogeneity of the magnetic field, leading to differences of the spin phase of individual neutrons and thus influencing the scattered beam polarization even in the absence of sample dynamics. The NRSE technique can overcome this limitation by reducing the dimensions of the field. However, all previously built NRSE spectrometers use transversal magnetic fields, making it impossible, in practice, to correct for additional limiting effects, such as the beam divergence. For the first time, we have built a NRSE setup with longitudinal field geometry, which does not have this disadvantage. In order to test this new approach, we combined a longitudinal NRSE setup in one spectrometer arm of the IN11 instrument at ILL with a (conventional) NSE setup in the other arm. This experiment demonstrates, how NRSE can be realized in longitudinal field geometry, and sheds light on the differences to previous NRSE setups. As a main result, we show that the effect of beam divergence could be corrected by means of standard Fresnel coils. The proper operation of this hybrid spectrometer is demonstrated by measurements of diffusive dynamics in a well-known micellar sample. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Neutron spin echo (NSE) [1] provides the highest energy resolution of all neutron scattering techniques. A fraction of 1 neV has been achieved in scattering experiments [2]. Soon after the first
*
Corresponding author. Tel.: +33-47620-7041; fax: +3347620-7688. E-mail address: [email protected] (W. H€aussler).
NSE instruments were running and available for experiments, a variety of phenomena has been studied [3]. The main parts of NSE instruments are magnetic coils in each of the two spectrometer arms before and behind the sample position. Using the neutron spin, which precesses in the magnetic fields produced by these coils, the velocities, and thereby the energies of the neutrons before and behind the sample are compared. The high energy resolution of the NSE method is based on the fact,
0301-0104/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00119-8
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that in high magnetic field the spin phase is very sensitive to small velocity differences due to inelastic sample scattering. In neutron resonance spin echo (NRSE) [4,5], the neutrons also accumulate spin phase in two spectrometer arms before and behind the sample region. Each NRSE arm consists of two NRSE coils and, in between, a flight region where no magnetic field is necessary. In both NSE and NRSE, the energy resolution is proportional to the magnitude of the field in the coils. Different from NSE, in NRSE the energy resolution also depends on the distance between the NRSE coils, but not on the length of an individual NRSE coil. Therefore, NRSE coils can be designed relatively compact creating only small stray fields which have to be corrected for. This is the main advantage of NRSE compared to conventional NSE, where high dynamic resolution requires long coils all the way between sample and detector. One can also build a multi-detector arm using several identical NRSE coils [6,7], or adjust NRSE coils to the shape of a dispersion relation to be observed in an inelastic experiment [8]. However, the available spin echo times of all earlier built NRSE spectrometers were effectively limited by the divergence of the neutron beam. Neutrons on trajectories which are not parallel to the symmetry axis stay longer in between the NRSE coils, hence accumulate more phase (‘‘divergence effect’’). This affects the resolution, because NRSE (as NSE) is based on the condition that neutrons of the same wavelength accumulate the same spin phase. All earlier NRSE instruments used coils producing a static magnetic field B0 directed perpendicularly to the flight path of the neutrons. Therefore, it was practically impossible to correct for the divergence effect. In this paper, we report on the test of a completely new NRSE setup with longitudinal B0 field. In NSE spectrometers, so-called ‘‘Fresnel’’ coils [3] are used to correct for both field inhomogeneities and the divergence effect. These Fresnel coils consisting of concentric current loops can only be used if the spin precession field is parallel to the flight path of the neutrons. This is the main motivation for building a longitudinal field NRSE, thus combining the advantages of conventional
NSE (Fresnel coils can be used for correcting for the divergence effect) and NRSE (resolution is decoupled from the length of the coils). In order to compare the performance of the new longitudinal NRSE setup with NSE, we combined one NRSE spectrometer arm with one NSE spectrometer arm of the instrument IN11 at the Institut Laue-Langevin (Grenoble) [9]. The whole arrangement represents a combined NSE–NRSE spectrometer. We demonstrate the functioning of this NSE–NRSE setup in a test experiment measuring the dynamics of a standard sample. First, we describe the special properties of our setup, which differs to some extent from other NRSE spectrometers. The main difference is that the field between the NRSE coils is significantly different from zero, whereas in NRSE up today this field was always as low as possible (‘‘zero field spin echo’’). Thereafter the performance of the NRSE coils in the longitudinal setup is examined. The main goal of the present study is to demonstrate that in NRSE with longitudinal B0 field, the inhomogeneities of the effective field integral due to the beam divergence can be corrected. We show that the polarization of an elastically scattered neutron beam can be increased from 25% to almost 80% using standard Fresnel coils, which makes it possible to improve the dynamic resolution of the spin echo method.
2. Theory 2.1. Neutron spin echo–neutron resonance spin echo Various theoretical descriptions of NSE and NRSE spectrometers have been developed [10,11]. Here, only a few important points are repeated. The spins ðs ¼ g=2Þ of neutrons traversing subsequently each arm of a NSE spectrometer precess in a magnetic field of magnitude B with Larmor frequency mL ¼ jcj=2pB, where c ¼ 2p 29:16 MHz/T is the gyromagnetic ratio of the neutron [12]. In the first arm, each neutron accumulates the spin phase /NSE which Ris proportional to the magnetic field integral JB ¼ B ds along its flight path s:
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/NSE ¼ c=2pðkm=hÞJB :
ð1Þ
Here k stands for the neutron wavelength and m for its mass. In the second arm, the precession is of opposite sense, and the spin phase decreases. For strictly elastic scattering, the final spin phase obtained by all neutrons after the second arm with field integral JB is / ¼ 0 (so-called ‘‘spin echo point’’). For inelastic scattering / ¼ sSE DE=g with the neutron energy change DE. The proportionality constant sSE , called spin echo time [2], determines the sensitivity of the spin phase to sample dynamics, sSE ¼ c=2pðm2 =h2 Þk3 JB :
ð2Þ
The measured parameter is the final beam polarization pSE , analyzed after the precession regions in the same direction in which the spin has been polarized originally: Z pSE ¼ cosð/ðkÞÞf ðkÞ dk: ð3Þ Here /ðkÞ is the final spin phase and the integration is performed over the whole wavelength spectrum f ðkÞ of the neutrons. Eq. (1) shows that the phase / depends on the neutron wavelength k. However, the NSE technique is well adapted to using a broad wavelength spectrum, because the final polarization depends only in second (and higher) order on the relative width Dk=k of the wavelength spectrum. The second arm of our experimental setup consists of two NRSE coils with a magnetic guide field in between. The NRSE coils produce a longitudinal static magnetic field B0 and additionally, a radio frequency (RF) field B1 perpendicular to B0 . The magnetic fields are tuned to get a resonant p-flip. The guide field between the NRSE coils has the task to avoid polarization losses due to nonadiabatic magnetic field changes along the neutron flight paths. The spin phase /NRSE accumulated in one NRSE arm is proportional to the field strength B0 and the effective distance between the centers of the NRSE coils L [4]: /NRSE ¼ c=2pðkm=hÞ2B0 L:
ð4Þ
Consequently, the NRSE spin echo time is given by sNRSE ¼ c=2pðm2 =h2 Þk3 2B0 L:
ð5Þ
503
Comparing with Eq. (2) for (conventional) NSE, we see that for identical resolution NRSE only needs half the static magnetic fields. Furthermore, this factor 2 can be increased by bootstrap methods to 2n [13]. The maximum accessible spin echo time in NRSE depends on the field strength B0 and distance between the NRSE coils, but not on the length of an individual NRSE coil, in contrast to NSE. Nevertheless, the effect of one NRSE arm on the neutron spin is identical to the action of an effective static magnetic filed, in analogy to NSE. Normally, a complete spin echo spectrometer consists of a symmetric setup of either two NSE or NRSE arms. However, a spectrometer can also be built by combining the two methods. The final beam polarization obtained in a spectrometer consisting of a NSE and a NRSE arm (NSE– NRSE) depends on the difference of the (effective) field integral in both arms, as can be seen by Eqs. (1), (3) and (4). If they are identical, the spin echo condition is fulfilled, and the original polarization value is recovered. However, if the field integrals differ, a non-zero spin phase remains which is proportional toRthe difference of the field integrals R DJB ¼ B ds 2B0 ds. Consequently, varying the NSE armÕs field integral, the polarization oscillates: Z pSE ¼ ð1 dÞ cosðc=2pðkm=hÞ DJB Þf ðkÞ dk: ð6Þ In this expression, the depolarization d is added taking into account polarization losses. While in an ideal spin echo setup, all neutrons end up with identical phase and d ¼ 0, deviations of the spin phase from zero arise due to inhomogeneities of the (effective) field integral. Mainly, there are two origins of field integral inhomogeneity: (1) The magnetic field itself is inhomogeneous (in long solenoids mainly at entrance and exit). This leads to different JB even for trajectories of the same length. (2) The length of the flight path is not the same for all trajectories. Neutrons with high divergence gain more phase, even in an ideally homogeneous magnetic field (divergence effect). Fresnel coils can be used to correct for both effects [2,14]. They are fixed within the precession field and consist of concentric current loops
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adding more field integral near the cylinder axis than off-center. The current density j in our Fresnel coil of outer radius R increases linearly with the distance r from the center. This leads to a quadratic field integral correction Jcorr , suited to cancel the leading order of inhomogeneities of the static field integral: 2
j r ) Jcorr ðr RÞ :
ð7Þ
2.2. Resonant spin flip and magnetic fields A resonant spin flip can be obtained by the combined action of a static magnetic and a circularly polarized RF field perpendicular to the B0 field. A fully polarized neutron beam (polarization pz ¼ 1), leaves the NRSE coil with polarization p [15]: p ¼ g2 þ c2 exp
l2 m2 r2 x2R 2h2
cos
lmkxR h
ð8Þ
with the prefactors g ¼ ðcB0 xÞ=xR and c ¼ 2 ðcB1 Þ=xR , the Rabi frequency xR ¼ ½ðcB0 xÞ þ 2 0:5 cB1 Þ , the RF frequency x, the path length l of the neutron within the RF field B1 , the neutron mass m, the mean neutron wavelength k and the standard deviation of the neutron wavelength distribution r. Eq. (8) was used to describe test measurements of the NRSE coils with initial polarization parallel to B0 . However, it has to be mentioned that for spin echo measurements, the initial polarization is perpendicular to B0 , and instead of Eq. (8), a more complicated expression is needed to describe the polarization. The RF fields used in practice are linearly, rather than circularly polarized (Fig. 2). Each linearly polarized field can be described as sum of two oppositely directed circularly polarized components. One component is used for the resonant flip. If the ratio between the strength of B0 and the strength of the RF field B1 is larger than 10, the other component of the RF field has only negligible small influence on the spin motion (rotating wave approximation) [7,15,16]. From Eq. (8) it can be easily seen that two conditions must be met in order to achieve a resonant p-flip:
cB0 x;
ð9aÞ
ðlmkhÞcB1 ¼ p:
ð9bÞ
ðaÞ the resonance condition : ðbÞ the p-flip condition :
The first condition requires that the RF field oscillates with Larmor frequency. The second condition means that the neutron wavelength has to be matched to the Rabi frequency in resonance cB1 for being flipped by p in the RF field. In order to produce static magnetic fields parallel to the neutron trajectory we use solenoids. The magnetic field is calculated starting from the law of Biot–Savart d~ j ~ r d~ B¼ r3
ð10Þ
with the infinitesimal linear current density element d~ j and ~ r pointing from the current to the point calculated. Eq. (10) is numerically integrated over a circular loop, and both length and thickness of a solenoid are taken into account by summation over equidistant loops. The static magnetic fields of all coils are taken into account (Fig. 1), which is important for predicting the tuning for the spin echo point (see below) as well as for optimizing the homogeneity of the static field inside the NRSE coils (see Section 3). The field created by the RF coils is analytically calculated starting from the law of Biot–Savart (Eq. (10)). The details of this calculation have been described elsewhere [7,15]. The RF field created by an oscillating current is different from the field in the static case, first due to eddy currents induced in all metallic parts surrounding the RF coils. In our setup, we can neglect this effect, because the coils are much smaller than the inner diameter of the B0 coils (see Section 3). Second, the skin effect pushes the oscillating current to the surface of the wire, influencing the generated RF field. The RF coils were made of wire of diameter as thin as 1 mm to minimize this effect.
3. Experimental section An overview of the experimental NSE–NRSE setup is given in Fig. 1. Neutrons coming from the
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Fig. 1. Experimental setup (top, details are described in the text). The profile of the longitudinal static magnetic field created by all solenoids (bottom).
left side (beam diameter 5 cm) enter the NSE precession coil with spin polarized perpendicularly to the B0 field after having passed the polarizer and the p=2-flipper. In the NSE coil (length 2 m), spin phase is accumulated by Larmor precession. Behind the NSE coil, the p-flipper provides a turn of the neutron spin by 180° around an axis perpendicular to the beam. This is necessary because the precession sense in the second arm is equal to the first arm. The p-flip ensures that the spin phase accumulated in the second arm is subtracted from the phase accumulated in the first arm. After having passed the sample position, the beam passes the NRSE setup (the distance between the NRSE coils is 2 m), another p=2-flipper, analyzer and detector. In the following, we describe some special aspects of the longitudinal NRSE setup used in this test experiment. The resonance condition (Eq. (9a)) has to be fulfilled by all neutrons. Consequently, the B0 field has to be homogeneous over the whole volume of the RF coil passed by neutrons. Magnetic field calculations (Fig. 1, proved by hall probe measurements) revealed that the field
homogeneity provided by the standard IN11 coils (double-helical solenoid, length 65 cm, inner/outer diameter 20/30 cm) is not sufficient (relative inhomogeneity DB0 =B0 3‰). Therefore, we added correction coils fixed inside the B0 solenoids next to the RF coils. As result, we achieve DB0 =B0 0:2‰ over a path length of 10 cm and diameter of 5 cm. The RF coils are pictured in Fig. 2. They are fixed in the beam, so that they are penetrated by the neutrons. This is necessary in order to provide a RF field pointing perpendicular to the static field B0 (which is equal to the direction of the neutron flight paths). We use cuboid-shaped RF coils (60 60 30 mm3 ) winded by aluminum wire which is almost transparent to neutrons (Fig. 2). At the top and bottom, additional correction coils have been added (25 25 10 mm3 ), in order to make the RF field sufficiently homogeneous (DB1 =B1 4:2% (standard deviation) calculated for a beam diameter of 5 cm). The cuboid coil shape provides identical RF field integrals for all neutrons of the beam. Different path lengths in these coils are not an issue: given the maximum
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Fig. 2. The design of the RF coil (left). The coil is cuboid-shaped, while at both ends, additional correction coils are added. The appropriate functioning of the RF coil together with the B0 coil is demonstrated by B0 - and B1 -resonance scans (center and right).
beam divergence of 20 mrad due to the use of diaphragms in different places in the spectrometer, the maximum relative path length difference in the B1 coil is less than 1& and can be neglected. We performed resonant p-flip test measurements using the neutron beam polarized parallel to the static magnetic field B0 (p=2-flippers off, only one NRSE coil switched on). Fig. 2 shows scans of the magnitude of both the B0 and the B1 field. The B1 field was supplied by a frequency generator (HP) combined with an amplifier (ENI 400 B, 80– 2700 kHz). Comparison with theory (Eq. (8), Fig. 2) shows that the main influence on the resonant pflip originates from the broad wavelength spectrum of the neutrons. Taking the width of Dk=k 16:4% (well known, see below), as input, the field inhomogeneities can be detected by comparison of data and theory. As a result, the spin polarization in the maximum of the p-flip (96%, Fig. 2) is slightly decreased due to field inhomogeneities. The inhomogeneity of B1 (6.4%) obtained by the fit is higher than the calculated value, most likely due to the idealizations made in the calculation (eddy currents, B0 inhomogeneity neglected). Moreover, the analysis of the B0 scan shows that in addition to the magnetic field inside the RF coil, also the stray fields outside the RF coil passed by the neutron beam have to be taken into account. A detailed analysis of this effect will be presented elsewhere. In Fig. 3, the first measured spin echo groups are shown. From the width of the echo group, the neutron wavelength spread Dk=k ¼ 16:4% (mean ) is found by fitting the theory wavelength 5.5 A
(Eq. (6)) to the data. A particular aspect of our NSE–NRSE arrangement is that it is actually not trivial to find the settings that match the Fourier times on both sides exactly. The main precession current in the NSE coil (196 A at the scans shown in Fig. 3) was calculated from the settings in the NRSE arm (RF-frequency, B0 and the NRSE coil distance) via Eqs. (1) and (4) (Fig. 1). The spin echo point was found near the predicted value (Fig. 3) confirming the correct calculation of all field integrals. This first echo was detected in the direct beam without sample. We used two Fresnel coils (Section 2) at entrance and exit of the NSE coil, in order to correct for field inhomogenities. However, in the NRSE arm, no correction coils were needed because the divergence effect is negligible (in the direct beam the angle of every neutron to the symmetry axis has the same size in both spectrometer arms, and therefore, the path length is identical). Consequently, the echo polarization is an exact measure for the quality of the p=2-flippers, p-flippers and NRSE coils. Echo polarization values higher than 80% are found already in the test measurements without much time spent on optimal tuning. In addition, Fig. 3 shows echo groups measured while performing an elastic scattering experiment (graphite sample). The first measurement was performed with the same adjustment as used for the direct beam measurement (without Fresnel coils in the NRSE arm). The echo polarization was much lower than in the direct beam case, because due to the sample scattering, the directions of the
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Fig. 3. The first NSE–NRSE spin echo group measured in the direct beam (circles). Spin echo groups measured in elastically scattered , the wavelength spread determined by intensity (triangles: Fresnel coils off, squares: Fresnel coils on). The mean wavelength was 5.5 A fitting to Eq. (6) was 16.4% (FWHM). The inset shows the echo polarization at three spin echo times.
neutron trajectories in the two spectrometer arms are decoupled, and the path lengths are different. With two Fresnel coils supplemented by an additional coil in between, we could add spin phase to straight-on neutrons, while the spin phase of divergent moving neutrons was reduced. This correction scheme was used for the first time in NRSE and works only, if the static magnetic field is longitudinal. Fig. 3 demonstrates the effect of the Fresnel coils on the NRSE side (spin echo groups on lefthand side). With Fresnel coils turned on, the final polarization increases and the position of the spin echo point shifts because the Fresnel coils add a supplementary magnetic field integral. In the framework of this test experiment the currents of the Fresnel coils were optimized for three different spin echo times. The respective maximum polarization values are shown in the inset of Fig. 3. The maximum spin echo time (sSE ¼ 4:6 ns) was only limited by the available power supply, but not by the depolarization. Although the line added in the inset serves primarily as a guide to the eye, it may be used also to obtain a rough estimate of the maximum resolution. Assuming that the depolarization is only due to remaining field integral inhomogeneities, the decay of the polarization was
fitted in first approximation to a quadratic function. From this fit we get a maximum spin echo time of s ¼ 110 ns. This value is more than one order of magnitude higher than the value we could reach without Fresnel coils. The possibility to reach high spin echo times was the main motivation for building the longitudinal NRSE setup. We would like to mention here, that it is mainly the usage of Fresnel correction coils which makes it possible to reach high spin echo times, but also the fact, that the resolution in longitudinal NRSE is only weakly sensitive to mechanical tolerances. The precession regions are limited by the p=2- and p-spin flippers as in NSE, but not by the NRSE coils, because the neutrons do not penetrate the wire of the B0 solenoids (in contrast to transversal NRSE). Moreover, the RF field integral inhomogeneity due to mechanical tolerances influences primarily the quality of the resonant p-flip (like the beam divergence as discussed above), but not the spin phase. In summary, mechanical tolerances of both the B0 solenoids and of the RF coils have only weak impact on the spin phase. (Only mechanical tolerances of the p=2- and p-spin flippers could, as in NSE, deteriorate the polarization. However, the precession field around these spin flippers is low as in NSE. Consequently,
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mechanical tolerances lead only to small deviations of the spin phase of neutrons passing the flippers on different trajectories.) This is another advantage of the longitudinal NRSE technique compared to transversal NRSE where the precession regions are limited by the NRSE coils with high magnetic field present, so that mechanical tolerances can lead to large spin phase deviations of neutrons traversing the coils on different trajectories. The impact of the Fresnel coils is illustrated in Fig. 4 in detail. At the highest spin echo time we could reach with our power supply, the Fresnel currents were stepwise varied between the optimum setting and zero. With the Fresnel coils switched off, the variation of the spin phase over the detector area amounts to more than one full period of the spin phase. At optimal Fresnel currents setting, resulting in maximum polarization, no variations of the spin phase are visible on the same scale. The best polarization value in the spin echo point is comparable to that of standard NSE measurements at IN11. Moreover, the current needed in the Fresnel coils of the NRSE arm is several times smaller than in the NSE arm, as expected. The higher homogeneity of the effective NRSE field integral requires less compensation than in the NSE arm. Finally, we tested the appropriate functioning of the NSE–NRSE setup by performing a quasielastic
neutron scattering (QENS) experiment. In QENS, the energy shift experienced by the scattered neutrons is centered around zero. The spin phase of each particular neutron is slightly shifted according to the energy change experienced and therefore, the polarization is decreased. Thus, the spin polarization of the scattered intensity is a sensitive measure of the dynamics present in the sample. We used a solution of sodium dodecyl sulfate (SDS) in salt-free D2 O as sample. The SDS content (65 mg/ml corresponding to the number concentration 0.23 M) was about two orders of magnitude above the critical micelle concentration. Fig. 5 shows the echo polarization as a function of the spin echo time. For this measurement, additional tuning of different spin echo times was required, in order to get a sufficient number of data points. Three zones of different mean momentum transfer Q were defined on the position sensitive multi-detector. The spin echo data were fitted to a single exponential function for each Qvalue. In the given SDS concentration range, the shape of the micelles is reported to resemble ellipsoids with short axis slightly below 2 nm and long axis about 3 nm and small polydispersity [17]. The applied single exponential fit neglects the (small) expected micellar polydispersity and allows to determine the mean relaxation times s. Due to the scattering contrast between hydrogen containing micelles and the coherently scattering D2 O,
Fig. 4. The phase of spin echo groups measured in each detector pixel, dependent on the Fresnel coil current (top). The polarization depends strongly on the tuning of the Fresnel coils (bottom, every three polarization values belong to one phase image).
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Fig. 5. The sample measurement on SDS (65 mg/ml). The relaxation time depends on the Q-value. Lines: Curves obtained by a fit to a single exponential function.
NSE measures the collective diffusion of micelles. Moreover, at the SDS concentration given (volume fraction about 1%), interactions between micelles have been reported. However, within the narrow Q-range investigated in our test experiment, deviations from the dependency 1=s Q2 are not verifiable. The obtained value for the apparent diffusion coefficient D ¼ ð8:1 0:1Þ 107 cm2 /s corresponds to the apparent hydrodynamic radius rH around 2 nm (some uncertainty because the sample was at room temperature) and agrees well with NSE and PCS data from literature [2,17] demonstrating the appropriate performance of the NSE–NRSE setup.
4. Conclusions We have built and tested a new NRSE setup with longitudinal magnetic field geometry. This setup combined with a standard NSE setup represents a NSE–NRSE spectrometer. The static magnetic field of the NRSE coils points in the same direction as the precession field of the NSE coil, i.e. parallel to the symmetry axis of the solenoids. This NRSE field geometry allows for correcting path length differences of divergent neutrons by means of Fresnel coils. This was the
first NRSE experiment using correction coils, because Fresnel coils cannot be used for a NRSE setup with transversal static magnetic fields due to the geometric properties of their magnetic fields. Thus, in the presented NSE–NRSE setup, the wellknown advantages of NRSE have been unified with features used in NSE. The design of the magnetic coils was determined by magnetic field calculations which take the special properties of a NRSE setup with longitudinal static fields into account. Consequently, already the preliminary setup used for the test experiment showed good performance. In particular, by using a set of correction coils the field inhomogeneities could be minimized, as experimentally demonstrated by resonant p-flip scans. The spin echo groups measured by our NSE–NRSE setup can be described in analogy to a classical NSE setup. The echo amplitudes achieved in the test experiment already reached the values typically found at the NSE spectrometer IN11. The performance of the setup in elastic scattering experiments was optimized using Fresnel correction coils belonging to the NSE standard equipment. As a central goal of the test experiment, the correct functioning of the Fresnel coils in the NRSE setup was demonstrated. We homogenized the spin phase pattern by tuning the currents of the
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Fresnel coils and, consequently, increased the resolution of the instrument. Moreover, the NSE– NRSE spectrometer allows to compare directly the correction currents needed in the two different arms. We find that in the NRSE setup, only small Fresnel currents are necessary. This reflects the fact that a NRSE setup intrinsically leads to small inhomogenites of the static magnetic field integral. Therefore a NRSE setup requires several times less material of the Fresnel coils in the beam. Thus, neutron absorption in the Fresnel coils, a limiting factor of conventional NSE, can be overcome using a longitudinal NRSE setup. Finally, the NSE–NRSE spin echo spectrometer was tested in a quasielastic scattering experiment. The dynamics of micellar SDS in solution was examined. As expected, the NSE–NRSE spectrometer revealed classical diffusive behavior described by a diffusion constant which was in agreement with the literature value. In future, several aspects of the longitudinal NRSE setup can still be improved, in order to reach higher spin echo times. The use of the IN11 coils restricted the homogeneity of the static magnetic fields. Coils specifically designed for NRSE with optimized homogeneity will allow to increase the static magnetic fields without loss of polarization in the resonant spin flips. Moreover, a new coil design would make it also possible to use RF coils of higher field integral homogeneity and reduced stray fields. Finally, Fresnel coils adapted especially to NRSE would reduce the divergence effect more effectively. Thus, an optimized longitudinal NRSE setup promises to reach at least the same dynamic resolution as existing NSE spectrometers, but with less absorbing material in the beam.
Acknowledgements We thank R. G€ahler for discussions, E. Thaveron for building the additional coils used in the NRSE setup and B. Farago for help with the sample.
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Neutron resonance spin echo using spin echo correction coils Wolfgang H€ aussler a,*, Ulrich Schmidt b, Georg Ehlers a, Ferenc Mezei c b
a Institut Laue-Langevin, 6 rue Jules Horowitz, BP 156, 38042 Grenoble Cedex 9, France Physikalisches Institut der Universit€at Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany c Hahn-Meitner-Institut, Glienicker Straße 100, 14109 Berlin, Germany
Received 31 October 2002; in final form 10 March 2003
Abstract We report on a recent experiment which was performed in order to examine a new experimental realization of the neutron resonance spin echo (NRSE) technique. In neutron spin echo (NSE) neutrons accumulate spin phase in long magnetic fields before and after the sample. The measured quantity is the final scattered beam polarization, which contains information on the sample dynamics (the intermediate scattering function SðQ; tÞ). This method is limited by the inhomogeneity of the magnetic field, leading to differences of the spin phase of individual neutrons and thus influencing the scattered beam polarization even in the absence of sample dynamics. The NRSE technique can overcome this limitation by reducing the dimensions of the field. However, all previously built NRSE spectrometers use transversal magnetic fields, making it impossible, in practice, to correct for additional limiting effects, such as the beam divergence. For the first time, we have built a NRSE setup with longitudinal field geometry, which does not have this disadvantage. In order to test this new approach, we combined a longitudinal NRSE setup in one spectrometer arm of the IN11 instrument at ILL with a (conventional) NSE setup in the other arm. This experiment demonstrates, how NRSE can be realized in longitudinal field geometry, and sheds light on the differences to previous NRSE setups. As a main result, we show that the effect of beam divergence could be corrected by means of standard Fresnel coils. The proper operation of this hybrid spectrometer is demonstrated by measurements of diffusive dynamics in a well-known micellar sample. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Neutron spin echo (NSE) [1] provides the highest energy resolution of all neutron scattering techniques. A fraction of 1 neV has been achieved in scattering experiments [2]. Soon after the first
*
Corresponding author. Tel.: +33-47620-7041; fax: +3347620-7688. E-mail address: [email protected] (W. H€aussler).
NSE instruments were running and available for experiments, a variety of phenomena has been studied [3]. The main parts of NSE instruments are magnetic coils in each of the two spectrometer arms before and behind the sample position. Using the neutron spin, which precesses in the magnetic fields produced by these coils, the velocities, and thereby the energies of the neutrons before and behind the sample are compared. The high energy resolution of the NSE method is based on the fact,
0301-0104/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00119-8
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that in high magnetic field the spin phase is very sensitive to small velocity differences due to inelastic sample scattering. In neutron resonance spin echo (NRSE) [4,5], the neutrons also accumulate spin phase in two spectrometer arms before and behind the sample region. Each NRSE arm consists of two NRSE coils and, in between, a flight region where no magnetic field is necessary. In both NSE and NRSE, the energy resolution is proportional to the magnitude of the field in the coils. Different from NSE, in NRSE the energy resolution also depends on the distance between the NRSE coils, but not on the length of an individual NRSE coil. Therefore, NRSE coils can be designed relatively compact creating only small stray fields which have to be corrected for. This is the main advantage of NRSE compared to conventional NSE, where high dynamic resolution requires long coils all the way between sample and detector. One can also build a multi-detector arm using several identical NRSE coils [6,7], or adjust NRSE coils to the shape of a dispersion relation to be observed in an inelastic experiment [8]. However, the available spin echo times of all earlier built NRSE spectrometers were effectively limited by the divergence of the neutron beam. Neutrons on trajectories which are not parallel to the symmetry axis stay longer in between the NRSE coils, hence accumulate more phase (‘‘divergence effect’’). This affects the resolution, because NRSE (as NSE) is based on the condition that neutrons of the same wavelength accumulate the same spin phase. All earlier NRSE instruments used coils producing a static magnetic field B0 directed perpendicularly to the flight path of the neutrons. Therefore, it was practically impossible to correct for the divergence effect. In this paper, we report on the test of a completely new NRSE setup with longitudinal B0 field. In NSE spectrometers, so-called ‘‘Fresnel’’ coils [3] are used to correct for both field inhomogeneities and the divergence effect. These Fresnel coils consisting of concentric current loops can only be used if the spin precession field is parallel to the flight path of the neutrons. This is the main motivation for building a longitudinal field NRSE, thus combining the advantages of conventional
NSE (Fresnel coils can be used for correcting for the divergence effect) and NRSE (resolution is decoupled from the length of the coils). In order to compare the performance of the new longitudinal NRSE setup with NSE, we combined one NRSE spectrometer arm with one NSE spectrometer arm of the instrument IN11 at the Institut Laue-Langevin (Grenoble) [9]. The whole arrangement represents a combined NSE–NRSE spectrometer. We demonstrate the functioning of this NSE–NRSE setup in a test experiment measuring the dynamics of a standard sample. First, we describe the special properties of our setup, which differs to some extent from other NRSE spectrometers. The main difference is that the field between the NRSE coils is significantly different from zero, whereas in NRSE up today this field was always as low as possible (‘‘zero field spin echo’’). Thereafter the performance of the NRSE coils in the longitudinal setup is examined. The main goal of the present study is to demonstrate that in NRSE with longitudinal B0 field, the inhomogeneities of the effective field integral due to the beam divergence can be corrected. We show that the polarization of an elastically scattered neutron beam can be increased from 25% to almost 80% using standard Fresnel coils, which makes it possible to improve the dynamic resolution of the spin echo method.
2. Theory 2.1. Neutron spin echo–neutron resonance spin echo Various theoretical descriptions of NSE and NRSE spectrometers have been developed [10,11]. Here, only a few important points are repeated. The spins ðs ¼ g=2Þ of neutrons traversing subsequently each arm of a NSE spectrometer precess in a magnetic field of magnitude B with Larmor frequency mL ¼ jcj=2pB, where c ¼ 2p 29:16 MHz/T is the gyromagnetic ratio of the neutron [12]. In the first arm, each neutron accumulates the spin phase /NSE which Ris proportional to the magnetic field integral JB ¼ B ds along its flight path s:
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/NSE ¼ c=2pðkm=hÞJB :
ð1Þ
Here k stands for the neutron wavelength and m for its mass. In the second arm, the precession is of opposite sense, and the spin phase decreases. For strictly elastic scattering, the final spin phase obtained by all neutrons after the second arm with field integral JB is / ¼ 0 (so-called ‘‘spin echo point’’). For inelastic scattering / ¼ sSE DE=g with the neutron energy change DE. The proportionality constant sSE , called spin echo time [2], determines the sensitivity of the spin phase to sample dynamics, sSE ¼ c=2pðm2 =h2 Þk3 JB :
ð2Þ
The measured parameter is the final beam polarization pSE , analyzed after the precession regions in the same direction in which the spin has been polarized originally: Z pSE ¼ cosð/ðkÞÞf ðkÞ dk: ð3Þ Here /ðkÞ is the final spin phase and the integration is performed over the whole wavelength spectrum f ðkÞ of the neutrons. Eq. (1) shows that the phase / depends on the neutron wavelength k. However, the NSE technique is well adapted to using a broad wavelength spectrum, because the final polarization depends only in second (and higher) order on the relative width Dk=k of the wavelength spectrum. The second arm of our experimental setup consists of two NRSE coils with a magnetic guide field in between. The NRSE coils produce a longitudinal static magnetic field B0 and additionally, a radio frequency (RF) field B1 perpendicular to B0 . The magnetic fields are tuned to get a resonant p-flip. The guide field between the NRSE coils has the task to avoid polarization losses due to nonadiabatic magnetic field changes along the neutron flight paths. The spin phase /NRSE accumulated in one NRSE arm is proportional to the field strength B0 and the effective distance between the centers of the NRSE coils L [4]: /NRSE ¼ c=2pðkm=hÞ2B0 L:
ð4Þ
Consequently, the NRSE spin echo time is given by sNRSE ¼ c=2pðm2 =h2 Þk3 2B0 L:
ð5Þ
503
Comparing with Eq. (2) for (conventional) NSE, we see that for identical resolution NRSE only needs half the static magnetic fields. Furthermore, this factor 2 can be increased by bootstrap methods to 2n [13]. The maximum accessible spin echo time in NRSE depends on the field strength B0 and distance between the NRSE coils, but not on the length of an individual NRSE coil, in contrast to NSE. Nevertheless, the effect of one NRSE arm on the neutron spin is identical to the action of an effective static magnetic filed, in analogy to NSE. Normally, a complete spin echo spectrometer consists of a symmetric setup of either two NSE or NRSE arms. However, a spectrometer can also be built by combining the two methods. The final beam polarization obtained in a spectrometer consisting of a NSE and a NRSE arm (NSE– NRSE) depends on the difference of the (effective) field integral in both arms, as can be seen by Eqs. (1), (3) and (4). If they are identical, the spin echo condition is fulfilled, and the original polarization value is recovered. However, if the field integrals differ, a non-zero spin phase remains which is proportional toRthe difference of the field integrals R DJB ¼ B ds 2B0 ds. Consequently, varying the NSE armÕs field integral, the polarization oscillates: Z pSE ¼ ð1 dÞ cosðc=2pðkm=hÞ DJB Þf ðkÞ dk: ð6Þ In this expression, the depolarization d is added taking into account polarization losses. While in an ideal spin echo setup, all neutrons end up with identical phase and d ¼ 0, deviations of the spin phase from zero arise due to inhomogeneities of the (effective) field integral. Mainly, there are two origins of field integral inhomogeneity: (1) The magnetic field itself is inhomogeneous (in long solenoids mainly at entrance and exit). This leads to different JB even for trajectories of the same length. (2) The length of the flight path is not the same for all trajectories. Neutrons with high divergence gain more phase, even in an ideally homogeneous magnetic field (divergence effect). Fresnel coils can be used to correct for both effects [2,14]. They are fixed within the precession field and consist of concentric current loops
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adding more field integral near the cylinder axis than off-center. The current density j in our Fresnel coil of outer radius R increases linearly with the distance r from the center. This leads to a quadratic field integral correction Jcorr , suited to cancel the leading order of inhomogeneities of the static field integral: 2
j r ) Jcorr ðr RÞ :
ð7Þ
2.2. Resonant spin flip and magnetic fields A resonant spin flip can be obtained by the combined action of a static magnetic and a circularly polarized RF field perpendicular to the B0 field. A fully polarized neutron beam (polarization pz ¼ 1), leaves the NRSE coil with polarization p [15]: p ¼ g2 þ c2 exp
l2 m2 r2 x2R 2h2
cos
lmkxR h
ð8Þ
with the prefactors g ¼ ðcB0 xÞ=xR and c ¼ 2 ðcB1 Þ=xR , the Rabi frequency xR ¼ ½ðcB0 xÞ þ 2 0:5 cB1 Þ , the RF frequency x, the path length l of the neutron within the RF field B1 , the neutron mass m, the mean neutron wavelength k and the standard deviation of the neutron wavelength distribution r. Eq. (8) was used to describe test measurements of the NRSE coils with initial polarization parallel to B0 . However, it has to be mentioned that for spin echo measurements, the initial polarization is perpendicular to B0 , and instead of Eq. (8), a more complicated expression is needed to describe the polarization. The RF fields used in practice are linearly, rather than circularly polarized (Fig. 2). Each linearly polarized field can be described as sum of two oppositely directed circularly polarized components. One component is used for the resonant flip. If the ratio between the strength of B0 and the strength of the RF field B1 is larger than 10, the other component of the RF field has only negligible small influence on the spin motion (rotating wave approximation) [7,15,16]. From Eq. (8) it can be easily seen that two conditions must be met in order to achieve a resonant p-flip:
cB0 x;
ð9aÞ
ðlmkhÞcB1 ¼ p:
ð9bÞ
ðaÞ the resonance condition : ðbÞ the p-flip condition :
The first condition requires that the RF field oscillates with Larmor frequency. The second condition means that the neutron wavelength has to be matched to the Rabi frequency in resonance cB1 for being flipped by p in the RF field. In order to produce static magnetic fields parallel to the neutron trajectory we use solenoids. The magnetic field is calculated starting from the law of Biot–Savart d~ j ~ r d~ B¼ r3
ð10Þ
with the infinitesimal linear current density element d~ j and ~ r pointing from the current to the point calculated. Eq. (10) is numerically integrated over a circular loop, and both length and thickness of a solenoid are taken into account by summation over equidistant loops. The static magnetic fields of all coils are taken into account (Fig. 1), which is important for predicting the tuning for the spin echo point (see below) as well as for optimizing the homogeneity of the static field inside the NRSE coils (see Section 3). The field created by the RF coils is analytically calculated starting from the law of Biot–Savart (Eq. (10)). The details of this calculation have been described elsewhere [7,15]. The RF field created by an oscillating current is different from the field in the static case, first due to eddy currents induced in all metallic parts surrounding the RF coils. In our setup, we can neglect this effect, because the coils are much smaller than the inner diameter of the B0 coils (see Section 3). Second, the skin effect pushes the oscillating current to the surface of the wire, influencing the generated RF field. The RF coils were made of wire of diameter as thin as 1 mm to minimize this effect.
3. Experimental section An overview of the experimental NSE–NRSE setup is given in Fig. 1. Neutrons coming from the
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Fig. 1. Experimental setup (top, details are described in the text). The profile of the longitudinal static magnetic field created by all solenoids (bottom).
left side (beam diameter 5 cm) enter the NSE precession coil with spin polarized perpendicularly to the B0 field after having passed the polarizer and the p=2-flipper. In the NSE coil (length 2 m), spin phase is accumulated by Larmor precession. Behind the NSE coil, the p-flipper provides a turn of the neutron spin by 180° around an axis perpendicular to the beam. This is necessary because the precession sense in the second arm is equal to the first arm. The p-flip ensures that the spin phase accumulated in the second arm is subtracted from the phase accumulated in the first arm. After having passed the sample position, the beam passes the NRSE setup (the distance between the NRSE coils is 2 m), another p=2-flipper, analyzer and detector. In the following, we describe some special aspects of the longitudinal NRSE setup used in this test experiment. The resonance condition (Eq. (9a)) has to be fulfilled by all neutrons. Consequently, the B0 field has to be homogeneous over the whole volume of the RF coil passed by neutrons. Magnetic field calculations (Fig. 1, proved by hall probe measurements) revealed that the field
homogeneity provided by the standard IN11 coils (double-helical solenoid, length 65 cm, inner/outer diameter 20/30 cm) is not sufficient (relative inhomogeneity DB0 =B0 3‰). Therefore, we added correction coils fixed inside the B0 solenoids next to the RF coils. As result, we achieve DB0 =B0 0:2‰ over a path length of 10 cm and diameter of 5 cm. The RF coils are pictured in Fig. 2. They are fixed in the beam, so that they are penetrated by the neutrons. This is necessary in order to provide a RF field pointing perpendicular to the static field B0 (which is equal to the direction of the neutron flight paths). We use cuboid-shaped RF coils (60 60 30 mm3 ) winded by aluminum wire which is almost transparent to neutrons (Fig. 2). At the top and bottom, additional correction coils have been added (25 25 10 mm3 ), in order to make the RF field sufficiently homogeneous (DB1 =B1 4:2% (standard deviation) calculated for a beam diameter of 5 cm). The cuboid coil shape provides identical RF field integrals for all neutrons of the beam. Different path lengths in these coils are not an issue: given the maximum
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Fig. 2. The design of the RF coil (left). The coil is cuboid-shaped, while at both ends, additional correction coils are added. The appropriate functioning of the RF coil together with the B0 coil is demonstrated by B0 - and B1 -resonance scans (center and right).
beam divergence of 20 mrad due to the use of diaphragms in different places in the spectrometer, the maximum relative path length difference in the B1 coil is less than 1& and can be neglected. We performed resonant p-flip test measurements using the neutron beam polarized parallel to the static magnetic field B0 (p=2-flippers off, only one NRSE coil switched on). Fig. 2 shows scans of the magnitude of both the B0 and the B1 field. The B1 field was supplied by a frequency generator (HP) combined with an amplifier (ENI 400 B, 80– 2700 kHz). Comparison with theory (Eq. (8), Fig. 2) shows that the main influence on the resonant pflip originates from the broad wavelength spectrum of the neutrons. Taking the width of Dk=k 16:4% (well known, see below), as input, the field inhomogeneities can be detected by comparison of data and theory. As a result, the spin polarization in the maximum of the p-flip (96%, Fig. 2) is slightly decreased due to field inhomogeneities. The inhomogeneity of B1 (6.4%) obtained by the fit is higher than the calculated value, most likely due to the idealizations made in the calculation (eddy currents, B0 inhomogeneity neglected). Moreover, the analysis of the B0 scan shows that in addition to the magnetic field inside the RF coil, also the stray fields outside the RF coil passed by the neutron beam have to be taken into account. A detailed analysis of this effect will be presented elsewhere. In Fig. 3, the first measured spin echo groups are shown. From the width of the echo group, the neutron wavelength spread Dk=k ¼ 16:4% (mean ) is found by fitting the theory wavelength 5.5 A
(Eq. (6)) to the data. A particular aspect of our NSE–NRSE arrangement is that it is actually not trivial to find the settings that match the Fourier times on both sides exactly. The main precession current in the NSE coil (196 A at the scans shown in Fig. 3) was calculated from the settings in the NRSE arm (RF-frequency, B0 and the NRSE coil distance) via Eqs. (1) and (4) (Fig. 1). The spin echo point was found near the predicted value (Fig. 3) confirming the correct calculation of all field integrals. This first echo was detected in the direct beam without sample. We used two Fresnel coils (Section 2) at entrance and exit of the NSE coil, in order to correct for field inhomogenities. However, in the NRSE arm, no correction coils were needed because the divergence effect is negligible (in the direct beam the angle of every neutron to the symmetry axis has the same size in both spectrometer arms, and therefore, the path length is identical). Consequently, the echo polarization is an exact measure for the quality of the p=2-flippers, p-flippers and NRSE coils. Echo polarization values higher than 80% are found already in the test measurements without much time spent on optimal tuning. In addition, Fig. 3 shows echo groups measured while performing an elastic scattering experiment (graphite sample). The first measurement was performed with the same adjustment as used for the direct beam measurement (without Fresnel coils in the NRSE arm). The echo polarization was much lower than in the direct beam case, because due to the sample scattering, the directions of the
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Fig. 3. The first NSE–NRSE spin echo group measured in the direct beam (circles). Spin echo groups measured in elastically scattered , the wavelength spread determined by intensity (triangles: Fresnel coils off, squares: Fresnel coils on). The mean wavelength was 5.5 A fitting to Eq. (6) was 16.4% (FWHM). The inset shows the echo polarization at three spin echo times.
neutron trajectories in the two spectrometer arms are decoupled, and the path lengths are different. With two Fresnel coils supplemented by an additional coil in between, we could add spin phase to straight-on neutrons, while the spin phase of divergent moving neutrons was reduced. This correction scheme was used for the first time in NRSE and works only, if the static magnetic field is longitudinal. Fig. 3 demonstrates the effect of the Fresnel coils on the NRSE side (spin echo groups on lefthand side). With Fresnel coils turned on, the final polarization increases and the position of the spin echo point shifts because the Fresnel coils add a supplementary magnetic field integral. In the framework of this test experiment the currents of the Fresnel coils were optimized for three different spin echo times. The respective maximum polarization values are shown in the inset of Fig. 3. The maximum spin echo time (sSE ¼ 4:6 ns) was only limited by the available power supply, but not by the depolarization. Although the line added in the inset serves primarily as a guide to the eye, it may be used also to obtain a rough estimate of the maximum resolution. Assuming that the depolarization is only due to remaining field integral inhomogeneities, the decay of the polarization was
fitted in first approximation to a quadratic function. From this fit we get a maximum spin echo time of s ¼ 110 ns. This value is more than one order of magnitude higher than the value we could reach without Fresnel coils. The possibility to reach high spin echo times was the main motivation for building the longitudinal NRSE setup. We would like to mention here, that it is mainly the usage of Fresnel correction coils which makes it possible to reach high spin echo times, but also the fact, that the resolution in longitudinal NRSE is only weakly sensitive to mechanical tolerances. The precession regions are limited by the p=2- and p-spin flippers as in NSE, but not by the NRSE coils, because the neutrons do not penetrate the wire of the B0 solenoids (in contrast to transversal NRSE). Moreover, the RF field integral inhomogeneity due to mechanical tolerances influences primarily the quality of the resonant p-flip (like the beam divergence as discussed above), but not the spin phase. In summary, mechanical tolerances of both the B0 solenoids and of the RF coils have only weak impact on the spin phase. (Only mechanical tolerances of the p=2- and p-spin flippers could, as in NSE, deteriorate the polarization. However, the precession field around these spin flippers is low as in NSE. Consequently,
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mechanical tolerances lead only to small deviations of the spin phase of neutrons passing the flippers on different trajectories.) This is another advantage of the longitudinal NRSE technique compared to transversal NRSE where the precession regions are limited by the NRSE coils with high magnetic field present, so that mechanical tolerances can lead to large spin phase deviations of neutrons traversing the coils on different trajectories. The impact of the Fresnel coils is illustrated in Fig. 4 in detail. At the highest spin echo time we could reach with our power supply, the Fresnel currents were stepwise varied between the optimum setting and zero. With the Fresnel coils switched off, the variation of the spin phase over the detector area amounts to more than one full period of the spin phase. At optimal Fresnel currents setting, resulting in maximum polarization, no variations of the spin phase are visible on the same scale. The best polarization value in the spin echo point is comparable to that of standard NSE measurements at IN11. Moreover, the current needed in the Fresnel coils of the NRSE arm is several times smaller than in the NSE arm, as expected. The higher homogeneity of the effective NRSE field integral requires less compensation than in the NSE arm. Finally, we tested the appropriate functioning of the NSE–NRSE setup by performing a quasielastic
neutron scattering (QENS) experiment. In QENS, the energy shift experienced by the scattered neutrons is centered around zero. The spin phase of each particular neutron is slightly shifted according to the energy change experienced and therefore, the polarization is decreased. Thus, the spin polarization of the scattered intensity is a sensitive measure of the dynamics present in the sample. We used a solution of sodium dodecyl sulfate (SDS) in salt-free D2 O as sample. The SDS content (65 mg/ml corresponding to the number concentration 0.23 M) was about two orders of magnitude above the critical micelle concentration. Fig. 5 shows the echo polarization as a function of the spin echo time. For this measurement, additional tuning of different spin echo times was required, in order to get a sufficient number of data points. Three zones of different mean momentum transfer Q were defined on the position sensitive multi-detector. The spin echo data were fitted to a single exponential function for each Qvalue. In the given SDS concentration range, the shape of the micelles is reported to resemble ellipsoids with short axis slightly below 2 nm and long axis about 3 nm and small polydispersity [17]. The applied single exponential fit neglects the (small) expected micellar polydispersity and allows to determine the mean relaxation times s. Due to the scattering contrast between hydrogen containing micelles and the coherently scattering D2 O,
Fig. 4. The phase of spin echo groups measured in each detector pixel, dependent on the Fresnel coil current (top). The polarization depends strongly on the tuning of the Fresnel coils (bottom, every three polarization values belong to one phase image).
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Fig. 5. The sample measurement on SDS (65 mg/ml). The relaxation time depends on the Q-value. Lines: Curves obtained by a fit to a single exponential function.
NSE measures the collective diffusion of micelles. Moreover, at the SDS concentration given (volume fraction about 1%), interactions between micelles have been reported. However, within the narrow Q-range investigated in our test experiment, deviations from the dependency 1=s Q2 are not verifiable. The obtained value for the apparent diffusion coefficient D ¼ ð8:1 0:1Þ 107 cm2 /s corresponds to the apparent hydrodynamic radius rH around 2 nm (some uncertainty because the sample was at room temperature) and agrees well with NSE and PCS data from literature [2,17] demonstrating the appropriate performance of the NSE–NRSE setup.
4. Conclusions We have built and tested a new NRSE setup with longitudinal magnetic field geometry. This setup combined with a standard NSE setup represents a NSE–NRSE spectrometer. The static magnetic field of the NRSE coils points in the same direction as the precession field of the NSE coil, i.e. parallel to the symmetry axis of the solenoids. This NRSE field geometry allows for correcting path length differences of divergent neutrons by means of Fresnel coils. This was the
first NRSE experiment using correction coils, because Fresnel coils cannot be used for a NRSE setup with transversal static magnetic fields due to the geometric properties of their magnetic fields. Thus, in the presented NSE–NRSE setup, the wellknown advantages of NRSE have been unified with features used in NSE. The design of the magnetic coils was determined by magnetic field calculations which take the special properties of a NRSE setup with longitudinal static fields into account. Consequently, already the preliminary setup used for the test experiment showed good performance. In particular, by using a set of correction coils the field inhomogeneities could be minimized, as experimentally demonstrated by resonant p-flip scans. The spin echo groups measured by our NSE–NRSE setup can be described in analogy to a classical NSE setup. The echo amplitudes achieved in the test experiment already reached the values typically found at the NSE spectrometer IN11. The performance of the setup in elastic scattering experiments was optimized using Fresnel correction coils belonging to the NSE standard equipment. As a central goal of the test experiment, the correct functioning of the Fresnel coils in the NRSE setup was demonstrated. We homogenized the spin phase pattern by tuning the currents of the
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Fresnel coils and, consequently, increased the resolution of the instrument. Moreover, the NSE– NRSE spectrometer allows to compare directly the correction currents needed in the two different arms. We find that in the NRSE setup, only small Fresnel currents are necessary. This reflects the fact that a NRSE setup intrinsically leads to small inhomogenites of the static magnetic field integral. Therefore a NRSE setup requires several times less material of the Fresnel coils in the beam. Thus, neutron absorption in the Fresnel coils, a limiting factor of conventional NSE, can be overcome using a longitudinal NRSE setup. Finally, the NSE–NRSE spin echo spectrometer was tested in a quasielastic scattering experiment. The dynamics of micellar SDS in solution was examined. As expected, the NSE–NRSE spectrometer revealed classical diffusive behavior described by a diffusion constant which was in agreement with the literature value. In future, several aspects of the longitudinal NRSE setup can still be improved, in order to reach higher spin echo times. The use of the IN11 coils restricted the homogeneity of the static magnetic fields. Coils specifically designed for NRSE with optimized homogeneity will allow to increase the static magnetic fields without loss of polarization in the resonant spin flips. Moreover, a new coil design would make it also possible to use RF coils of higher field integral homogeneity and reduced stray fields. Finally, Fresnel coils adapted especially to NRSE would reduce the divergence effect more effectively. Thus, an optimized longitudinal NRSE setup promises to reach at least the same dynamic resolution as existing NSE spectrometers, but with less absorbing material in the beam.
Acknowledgements We thank R. G€ahler for discussions, E. Thaveron for building the additional coils used in the NRSE setup and B. Farago for help with the sample.
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