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Diffraction enhancement and new way to measure neutron electric charge and the ratio of inertial to gravitational mass
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 593 (2008) 505– 509
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Diffraction enhancement and new way to measure neutron electric charge and the ratio of inertial to gravitational mass V.V. Fedorov, I.A. Kuznetsov, E.G. Lapin, S.Yu. Semenikhin, V.V. Voronin Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia
a r t i c l e in fo
abstract
Article history: Received 21 January 2008 Received in revised form 24 March 2008 Accepted 27 May 2008 Available online 23 June 2008
The essential magnification of an external affect on neutron diffracting by Laue for the Bragg angles yB close to right one has been experimentally observed. The factor of diffraction enhancement for the neutron trajectory ‘‘curvature’’ due to external field is proportional to tan2 yB and can reach magnitude 108 2109 in comparison with a free neutron. The new method to measure a neutron electric charge and the ratio of inertial to gravitational masses has been proposed. The estimations have shown a feasibility to improve the accuracy of a neutron electric charge measurement by about two orders of magnitude in comparison with the current value and measure the ratio of inertial to gravitational masses with the accuracy about sðmi =mG Þ106 . & 2008 Elsevier B.V. All rights reserved.
Keywords: Perfect crystal Neutron Diffraction Electric charge Weak equivalence principle
1. Introduction The propagation of neutrons or X-rays in an elastically deformed crystal essentially differs from that in an undeformed one [1–3]. There is a well-known effect of diffraction enhancement when a small variation of the incident beam direction leads to a considerable deflection of a neutron trajectory inside a crystal [4]. The neutron in the crystal changes the momentum direction by the angle of 2yB (by several tens degrees) while the incident neutron beam deflects by the Bragg width (within a few arc seconds) [4]. There are some proposals to utilize this effect for measuring a neutron charge [5] and for the investigation of gravitational properties of a neutron falling in the Earth field [6]. The influence of gravity on a neutron diffraction in a deformed single crystal was firstly observed in Ref. [3]. The neutron gravity was investigated before by using the neutron interferometer [7,8] and diffraction grating [9]. Recently an essential increase of the neutron stay time in a crystal for the Bragg angles close to p=2 has been observed [10]. Any effect connected with an external affect on a neutron should depend on the interaction time; so we expect a considerable increase of sensitivity to a small external affect for Bragg angles close to the right one.
Corresponding author.
E-mail address: [email protected] (V.V. Voronin). 0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.05.048
It is known that an external field affecting the diffracting neutron is equivalent to some kind of a crystal elastic deformation, see Refs. [3,6]. So, at the first stage, we have carried out a simple test of a small elastic crystal deformation influence upon a neutron Laue diffraction for the Bragg angles close to p=2 [11].
2. Neutron trajectories in the crystal Here we will consider the symmetrical Laue diffraction scheme in a transparent crystal with the system of crystallographic planes described by the reciprocal lattice vector g normal to the planes (see Fig. 1), g ¼ 2p=d, d is the interplanar distance. In this case, the neutron wave function in a crystal will be a superposition of two ð1Þ ð2Þ Bloch waves c and c corresponding to two branches of the dispersion surface [12]. Any deformation of a crystal means that the value and/or direction of vector g are different for the crystal different points, i.e. g depends on the spatial coordinates Y and Z. Neutron Laue diffraction in a weakly deformed crystal can be described by the so-called ‘‘Kato forces’’ [13], which are directed along the reciprocal lattice vector g and their values are determined by the value of crystal deformation [3]: f k ðy; zÞ ¼
k0 q 1 q þ aðy; zÞ 4 cos yB qz c qy
(1)
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crystal in a certain field of the force affecting the neutron along reciprocal lattice vector g, we will have the same result as for the deformed crystal. The external field affect on a diffracting neutron was considered in Ref. [6] and influence of gravity on a neutron diffraction in an elastically bent crystal was experimentally observed [3]. It is easy to show that the external force Fext affecting the neutron along the vector g (Z-axis, see Fig. 1) is equivalent to the gradient of interplanar distance with the value
xf ¼
F ext 2En
(6)
where En is the neutron energy. Therefore the neutron trajectory equation in the crystal in the presence of external field will have the form
q2 z c2 g F ext ¼ . 2 2m0 2En qy
(7)
Let us compare this equation for the ‘‘Kato trajectory’’ with that for a usual trajectory of a neutron under the same external field in free space. The last one is described by usual Newtonian equation which has the form
q2 z F ext ¼ . qy2 2En
Fig. 1. Symmetrical Laue diffraction scheme for finite perfect crystal. jcð1Þ and jcð2Þ are the neutron fluxes (‘‘Kato trajectories’’) for two Bloch waves.
As it follows from Eqs. (7) and (8) the ‘‘curvature’’ of the diffracting neutron trajectory in the crystal is magnified by the factor Ke ¼
where k0 is the neutron wave vector in the crystal, yB is the Bragg angle,
aðy; zÞ ¼
g 2 þ 2ðk0 gÞ 2
k0
(2)
(3)
where c tan yB and m0 2F g d=V is the so-called ‘‘mass Kato’’, F g is the neutron structure amplitude, V is the unit cell volume. The sign in Eq. (3) corresponds to different Bloch waves. Consider the most simple case of a crystal deformation. So let the gradient of the interplanar distance be constant and d ¼ d0 ð1 þ xzÞ, see Fig. 1. For this case the ‘‘Kato force’’ will be equal to f k ðy; zÞ ¼
1 2cg x
(9)
This factor depends on the Bragg angle as c2 tan2 yB . The numerical calculation of factor K e for (11 0) and (2 0 0) quartz planes gives (10)
0 0Þ ¼ 1:4 106 c2 . K ð2 e
(11)
So, for the Bragg angle yB ¼ 87 ðc ¼ 20Þ, this factor will be equal to 1 0Þ ¼ 0:7 108 K ð1 e
(12)
0 0Þ ¼ 0:6 109 . K ð2 e
(13)
Therefore, the 10 cm crystal is for the trajectory curvature equivalent to 1 km of free flight. The diffraction enhancement of the angular deflection of a neutron trajectory inside a crystal is well known, see for instance Ref. [4], but we have to note that such an effect can be considerably magnified by an additional gain factor proportional to tan2 yB for the Bragg angles close to p=2.
3. Experiment
(4)
and the corresponding equation of neutron trajectories will be
q2 z c2 g x ¼ . 2m0 qy2
c2 g . 2m0
1 0Þ K ð1 ¼ 1:8 105 c2 e
is the parameter of deviation from the exact Bragg condition. Eq. (1) is valid for crystal deformations small as compared with the diffraction Bragg width. The neutron ‘‘Kato trajectories’’ for two kinds of Bloch waves in the crystal (describing a behavior of the neutron density fluxes jcð1Þ and jcð2Þ ) are determined by the equation
q2 z c ¼ f ðy; zÞ m0 k qy2
(8)
(5)
The right part of Eq. (5) is proportional to the square of c ¼ tan yB . So for the Bragg angles yB ð84288Þ influence of deformation can be intensified by a factor 10021000 as compared with the Bragg angle 45 . As it follows from Eqs. (1) and (2) the ‘‘Kato force’’ arises due to the dependence of the deviation parameter a on the spatial coordinates Z and Y. Therefore if we put the undeformed perfect
The experiment has been carried out at the cold neutron beam ˚ of WWR-M reactor (PNPI, Gatchina). The beam size was (l 5 A) about 50 10 cm2 , the angular divergence was about 0:2 . The scheme of the experiment is shown in Fig. 2. The neutron ˚ of quartz crystals with diffraction of the (11 0) plane (d ¼ 2:45 A) the sizes 140 35 and 140 140 mm2 has been studied. We have been using high perfect quartz crystals with the mosaic o1 s of arc and homogeneity of interplanar distance less than the Bragg width of the reflex (Dd=do105 ). We deformed the crystal by producing a temperature gradient. The coefficient of the thermal expansion for the quartz crystal along the direction perpendicular to the (11 0) plane is
ARTICLE IN PRESS V.V. Fedorov et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 505–509
507
1.6
N
W =5 10-7/cm
1.2
Fig. 2. Scheme of the experiment. (1), (2), and (3) are the positions of the temperature probes. g is the direction of gravity.
0.8
-0.4
0.0 -6
0.4 /cm
Fig. 4. Example of the dependence of diffraction beam intensity on parameter x. Bragg angle was equal to 76 , crystal thickness L ¼ 14 cm and height h ¼ 14 cm.
Fig. 3. Example of the neutron trajectories in the deformed quartz crystal for the yB ¼ 84 and x ¼ 6 108 =cm.
wð1 1 0Þ ¼ 1:3 105 =K. Therefore, the parameter of deformation will be
x ¼ wð1 1 0Þ DT=h ¼ 0:93 106 DT
(14)
where DT is the crystal temperature difference between the points (1) and (3), see Fig. 2, h ¼ 14 cm is the crystal size along Z-axis, see Fig. 1. The Peltier elements were used to control the crystal temperature gradient. The PT-100 probes in combination with the KETHLEY multimeter have been used to measure the
temperatures of crystal. The accuracy to measure a temperature variation was 0:002 K and to measure the absolute value of temperature difference in the different point of the crystal was 0:01 K. The crystal deformation increase will result in decreasing the diffraction beam intensity because some part of neutrons will not reach the exit surface of the crystal due to strong bending of the Kato trajectories. The neutron trajectories calculated for Bragg angle equal to 84 and parameter of deformation x ¼ 6 108 =cm, which corresponds to DT ¼ 0:07 K, are shown in Fig. 3. The trajectories are shown in the scale, corresponding to length of crystal L ¼ 3:5 cm (along Y-axis) and height h ¼ 14 cm (along Z-axis). Each couple of the trajectories symmetrical relative to Y-axis in Fig. 3 corresponds to the definite direction of the incident beam. The different couples correspond to slightly different incident beam directions within the angle interval which is much less than the Bragg width. One can see that even for such a small crystal deformation a splitting Ds between the different Bloch wave trajectories can reach a few centimeters. When this split becomes equal to the height of crystal h, none of the neutrons can reach the exit crystal surface, because their trajectories as well as neutrons themselves go out from the crystal through its lateral faces. An example of the diffraction beam intensity dependence on the deformation parameter x is shown in Fig. 4. One can see that the width of this line W x is 5 107 =cm. Using Eq. (6) we can calculate the value of the line width in units of external force F ext and get W F ¼ 3 109 eV=cm. It should be noted that the gravity interaction of a neutron with the Earth is only a few times less (109 eV=cm). So, even such a simple experiment has the resolution at the level of neutron gravitational interaction with the Earth. The measured dependence of W x on tan yB for crystal sizes L ¼ 3:5 cm and h ¼ 14 cm is shown in Fig. 5. One can calculate width W x using the equation of neutron trajectory (5) Wx ¼
h 8m0 d L2 c2
p
.
(15)
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Fig. 5. Dependence of W x on tangent of Bragg angle. Crystal sizes L ¼ 3:5 cm and h ¼ 14 cm.
Fig. 7. The effect of two crystal diffraction focusing. c different Bloch waves exited in the crystals.
Fext, 10-11eV/cm -3.5
-1.7
0.0
1.7 1.5
0.010
1.0
0.005
0.5
0.000
0.0
-0.005
-0.5 -3
-2
-1
0
1
Fig. 6. Dependence of the line position, see Fig. 4, on an angular inclination of the crystal j. The corresponding value of F ext is shown on the top axis. The line positions in DT and x units are connected by Eq. (14).
The solid curve in Fig. 5 is the theoretical dependence (15). The discrepancy between the calculated and experimental results for the tan yB 410 may be connected with the reflection of neutrons on the lateral faces of a crystal. The present experimental scheme allows to incline the plane of neutron diffraction by some angle j to have the non-zero projection of gravity to the crystal reciprocal lattice vector, see Fig. 2. So, for a small angle j the effective force acting on a neutron has the form F ext ¼ mg sinðjÞ mg j
are the trajectories of
4. Discussion
10-8/cm
Tc,K
-5.2 0.015
ði;jÞ
The observed effects give us a chance to build up a device with unprecedented sensitivity to an external force affected the neutron. To make the setup more sensitive to a small curvature of neutron trajectories we have proposed to use the effect of diffraction focusing on the two crystal Laue diffraction scheme with the crystals by the same thickness [14,15]. In such a scheme half of neutron intensity will be focused at the exit of the second crystal; in others words, at the exit of the second crystal we will see the image of the slit placed at the entrance face of the first crystal, see Fig. 7. Its own spatial width of the focus is equal to [14] xw ¼
2p
¼
1 2m0
(17)
where L ¼ p=ðm0 tanðyB ÞÞ is the extinction length. We should note that the value xw does not depend on a Bragg angle. The calculated value of xw lies in the region 10250 mm for different crystallographic planes. The presence of the external field will move this focus and, if we place the slits both at the first crystal entrance and at the second crystal exit, we will see the intensity variation correlated with the external force value. More detailed considerations have shown that it is necessary to apply some small constant field to move the focus position by half of the slit width to have the sensitivity maximum. As it follows from Eq. (7) the focus shift value due to applying the external field is equal to
(16)
DF ¼ where m is a neutron mass, g is the gravity acceleration. It is clear from Eq. (6) that the applied field can be compensated by the crystal deformation, so we should observe the dependence of the line position, see Fig. 4, on the angle j. The experimentally measured dependence is shown in Fig. 6. The solid line is a result of the experimental data fitting. One can see that the sensitivity to the external field, see the top axis, is already a few 1011 eV=cm that corresponds to a few percent of the neutron gravity. Unfortunately, we could not increase the effect by using a larger angle j of the crystal inclination in the current experiment layout.
L tanðyB Þ
pc2 L2 F ext . 2m0 dEn
(18)
Therefore, the external field increase should result in decreasing the registered neutron intensity as it was for the single crystal scheme, see Fig. 4. The width of the line W F will be determined by the size of slits Ds WF ¼
2m0 dEn
pc2 L2
Ds .
(19)
The calculation have shown that for quartz (11 0) and silicon (2 2 0) crystallographic planes, the value of W F can be 1013 eV=cm for Ds ¼ 0:01 cm, and the sensitivity to the external
ARTICLE IN PRESS V.V. Fedorov et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 505–509
˚ force for the available cold neutron beam flux 109 n=ðcm2 s AÞ and crystal size 10 10 10 cm3 can reach
sðF ext Þ5 1019 eV=cm
(20)
per 100 day of statistic accumulation. This device can be applied, for example, to a precise measurement of neutron electric charge qn . If we place a diffracting neutron into the electric field E, the presence of a non-zero electric charge will lead to the force acting on the neutron F q ¼ Eqn . For the value of electric field E ¼ 50 kV=cm, the sensitivity to qn will be
sðqn ÞsðF ext Þ=E 1023 e
(21)
where e is the electron charge. Such an accuracy is about by two orders of magnitude better than the current precision for qn [18]. The second application can be connected with the measurement of the ratio of inertial and gravitational neutron masses. Our Earth is moving at the stationary orbit around the Sun; it means that the gravitational force proportional to the gravitational mass is in balance with the centrifugal force proportional to the inertial mass. If it is not the case for neutron, then in the coordinate system connected with the Earth the neutron will see the nonzero force1 F ðGiÞ ¼
ðmi mG ÞGMS R2S
(22)
where mG and mi are the neutron gravitational and inertial masses, G is the gravitational constant, M S is the Sun mass and RS is the distance to the Sun. Moreover, this force will oscillate in the laboratory coordinate system with a one day period due to the Earth spinning motion. The gravity force between the Sun and neutron is F G ¼ GmG M S =R2S ¼ 6 1013 eV=cm, so the accuracy (20) corresponds to the
sðmi =mG Þ106
(23)
that is better two orders more than the current accuracy [17].
5. Conclusions Effect of strong influence of a small crystal deformation on a neutron diffraction beam intensity has been observed for the case of Laue diffraction in a transparent crystal. It has been shown for
1 The idea of this experiment is an analogue to the well-known Eo˜tvo˜s experiment for the equivalence principle checking [16].
509
the first time that the effect considerably increases for the Bragg angles close to p=2 as tan2 yB . The additional gain factor determined by the large Bragg angles can reach 103 . So the joint factor of a diffraction enhancement for the neutron trajectory bending due to the external field can reach 109 (not 105 2106 as for Bragg angles 45 ). The observed effect gives new possibilities for the precise investigation of neutron interaction with the external fields. The estimation has shown that for the specially designed two crystal setup, the resolution to the external force (width of the line, see Fig. 4) can reach 1013 eV=cm and the sensitivity can be better than 1018 eV=cm. The proposed scheme of the experiment can improve the accuracy of a neutron electric charge measurement by about two orders of magnitude and carry out measurement of the ratio of the neutron inertial and gravitational masses at level sðmi =mG Þ106 . The authors would like to thank the personnel of the WWR-M reactor (Gatchina, Russia). This work was supported by RFBR (Grant Nos. 05-02-16241 and 06-02-81011). References [1] Yu.S. Grushko, E.G. Lapin, O.I. Sumbaev, A.V. Tyunis, JETP 47 (1978) 1185. [2] O.I. Sumbaev, E.G. Lapin, JETP 51 (1980) 403. [3] V.L. Alexeev, E.G. Lapin, E.K. Leushkin, V.L. Rumiantsev, O.I. Sumbaev, V.V. Fedorov, JETP 94 (1988) 371. [4] A. Zeilinger, C.G. Shull, M.A. Horne, K.G. Finkelstein, Phys. Rev. Lett. 57 (1986) 3089. [5] Yu. Alexandrov, in: Proceeding of ISINN-XIII, Dubna, 2006, p. 166. [6] S.A. Werner, Phys. Rev. B 21 (1980) 1774. [7] R. Colella, A.W. Overhauser, S.A. Werner, Phys. Rev. Lett. 34 (1975) 1472. [8] J.-L. Staudenmann, S.A. Werner, R. Colella, A.W. Overhauser, Phys. Rev. A. 21 (1980) 1419. [9] A.I. Frank, P. Geltenbort, M. Jentchel, G.V. Kulin, D.V. Kustov, V.G. Nosov, A.I. Strepetov, JETP Lett. 86 (2007) 255. [10] V.V. Voronin, E.G. Lapin, S.Yu. Semenikhin, V.V. Fedorov, JETP Lett. 71 (2000) 76. [11] V.V. Fedorov, I.A. Kuznetsov, E.G. Lapin, S.Yu. Semenikhin, V.V. Voronin, JETP Lett. 85 (1) (2007) 82. [12] H. Rauch, D. Petrachek, in: H. Duchs (Ed.), Neutron Diffraction, Springer, Berlin, 1978, pp. 303–351. [13] N. Kato, J. Phys. Soc. Jpn. 19 (1964) 971. [14] V.I. Indenbom, I.Sh. Slobodetsky, K.G. Truni, ZhETF 66 (3) (1974) 1110. [15] J. Arthur, C.G. Shull, A. Zeilinger, Phys. Rev. B 32 (9) (1985) 5753. [16] R.v. Eotvos, D. Pekar, E. Fekete, Ann. Phys. (Leipzig) 68 (1922) 11. [17] J. Schmiedmayer, Nucl. Instr. and Meth. A 284 (1989) 59. [18] J. Baumann, R. Gahler, J. Kalus, W. Mampe, Phys. Rev. D 37 (1988) 3107.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Diffraction enhancement and new way to measure neutron electric charge and the ratio of inertial to gravitational mass V.V. Fedorov, I.A. Kuznetsov, E.G. Lapin, S.Yu. Semenikhin, V.V. Voronin Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia
a r t i c l e in fo
abstract
Article history: Received 21 January 2008 Received in revised form 24 March 2008 Accepted 27 May 2008 Available online 23 June 2008
The essential magnification of an external affect on neutron diffracting by Laue for the Bragg angles yB close to right one has been experimentally observed. The factor of diffraction enhancement for the neutron trajectory ‘‘curvature’’ due to external field is proportional to tan2 yB and can reach magnitude 108 2109 in comparison with a free neutron. The new method to measure a neutron electric charge and the ratio of inertial to gravitational masses has been proposed. The estimations have shown a feasibility to improve the accuracy of a neutron electric charge measurement by about two orders of magnitude in comparison with the current value and measure the ratio of inertial to gravitational masses with the accuracy about sðmi =mG Þ106 . & 2008 Elsevier B.V. All rights reserved.
Keywords: Perfect crystal Neutron Diffraction Electric charge Weak equivalence principle
1. Introduction The propagation of neutrons or X-rays in an elastically deformed crystal essentially differs from that in an undeformed one [1–3]. There is a well-known effect of diffraction enhancement when a small variation of the incident beam direction leads to a considerable deflection of a neutron trajectory inside a crystal [4]. The neutron in the crystal changes the momentum direction by the angle of 2yB (by several tens degrees) while the incident neutron beam deflects by the Bragg width (within a few arc seconds) [4]. There are some proposals to utilize this effect for measuring a neutron charge [5] and for the investigation of gravitational properties of a neutron falling in the Earth field [6]. The influence of gravity on a neutron diffraction in a deformed single crystal was firstly observed in Ref. [3]. The neutron gravity was investigated before by using the neutron interferometer [7,8] and diffraction grating [9]. Recently an essential increase of the neutron stay time in a crystal for the Bragg angles close to p=2 has been observed [10]. Any effect connected with an external affect on a neutron should depend on the interaction time; so we expect a considerable increase of sensitivity to a small external affect for Bragg angles close to the right one.
Corresponding author.
E-mail address: [email protected] (V.V. Voronin). 0168-9002/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2008.05.048
It is known that an external field affecting the diffracting neutron is equivalent to some kind of a crystal elastic deformation, see Refs. [3,6]. So, at the first stage, we have carried out a simple test of a small elastic crystal deformation influence upon a neutron Laue diffraction for the Bragg angles close to p=2 [11].
2. Neutron trajectories in the crystal Here we will consider the symmetrical Laue diffraction scheme in a transparent crystal with the system of crystallographic planes described by the reciprocal lattice vector g normal to the planes (see Fig. 1), g ¼ 2p=d, d is the interplanar distance. In this case, the neutron wave function in a crystal will be a superposition of two ð1Þ ð2Þ Bloch waves c and c corresponding to two branches of the dispersion surface [12]. Any deformation of a crystal means that the value and/or direction of vector g are different for the crystal different points, i.e. g depends on the spatial coordinates Y and Z. Neutron Laue diffraction in a weakly deformed crystal can be described by the so-called ‘‘Kato forces’’ [13], which are directed along the reciprocal lattice vector g and their values are determined by the value of crystal deformation [3]: f k ðy; zÞ ¼
k0 q 1 q þ aðy; zÞ 4 cos yB qz c qy
(1)
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V.V. Fedorov et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 505–509
crystal in a certain field of the force affecting the neutron along reciprocal lattice vector g, we will have the same result as for the deformed crystal. The external field affect on a diffracting neutron was considered in Ref. [6] and influence of gravity on a neutron diffraction in an elastically bent crystal was experimentally observed [3]. It is easy to show that the external force Fext affecting the neutron along the vector g (Z-axis, see Fig. 1) is equivalent to the gradient of interplanar distance with the value
xf ¼
F ext 2En
(6)
where En is the neutron energy. Therefore the neutron trajectory equation in the crystal in the presence of external field will have the form
q2 z c2 g F ext ¼ . 2 2m0 2En qy
(7)
Let us compare this equation for the ‘‘Kato trajectory’’ with that for a usual trajectory of a neutron under the same external field in free space. The last one is described by usual Newtonian equation which has the form
q2 z F ext ¼ . qy2 2En
Fig. 1. Symmetrical Laue diffraction scheme for finite perfect crystal. jcð1Þ and jcð2Þ are the neutron fluxes (‘‘Kato trajectories’’) for two Bloch waves.
As it follows from Eqs. (7) and (8) the ‘‘curvature’’ of the diffracting neutron trajectory in the crystal is magnified by the factor Ke ¼
where k0 is the neutron wave vector in the crystal, yB is the Bragg angle,
aðy; zÞ ¼
g 2 þ 2ðk0 gÞ 2
k0
(2)
(3)
where c tan yB and m0 2F g d=V is the so-called ‘‘mass Kato’’, F g is the neutron structure amplitude, V is the unit cell volume. The sign in Eq. (3) corresponds to different Bloch waves. Consider the most simple case of a crystal deformation. So let the gradient of the interplanar distance be constant and d ¼ d0 ð1 þ xzÞ, see Fig. 1. For this case the ‘‘Kato force’’ will be equal to f k ðy; zÞ ¼
1 2cg x
(9)
This factor depends on the Bragg angle as c2 tan2 yB . The numerical calculation of factor K e for (11 0) and (2 0 0) quartz planes gives (10)
0 0Þ ¼ 1:4 106 c2 . K ð2 e
(11)
So, for the Bragg angle yB ¼ 87 ðc ¼ 20Þ, this factor will be equal to 1 0Þ ¼ 0:7 108 K ð1 e
(12)
0 0Þ ¼ 0:6 109 . K ð2 e
(13)
Therefore, the 10 cm crystal is for the trajectory curvature equivalent to 1 km of free flight. The diffraction enhancement of the angular deflection of a neutron trajectory inside a crystal is well known, see for instance Ref. [4], but we have to note that such an effect can be considerably magnified by an additional gain factor proportional to tan2 yB for the Bragg angles close to p=2.
3. Experiment
(4)
and the corresponding equation of neutron trajectories will be
q2 z c2 g x ¼ . 2m0 qy2
c2 g . 2m0
1 0Þ K ð1 ¼ 1:8 105 c2 e
is the parameter of deviation from the exact Bragg condition. Eq. (1) is valid for crystal deformations small as compared with the diffraction Bragg width. The neutron ‘‘Kato trajectories’’ for two kinds of Bloch waves in the crystal (describing a behavior of the neutron density fluxes jcð1Þ and jcð2Þ ) are determined by the equation
q2 z c ¼ f ðy; zÞ m0 k qy2
(8)
(5)
The right part of Eq. (5) is proportional to the square of c ¼ tan yB . So for the Bragg angles yB ð84288Þ influence of deformation can be intensified by a factor 10021000 as compared with the Bragg angle 45 . As it follows from Eqs. (1) and (2) the ‘‘Kato force’’ arises due to the dependence of the deviation parameter a on the spatial coordinates Z and Y. Therefore if we put the undeformed perfect
The experiment has been carried out at the cold neutron beam ˚ of WWR-M reactor (PNPI, Gatchina). The beam size was (l 5 A) about 50 10 cm2 , the angular divergence was about 0:2 . The scheme of the experiment is shown in Fig. 2. The neutron ˚ of quartz crystals with diffraction of the (11 0) plane (d ¼ 2:45 A) the sizes 140 35 and 140 140 mm2 has been studied. We have been using high perfect quartz crystals with the mosaic o1 s of arc and homogeneity of interplanar distance less than the Bragg width of the reflex (Dd=do105 ). We deformed the crystal by producing a temperature gradient. The coefficient of the thermal expansion for the quartz crystal along the direction perpendicular to the (11 0) plane is
ARTICLE IN PRESS V.V. Fedorov et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 505–509
507
1.6
N
W =5 10-7/cm
1.2
Fig. 2. Scheme of the experiment. (1), (2), and (3) are the positions of the temperature probes. g is the direction of gravity.
0.8
-0.4
0.0 -6
0.4 /cm
Fig. 4. Example of the dependence of diffraction beam intensity on parameter x. Bragg angle was equal to 76 , crystal thickness L ¼ 14 cm and height h ¼ 14 cm.
Fig. 3. Example of the neutron trajectories in the deformed quartz crystal for the yB ¼ 84 and x ¼ 6 108 =cm.
wð1 1 0Þ ¼ 1:3 105 =K. Therefore, the parameter of deformation will be
x ¼ wð1 1 0Þ DT=h ¼ 0:93 106 DT
(14)
where DT is the crystal temperature difference between the points (1) and (3), see Fig. 2, h ¼ 14 cm is the crystal size along Z-axis, see Fig. 1. The Peltier elements were used to control the crystal temperature gradient. The PT-100 probes in combination with the KETHLEY multimeter have been used to measure the
temperatures of crystal. The accuracy to measure a temperature variation was 0:002 K and to measure the absolute value of temperature difference in the different point of the crystal was 0:01 K. The crystal deformation increase will result in decreasing the diffraction beam intensity because some part of neutrons will not reach the exit surface of the crystal due to strong bending of the Kato trajectories. The neutron trajectories calculated for Bragg angle equal to 84 and parameter of deformation x ¼ 6 108 =cm, which corresponds to DT ¼ 0:07 K, are shown in Fig. 3. The trajectories are shown in the scale, corresponding to length of crystal L ¼ 3:5 cm (along Y-axis) and height h ¼ 14 cm (along Z-axis). Each couple of the trajectories symmetrical relative to Y-axis in Fig. 3 corresponds to the definite direction of the incident beam. The different couples correspond to slightly different incident beam directions within the angle interval which is much less than the Bragg width. One can see that even for such a small crystal deformation a splitting Ds between the different Bloch wave trajectories can reach a few centimeters. When this split becomes equal to the height of crystal h, none of the neutrons can reach the exit crystal surface, because their trajectories as well as neutrons themselves go out from the crystal through its lateral faces. An example of the diffraction beam intensity dependence on the deformation parameter x is shown in Fig. 4. One can see that the width of this line W x is 5 107 =cm. Using Eq. (6) we can calculate the value of the line width in units of external force F ext and get W F ¼ 3 109 eV=cm. It should be noted that the gravity interaction of a neutron with the Earth is only a few times less (109 eV=cm). So, even such a simple experiment has the resolution at the level of neutron gravitational interaction with the Earth. The measured dependence of W x on tan yB for crystal sizes L ¼ 3:5 cm and h ¼ 14 cm is shown in Fig. 5. One can calculate width W x using the equation of neutron trajectory (5) Wx ¼
h 8m0 d L2 c2
p
.
(15)
ARTICLE IN PRESS 508
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Fig. 5. Dependence of W x on tangent of Bragg angle. Crystal sizes L ¼ 3:5 cm and h ¼ 14 cm.
Fig. 7. The effect of two crystal diffraction focusing. c different Bloch waves exited in the crystals.
Fext, 10-11eV/cm -3.5
-1.7
0.0
1.7 1.5
0.010
1.0
0.005
0.5
0.000
0.0
-0.005
-0.5 -3
-2
-1
0
1
Fig. 6. Dependence of the line position, see Fig. 4, on an angular inclination of the crystal j. The corresponding value of F ext is shown on the top axis. The line positions in DT and x units are connected by Eq. (14).
The solid curve in Fig. 5 is the theoretical dependence (15). The discrepancy between the calculated and experimental results for the tan yB 410 may be connected with the reflection of neutrons on the lateral faces of a crystal. The present experimental scheme allows to incline the plane of neutron diffraction by some angle j to have the non-zero projection of gravity to the crystal reciprocal lattice vector, see Fig. 2. So, for a small angle j the effective force acting on a neutron has the form F ext ¼ mg sinðjÞ mg j
are the trajectories of
4. Discussion
10-8/cm
Tc,K
-5.2 0.015
ði;jÞ
The observed effects give us a chance to build up a device with unprecedented sensitivity to an external force affected the neutron. To make the setup more sensitive to a small curvature of neutron trajectories we have proposed to use the effect of diffraction focusing on the two crystal Laue diffraction scheme with the crystals by the same thickness [14,15]. In such a scheme half of neutron intensity will be focused at the exit of the second crystal; in others words, at the exit of the second crystal we will see the image of the slit placed at the entrance face of the first crystal, see Fig. 7. Its own spatial width of the focus is equal to [14] xw ¼
2p
¼
1 2m0
(17)
where L ¼ p=ðm0 tanðyB ÞÞ is the extinction length. We should note that the value xw does not depend on a Bragg angle. The calculated value of xw lies in the region 10250 mm for different crystallographic planes. The presence of the external field will move this focus and, if we place the slits both at the first crystal entrance and at the second crystal exit, we will see the intensity variation correlated with the external force value. More detailed considerations have shown that it is necessary to apply some small constant field to move the focus position by half of the slit width to have the sensitivity maximum. As it follows from Eq. (7) the focus shift value due to applying the external field is equal to
(16)
DF ¼ where m is a neutron mass, g is the gravity acceleration. It is clear from Eq. (6) that the applied field can be compensated by the crystal deformation, so we should observe the dependence of the line position, see Fig. 4, on the angle j. The experimentally measured dependence is shown in Fig. 6. The solid line is a result of the experimental data fitting. One can see that the sensitivity to the external field, see the top axis, is already a few 1011 eV=cm that corresponds to a few percent of the neutron gravity. Unfortunately, we could not increase the effect by using a larger angle j of the crystal inclination in the current experiment layout.
L tanðyB Þ
pc2 L2 F ext . 2m0 dEn
(18)
Therefore, the external field increase should result in decreasing the registered neutron intensity as it was for the single crystal scheme, see Fig. 4. The width of the line W F will be determined by the size of slits Ds WF ¼
2m0 dEn
pc2 L2
Ds .
(19)
The calculation have shown that for quartz (11 0) and silicon (2 2 0) crystallographic planes, the value of W F can be 1013 eV=cm for Ds ¼ 0:01 cm, and the sensitivity to the external
ARTICLE IN PRESS V.V. Fedorov et al. / Nuclear Instruments and Methods in Physics Research A 593 (2008) 505–509
˚ force for the available cold neutron beam flux 109 n=ðcm2 s AÞ and crystal size 10 10 10 cm3 can reach
sðF ext Þ5 1019 eV=cm
(20)
per 100 day of statistic accumulation. This device can be applied, for example, to a precise measurement of neutron electric charge qn . If we place a diffracting neutron into the electric field E, the presence of a non-zero electric charge will lead to the force acting on the neutron F q ¼ Eqn . For the value of electric field E ¼ 50 kV=cm, the sensitivity to qn will be
sðqn ÞsðF ext Þ=E 1023 e
(21)
where e is the electron charge. Such an accuracy is about by two orders of magnitude better than the current precision for qn [18]. The second application can be connected with the measurement of the ratio of inertial and gravitational neutron masses. Our Earth is moving at the stationary orbit around the Sun; it means that the gravitational force proportional to the gravitational mass is in balance with the centrifugal force proportional to the inertial mass. If it is not the case for neutron, then in the coordinate system connected with the Earth the neutron will see the nonzero force1 F ðGiÞ ¼
ðmi mG ÞGMS R2S
(22)
where mG and mi are the neutron gravitational and inertial masses, G is the gravitational constant, M S is the Sun mass and RS is the distance to the Sun. Moreover, this force will oscillate in the laboratory coordinate system with a one day period due to the Earth spinning motion. The gravity force between the Sun and neutron is F G ¼ GmG M S =R2S ¼ 6 1013 eV=cm, so the accuracy (20) corresponds to the
sðmi =mG Þ106
(23)
that is better two orders more than the current accuracy [17].
5. Conclusions Effect of strong influence of a small crystal deformation on a neutron diffraction beam intensity has been observed for the case of Laue diffraction in a transparent crystal. It has been shown for
1 The idea of this experiment is an analogue to the well-known Eo˜tvo˜s experiment for the equivalence principle checking [16].
509
the first time that the effect considerably increases for the Bragg angles close to p=2 as tan2 yB . The additional gain factor determined by the large Bragg angles can reach 103 . So the joint factor of a diffraction enhancement for the neutron trajectory bending due to the external field can reach 109 (not 105 2106 as for Bragg angles 45 ). The observed effect gives new possibilities for the precise investigation of neutron interaction with the external fields. The estimation has shown that for the specially designed two crystal setup, the resolution to the external force (width of the line, see Fig. 4) can reach 1013 eV=cm and the sensitivity can be better than 1018 eV=cm. The proposed scheme of the experiment can improve the accuracy of a neutron electric charge measurement by about two orders of magnitude and carry out measurement of the ratio of the neutron inertial and gravitational masses at level sðmi =mG Þ106 . The authors would like to thank the personnel of the WWR-M reactor (Gatchina, Russia). This work was supported by RFBR (Grant Nos. 05-02-16241 and 06-02-81011). References [1] Yu.S. Grushko, E.G. Lapin, O.I. Sumbaev, A.V. Tyunis, JETP 47 (1978) 1185. [2] O.I. Sumbaev, E.G. Lapin, JETP 51 (1980) 403. [3] V.L. Alexeev, E.G. Lapin, E.K. Leushkin, V.L. Rumiantsev, O.I. Sumbaev, V.V. Fedorov, JETP 94 (1988) 371. [4] A. Zeilinger, C.G. Shull, M.A. Horne, K.G. Finkelstein, Phys. Rev. Lett. 57 (1986) 3089. [5] Yu. Alexandrov, in: Proceeding of ISINN-XIII, Dubna, 2006, p. 166. [6] S.A. Werner, Phys. Rev. B 21 (1980) 1774. [7] R. Colella, A.W. Overhauser, S.A. Werner, Phys. Rev. Lett. 34 (1975) 1472. [8] J.-L. Staudenmann, S.A. Werner, R. Colella, A.W. Overhauser, Phys. Rev. A. 21 (1980) 1419. [9] A.I. Frank, P. Geltenbort, M. Jentchel, G.V. Kulin, D.V. Kustov, V.G. Nosov, A.I. Strepetov, JETP Lett. 86 (2007) 255. [10] V.V. Voronin, E.G. Lapin, S.Yu. Semenikhin, V.V. Fedorov, JETP Lett. 71 (2000) 76. [11] V.V. Fedorov, I.A. Kuznetsov, E.G. Lapin, S.Yu. Semenikhin, V.V. Voronin, JETP Lett. 85 (1) (2007) 82. [12] H. Rauch, D. Petrachek, in: H. Duchs (Ed.), Neutron Diffraction, Springer, Berlin, 1978, pp. 303–351. [13] N. Kato, J. Phys. Soc. Jpn. 19 (1964) 971. [14] V.I. Indenbom, I.Sh. Slobodetsky, K.G. Truni, ZhETF 66 (3) (1974) 1110. [15] J. Arthur, C.G. Shull, A. Zeilinger, Phys. Rev. B 32 (9) (1985) 5753. [16] R.v. Eotvos, D. Pekar, E. Fekete, Ann. Phys. (Leipzig) 68 (1922) 11. [17] J. Schmiedmayer, Nucl. Instr. and Meth. A 284 (1989) 59. [18] J. Baumann, R. Gahler, J. Kalus, W. Mampe, Phys. Rev. D 37 (1988) 3107.