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Precision measurement of single slit diffraction with very cold neutrons
Physics Letters A 164 (1992) 365-368 North-Holland
PHYSICS
LETTERS
A
Precision measurement of single slit diffraction with very cold neutrons M. Tschemitz ‘, R. G%ler a, W. Mampe b, B. Schillinger a and A. Zeilinger ’ a Technische lJniversit& Miinchen. W-8046 Garching, Germany b Institut Laue-Langevin, F-38042 Grenoble Cedex, France c Universitiit Innsbruck, A-6020 Innsbruck, Austria Received 13 February 1992; accepted for publication 26 February 1992 Communicated by J.P. Vigier
We performed experiments on diffraction of very slow neutrons (20 A wavelength) from a slit of 90 urn width. A careful comparison was made between the slit width measured by light optical methods and the slit width deduced from the neutron diffraction pattern. We get satisfactory agreement between both values - possible explanations of a discrepancy, which had ap peared in former measurements, are discussed.
1. Introduction Slow neutrons probably constitute the most suitable massive particles for precision experiments on diffraction from macroscopic objects: compared to electrons, electric fields have very little effect on the neutrons’ flight path, compared to atoms their interaction with matter is comparatively simple from the theoretical point of view. Comprehensive reviews of diffraction experiments with very slow neutrons were given by Zeilinger et al. [ 11, Gruber [ 2 ] and G&hler et al. [ 3 1. These earlier experiments (diffraction from two single slit and two double slit assemblies) did demonstrate to a very high precision an agreement of the experimentally determined neutron distributions with the theoretical prediction. There was only one significant discrepancy: the diffraction of neutrons with 20 8, wavelength from a single slit of 90 pm width revealed a difference between the light optical and neutron optical slit width of several standard deviations [ 1,2 1. The aim of the present experiment was to clarify this unsatisfactory situation.
Elsevier Science Publishers B.V.
2. Experiment Except for an improved slit assembly, the present experimental setup was rather similar to the one used in the earlier experiments [ l-31, and we briefly summarize the basic components and the changes compared to the old setup. The neutron optical bench (already described in ref. [ 41) was installed at the long wavelength neutron guide H18 (wavelength range: 1O-40 A) at the high flux reactor in Grenoble. Slits Sz and Ss (see fig. 1) define the beam divergence of N 10e4 rad, slits Si, S2 and S3 together with the diffracting prism P constitute the monochromator, selecting a wavelength band with a mean value of 18.10 8, and a bandwidth (PWHM) of 2.12 8, determined by time of flight measurement. The uncertainties in the mean wavelength and the bandwidth are 0.05 and 0.06 A respectively. We performed independent determinations of these values before and after the actual diffraction measurements, which agreed within 0.01 A. Because of geometrical reasons, we had to deflect the neutron beam by about 3 x 10m2 rad with the help of a mirror positioned in front of the monochromator. The narrow width of 20 urn for the entrance slit Ss leads to an illumination of the actual diffracting object slit S4 with fairly plane waves. This is due to 365
Volume
164, number
5,6
PHYSICS
06m
I Optlcal
LETTERS
4 96 m bench
A
27 April 1992
+__X!m__
__/
evacuated
flight
tubes
Monochromator slow from
neutron beam cold source
Fig. 1. Setup of the diffraction apparatus. Slit S, is the diffracting object of 89.6 * 0.6 pm width. It is illuminated by fairly plane waves from slit S3. The diffraction pattern is scanned by slit Ss. The slits S, to S, together with the prism define the monochromator, selecting a wavelength band of 18.1 f 2.1 A.
the fact that the zero order diffraction maximum of slit S3 is about 500 urn wide in the plane of slit S4 much broader than slit S4 itself ( N 90 urn width). For this geometry, it is easy to show that the Fresnel approximation is sufficient for calculating the diffraction pattern. The distance between S3 and the scanning slit S5 (20 pm wide) is about 10 m, with slit S4 in between. The distances between these three slits are given in fig. 1, the uncertainties are less than 5 mm. All three slits are installed on a stabilized optical bench with alignment errors of less than lo-’ rad with respect to each other. The stability of our setup was controlled by electronic levels on top of all three slits; we observed long time drifts of the relative angles between them of less than 3 x 1Oe6 rad over a 5 day run. We mention without proof that possible changes of the diffraction pattern due to these small alignment errors will only have a negligible influence on the precision of the experiment. The lateral position of scanning slit Ss was controlled by a motor-driven differential micrometer, with an optical displacement sensor as reference. Its overall precision is better than 1 urn. We scanned the diffraction pattern (see fig. 2) over a total width of 600 urn with a 10 urn step width and a total measuring time of 6400 s per point. Much effort was put into the construction of the actual diffracting object (slit Sq, see figs. 3a, 3b), as we suspected here the possibility of a systematic error of the former measurement. The slit edges were made of boron enriched glass blocks (cross section 30x 60 mm, length 300 mm), with 400 nm of Gd evaporated on the surfaces. The blocks were tilted by 366
IUU
4 JIJU
LOO PosItIon
500
600
700
800
( pm )
Fig. 2. Measured diffraction pattern, including background (points) and calculated pattern (full line) using standard optical theory. The width of slit S4 and the background were free parameters in the tit.
one degree with respect to each other (see fig. 3b). Due to this tilt, the slit width is defined by the distance between the top edges of the two glass blocks and the measurement of its width is possible by standard high resolution microscopes. Furthermore this arrangement of the two blocks avoids neutron reflection from their inner surfaces, because the mean divergence of the incoming neutron beam as well as the mean divergence of the diffracted beam are much smaller than one degree. In this geometry, due to the highly absorbing gadolinium film, an attenuation of 1O4 is achieved inside each glass block only 0.1 pm apart from the nominal edge. The surface flatness of
I Glass
blocks .
Interferometer
II
r\
I 133-l
1 Diffraction
stlt
Interferometer
II
Differential
He-Ne laser
1
Intersection
DIrectIon of neutron
A
beam with
retirule
Olfferentlal
Gadollntum
block
Steei
b
on one of the glass blocks with a retroreflector on the opposite block in each case (see fig. 3a). The step width of the digital readout was 0.15 pm. Tilts of the glass block during moving might lead to erroneous slit widths, these tilts were controlled by autocollimation during moving of the blocks. (2) Three microscopes with a magnification of 600 were positioned along the slit (see fig. 3b). They were mounted on precise linear drives equipped with differential micrometers. The displacements from edge to edge were controlled by optical sensors (0.1 urn digital readout, calibrated by laser interferometers), mounted close to the microscope’s objective. The whole assembly was embedded in a steel frame made of 20 mm thick plates. We claim a precision of f0.6 pm (maximum value) for the determination of the mean slit width of 89.6 l.trn. This was the maximum error for a sequence of measurements at different times and at different positions of the slit, including the systematic errors of the measuring devices.
the Hlcroscope
Glass
27 April 1992
PHYSICS LETTERS A
Volume 164, number $6
u
3. Numerical calculation screw
film
frame
u
Fig. 3. (a) Diffraction slit S., (front view). The two glass blocks defining the slit itself, are shaded. The two laser interferometers on top of them serve for a measurement of the slit width. (b) Diffraction slit S4 (top view) with one microscope. Three such microscopes placed along the slit (30 cm long) allow for an independent determination of the slit width. The glass blocks are tilted by one degree with respect to each other. The width of slit S4 is defined by the two top edges of the blocks.
the glass blocks was specified to better than 0.3 pm. This wavyness is one of the main errors in the determination of the actual slit width. The slit width itself was controlled by two independent methods: ( 1) Two Michelson interferometers were placed
For calculating the intensity J of the diffraction pattern, we used the following equation, based on first principle wave optical theory, J(P)=
j-j-j- I U(P-
VA)
) 12WP(fl)
Cu @dS,
,
with U(p)=
jf(S)
exp[Zirc/l(a+b)]
dSJ dS,.
p is the point of observation in the plane of the scan-
ning slit S5 (lateral position relative to the optical axis). The amplitude U(p) is derived by summing over all possible paths a and b, assuming an incoming plane wave at slit Ss. Path a counts from a point in the plane of slit SJ to the diffracting object S,, path b from S, to S5. In f(S) we take into account the variation of the neutron phase over the entrance slit width of !$ for an off-axial angle of incidence. The integration over the wavelength distribution s(A) and over the divergence D(t9) of the incoming beam can be performed incoherently, because we can assume all neutrons as plane waves, independent of 367
Volume 164, number 5,6
PHYSICS LETTERS A
each other [ 61. J(p) denotes the intensity of the pattern and V(1) describes the Coriolis deflection of the beam, which is proportional to I and has a value of 13 urn for 20 A wavelength in our setup.
4. Results Any disagreement between calculated and measured diffraction patterns will be expressed in terms of different slit widths S.+,because the measurement of this width has a much bigger uncertainty than all other relevant parameters, mainly the wavelength distribution and the distances between the slits. Numerical calculations show that the divergence of the incoming beam, the Coriolis deflection and the exact values of the widths of slits S, and S5 have little influence on the diffraction pattern. In order to deduce the slit width S, from the neutron diffraction pattern, a least-squares fit of the calculated to the measured diffraction pattern was made, leaving only the slit width itself and the background (assumed to be flat) as free parameters. From the relation between this slit width and the variance of the tit (see fig. 4)) we finally obtained for the slit width Sq S,=88.3f0.5
urn (68%confidence),
with the uncertainty in 1 being the main contribution ( f0.3 urn) to the statistical error. This should be compared with the direct measurement of the width of slit S4 (89.6 k 0.6 urn, see
27 April 1992
before). Both values show reasonable agreement, and we claim that standard optical theory describes neutron diffraction from macroscopic objects within the precision of our measurement. In the former similar experiments [ 1,2], the discrepancy between experiment and calculation was 3-4 urn, corresponding to 6-8 standard deviations. Two hidden systematic errors might have been the reason for this: (a) A wrong determination or undetected change of the width of Sq, which was controlled by spacers and by one microscope; (b) Any error or drift in the mean wavelength due to an unrecognized tilt of the slits (of about 1Oe4 rad) of the prism monochromator with respect to each other, which could lead to a variation of the mean transmitted wavelength of up to 4% over the beam height. As the chopper for the time-of-flight measurement covered only one fixed section of the beam height, this possible variation of the mean wavelength might have stayed unrecognized.
Acknowledgement We want to thank our colleagues from the FakultHt fur Physik for their support and for many valuable discussions and C. Herzog for the precise production of the mechanical components. We are grateful to Professor Glaser and to the director of the ILL for their interest. Financial support was provided by the Bundesministerium fur Forschung und Technologie and the Austrian Fonds zur Forderung der wissenschaftlichen Forschung (Projekt P6635 ).
References [ 1] A. Zeilinger, R. Ggihler, C.G. Shull, W. Treimer and W.
Slit
width
S4 (pm)
Fig. 4. Variance ofthe fitted diffraction pattern for different widths of slit S.,. From the curvature of the parabola fitted to these points, the error for the calculated slit width was deduced.
368
Mampe, Rev. Mod. Phys. 60 (1988) 1067. [2] M. Gruber, Beugung und Interferometrie sehr kalter Neutronen, thesis, Vienna ( 1988 ) [ 3 ] R. Giihler and A. Zeilinger, Am. J. Phys. 59 ( 199 1) 4. [4] J. Baumann, R. Glhler, J. Kalus and W. Mampe, J. Phys. E 20 (1987) 448. [ 51 H. Maier-Leibnitz and T. Springer, Z. Phys. 167 ( 1962) 386. [6] A.G. Klein, G.I. Opat and W.A. Hamilton, Phys. Rev. Lett. 50 (1983) 563.
PHYSICS
LETTERS
A
Precision measurement of single slit diffraction with very cold neutrons M. Tschemitz ‘, R. G%ler a, W. Mampe b, B. Schillinger a and A. Zeilinger ’ a Technische lJniversit& Miinchen. W-8046 Garching, Germany b Institut Laue-Langevin, F-38042 Grenoble Cedex, France c Universitiit Innsbruck, A-6020 Innsbruck, Austria Received 13 February 1992; accepted for publication 26 February 1992 Communicated by J.P. Vigier
We performed experiments on diffraction of very slow neutrons (20 A wavelength) from a slit of 90 urn width. A careful comparison was made between the slit width measured by light optical methods and the slit width deduced from the neutron diffraction pattern. We get satisfactory agreement between both values - possible explanations of a discrepancy, which had ap peared in former measurements, are discussed.
1. Introduction Slow neutrons probably constitute the most suitable massive particles for precision experiments on diffraction from macroscopic objects: compared to electrons, electric fields have very little effect on the neutrons’ flight path, compared to atoms their interaction with matter is comparatively simple from the theoretical point of view. Comprehensive reviews of diffraction experiments with very slow neutrons were given by Zeilinger et al. [ 11, Gruber [ 2 ] and G&hler et al. [ 3 1. These earlier experiments (diffraction from two single slit and two double slit assemblies) did demonstrate to a very high precision an agreement of the experimentally determined neutron distributions with the theoretical prediction. There was only one significant discrepancy: the diffraction of neutrons with 20 8, wavelength from a single slit of 90 pm width revealed a difference between the light optical and neutron optical slit width of several standard deviations [ 1,2 1. The aim of the present experiment was to clarify this unsatisfactory situation.
Elsevier Science Publishers B.V.
2. Experiment Except for an improved slit assembly, the present experimental setup was rather similar to the one used in the earlier experiments [ l-31, and we briefly summarize the basic components and the changes compared to the old setup. The neutron optical bench (already described in ref. [ 41) was installed at the long wavelength neutron guide H18 (wavelength range: 1O-40 A) at the high flux reactor in Grenoble. Slits Sz and Ss (see fig. 1) define the beam divergence of N 10e4 rad, slits Si, S2 and S3 together with the diffracting prism P constitute the monochromator, selecting a wavelength band with a mean value of 18.10 8, and a bandwidth (PWHM) of 2.12 8, determined by time of flight measurement. The uncertainties in the mean wavelength and the bandwidth are 0.05 and 0.06 A respectively. We performed independent determinations of these values before and after the actual diffraction measurements, which agreed within 0.01 A. Because of geometrical reasons, we had to deflect the neutron beam by about 3 x 10m2 rad with the help of a mirror positioned in front of the monochromator. The narrow width of 20 urn for the entrance slit Ss leads to an illumination of the actual diffracting object slit S4 with fairly plane waves. This is due to 365
Volume
164, number
5,6
PHYSICS
06m
I Optlcal
LETTERS
4 96 m bench
A
27 April 1992
+__X!m__
__/
evacuated
flight
tubes
Monochromator slow from
neutron beam cold source
Fig. 1. Setup of the diffraction apparatus. Slit S, is the diffracting object of 89.6 * 0.6 pm width. It is illuminated by fairly plane waves from slit S3. The diffraction pattern is scanned by slit Ss. The slits S, to S, together with the prism define the monochromator, selecting a wavelength band of 18.1 f 2.1 A.
the fact that the zero order diffraction maximum of slit S3 is about 500 urn wide in the plane of slit S4 much broader than slit S4 itself ( N 90 urn width). For this geometry, it is easy to show that the Fresnel approximation is sufficient for calculating the diffraction pattern. The distance between S3 and the scanning slit S5 (20 pm wide) is about 10 m, with slit S4 in between. The distances between these three slits are given in fig. 1, the uncertainties are less than 5 mm. All three slits are installed on a stabilized optical bench with alignment errors of less than lo-’ rad with respect to each other. The stability of our setup was controlled by electronic levels on top of all three slits; we observed long time drifts of the relative angles between them of less than 3 x 1Oe6 rad over a 5 day run. We mention without proof that possible changes of the diffraction pattern due to these small alignment errors will only have a negligible influence on the precision of the experiment. The lateral position of scanning slit Ss was controlled by a motor-driven differential micrometer, with an optical displacement sensor as reference. Its overall precision is better than 1 urn. We scanned the diffraction pattern (see fig. 2) over a total width of 600 urn with a 10 urn step width and a total measuring time of 6400 s per point. Much effort was put into the construction of the actual diffracting object (slit Sq, see figs. 3a, 3b), as we suspected here the possibility of a systematic error of the former measurement. The slit edges were made of boron enriched glass blocks (cross section 30x 60 mm, length 300 mm), with 400 nm of Gd evaporated on the surfaces. The blocks were tilted by 366
IUU
4 JIJU
LOO PosItIon
500
600
700
800
( pm )
Fig. 2. Measured diffraction pattern, including background (points) and calculated pattern (full line) using standard optical theory. The width of slit S4 and the background were free parameters in the tit.
one degree with respect to each other (see fig. 3b). Due to this tilt, the slit width is defined by the distance between the top edges of the two glass blocks and the measurement of its width is possible by standard high resolution microscopes. Furthermore this arrangement of the two blocks avoids neutron reflection from their inner surfaces, because the mean divergence of the incoming neutron beam as well as the mean divergence of the diffracted beam are much smaller than one degree. In this geometry, due to the highly absorbing gadolinium film, an attenuation of 1O4 is achieved inside each glass block only 0.1 pm apart from the nominal edge. The surface flatness of
I Glass
blocks .
Interferometer
II
r\
I 133-l
1 Diffraction
stlt
Interferometer
II
Differential
He-Ne laser
1
Intersection
DIrectIon of neutron
A
beam with
retirule
Olfferentlal
Gadollntum
block
Steei
b
on one of the glass blocks with a retroreflector on the opposite block in each case (see fig. 3a). The step width of the digital readout was 0.15 pm. Tilts of the glass block during moving might lead to erroneous slit widths, these tilts were controlled by autocollimation during moving of the blocks. (2) Three microscopes with a magnification of 600 were positioned along the slit (see fig. 3b). They were mounted on precise linear drives equipped with differential micrometers. The displacements from edge to edge were controlled by optical sensors (0.1 urn digital readout, calibrated by laser interferometers), mounted close to the microscope’s objective. The whole assembly was embedded in a steel frame made of 20 mm thick plates. We claim a precision of f0.6 pm (maximum value) for the determination of the mean slit width of 89.6 l.trn. This was the maximum error for a sequence of measurements at different times and at different positions of the slit, including the systematic errors of the measuring devices.
the Hlcroscope
Glass
27 April 1992
PHYSICS LETTERS A
Volume 164, number $6
u
3. Numerical calculation screw
film
frame
u
Fig. 3. (a) Diffraction slit S., (front view). The two glass blocks defining the slit itself, are shaded. The two laser interferometers on top of them serve for a measurement of the slit width. (b) Diffraction slit S4 (top view) with one microscope. Three such microscopes placed along the slit (30 cm long) allow for an independent determination of the slit width. The glass blocks are tilted by one degree with respect to each other. The width of slit S4 is defined by the two top edges of the blocks.
the glass blocks was specified to better than 0.3 pm. This wavyness is one of the main errors in the determination of the actual slit width. The slit width itself was controlled by two independent methods: ( 1) Two Michelson interferometers were placed
For calculating the intensity J of the diffraction pattern, we used the following equation, based on first principle wave optical theory, J(P)=
j-j-j- I U(P-
VA)
) 12WP(fl)
Cu @dS,
,
with U(p)=
jf(S)
exp[Zirc/l(a+b)]
dSJ dS,.
p is the point of observation in the plane of the scan-
ning slit S5 (lateral position relative to the optical axis). The amplitude U(p) is derived by summing over all possible paths a and b, assuming an incoming plane wave at slit Ss. Path a counts from a point in the plane of slit SJ to the diffracting object S,, path b from S, to S5. In f(S) we take into account the variation of the neutron phase over the entrance slit width of !$ for an off-axial angle of incidence. The integration over the wavelength distribution s(A) and over the divergence D(t9) of the incoming beam can be performed incoherently, because we can assume all neutrons as plane waves, independent of 367
Volume 164, number 5,6
PHYSICS LETTERS A
each other [ 61. J(p) denotes the intensity of the pattern and V(1) describes the Coriolis deflection of the beam, which is proportional to I and has a value of 13 urn for 20 A wavelength in our setup.
4. Results Any disagreement between calculated and measured diffraction patterns will be expressed in terms of different slit widths S.+,because the measurement of this width has a much bigger uncertainty than all other relevant parameters, mainly the wavelength distribution and the distances between the slits. Numerical calculations show that the divergence of the incoming beam, the Coriolis deflection and the exact values of the widths of slits S, and S5 have little influence on the diffraction pattern. In order to deduce the slit width S, from the neutron diffraction pattern, a least-squares fit of the calculated to the measured diffraction pattern was made, leaving only the slit width itself and the background (assumed to be flat) as free parameters. From the relation between this slit width and the variance of the tit (see fig. 4)) we finally obtained for the slit width Sq S,=88.3f0.5
urn (68%confidence),
with the uncertainty in 1 being the main contribution ( f0.3 urn) to the statistical error. This should be compared with the direct measurement of the width of slit S4 (89.6 k 0.6 urn, see
27 April 1992
before). Both values show reasonable agreement, and we claim that standard optical theory describes neutron diffraction from macroscopic objects within the precision of our measurement. In the former similar experiments [ 1,2], the discrepancy between experiment and calculation was 3-4 urn, corresponding to 6-8 standard deviations. Two hidden systematic errors might have been the reason for this: (a) A wrong determination or undetected change of the width of Sq, which was controlled by spacers and by one microscope; (b) Any error or drift in the mean wavelength due to an unrecognized tilt of the slits (of about 1Oe4 rad) of the prism monochromator with respect to each other, which could lead to a variation of the mean transmitted wavelength of up to 4% over the beam height. As the chopper for the time-of-flight measurement covered only one fixed section of the beam height, this possible variation of the mean wavelength might have stayed unrecognized.
Acknowledgement We want to thank our colleagues from the FakultHt fur Physik for their support and for many valuable discussions and C. Herzog for the precise production of the mechanical components. We are grateful to Professor Glaser and to the director of the ILL for their interest. Financial support was provided by the Bundesministerium fur Forschung und Technologie and the Austrian Fonds zur Forderung der wissenschaftlichen Forschung (Projekt P6635 ).
References [ 1] A. Zeilinger, R. Ggihler, C.G. Shull, W. Treimer and W.
Slit
width
S4 (pm)
Fig. 4. Variance ofthe fitted diffraction pattern for different widths of slit S.,. From the curvature of the parabola fitted to these points, the error for the calculated slit width was deduced.
368
Mampe, Rev. Mod. Phys. 60 (1988) 1067. [2] M. Gruber, Beugung und Interferometrie sehr kalter Neutronen, thesis, Vienna ( 1988 ) [ 3 ] R. Giihler and A. Zeilinger, Am. J. Phys. 59 ( 199 1) 4. [4] J. Baumann, R. Glhler, J. Kalus and W. Mampe, J. Phys. E 20 (1987) 448. [ 51 H. Maier-Leibnitz and T. Springer, Z. Phys. 167 ( 1962) 386. [6] A.G. Klein, G.I. Opat and W.A. Hamilton, Phys. Rev. Lett. 50 (1983) 563.