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Detector area expansion at iNSE neutron spin echo spectrometer
ARTICLE IN PRESS Physica B 404 (2009) 2607–2610
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Physica B journal homepage: www.elsevier.com/locate/physb
Detector area expansion at iNSE neutron spin echo spectrometer N.L. Yamada a,, H. Endo b, N. Osaka b, Y. Kawabata c, M. Nagao b,1, T. Takeda d, H. Seto a, M. Shibayama b a
Neutron Science Laboratory, High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan Institute for Solid State Physics, The University of Tokyo, Tokai 319-1106, Japan Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan d Graduate School of Integrated Arts and Sciences, Hiroshima University, Higashihiroshima 739-8521, Japan b c
a r t i c l e in fo
PACS: 61.05.fg 83.85.Hf 42.25.Fx Keywords: Neutron spin echo (NSE) Detector area expansion Phase correction method
abstract Expansion of a detection area is an effective method to increase the measurement efficiency of a neutron spin echo (NSE) spectrometer as well as other spectrometers. For this purpose, we installed a new p=2 spin flipper and Fresnel coil of iNSE spectrometer at JRR-3, Tokai, Japan, for wide-area data acquisition. In this study, we propose a data reduction method to correct the phase inhomogeneity due to the path difference of neutrons on the large detection area. This method can convert many NSE signals at small areas to one NSE signal at a large area with taking the phase offset due to the phase inhomogeneity into account. The measurement efficiency increased by approximately one order of magnitude as a result of the detection area expansion. & 2009 Elsevier B.V. All rights reserved.
1. Introduction In Neutron spin echo (NSE) spectroscopy proposed by Mezei [1], a phase shift in the Larmor precession of neutron spin in a magnetic field is a measure of the energy transfer of a sample. This technique provides an intermediate correlation function IðQ ; tÞ with high energy resolution in the time range from ps to sub-ms. Therefore, NSE is appropriate to observe the relaxation mode of soft-condensed matters, such as microemulsions, polymers, glasses, and so on. The first NSE spectrometer in Japan (ISSP-NSE) was installed at a branch port of the C2 guide of JRR-3, JAERI (now renamed as JAEA), Tokai, Japan [2]. In 2004, the ISSPNSE spectrometer was relocated to the end of the C2 guide and renamed as iNSE. Then, some new components, e.g. a neutron velocity selector, polarizer and analyzer mirrors, and a wide area two-dimensional detector, were installed [3]. Since the expansion of a detection area is an effective method to increase the measurement efficiency of NSE spectrometers, almost all NSE spectrometers in the world have a large position sensitive detector [4]. After this relocation, the instrumental performance was effectively improved by a factor 2–3 because of an increase of the incident beam flux. Although the new large area two-dimensional detector and analyzer were installed on iNSE spectrometer for wide-area data
Corresponding author.
E-mail address: [email protected] (N.L. Yamada). Present address: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899-6102, USA, and Cyclotron Facility, Indiana University, Bloomington, IN 47408-1398, USA. 1
0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.06.033
acquisition, spin flippers and Fresnel coils were unchanged. These components are required for NSE spectrometers as well as the detector and analyzer: spin flippers control the Larmor precession of neutrons, and Fresnel coils correct the inhomogeneity of magnetic field integral due to the path difference of neutrons. In this study, we renew a p=2 flipper and Fresnel coil to expand the detection area, and propose a novel method for the phase correction of NSE signals on the large area.
2. Hardware upgrade Fig. 1 shows a schematic illustration of iNSE spectrometer. Neutrons are transported through the C2 guide at JRR-3, attenuated by B2O3–SiO2 glass plates if required, monochromatized by a neutron velocity selector, and polarized by a FeCoV/TiNx supermirror bender. At the p=2 flipper, the neutrons start Larmor precession in the magnetic field generated by an optimal field shape coil (precession coil), in which the phase shift R is proportional to the magnetic field integral, B dl. The neutron beam is reshaped by a beam narrower, and scattered at a sample. After that, the phase shift of the scattered neutrons is reversed at the p flipper, starts to decrease in the magnetic field of a second precession coil, and stops at the second p=2 flipper. In front of the two-dimensional detector, a multichannel supermirror bender is installed as an analyzer to measure the polarization of neutrons. If the magnetic field integrals at the first and second precession coils are the same, the polarization of the neutrons before and after the precession region remains to be identical for all wavelength because the phase shifts are symmetric. On the other hand, the
ARTICLE IN PRESS 2608
N.L. Yamada et al. / Physica B 404 (2009) 2607–2610
precession coil C2 guide
NVS
Fresnel coil
polarizing bender
attenuator
π/2 flipper
symmetry coil
sample holder precession coil Fresnel coil Fresnel coil
beam narrower
π flipper
symmetry coil
Fresnel coil
π/2 flipper
analyzer
2D detector
Fig. 1. Schematic illustration of iNSE spectrometer. For detection area expansion, the Fresnel coil and p=2 flipper in front of the analyzer have been renewed. By the replacement, the detection area increases from 85 to 240 mm+.
phase shifts become asymmetric in the case of quasi-elastic scattering, because the neutron velocity changes due to energy transfer at the scattering event. Therefore, a small energy change at quasi-elastic scattering can be detected as the polarization change due to the phase shift in the Larmor precession. To expand the detection area we have installed a large area p=2 flipper, which works in the area of about 250 mm+. As mentioned above, the phase shift in the Larmor precession is proportional to the magnetic field integral. This means that the path difference of scattered neutrons induces inhomogeneity of the phase shift on the large detection area. To recover this phase inhomogeneity, Fresnel coils are installed at the ends of precession coils. Therefore, a new large Fresnel coil is also required to cover the large area at the end of the second precession coil. For making the new Fresnel coil, Al alloy plates (150 mm in radius) were spirally cut by electrospark machining as the previous ones [2]. The distance of the cut position from the center r 0 is proportional p toffiffiffi the square root of the azimuth y in each plate, that is r 0 ¼ a y. By the replacement of the Fresnel coil and p=2 flipper, the valid detection area increased from 85 to 240 mm+.
3. Phase inhomogeneity problem As mentioned above, NSE detects small energy changes on scattering by analyzing the phase shift in the Larmor precession, which is measured as neutron spin polarization. Here, NSE signal INSE can be described as INSE ðQ ; t; DfÞ ¼ AðQ ; tÞ cos½Df exp½ðDf=sÞ2 þ BðQ Þ,
(1)
where Q is the momentum transfer of the scattered neutrons, t the Fourier time which is proportional to the phase shift of the incident neutrons, Df the phase shift from the spin echo condition (Df ¼ 0 is so-called an echo center), AðQ ; tÞ the amplitude of the NSE signal, s the decay parameter due to the distribution of wavelength, and BðQ Þ the average intensity of the NSE signal. Ideally, an intermediate correlation function IðQ ; tÞ=IðQ ; 0Þ can be evaluated as the normalized NSE signal amplitude P NSE : P NSE ðQ ; tÞ ¼
AðQ ; tÞ . BðQ ÞP
(2)
Here, P is the polarization of the scattered neutrons, which can be evaluated as P¼
I" I# , I" þ I#
(3)
where I# and I" are the scattering intensities of up and down spin state neutrons, respectively, which is measured on the same sample with the p=2 flippers turned off. Also, BðQ Þ is described by I# and I" as BðQ Þ ¼
I" þ I# . 2
(4)
The magnetic field integral inhomogeneity for precession due to the path difference of neutrons is proportional to the square of the
distance from the center line of the optimal field shape coil to the neutron trajectory at the Fresnel correction coil [2]. Although Fresnel coils can recover such the inhomogeneity by generating an effective magnetic field, it is difficult to correct the inhomogeneity completely. Since synthesis of waves with different phases decreases the amplitude of a synthesized wave, the phase inhomogeneity due to the field integral inhomogeneity decreases the amplitude of an NSE signal. Therefore, we need to normalize PNSE by a resolution function to evaluate IðQ ; tÞ=IðQ ; 0Þ as IðQ ; tÞ PNSE ðQ ; tÞ , ¼ IðQ ; 0Þ P0NSE ðQ ; tÞ
(5)
where P0NSE is the PNSE for the elastic scattering by a standard sample. In the same way, the magnetic field integral is distributed depending on the detection position. Since the radius of the detection area, R, is about 10 times larger than that of the sample, r, the field integral distribution on the detection area, aR2 , is about a hundred times larger than that of the sample, ar 2 , where a is a proportionality factor depending on the precession coil current. Therefore, the obtained two-dimensional data are usually divided into small areas and separately analyzed to suppress the phase inhomogeneity effect. It should be noted here that the phase distribution on the detector is simple, because the field integral is proportional to the square of the distance from the center. If the phase distribution on the detector is well-calculated, we can sum up many NSE signals at small pixels to obtain an averaged NSE signal by taking the phase offset of the NSE signals into account. Hence, we attempt to evaluate the phase inhomogeneity on a detector as follows.
4. Phase evaluation Gray-scale images in Fig. 2 show two-dimensional intensity profiles depending on the precession coil current which is proportional to Fourier time t. The raw intensity profiles were divided by BðQ Þ at each pixel evaluated by Eq. (4) to normalize the intensity depending on Q . The sample was a stacking of carbon sheets (Grafoil), which is a strong scatterer for neutrons and used as a standard sample for elastic scattering. The Q value at 1 ˚ and the wavelength of the detector center, Q c , was 0:090 A ˚ the incident neutron, l, was 7:3 A ðDl=l ¼ 15%Þ. In this condition, t ¼ 15:9 ns is the maximum Fourier time, because the maximum value of the magnetic field integral is 0.22 Tm for iNSE. All the profiles show the modulation of the scattering intensity due to the phase inhomogeneity. Note that, the current of the Fresnel coil installed at the end of the second precession coil was adjusted to be slightly higher than the optimal value to observe the modulation clearly. To take the phase inhomogeneity depending on the detection position, we divide Df into two terms as
Df ¼ Df0 þ Df00 , 0
(6)
where Df is the phase shift on the detector due to the path 00 difference of scattered neutrons, and Df is that due to symmetry
ARTICLE IN PRESS N.L. Yamada et al. / Physica B 404 (2009) 2607–2610
159 ps
3.19 ns
2609
2.0
5. Data consistency check
1.5
To check the reliability of the data reduced by the method, further NSE experiment was performed in a non-ionic surfactant C12E5 and D2O mixture system [5]. In this system, a lamellar phase appears above 52 1C [6], and the theory given by Zilman and Granek [7] is well established to describe the undulation motion in the lamellar phase as follows:
1.0 0.5 0.0
IðQ ; tÞ=IðQ ; 0Þ ¼ exp½ðGðQ ÞtÞ2=3 ,
7.97 ns
15.9 ns
Fig. 2. Two-dimensional intensity profiles near the echo center. The image plot shows the intensity profile normalized by BðQ Þ, and the contour lines show the fitting result to Eq. (1).
coil attached to the center of a precession coil. In the measurement, Df was scanned by changing the symmetry coil current to obtain an echo signal in the wide range of Df. 0 Under the assumption that Df is basically proportional to the square of the distance from the center of the phase inhomogeneity as
Df0 ¼ k0 fðx x0 Þ2 þ aðy y0 Þ2 þ bðx x0 Þðy y0 Þg,
(9)
GðQ Þ ¼ 0:025gðkB TÞ3=2 k1=2 Z1 Q 3 .
(10)
Here, k is the bending modulus of the membranes, and Z the viscosity of the solvent. The factor g originates from averaging over the angle between the wave vector and the membrane surface normal in the calculation of IðQ ; tÞ, and it approaches unity for k kB T. Fig. 3 shows the NSE signals of the Grafoil and sample at t ¼ 15:9 ns and IðQ ; tÞ=IðQ ; 0Þ, which are evaluated at the center area (10 40 pixels) of the detector. Both the NSE signals are plotted as a function of Df calculated by Eqs. (6)–(8), and averaged over a certain range of Df (0.5 rad) for better visualization. The difference of the amplitude between the Grafoil and sample indicates the decrease of IðQ ; tÞ=IðQ ; 0Þ due to the quasi-elastic scattering at the sample. Therefore, we evaluated IðQ ; tÞ=IðQ ; 0Þ by analyzing NSE signal with Eq. (1), in which the parameter s of the sample was fixed to be that of the Grafoil because the wavelength distribution is almost the same in the case of a quasi-elastic scattering. On the other hand, we also
(7)
2.0
where k a proportional factor of the phase shift, ðx; yÞ the pixel position on the detector, ðx0 ; y0 Þ the center pixel of the phase inhomogeneity. As seen in Fig. 2, the phase shift on a detector elliptically spreads in the case of long Fourier time, which originates from the misalignment of the Fresnel coil positions. Hence, we approximately apply the parameters of ellipsoid, a and 0 00 b, to describe Df as Eq. (7). On the other hand, Df can be described as 00
Df ¼ k ði i0 Þ,
1.5
1.0
0.5 Grafoil Sample
(8)
00
where k is a proportional factor of the phase shift due to the difference between a symmetry coil current, i, and that of the echo center, i0 . It should be noted here that the distance from the sample to a pixel of the detector has a distribution due to the finite sample size. Thus, we need to consider the phase inhomogeneity at the sample as well as that on the detector. The phase inhomogeneity on the sample is, however, not so effective comparing with that on the detector, since the size of the sample is much smaller than that of the detector as mentioned above. Moreover, the current of the Fresnel coil to correct the inhomogeneity on the detector is slightly higher than the optimal value, in which the phase inhomogeneity is enhanced. Therefore, the phase distribution on the sample was neglected. The contour lines in Fig. 2 show the fitting result to Eq. (1) with Eqs. (6)–(8), in which the fitting was performed with changing the symmetry coil current simultaneously. Eq. (7) well reproduces the phase inhomogeneity on the detector. This means that we can evaluate Df as the function of the detector pixel ðx; yÞ and 0 symmetry coil current i. Since the fitting parameters, k , x0 , y0 , a, b, 00 k , and i0 , at each Fourier time are regarded as instrumental parameters, we use these parameters to calculate Df in all the following analysis.
0.0 -20
-15
-10
-5
0
5
10
15
[rad.] 1.0 0.8 I(Q,t)/I(Q,0)
00
INSE (Q,t,)/B(Q)
0
0.6 present method fitting (e-[(0.0323±0.0002)t]
0.4
-2/3
)
conventional method fitting (e-[(0.0329±0.0002)t]
0.2
-2/3
)
0.0 0
2
4
6 8 10 12 Fourier Time [ns]
14
16
Fig. 3. Reduced data at the center area (10 40 pixels) of the detector. (a) NSE signals of a Grafoil and sample at t ¼ 15:9 ns. (b) IðQ ; tÞ=IðQ ; 0Þ’s reduced by the present and conventional methods.
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(Q) [ns-1]
10-1
4
6. Conclusion
2
In this study, we renewed a p=2 spin flipper and Fresnel coil to expand the detection area. Since the detection area was expanded from 85 to 240 mm+, the efficiency of the experiment increased by one order of magnitude as a result of the expansion of the detection area. Also, we proposed a method to reduce NSE signals with taking the phase inhomogeneity due to the difference of the scattering path from a sample to a detector into account. This method enables us to analyze NSE signals at a wide detection area at the same time, whereas the wide area is divided into many small areas and many NSE signals are separately analyzed in a conventional method. The result shows the feasibility of the present analysis method. iNSE now has much better capabilities to do an experiment than before, about 30–40 times higher efficiency than that before the relocation, and our method would reduce the dulation time at high-Q region in the future because many pixels at large area can be converted to one echo signal thanks to large DQ .
8 6 4 2
10-2
current data previous data
8 6 4 4
5
6
7
8
9
0.1 Q [Å-1] Fig. 4. Q -dependence of G evaluated by the present method. GðQ Þ well agrees with the Zilman–Granek theory, that is, GðQ Þ is proportional to Q 3 .
evaluated IðQ ; tÞ=IðQ ; 0Þ by using a conventional NSE analysis, that is, we divided the center area into 16 small areas (5 5 pixels), analyze the NSE signals with Eq. (1) (s of the sample was fixed to be that of the Grafoil), and averaged all the IðQ ; tÞ=IðQ ; 0Þ’s at the small areas. As shown in Fig. 3(b), IðQ ; tÞ=IðQ ; 0Þ’s reduced by the present and conventional methods are almost the same, and G’s evaluated by the fitting with Eq. (9) are consistent between the two methods. Next, we extracted the pixels in the range of Q 0 DQ =2o Q oQ 0 þ DQ =2 (DQ is the Q -resolution) for Q 0 ¼ 0:070, 0.080, 1 ˚ to perform multi-Q region analysis. 0.090, 0.100, and 0:110 A The GðQ Þ obtained from the NSE analysis by the present method were shown in Fig. 4 with the literature values [5]. Although the present values are a little less than those of the literature, the Q dependence is essentially the same as Eq. (10) [8]. Therefore, we conclude that our data reduction method well works to evaluate the Q -dependence of IðQ ; tÞ=IðQ ; 0Þ.
References [1] F. Mezei (Ed.), Neutron Spin Echo, Lecture Notes in Physics, vol. 128, Springer, Berlin, 1980. [2] T. Takeda, S. Komura, H. Seto, M. Nagai, H. Kobayashi, E. yokoi, C.M.E. Zeyen, T. Ebisawa, S. Tasaki, Y. Ito, S. Takahashi, H. Yoshizawa, Nucl. Instrum. Methods A 364 (1995) 186. [3] M. Nagao, N.L. Yamada, Y. Kawabata, H. Seto, H. Yoshizawa, T. Takeda, Physica B 385–386 (2006) 1118. [4] B. Farago, in: F. Mezei, C. Pappas, T. Gutberlet (Eds.), Neutron Spin Echo, Lecture Notes in Physics, vol. 601, Springer, Berlin, 2003. [5] N.L. Yamada, T. Takeda, K. Kato, M. Nagao, H. Seto, J. Phys. Soc. Japan 74 (2005) 875. [6] R. Strey, R. Schomacker, D. Roux, F. Nallet, U. Olsson, J. Chem. Soc. Faraday Trans. 86 (1990) 2253. [7] A.G. Zilman, R. Granek, Phys. Rev. Lett. 77 (1996) 4788. [8] This inconsistency would be originated from the sample preparation, because the effective Z value depends on the surfactant concentration due to the interaction between membranes.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Detector area expansion at iNSE neutron spin echo spectrometer N.L. Yamada a,, H. Endo b, N. Osaka b, Y. Kawabata c, M. Nagao b,1, T. Takeda d, H. Seto a, M. Shibayama b a
Neutron Science Laboratory, High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan Institute for Solid State Physics, The University of Tokyo, Tokai 319-1106, Japan Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan d Graduate School of Integrated Arts and Sciences, Hiroshima University, Higashihiroshima 739-8521, Japan b c
a r t i c l e in fo
PACS: 61.05.fg 83.85.Hf 42.25.Fx Keywords: Neutron spin echo (NSE) Detector area expansion Phase correction method
abstract Expansion of a detection area is an effective method to increase the measurement efficiency of a neutron spin echo (NSE) spectrometer as well as other spectrometers. For this purpose, we installed a new p=2 spin flipper and Fresnel coil of iNSE spectrometer at JRR-3, Tokai, Japan, for wide-area data acquisition. In this study, we propose a data reduction method to correct the phase inhomogeneity due to the path difference of neutrons on the large detection area. This method can convert many NSE signals at small areas to one NSE signal at a large area with taking the phase offset due to the phase inhomogeneity into account. The measurement efficiency increased by approximately one order of magnitude as a result of the detection area expansion. & 2009 Elsevier B.V. All rights reserved.
1. Introduction In Neutron spin echo (NSE) spectroscopy proposed by Mezei [1], a phase shift in the Larmor precession of neutron spin in a magnetic field is a measure of the energy transfer of a sample. This technique provides an intermediate correlation function IðQ ; tÞ with high energy resolution in the time range from ps to sub-ms. Therefore, NSE is appropriate to observe the relaxation mode of soft-condensed matters, such as microemulsions, polymers, glasses, and so on. The first NSE spectrometer in Japan (ISSP-NSE) was installed at a branch port of the C2 guide of JRR-3, JAERI (now renamed as JAEA), Tokai, Japan [2]. In 2004, the ISSPNSE spectrometer was relocated to the end of the C2 guide and renamed as iNSE. Then, some new components, e.g. a neutron velocity selector, polarizer and analyzer mirrors, and a wide area two-dimensional detector, were installed [3]. Since the expansion of a detection area is an effective method to increase the measurement efficiency of NSE spectrometers, almost all NSE spectrometers in the world have a large position sensitive detector [4]. After this relocation, the instrumental performance was effectively improved by a factor 2–3 because of an increase of the incident beam flux. Although the new large area two-dimensional detector and analyzer were installed on iNSE spectrometer for wide-area data
Corresponding author.
E-mail address: [email protected] (N.L. Yamada). Present address: NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD 20899-6102, USA, and Cyclotron Facility, Indiana University, Bloomington, IN 47408-1398, USA. 1
0921-4526/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2009.06.033
acquisition, spin flippers and Fresnel coils were unchanged. These components are required for NSE spectrometers as well as the detector and analyzer: spin flippers control the Larmor precession of neutrons, and Fresnel coils correct the inhomogeneity of magnetic field integral due to the path difference of neutrons. In this study, we renew a p=2 flipper and Fresnel coil to expand the detection area, and propose a novel method for the phase correction of NSE signals on the large area.
2. Hardware upgrade Fig. 1 shows a schematic illustration of iNSE spectrometer. Neutrons are transported through the C2 guide at JRR-3, attenuated by B2O3–SiO2 glass plates if required, monochromatized by a neutron velocity selector, and polarized by a FeCoV/TiNx supermirror bender. At the p=2 flipper, the neutrons start Larmor precession in the magnetic field generated by an optimal field shape coil (precession coil), in which the phase shift R is proportional to the magnetic field integral, B dl. The neutron beam is reshaped by a beam narrower, and scattered at a sample. After that, the phase shift of the scattered neutrons is reversed at the p flipper, starts to decrease in the magnetic field of a second precession coil, and stops at the second p=2 flipper. In front of the two-dimensional detector, a multichannel supermirror bender is installed as an analyzer to measure the polarization of neutrons. If the magnetic field integrals at the first and second precession coils are the same, the polarization of the neutrons before and after the precession region remains to be identical for all wavelength because the phase shifts are symmetric. On the other hand, the
ARTICLE IN PRESS 2608
N.L. Yamada et al. / Physica B 404 (2009) 2607–2610
precession coil C2 guide
NVS
Fresnel coil
polarizing bender
attenuator
π/2 flipper
symmetry coil
sample holder precession coil Fresnel coil Fresnel coil
beam narrower
π flipper
symmetry coil
Fresnel coil
π/2 flipper
analyzer
2D detector
Fig. 1. Schematic illustration of iNSE spectrometer. For detection area expansion, the Fresnel coil and p=2 flipper in front of the analyzer have been renewed. By the replacement, the detection area increases from 85 to 240 mm+.
phase shifts become asymmetric in the case of quasi-elastic scattering, because the neutron velocity changes due to energy transfer at the scattering event. Therefore, a small energy change at quasi-elastic scattering can be detected as the polarization change due to the phase shift in the Larmor precession. To expand the detection area we have installed a large area p=2 flipper, which works in the area of about 250 mm+. As mentioned above, the phase shift in the Larmor precession is proportional to the magnetic field integral. This means that the path difference of scattered neutrons induces inhomogeneity of the phase shift on the large detection area. To recover this phase inhomogeneity, Fresnel coils are installed at the ends of precession coils. Therefore, a new large Fresnel coil is also required to cover the large area at the end of the second precession coil. For making the new Fresnel coil, Al alloy plates (150 mm in radius) were spirally cut by electrospark machining as the previous ones [2]. The distance of the cut position from the center r 0 is proportional p toffiffiffi the square root of the azimuth y in each plate, that is r 0 ¼ a y. By the replacement of the Fresnel coil and p=2 flipper, the valid detection area increased from 85 to 240 mm+.
3. Phase inhomogeneity problem As mentioned above, NSE detects small energy changes on scattering by analyzing the phase shift in the Larmor precession, which is measured as neutron spin polarization. Here, NSE signal INSE can be described as INSE ðQ ; t; DfÞ ¼ AðQ ; tÞ cos½Df exp½ðDf=sÞ2 þ BðQ Þ,
(1)
where Q is the momentum transfer of the scattered neutrons, t the Fourier time which is proportional to the phase shift of the incident neutrons, Df the phase shift from the spin echo condition (Df ¼ 0 is so-called an echo center), AðQ ; tÞ the amplitude of the NSE signal, s the decay parameter due to the distribution of wavelength, and BðQ Þ the average intensity of the NSE signal. Ideally, an intermediate correlation function IðQ ; tÞ=IðQ ; 0Þ can be evaluated as the normalized NSE signal amplitude P NSE : P NSE ðQ ; tÞ ¼
AðQ ; tÞ . BðQ ÞP
(2)
Here, P is the polarization of the scattered neutrons, which can be evaluated as P¼
I" I# , I" þ I#
(3)
where I# and I" are the scattering intensities of up and down spin state neutrons, respectively, which is measured on the same sample with the p=2 flippers turned off. Also, BðQ Þ is described by I# and I" as BðQ Þ ¼
I" þ I# . 2
(4)
The magnetic field integral inhomogeneity for precession due to the path difference of neutrons is proportional to the square of the
distance from the center line of the optimal field shape coil to the neutron trajectory at the Fresnel correction coil [2]. Although Fresnel coils can recover such the inhomogeneity by generating an effective magnetic field, it is difficult to correct the inhomogeneity completely. Since synthesis of waves with different phases decreases the amplitude of a synthesized wave, the phase inhomogeneity due to the field integral inhomogeneity decreases the amplitude of an NSE signal. Therefore, we need to normalize PNSE by a resolution function to evaluate IðQ ; tÞ=IðQ ; 0Þ as IðQ ; tÞ PNSE ðQ ; tÞ , ¼ IðQ ; 0Þ P0NSE ðQ ; tÞ
(5)
where P0NSE is the PNSE for the elastic scattering by a standard sample. In the same way, the magnetic field integral is distributed depending on the detection position. Since the radius of the detection area, R, is about 10 times larger than that of the sample, r, the field integral distribution on the detection area, aR2 , is about a hundred times larger than that of the sample, ar 2 , where a is a proportionality factor depending on the precession coil current. Therefore, the obtained two-dimensional data are usually divided into small areas and separately analyzed to suppress the phase inhomogeneity effect. It should be noted here that the phase distribution on the detector is simple, because the field integral is proportional to the square of the distance from the center. If the phase distribution on the detector is well-calculated, we can sum up many NSE signals at small pixels to obtain an averaged NSE signal by taking the phase offset of the NSE signals into account. Hence, we attempt to evaluate the phase inhomogeneity on a detector as follows.
4. Phase evaluation Gray-scale images in Fig. 2 show two-dimensional intensity profiles depending on the precession coil current which is proportional to Fourier time t. The raw intensity profiles were divided by BðQ Þ at each pixel evaluated by Eq. (4) to normalize the intensity depending on Q . The sample was a stacking of carbon sheets (Grafoil), which is a strong scatterer for neutrons and used as a standard sample for elastic scattering. The Q value at 1 ˚ and the wavelength of the detector center, Q c , was 0:090 A ˚ the incident neutron, l, was 7:3 A ðDl=l ¼ 15%Þ. In this condition, t ¼ 15:9 ns is the maximum Fourier time, because the maximum value of the magnetic field integral is 0.22 Tm for iNSE. All the profiles show the modulation of the scattering intensity due to the phase inhomogeneity. Note that, the current of the Fresnel coil installed at the end of the second precession coil was adjusted to be slightly higher than the optimal value to observe the modulation clearly. To take the phase inhomogeneity depending on the detection position, we divide Df into two terms as
Df ¼ Df0 þ Df00 , 0
(6)
where Df is the phase shift on the detector due to the path 00 difference of scattered neutrons, and Df is that due to symmetry
ARTICLE IN PRESS N.L. Yamada et al. / Physica B 404 (2009) 2607–2610
159 ps
3.19 ns
2609
2.0
5. Data consistency check
1.5
To check the reliability of the data reduced by the method, further NSE experiment was performed in a non-ionic surfactant C12E5 and D2O mixture system [5]. In this system, a lamellar phase appears above 52 1C [6], and the theory given by Zilman and Granek [7] is well established to describe the undulation motion in the lamellar phase as follows:
1.0 0.5 0.0
IðQ ; tÞ=IðQ ; 0Þ ¼ exp½ðGðQ ÞtÞ2=3 ,
7.97 ns
15.9 ns
Fig. 2. Two-dimensional intensity profiles near the echo center. The image plot shows the intensity profile normalized by BðQ Þ, and the contour lines show the fitting result to Eq. (1).
coil attached to the center of a precession coil. In the measurement, Df was scanned by changing the symmetry coil current to obtain an echo signal in the wide range of Df. 0 Under the assumption that Df is basically proportional to the square of the distance from the center of the phase inhomogeneity as
Df0 ¼ k0 fðx x0 Þ2 þ aðy y0 Þ2 þ bðx x0 Þðy y0 Þg,
(9)
GðQ Þ ¼ 0:025gðkB TÞ3=2 k1=2 Z1 Q 3 .
(10)
Here, k is the bending modulus of the membranes, and Z the viscosity of the solvent. The factor g originates from averaging over the angle between the wave vector and the membrane surface normal in the calculation of IðQ ; tÞ, and it approaches unity for k kB T. Fig. 3 shows the NSE signals of the Grafoil and sample at t ¼ 15:9 ns and IðQ ; tÞ=IðQ ; 0Þ, which are evaluated at the center area (10 40 pixels) of the detector. Both the NSE signals are plotted as a function of Df calculated by Eqs. (6)–(8), and averaged over a certain range of Df (0.5 rad) for better visualization. The difference of the amplitude between the Grafoil and sample indicates the decrease of IðQ ; tÞ=IðQ ; 0Þ due to the quasi-elastic scattering at the sample. Therefore, we evaluated IðQ ; tÞ=IðQ ; 0Þ by analyzing NSE signal with Eq. (1), in which the parameter s of the sample was fixed to be that of the Grafoil because the wavelength distribution is almost the same in the case of a quasi-elastic scattering. On the other hand, we also
(7)
2.0
where k a proportional factor of the phase shift, ðx; yÞ the pixel position on the detector, ðx0 ; y0 Þ the center pixel of the phase inhomogeneity. As seen in Fig. 2, the phase shift on a detector elliptically spreads in the case of long Fourier time, which originates from the misalignment of the Fresnel coil positions. Hence, we approximately apply the parameters of ellipsoid, a and 0 00 b, to describe Df as Eq. (7). On the other hand, Df can be described as 00
Df ¼ k ði i0 Þ,
1.5
1.0
0.5 Grafoil Sample
(8)
00
where k is a proportional factor of the phase shift due to the difference between a symmetry coil current, i, and that of the echo center, i0 . It should be noted here that the distance from the sample to a pixel of the detector has a distribution due to the finite sample size. Thus, we need to consider the phase inhomogeneity at the sample as well as that on the detector. The phase inhomogeneity on the sample is, however, not so effective comparing with that on the detector, since the size of the sample is much smaller than that of the detector as mentioned above. Moreover, the current of the Fresnel coil to correct the inhomogeneity on the detector is slightly higher than the optimal value, in which the phase inhomogeneity is enhanced. Therefore, the phase distribution on the sample was neglected. The contour lines in Fig. 2 show the fitting result to Eq. (1) with Eqs. (6)–(8), in which the fitting was performed with changing the symmetry coil current simultaneously. Eq. (7) well reproduces the phase inhomogeneity on the detector. This means that we can evaluate Df as the function of the detector pixel ðx; yÞ and 0 symmetry coil current i. Since the fitting parameters, k , x0 , y0 , a, b, 00 k , and i0 , at each Fourier time are regarded as instrumental parameters, we use these parameters to calculate Df in all the following analysis.
0.0 -20
-15
-10
-5
0
5
10
15
[rad.] 1.0 0.8 I(Q,t)/I(Q,0)
00
INSE (Q,t,)/B(Q)
0
0.6 present method fitting (e-[(0.0323±0.0002)t]
0.4
-2/3
)
conventional method fitting (e-[(0.0329±0.0002)t]
0.2
-2/3
)
0.0 0
2
4
6 8 10 12 Fourier Time [ns]
14
16
Fig. 3. Reduced data at the center area (10 40 pixels) of the detector. (a) NSE signals of a Grafoil and sample at t ¼ 15:9 ns. (b) IðQ ; tÞ=IðQ ; 0Þ’s reduced by the present and conventional methods.
ARTICLE IN PRESS 2610
N.L. Yamada et al. / Physica B 404 (2009) 2607–2610
(Q) [ns-1]
10-1
4
6. Conclusion
2
In this study, we renewed a p=2 spin flipper and Fresnel coil to expand the detection area. Since the detection area was expanded from 85 to 240 mm+, the efficiency of the experiment increased by one order of magnitude as a result of the expansion of the detection area. Also, we proposed a method to reduce NSE signals with taking the phase inhomogeneity due to the difference of the scattering path from a sample to a detector into account. This method enables us to analyze NSE signals at a wide detection area at the same time, whereas the wide area is divided into many small areas and many NSE signals are separately analyzed in a conventional method. The result shows the feasibility of the present analysis method. iNSE now has much better capabilities to do an experiment than before, about 30–40 times higher efficiency than that before the relocation, and our method would reduce the dulation time at high-Q region in the future because many pixels at large area can be converted to one echo signal thanks to large DQ .
8 6 4 2
10-2
current data previous data
8 6 4 4
5
6
7
8
9
0.1 Q [Å-1] Fig. 4. Q -dependence of G evaluated by the present method. GðQ Þ well agrees with the Zilman–Granek theory, that is, GðQ Þ is proportional to Q 3 .
evaluated IðQ ; tÞ=IðQ ; 0Þ by using a conventional NSE analysis, that is, we divided the center area into 16 small areas (5 5 pixels), analyze the NSE signals with Eq. (1) (s of the sample was fixed to be that of the Grafoil), and averaged all the IðQ ; tÞ=IðQ ; 0Þ’s at the small areas. As shown in Fig. 3(b), IðQ ; tÞ=IðQ ; 0Þ’s reduced by the present and conventional methods are almost the same, and G’s evaluated by the fitting with Eq. (9) are consistent between the two methods. Next, we extracted the pixels in the range of Q 0 DQ =2o Q oQ 0 þ DQ =2 (DQ is the Q -resolution) for Q 0 ¼ 0:070, 0.080, 1 ˚ to perform multi-Q region analysis. 0.090, 0.100, and 0:110 A The GðQ Þ obtained from the NSE analysis by the present method were shown in Fig. 4 with the literature values [5]. Although the present values are a little less than those of the literature, the Q dependence is essentially the same as Eq. (10) [8]. Therefore, we conclude that our data reduction method well works to evaluate the Q -dependence of IðQ ; tÞ=IðQ ; 0Þ.
References [1] F. Mezei (Ed.), Neutron Spin Echo, Lecture Notes in Physics, vol. 128, Springer, Berlin, 1980. [2] T. Takeda, S. Komura, H. Seto, M. Nagai, H. Kobayashi, E. yokoi, C.M.E. Zeyen, T. Ebisawa, S. Tasaki, Y. Ito, S. Takahashi, H. Yoshizawa, Nucl. Instrum. Methods A 364 (1995) 186. [3] M. Nagao, N.L. Yamada, Y. Kawabata, H. Seto, H. Yoshizawa, T. Takeda, Physica B 385–386 (2006) 1118. [4] B. Farago, in: F. Mezei, C. Pappas, T. Gutberlet (Eds.), Neutron Spin Echo, Lecture Notes in Physics, vol. 601, Springer, Berlin, 2003. [5] N.L. Yamada, T. Takeda, K. Kato, M. Nagao, H. Seto, J. Phys. Soc. Japan 74 (2005) 875. [6] R. Strey, R. Schomacker, D. Roux, F. Nallet, U. Olsson, J. Chem. Soc. Faraday Trans. 86 (1990) 2253. [7] A.G. Zilman, R. Granek, Phys. Rev. Lett. 77 (1996) 4788. [8] This inconsistency would be originated from the sample preparation, because the effective Z value depends on the surfactant concentration due to the interaction between membranes.