- Email: [email protected]
Development of polarized neutron optics
ARTICLE IN PRESS
Physica B 356 (2005) 121–125 www.elsevier.com/locate/physb
Development of polarized neutron optics Hirohiko M. Shimizua,b,, Takayuki Okub, Jun-ichi Suzukib, Michihiro Furusakac,b, Yoshiaki Kiyanagid a RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Japan Atomic Energy Research Institute, 2-4 Tokai, Ibaraki, Japan c KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan d Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
b
Abstract Neutron magnetic optics is a spin-selective optics and suitable for precise focusing since it is free from interactions with materials. We discuss its applications to measure the small angle neutron scattering in low-q regions. r 2004 Elsevier B.V. All rights reserved. PACS: 03.75.Be; 61.12 Keywords: Magnetic neutron lens; Sextupole magnet; Cold neutron; Beam focusing
1. Introduction Improvement of neutron optics enables us to increase the utilization efficacy of neutrons with limited beam intensity. However, neutron-focusing devices increase the neutron beam intensity together with the beam divergence at the focal point. Thus their applications are limited to a class of experiments, which do not require high accuracy
Corresponding author. RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan. Tel.: +81 48 462 1111; fax: +81 48 467 9721. E-mail address: [email protected] (H.M. Shimizu).
of momentum transfer (q-resolution), as long as the beam is focused at the sample position. Neutron beam is focused at the detector position in the focusing small angle neutron scattering (FSANS) to achieve a good q-resolution using the converging beam. The FSANS has been developed using Kumakov lens, toroidal mirror, compound refractive lens and magnetic lens [1–3]. The magnetic optics is based on electromagnetic interaction, which is precisely calculable and free from interactions with materials. It also delivers polarized neutrons to the sample and enables the measurement of magnetic scattering. In this report, we describe the FSANS with sextupole magnetic focusing lens and discuss further applications of magnetic optics.
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.10.061
ARTICLE IN PRESS 122
H.M. Shimizu et al. / Physica B 356 (2005) 121–125
2. Magnetic optics We consider the neutron motion in a sextupole magnetic field, which can be written as |B| ¼ (G/ 2)(x2+y2) taking the beam axis as the z-axis. The equation of motion can be written as ! ! x d2 x 2 ¼ o ; (1) y dt2 y where o2 ¼ aG and a ¼ |m/m| ¼ 5.77 T m2 as long as the magnetic field is sufficiently strong so that the angular velocity of Larmor precession is sufficiently larger than the angular velocity of the field rotation in the neutron rest frame. We assume that this equation of motion is always valid in this paper. The signs correspond to the cases where neutron polarization is parallel and antiparallel to the local magnetic field direction, respectively. The solution of Eq. (1) is given as 8 ! cos y sin y > > > > ! > > < sin y cos y x y ! ¼ > cosh y sinh y x Z > > > > > : sinh y cosh y ! x0 y 0 parallel case; x0 Z0 ð2Þ ! x0 y0 antiparallel case; x0 Z0 where y ¼ ot, x ¼ dx/dy, Z ¼ dy/dy and subscripts 0 indicate the value at y ¼ 0. x and Z represent the beam divergences normalized by o. A neutron beam is rotated by y on the xx-plane and yZ-plane for the polarization parallel case, while it is expanded by ey along (1,1) direction and compressed by ey along (1,1) direction for polarization anti-parallel case. In the same notation, the propagation without magnetic interaction is described as ! ! x0 y0 x y 1 y : (3) ¼ x0 Z0 x Z 0 1 These transformations conserve the phase space areas on xx- and yZ-planes, therefore satisfy the
Liouville’s theorem. We do not consider the gravitational interaction in this paper.
3. Magnetic focusing small angle neutron scattering The conventional small angle neutron scattering has a pair of pinhole collimators at z ¼ 0 and z ¼ L1 as shown in Fig. 1. We refer to the geometry as the pinhole small angle neutron scattering (P-SANS). For simplicity, we consider the one-dimensional case with a monochromatic neutron beam with the velocity vz. The incident beam is defined by the pinhole collimator at z ¼ 0 with a total width of 2x0. The beam profile in the phase space is transformed by ! x ! xðbÞ ðaÞ 1 y1 ¼ ; (4) xðbÞ xðaÞ 0 1 where y1 ¼ L1/vz. It is then shaped into the hatched parallelogram in Fig. 1(b) by the second collimator before the sample. Some neutrons are scattered by the sample with Dx ¼ _q=mo; where q is the momentum transfer, and propagates to the detector according to ! x ! xðdÞ ðcÞ 1 y2 ¼ ; (5) xðdÞ xðcÞ 0 1 where y2 ¼ L2/vz. The detector measures the neutron distribution on the x-axis. The scattered neutrons are completely separated from the transmitted neutrons, if the x-coordinate of the point A is larger than that of the point B in Fig. 1(d), which corresponds to _q y2 X2x0 1 þ 2 Dx ¼ : (6) mo y1 We refer to the minimum value of q, which satisfies the inequality (6), as qmin. We consider the case of L1 ¼ L2 (y1 ¼ y2) for simplicity. In this case, the qmin of P-SANS is obtained as qmin;P ¼
mo 6x0 ; _
(7)
The area of the parallelogram is proportional to the total number of neutrons delivered to the
ARTICLE IN PRESS H.M. Shimizu et al. / Physica B 356 (2005) 121–125
123
Fig. 1. Illustration of the geometry of P-SANS together with the beam shapes on the xx-plane (a) at the first pinhole collimator; (b) at the second pinhole collimator; (c) at the sample and (d) on the detector.
Fig. 2. Illustration of the geometry of FSANS together with the beam shapes on xx-plane (a) at the pinhole collimator; (b) at the entrance of the sextupole magnet; (c) at the exit of the sextupole magnet; (d) in the sample and (e) on the detector.
sample, which is given as S xx;P ¼
4x20 : y1
(8)
In the same manner, we describe the magnetic F-SANS in Fig. 2. The neutrons collimated by the first pinhole collimator propagate to the entrance of the sextupole magnet as shown in Fig. 2(b). The sextupole magnet has an aperture diameter 2a and length L. The phase space beam profile is rotated by y ¼ oL/vz on the xx-plane and cropped into the
rectangle-like hatched area in Fig. 2(c), assuming that the inner surface of the magnet aperture is perfectly absorptive. The focusing geometry dictates that the phase space beam profile is rotated across the x-axis. The scattered neutrons are separated from the transmitted neutrons, if Dx42x0/y1, which corresponds to qmin;F ¼
mo 2x0 : _ y1
(9)
ARTICLE IN PRESS H.M. Shimizu et al. / Physica B 356 (2005) 121–125
124
10 -1
10 -3 SF
10
10 -4 SP
qmin,P
-3
10 -5
qmin,F 10 -4 10 -5
Sxξ [m2]
qmin [Å-1]
10 -2
10 -6
1
2
3
4 5
10
10 -7
2x0 [mm]
Fig. 3. Values of qmin and Sxx of P-SANS and F-SANS for L1 ¼ L2 ¼ 10 m as functions of 2x0. We used the values of 2a ¼ 30 mm and G ¼ 23,500 T m2.
under development. At each pulse, the field gradient is expected to go from 0 to 7000 T m2 with a controlled pulse shape [5]. The tunable range of neutron wavelength is shown in Fig. 5(a) as a function as the maximum value of the field gradient Gmax for L1 ¼ 10 m and L ¼ 2 m. The Gmax ¼ 7000 T m2 enables the tunable range of lX6 A˚. The tunable range can be extended to shorter wavelength with sacrificing the reach to longer wavelength by adding a pair of permanent sextupole magnets at both ends of the pulse magnet, as shown in Fig. 4. The grey regions in Figs. 5(b) and (c) correspond to the tunable range with the permanent magnet length L’ ¼ 0.1 and 0.3 m, respectively, and assuming G ¼ 23,500 T m2 for these magnets. In the case of L’ ¼ 0.3 m, the tunable range can be extended to 4.4 A˚plp6.2 A˚.
Putting L1 ¼ L2, the focusing condition is given as tan y ¼
2y1 y21 1
5. Discussion (10)
and the phase space area of the beam delivered to the sample is approximately given as S xx;F ¼ 4ax0 :
(11)
Here we visualize the results using realistic numbers. We take that vz ¼ 609 m s1 (l ¼ 6.5 A˚), L1 ¼ L2 ¼ 10 m, and G ¼ 23,500 T m2 (o ¼ 368 rad s1) and 2a ¼ 30 mm which are available with an extended Halbach-type permanent sextupole magnet [4]. This results in y1 ¼ 6.0 and y ¼ 0.33 (L ¼ 0.54 m). Taking 2x0 ¼ 1 mm, we obtain qmin,P ¼ 1.8 103 A˚1 and qmin,F ¼ 9.7 105A˚1. qmin can be improved by one order of magnitude by employing the focusing geometry. The values of qmin and Sxx are shown as functions of 2x0 in Fig. 3.
4. Magnetic focusing small angle neutron scattering for pulsed neutron sources The magnetic F-SANS can be used at pulsed neutron sources by employing a pulsed magnetic lense. A 2-m-long pulsed sextupole magnet is
The performance of F-SANS can be further improved by delivering more neutrons into the aperture of the incident pinhole collimator of FSANS. The focusing to the aperture (primary focusing) is not required to have as high a quality as the focusing to the detector (main focusing). We therefore can employ focusing mirrors and compound refractive lenses for the ‘primary focusing’ device. For the best efficacy of F-SANS, we should then consider the integrated optimization of the beam transport from the moderator, ‘primary focusing’, ‘main focusing’, detector and data analysis algorithm. The magnetic F-SANS provides polarized neutrons at the exit of the magnet. However, the net polarization of the beam is zero since the neutrons are polarized along the local magnetic field. A focused polarized neutron beam, however, can be obtained by adiabatically connecting the sextupole field to a solenoid field between the sextupole magnet and the sample. The sextupole magnet may also be used downstream from the sample as an energy analyzer. The kinetic energy of individual neutrons from a pointlike source can be determined by measuring the
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Fig. 4. Schematic view of a symmetric arrangement of FSANS with a pulse magnet in combination with a pair of permanent magnets.
scattering and/or reflection from the inner surface should therefore be suppressed. If the beam path is evacuated and the magnet is designed to also compensate the neutron trajectory shift due to gravity, beyond y ¼ p, the neutrons confined along the beam path in the magnet will not be intercepted and scattered by any materials except for the residual gas. Therefore, neutrons can be transported without loss according to precisely calculated kinematics through electromagnetic interactions.
Acknowledgements
Fig. 5. Tunable wavelength regions as functions of the maximum field gradient Gmax for the cases (a) L’ ¼ 0 (no permanent magnets) (b) L’ ¼ 0.1 m and (c) L’ ¼ 0.3 m. We assume that L ¼ 10 m and the field gradient of the permanent magnets are G ¼ 23500 T m2.
focal length using a linear position sensitive detector placed on the sextupole magnet axis [6]. This method can be applied in inelastic neutron scattering provided the sample is sufficiently small. Furthermore, the sextupole magnet can be used as a magnetic neutron guide. The beam shape in the phase space is confined within a circle on the xx-plane and yZ-plane after propagating yXp, which corresponds to zX(34 A˚/l) m when G ¼ 23,500 T m2. The diffuse background from
Authors thank NEOMAX Co. Ltd. for their technical improvements in the fabrication of permanent sextupole magnets. This study was carried out as a part of the NOP project under the support of the Special Coordination Funds for Promoting the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government.
References [1] [2] [3] [4]
B. Alefeld, et al., Physica B 234–236 (1997) 1052. S.-M. Choi, et al., J. Appl. Cryst. 36 (2003) 795. H.M. Shimizu, et al., Nucl. Instrum. Methods A 529 (2004) 5. T. Oku, et al., Proceedings of the International Collaboration on Advanced Neutron Sources-XVI, 2003, p. 355. [5] J. Suzuki, et al., Proceedings of the International Collaboration on Advanced Neutron Sources-XVI, 2003, p. 427. [6] T. Oku, H.M. Shimizu, Physica B 276–278 (2000) 112.
Physica B 356 (2005) 121–125 www.elsevier.com/locate/physb
Development of polarized neutron optics Hirohiko M. Shimizua,b,, Takayuki Okub, Jun-ichi Suzukib, Michihiro Furusakac,b, Yoshiaki Kiyanagid a RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Japan Atomic Energy Research Institute, 2-4 Tokai, Ibaraki, Japan c KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan d Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
b
Abstract Neutron magnetic optics is a spin-selective optics and suitable for precise focusing since it is free from interactions with materials. We discuss its applications to measure the small angle neutron scattering in low-q regions. r 2004 Elsevier B.V. All rights reserved. PACS: 03.75.Be; 61.12 Keywords: Magnetic neutron lens; Sextupole magnet; Cold neutron; Beam focusing
1. Introduction Improvement of neutron optics enables us to increase the utilization efficacy of neutrons with limited beam intensity. However, neutron-focusing devices increase the neutron beam intensity together with the beam divergence at the focal point. Thus their applications are limited to a class of experiments, which do not require high accuracy
Corresponding author. RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan. Tel.: +81 48 462 1111; fax: +81 48 467 9721. E-mail address: [email protected] (H.M. Shimizu).
of momentum transfer (q-resolution), as long as the beam is focused at the sample position. Neutron beam is focused at the detector position in the focusing small angle neutron scattering (FSANS) to achieve a good q-resolution using the converging beam. The FSANS has been developed using Kumakov lens, toroidal mirror, compound refractive lens and magnetic lens [1–3]. The magnetic optics is based on electromagnetic interaction, which is precisely calculable and free from interactions with materials. It also delivers polarized neutrons to the sample and enables the measurement of magnetic scattering. In this report, we describe the FSANS with sextupole magnetic focusing lens and discuss further applications of magnetic optics.
0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.10.061
ARTICLE IN PRESS 122
H.M. Shimizu et al. / Physica B 356 (2005) 121–125
2. Magnetic optics We consider the neutron motion in a sextupole magnetic field, which can be written as |B| ¼ (G/ 2)(x2+y2) taking the beam axis as the z-axis. The equation of motion can be written as ! ! x d2 x 2 ¼ o ; (1) y dt2 y where o2 ¼ aG and a ¼ |m/m| ¼ 5.77 T m2 as long as the magnetic field is sufficiently strong so that the angular velocity of Larmor precession is sufficiently larger than the angular velocity of the field rotation in the neutron rest frame. We assume that this equation of motion is always valid in this paper. The signs correspond to the cases where neutron polarization is parallel and antiparallel to the local magnetic field direction, respectively. The solution of Eq. (1) is given as 8 ! cos y sin y > > > > ! > > < sin y cos y x y ! ¼ > cosh y sinh y x Z > > > > > : sinh y cosh y ! x0 y 0 parallel case; x0 Z0 ð2Þ ! x0 y0 antiparallel case; x0 Z0 where y ¼ ot, x ¼ dx/dy, Z ¼ dy/dy and subscripts 0 indicate the value at y ¼ 0. x and Z represent the beam divergences normalized by o. A neutron beam is rotated by y on the xx-plane and yZ-plane for the polarization parallel case, while it is expanded by ey along (1,1) direction and compressed by ey along (1,1) direction for polarization anti-parallel case. In the same notation, the propagation without magnetic interaction is described as ! ! x0 y0 x y 1 y : (3) ¼ x0 Z0 x Z 0 1 These transformations conserve the phase space areas on xx- and yZ-planes, therefore satisfy the
Liouville’s theorem. We do not consider the gravitational interaction in this paper.
3. Magnetic focusing small angle neutron scattering The conventional small angle neutron scattering has a pair of pinhole collimators at z ¼ 0 and z ¼ L1 as shown in Fig. 1. We refer to the geometry as the pinhole small angle neutron scattering (P-SANS). For simplicity, we consider the one-dimensional case with a monochromatic neutron beam with the velocity vz. The incident beam is defined by the pinhole collimator at z ¼ 0 with a total width of 2x0. The beam profile in the phase space is transformed by ! x ! xðbÞ ðaÞ 1 y1 ¼ ; (4) xðbÞ xðaÞ 0 1 where y1 ¼ L1/vz. It is then shaped into the hatched parallelogram in Fig. 1(b) by the second collimator before the sample. Some neutrons are scattered by the sample with Dx ¼ _q=mo; where q is the momentum transfer, and propagates to the detector according to ! x ! xðdÞ ðcÞ 1 y2 ¼ ; (5) xðdÞ xðcÞ 0 1 where y2 ¼ L2/vz. The detector measures the neutron distribution on the x-axis. The scattered neutrons are completely separated from the transmitted neutrons, if the x-coordinate of the point A is larger than that of the point B in Fig. 1(d), which corresponds to _q y2 X2x0 1 þ 2 Dx ¼ : (6) mo y1 We refer to the minimum value of q, which satisfies the inequality (6), as qmin. We consider the case of L1 ¼ L2 (y1 ¼ y2) for simplicity. In this case, the qmin of P-SANS is obtained as qmin;P ¼
mo 6x0 ; _
(7)
The area of the parallelogram is proportional to the total number of neutrons delivered to the
ARTICLE IN PRESS H.M. Shimizu et al. / Physica B 356 (2005) 121–125
123
Fig. 1. Illustration of the geometry of P-SANS together with the beam shapes on the xx-plane (a) at the first pinhole collimator; (b) at the second pinhole collimator; (c) at the sample and (d) on the detector.
Fig. 2. Illustration of the geometry of FSANS together with the beam shapes on xx-plane (a) at the pinhole collimator; (b) at the entrance of the sextupole magnet; (c) at the exit of the sextupole magnet; (d) in the sample and (e) on the detector.
sample, which is given as S xx;P ¼
4x20 : y1
(8)
In the same manner, we describe the magnetic F-SANS in Fig. 2. The neutrons collimated by the first pinhole collimator propagate to the entrance of the sextupole magnet as shown in Fig. 2(b). The sextupole magnet has an aperture diameter 2a and length L. The phase space beam profile is rotated by y ¼ oL/vz on the xx-plane and cropped into the
rectangle-like hatched area in Fig. 2(c), assuming that the inner surface of the magnet aperture is perfectly absorptive. The focusing geometry dictates that the phase space beam profile is rotated across the x-axis. The scattered neutrons are separated from the transmitted neutrons, if Dx42x0/y1, which corresponds to qmin;F ¼
mo 2x0 : _ y1
(9)
ARTICLE IN PRESS H.M. Shimizu et al. / Physica B 356 (2005) 121–125
124
10 -1
10 -3 SF
10
10 -4 SP
qmin,P
-3
10 -5
qmin,F 10 -4 10 -5
Sxξ [m2]
qmin [Å-1]
10 -2
10 -6
1
2
3
4 5
10
10 -7
2x0 [mm]
Fig. 3. Values of qmin and Sxx of P-SANS and F-SANS for L1 ¼ L2 ¼ 10 m as functions of 2x0. We used the values of 2a ¼ 30 mm and G ¼ 23,500 T m2.
under development. At each pulse, the field gradient is expected to go from 0 to 7000 T m2 with a controlled pulse shape [5]. The tunable range of neutron wavelength is shown in Fig. 5(a) as a function as the maximum value of the field gradient Gmax for L1 ¼ 10 m and L ¼ 2 m. The Gmax ¼ 7000 T m2 enables the tunable range of lX6 A˚. The tunable range can be extended to shorter wavelength with sacrificing the reach to longer wavelength by adding a pair of permanent sextupole magnets at both ends of the pulse magnet, as shown in Fig. 4. The grey regions in Figs. 5(b) and (c) correspond to the tunable range with the permanent magnet length L’ ¼ 0.1 and 0.3 m, respectively, and assuming G ¼ 23,500 T m2 for these magnets. In the case of L’ ¼ 0.3 m, the tunable range can be extended to 4.4 A˚plp6.2 A˚.
Putting L1 ¼ L2, the focusing condition is given as tan y ¼
2y1 y21 1
5. Discussion (10)
and the phase space area of the beam delivered to the sample is approximately given as S xx;F ¼ 4ax0 :
(11)
Here we visualize the results using realistic numbers. We take that vz ¼ 609 m s1 (l ¼ 6.5 A˚), L1 ¼ L2 ¼ 10 m, and G ¼ 23,500 T m2 (o ¼ 368 rad s1) and 2a ¼ 30 mm which are available with an extended Halbach-type permanent sextupole magnet [4]. This results in y1 ¼ 6.0 and y ¼ 0.33 (L ¼ 0.54 m). Taking 2x0 ¼ 1 mm, we obtain qmin,P ¼ 1.8 103 A˚1 and qmin,F ¼ 9.7 105A˚1. qmin can be improved by one order of magnitude by employing the focusing geometry. The values of qmin and Sxx are shown as functions of 2x0 in Fig. 3.
4. Magnetic focusing small angle neutron scattering for pulsed neutron sources The magnetic F-SANS can be used at pulsed neutron sources by employing a pulsed magnetic lense. A 2-m-long pulsed sextupole magnet is
The performance of F-SANS can be further improved by delivering more neutrons into the aperture of the incident pinhole collimator of FSANS. The focusing to the aperture (primary focusing) is not required to have as high a quality as the focusing to the detector (main focusing). We therefore can employ focusing mirrors and compound refractive lenses for the ‘primary focusing’ device. For the best efficacy of F-SANS, we should then consider the integrated optimization of the beam transport from the moderator, ‘primary focusing’, ‘main focusing’, detector and data analysis algorithm. The magnetic F-SANS provides polarized neutrons at the exit of the magnet. However, the net polarization of the beam is zero since the neutrons are polarized along the local magnetic field. A focused polarized neutron beam, however, can be obtained by adiabatically connecting the sextupole field to a solenoid field between the sextupole magnet and the sample. The sextupole magnet may also be used downstream from the sample as an energy analyzer. The kinetic energy of individual neutrons from a pointlike source can be determined by measuring the
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125
Fig. 4. Schematic view of a symmetric arrangement of FSANS with a pulse magnet in combination with a pair of permanent magnets.
scattering and/or reflection from the inner surface should therefore be suppressed. If the beam path is evacuated and the magnet is designed to also compensate the neutron trajectory shift due to gravity, beyond y ¼ p, the neutrons confined along the beam path in the magnet will not be intercepted and scattered by any materials except for the residual gas. Therefore, neutrons can be transported without loss according to precisely calculated kinematics through electromagnetic interactions.
Acknowledgements
Fig. 5. Tunable wavelength regions as functions of the maximum field gradient Gmax for the cases (a) L’ ¼ 0 (no permanent magnets) (b) L’ ¼ 0.1 m and (c) L’ ¼ 0.3 m. We assume that L ¼ 10 m and the field gradient of the permanent magnets are G ¼ 23500 T m2.
focal length using a linear position sensitive detector placed on the sextupole magnet axis [6]. This method can be applied in inelastic neutron scattering provided the sample is sufficiently small. Furthermore, the sextupole magnet can be used as a magnetic neutron guide. The beam shape in the phase space is confined within a circle on the xx-plane and yZ-plane after propagating yXp, which corresponds to zX(34 A˚/l) m when G ¼ 23,500 T m2. The diffuse background from
Authors thank NEOMAX Co. Ltd. for their technical improvements in the fabrication of permanent sextupole magnets. This study was carried out as a part of the NOP project under the support of the Special Coordination Funds for Promoting the Ministry of Education, Culture, Sports, Science and Technology of the Japanese Government.
References [1] [2] [3] [4]
B. Alefeld, et al., Physica B 234–236 (1997) 1052. S.-M. Choi, et al., J. Appl. Cryst. 36 (2003) 795. H.M. Shimizu, et al., Nucl. Instrum. Methods A 529 (2004) 5. T. Oku, et al., Proceedings of the International Collaboration on Advanced Neutron Sources-XVI, 2003, p. 355. [5] J. Suzuki, et al., Proceedings of the International Collaboration on Advanced Neutron Sources-XVI, 2003, p. 427. [6] T. Oku, H.M. Shimizu, Physica B 276–278 (2000) 112.