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The effect of gravity on the resolution for time-of-flight specular neutron reflectivity
Nuclear Instruments and Methods in Physics Research A 631 (2011) 121–124
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
The effect of gravity on the resolution for time-of-flight specular neutron reflectivity Irina Bodnarchuk a,n, Sergei Manoshin b, Sergei Yaradaikin b, Viacheslav Kazimirov b, Victor Bodnarchuk b a b
Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics, 119991 Moscow, Russia Joint Institute for Nuclear Research, Frank Laboratory of Neutron Physics, 141980 Moscow Region, Dubna, Russia
a r t i c l e in f o
abstract
Article history: Received 30 September 2010 Received in revised form 7 December 2010 Accepted 7 December 2010 Available online 17 December 2010
The effect of gravity on neutron scattering is negligible if a thermal spectrum of up to 10 A˚ is used. Modern cold sources produce spectra with an ample quantity of cold neutrons. Gravity may play a crucial role for the cold part of the spectrum in neutron scattering experiments demanding high angle resolution. This mainly concerns the reflectometry method where a small deviation in the angle distribution may lead to visible effects. We present a theoretical model that takes into account the effect of gravity on the resolution function as well as the flux distribution for the specular neutron reflectometry. Theoretically calculated reflectivity curves convoluted with the model resolution function were compared with MonteCarlo simulations, which imitated real measurements. The good agreement between the calculated and simulated curves allows one to apply the theoretical model to treat the real experimental data. & 2010 Elsevier B.V. All rights reserved.
Keywords: Resolution function Neutron reflectometry Monte-Carlo simulations
1. Introduction Neutron reflectometry is a widely used method to study surface and interface structures [1,2]. The instrumental resolution function for zero-gravity specular reflectivity has been described in Ref. [3]. The analytical beam-analysis method coupling position, angle and wavelength of a neutron [4] has been applied previously. However, the angular resolution in a vertical scattering plane (horizontal sample plane) depends on gravity due to the bending of neutron trajectories. The effect of gravity is negligible if neutron spectra of up to 10 A˚ are used [5], which is typical for moderators without cold sources. Reflectometry on the sources that generate spectra with a significant part of cold neutrons allows one to achieve low values of the scattering vector Q. The extended dynamical range provides considerably more detailed information on the scattering-length density profile perpendicular to the surface of fluids, membranes, polymers and biological objects. The gravity effect can be overcome if a monochromic incident neutron beam is used (steady-state reactors) and the center of the scattering pattern is corrected. In the case of the time-of-flight (TOF) technique (pulse neutron sources), the center of the scattering beam is smeared because neutrons with different wavelengths are deflected in the gravity field by different values. Presently, the new multifunctional TOF reflectometer, GRAINS, is under construction at the modernized high flux pulsed reactor
n
Corresponding author. E-mail address: [email protected] (I. Bodnarchuk).
0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.12.074
with the new cold moderator, IBR-2M, in Dubna (Russia) [6]. It is necessary to study the influence of gravity on the resolution function because the principal feature of this reflectometer is the horizontal sample plane. The main aim of this article is the development of the theoretical approach for the resolution function, which takes into account the effect of gravity. To test the approach, reflectivity curves smeared by the derived resolution function were compared with the reflectivity curves simulated by the VITESS Monte-Carlo software package [7]. The VITESS package allows users to carry out simulations of the neutron scattering with and without the influence of gravity and to identify the contribution of gravity on the reflectivity curves. An ideal small-angle neutron spectrometer with perfect collimation was considered and an analytical form for the gravity resolution function was calculated in the paper dedicated to the effect of gravity [8]. It was shown that the total resolution function of a real spectrometer can be treated as a convolution of the gravity resolution function with the geometric and wavelength resolution functions. In the present paper, the influences of gravity and geometrical factors on the resolution of a TOF neutron specular reflectometer were not separated. In the present work, the instrumental angular resolution function for an idealized TOF reflectometer was deduced by extending the analytical beam-analysis method, which takes into account the influence of gravity. This function was obtained for the fixed neutron wavelength and was then convoluted with the wavelength resolution function. Thus, the total resolution function of the TOF reflectometer was represented in the integral form. We demonstrate that the effect of gravity is recognizable and has to be considered in real experiments.
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2. Theoretical expression for the resolution function The general configuration of elements, which defines the reflectometry mode, was considered to estimate the effect of gravity on the reflectometer resolution. The source M, two slits D1 and D2 with widths w1 and w2, respectively, sample S of length LS and a detector are shown in Fig. 1. The centers of the slits and the sample are on the line inclined at an angle y to the horizontal plane. The distances between the corresponding elements are SMD1, SD1D2, SD2S and L0. Thus, the full neutron flight path can be calculated as L¼SMD1 +SD1D2 +SD2S +L0. The coordinate system was chosen as shown in Fig. 1. The center of the coordinate system is at the center of the first slit. The resolution function R(/QS,Q) describes the distribution of the real scattering vectors Q around the measured value /QS, and the convolution of R(/QS,Q) with the model reflectivity Int(Q) gives the experimentally measured reflectivity Int(/QS) [4] Z 1 Intð/Q SÞ ¼ IntðQ ÞRð/Q S,Q ÞdQ : ð1Þ 0
1
Rð/Q S,Q ÞdQ ¼ 1:
The distribution of neutron positions in the beam changes along the flight path due to the divergence of the beam. A neutron with angle z0 and position z will change its position after a distance SD1D2 to z þ zuSD1D2 þ ðg=2Þðlm=hÞ2 S2D1D2 , where g is the gravitational constant acceleration, h is the Planck constant and m is the neutron mass. The second slit D2 had the same transmission function as pffiffiffiffiffiffi the first slit (3) with the standard deviation sz2 ¼ w2 = 12 and with the center of the distribution at y SD1D2. Thus, the distribution after the second slit is given by the following expression: 8 2 !2 , 39 2 < 1 z2 = g lm 2 4 Iðz,zu, lÞp exp þ z þzuSD1D2 þ SD1D2 ySD1D2 s2z2 5 : : 2 s2z1 ; 2 h
The normalization of resolution function must be unity Z
divergence of the neutron beam after the slits. The first element of the instrument is the slit D1. The first slit passed only neutrons that fall into its aperture defined by width w1. The transmission function is a box function, which can be approximated as a Gaussian function pffiffiffiffiffiffi with standard deviation sz1 ¼ w1 = 12. Thus, the distribution right after the first slit can be written as: " # 1 z2 Iðz,zu, lÞp exp : ð3Þ 2 s2z1
ð2Þ
ð4Þ
0
Let us consider neutrons with a wavelength l. The distribution of the real scattering vector Q of these neutrons is determined by the angular divergence due to the finite collimation and the bending of the neutron trajectories in the gravity field. The analytical beam-line analysis method [4] has been employed to calculate the divergence at the sample position. In the original method, the distribution of neutrons is described as a function of the position, angle and corresponding wavelength. However, in our case, there was no crystal monochromator, and a white-beam pulsed-source instrument was considered. The position of any neutron with wavelength l can be described by the distribution I(z,z0 ,l), where z0 is the angle of the wavevector with respect to the horizontal plane. Each element of the reflectometer transforms the distribution I(z,z0 ,l). The transmission functions in each element of the instrument are approximated by Gaussian functions, which make it possible to calculate analytically the final neutron distribution I(z,z0 ,l). It was assumed that the neutron source radiates neutrons isotropically. This assumption is reasonable because the divergence of neutrons before the instrument was large in comparison with the
Let us consider the sample as a vertical slit of width wS ¼ y LS (the corresponding pffiffiffiffiffiffi standard deviation of the transmission function is szS ¼ wS = 12) at distance SD1S ¼SD1D2 +SD2S from the first slit D1 (the corresponding center of the distribution is y SD1S). Then, the distribution after the third slit is given by the following expression: Iðz,zu, lÞp 8 2 2 2 > < 1 z2 z þ zuSD1D2 þ 2g lhm S2D1D2 ySD1D2 6 exp 4 2 þ > s2z2 : 2 sz1 2 39 2 = z þ zuSD1S þ 2g lhm S2D1S ySD1S 7> þ 5 : 2 > szS ;
ð5Þ
The resulting distribution (5) consists of three Gaussian functions and has the following general form: ( " #) 1 ðzAÞ2 ðzBÞ2 ðzCÞ2 exp þ þ : ð6Þ 2 s2A s2B s2C The reduction of expression (6) to a form with one Gaussian function by extracting the perfect square within the exponent gives the following expression: " # " # 1 ðz/zSÞ2 1 ðABÞ2 s2C þ ðACÞ2 s2B þ ðBCÞ2 s2A exp , exp 2 2 s2z s2A s2B þ s2A s2C þ s2B s2C ð7Þ where /zS ¼
Fig. 1. Layout of reflectometer with a horizontal sample plane. The bent solid line demonstrates the real neutron trajectory in the gravity field. The dashed line depicts the axis passing through the centers of both slits and the sample center. The inset shows the ideal reflectivity as a function of neutron wavelengths from monolayer with a critical angle of 5.56 10 4 rad/A˚ and thickness of 1500 A˚ on a substrate with a critical angle of 4.17 10 4 rad/A˚ at a grazing angle of 15 mrad.
As2B s2C þBs2A s2C þ C s2A s2B , s2A s2B þ s2A s2C þ s2B s2C
sA sB sC ffi: sz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 sA sB þ s2A s2C þ s2B s2C
ð8Þ
Since the second exponent in expression (7) does not depend on z, the integration of expression (7) over z from N to + N gives the constant from the first exponent multiplied by the unchanged second exponent. The resulting angular distribution for the neutrons in the first slit D1 that will eventually be reflected by the sample can be obtained by extracting the perfect square within the second exponent " # " # 1 ðzu/zuSÞ2 1 k/zuS2 Iðzu, lÞpexp exp , ð9Þ 2 2 s2zu s2zu
I. Bodnarchuk et al. / Nuclear Instruments and Methods in Physics Research A 631 (2011) 121–124
123
where /zuS ¼
szu ¼
lm2 h
2 2 SD1D2 S2D1D2 s2zS þ y 2g lhm SD1S S2D1S s2z2 þ y 2g lhm ð2SD1D2 þSD2S Þ S2D2S s2z1 S2D1D2 s2zS þ S2D1S s2z2 þ S2D2S s2z1
,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2z1 s2z2 þ s2z1 s2zS þ s2z2 s2zS , 2 SD1D2 s2zS þ S2D1S s2z2 þ S2D2S s2z1
k¼
y 2g
y 2g
lm2 h
SD1D2
2
ð11Þ
2 2 2 2 S2D1D2 s2zS þ y 2g lhm SD1S S2D1S s2z2 þ y 2g lhm ð2SD1D2 þSD2S Þ S2D2S s2z1 S2D1D2 s2zS þ S2D1S s2z2 þ S2D2S s2z1
The scattering vector Q can be written as: ! 2 4p lm zu þg SD1S , Q¼ h l
ð13Þ
because the angle in the first slit z0 is increased by the factor g(lm/h)2SD1S due to the gravity bending at distance SD1S. Taking into account (13), the distribution (9) can be rewritten as: 2 6 1 IðQ , lÞpexp4 2
2 3 2 " # Q 4lp /zuSþ g lhm SD1S 1 k/zuS2 7 : 5exp 2 2 s2zu 4pszu
ð14Þ
l
The first exponent in Eq. (14) is the Gaussian function responsible for the smearing of the reflected intensity due to the angular divergence with a gravity shift of mean grazing angle /z0 S+ g(lm/h)2SD1S from the angle y. The value of the second exponent is equal to unity at zero gravity. At non-zero gravity, the exponent decreases when wavelength increases. This exponent is responsible for the reduction of the reflected flux due to the neutrons fall before the sample. In TOF experiments, the neutron wavelength can be determined by the neutron time-of-flight t from the source to the detector at distance L th : ð15Þ lt ¼ mL The distribution of wavelengths can be approximated by the following Gaussian function: " # 1 ðllt Þ2 Rl ðl, lt Þp exp 2 , ð16Þ 2 stm þ Dt 2 =12 ðh=mLÞ2 pffiffiffiffiffiffiffiffiffiffiffi where stm ¼ T=ð2 2 ln 2Þ is the uncertainty of the moment of the neutron emission from the source (T is the pulse width), and Dt is the time width of the analyzer channel. Thus, the measured value of /QS can be written as the following expression: /Q S ¼
4p
lt
y:
ð17Þ
In real experiments, y can be determined with a finite precision. In our VITESS simulations, y was determined precisely by the line passing through the centers of all reflectometer elements. The total resolution function of the reflectometer is the integration over all wavelengths l, given as follows: Rð/Q S,Q Þ ¼
Z
ð10Þ
IðQ , lÞRl ðl,4py=/Q SÞdl:
ð18Þ
3. Parameters of simulations in the VITESS package The VITESS software package is a Monte Carlo instrumentation tool for the simulation of neutron scattering at pulsed and steadystate sources [7]. It describes the motion of a neutron as a classical particle in real 3-D space, excluding the sample where neutrons have to be considered as a wave. The choice of instrumental parameters was based on the real beam line characteristics of the reflectometer GRAINS [6]. In all the
:
ð12Þ
simulations, the distance from the source to the first slit D1 was SMD1 ¼10 m, the distance from D1 to the second slit D2 was SD1D2 ¼2 m, the distance from the sample S to the detector was L0 ¼4 m and y ¼15 mrad. The widths of the slits were w1 ¼2 mm and w2 ¼1 mm. The analyzer time channel width was Dt¼64 ms and the neutron pulse width was T¼320 ms. To analyze the role of the sample size along the xaxis, two values that represented a ‘‘big’’ sample with length of 100 mm and a ‘‘small’’ one with length of 20 mm were chosen. A thin monolayer with a critical angle of 5.56 10 4 rad/A˚ and thickness of 1500 A˚ on a substrate with a critical angle of 4.17 10 4 rad/A˚ was used as an idealized sample. The wavelength dependence of reflectivity from such a monolayer consists of a sequence of narrow oscillations whose positions and shapes are very sensitive to the resolution factor (Fig. 1). An incident spectrum of constant intensity for all wavelengths was used to exclude the factor of spectrum shape from consideration.
4. Results and discussion The dependence of the gravity effect on the distance between the second slit and the sample SD2S and on the sample size LS was analyzed. The reflectivities for two sample lengths, LS ¼20 and 100 mm, and three sample positions, SD2S ¼0.033 or 0.05 m (the sample position right after the second slit for two sample lengths correspondingly), SD2S ¼0.5 and 1 m are shown on Fig. 2. It can be seen that the agreement between the theoretical calculations in the frame of proposed approach and the Monte-Carlo simulations by means of the VITESS package is very good. In all cases, the deviations of reflectivity with zero and non-zero gravity can be observed. An increase in the distance between the second slit and the sample leads to higher deviation. Two factors, both of which are due to the bending of neutron trajectories in the gravity field, contribute to the deviation. The first factor is the increase in the grazing incidence angle, which leads to the shift of reflectivity fringes to higher wavelengths. If the angular divergence increases, the fringes broaden and the gravity shift becomes less distinguishable. The second factor is that neutrons fall before the sample, which leads to the flux deficiency for longer wavelengths. This effect is greater for a smaller sample length. In the case of non-zero gravity, each element of the reflectometer acts as a wavelength and angular filter while in the case of zero gravity, each element only selects angles. Positioning the sample as close as possible to the second slit minimizes the effect of gravity on the scattering pattern, but this is not possible to do in particular cases. For example, if it is necessary to install some device for polarization control before the sample (polarizer, flipper, etc.), the influence of gravity on the resolution cannot be avoided.
5. Conclusions The analytical beam-line analysis method for deriving the resolution function of the neutron TOF reflectometer accounting for the effect
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Fig. 2. Reflectivities for two sample lengths and three sample positions with respect to the second slit. The solid lines are the theoretical calculations, and the triangles and circles are the Monte-Carlo simulations prepared by the VITESS package with zero and non-zero gravity, respectively. All plots are represented from 12 A˚ where all curves are distinguishable.
of gravity was applied. The theoretical calculations are in good agreement with Monte-Carlo simulations, which mimic the real measurements with idealized samples. The shape of the reflectivity curve with non-zero gravity is defined by two factors that depend on the neutron wavelengths: angle deviations due to the neutron trajectory bending and a part of neutrons miss the sample. Positioning the sample right after the second collimating slit minimizes the effect of gravity. However, the effect of gravity is still present even in this case. The proposed theoretical approach makes it possible to take the resolution into account correctly and allows one to carry out real measurements with a broad wavelength band.
References [1] [2] [3] [4] [5] [6]
J. Penfold, R.K. Thomas, J. Phys. C 2 (1990) 1369. T.P. Russell, Mater. Sci. Rep. 5 (1990) 171. W.G. Bouwman, J.S. Pedersen, J. Appl. Crystallogr. 29 (1996) 152. J.S. Pedersen, C. Riekel, J. Appl. Crystallogr. 24 (1991) 893. V.-O. de Haan, Thesis, Technische Universiteit Delft, 1995. M.V. Avdeev, V.I. Bodnarchuk, V.V. Lauter-Pasyuk, H. Lauter, V.L. Aksenov, S.P. Yaradaikin, V.A. Ulyanov, V.A. Trounov, S.I. Kalinin, J. Phys.: Conf. Ser. 251 (2010) 012060. [7] VITESS, /http://www.hmi.de/projects/ess/vitessS. [8] A.T. Boothroyd, J. Appl. Crystallogr. 22 (1989) 252.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
The effect of gravity on the resolution for time-of-flight specular neutron reflectivity Irina Bodnarchuk a,n, Sergei Manoshin b, Sergei Yaradaikin b, Viacheslav Kazimirov b, Victor Bodnarchuk b a b
Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics, 119991 Moscow, Russia Joint Institute for Nuclear Research, Frank Laboratory of Neutron Physics, 141980 Moscow Region, Dubna, Russia
a r t i c l e in f o
abstract
Article history: Received 30 September 2010 Received in revised form 7 December 2010 Accepted 7 December 2010 Available online 17 December 2010
The effect of gravity on neutron scattering is negligible if a thermal spectrum of up to 10 A˚ is used. Modern cold sources produce spectra with an ample quantity of cold neutrons. Gravity may play a crucial role for the cold part of the spectrum in neutron scattering experiments demanding high angle resolution. This mainly concerns the reflectometry method where a small deviation in the angle distribution may lead to visible effects. We present a theoretical model that takes into account the effect of gravity on the resolution function as well as the flux distribution for the specular neutron reflectometry. Theoretically calculated reflectivity curves convoluted with the model resolution function were compared with MonteCarlo simulations, which imitated real measurements. The good agreement between the calculated and simulated curves allows one to apply the theoretical model to treat the real experimental data. & 2010 Elsevier B.V. All rights reserved.
Keywords: Resolution function Neutron reflectometry Monte-Carlo simulations
1. Introduction Neutron reflectometry is a widely used method to study surface and interface structures [1,2]. The instrumental resolution function for zero-gravity specular reflectivity has been described in Ref. [3]. The analytical beam-analysis method coupling position, angle and wavelength of a neutron [4] has been applied previously. However, the angular resolution in a vertical scattering plane (horizontal sample plane) depends on gravity due to the bending of neutron trajectories. The effect of gravity is negligible if neutron spectra of up to 10 A˚ are used [5], which is typical for moderators without cold sources. Reflectometry on the sources that generate spectra with a significant part of cold neutrons allows one to achieve low values of the scattering vector Q. The extended dynamical range provides considerably more detailed information on the scattering-length density profile perpendicular to the surface of fluids, membranes, polymers and biological objects. The gravity effect can be overcome if a monochromic incident neutron beam is used (steady-state reactors) and the center of the scattering pattern is corrected. In the case of the time-of-flight (TOF) technique (pulse neutron sources), the center of the scattering beam is smeared because neutrons with different wavelengths are deflected in the gravity field by different values. Presently, the new multifunctional TOF reflectometer, GRAINS, is under construction at the modernized high flux pulsed reactor
n
Corresponding author. E-mail address: [email protected] (I. Bodnarchuk).
0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.12.074
with the new cold moderator, IBR-2M, in Dubna (Russia) [6]. It is necessary to study the influence of gravity on the resolution function because the principal feature of this reflectometer is the horizontal sample plane. The main aim of this article is the development of the theoretical approach for the resolution function, which takes into account the effect of gravity. To test the approach, reflectivity curves smeared by the derived resolution function were compared with the reflectivity curves simulated by the VITESS Monte-Carlo software package [7]. The VITESS package allows users to carry out simulations of the neutron scattering with and without the influence of gravity and to identify the contribution of gravity on the reflectivity curves. An ideal small-angle neutron spectrometer with perfect collimation was considered and an analytical form for the gravity resolution function was calculated in the paper dedicated to the effect of gravity [8]. It was shown that the total resolution function of a real spectrometer can be treated as a convolution of the gravity resolution function with the geometric and wavelength resolution functions. In the present paper, the influences of gravity and geometrical factors on the resolution of a TOF neutron specular reflectometer were not separated. In the present work, the instrumental angular resolution function for an idealized TOF reflectometer was deduced by extending the analytical beam-analysis method, which takes into account the influence of gravity. This function was obtained for the fixed neutron wavelength and was then convoluted with the wavelength resolution function. Thus, the total resolution function of the TOF reflectometer was represented in the integral form. We demonstrate that the effect of gravity is recognizable and has to be considered in real experiments.
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I. Bodnarchuk et al. / Nuclear Instruments and Methods in Physics Research A 631 (2011) 121–124
2. Theoretical expression for the resolution function The general configuration of elements, which defines the reflectometry mode, was considered to estimate the effect of gravity on the reflectometer resolution. The source M, two slits D1 and D2 with widths w1 and w2, respectively, sample S of length LS and a detector are shown in Fig. 1. The centers of the slits and the sample are on the line inclined at an angle y to the horizontal plane. The distances between the corresponding elements are SMD1, SD1D2, SD2S and L0. Thus, the full neutron flight path can be calculated as L¼SMD1 +SD1D2 +SD2S +L0. The coordinate system was chosen as shown in Fig. 1. The center of the coordinate system is at the center of the first slit. The resolution function R(/QS,Q) describes the distribution of the real scattering vectors Q around the measured value /QS, and the convolution of R(/QS,Q) with the model reflectivity Int(Q) gives the experimentally measured reflectivity Int(/QS) [4] Z 1 Intð/Q SÞ ¼ IntðQ ÞRð/Q S,Q ÞdQ : ð1Þ 0
1
Rð/Q S,Q ÞdQ ¼ 1:
The distribution of neutron positions in the beam changes along the flight path due to the divergence of the beam. A neutron with angle z0 and position z will change its position after a distance SD1D2 to z þ zuSD1D2 þ ðg=2Þðlm=hÞ2 S2D1D2 , where g is the gravitational constant acceleration, h is the Planck constant and m is the neutron mass. The second slit D2 had the same transmission function as pffiffiffiffiffiffi the first slit (3) with the standard deviation sz2 ¼ w2 = 12 and with the center of the distribution at y SD1D2. Thus, the distribution after the second slit is given by the following expression: 8 2 !2 , 39 2 < 1 z2 = g lm 2 4 Iðz,zu, lÞp exp þ z þzuSD1D2 þ SD1D2 ySD1D2 s2z2 5 : : 2 s2z1 ; 2 h
The normalization of resolution function must be unity Z
divergence of the neutron beam after the slits. The first element of the instrument is the slit D1. The first slit passed only neutrons that fall into its aperture defined by width w1. The transmission function is a box function, which can be approximated as a Gaussian function pffiffiffiffiffiffi with standard deviation sz1 ¼ w1 = 12. Thus, the distribution right after the first slit can be written as: " # 1 z2 Iðz,zu, lÞp exp : ð3Þ 2 s2z1
ð2Þ
ð4Þ
0
Let us consider neutrons with a wavelength l. The distribution of the real scattering vector Q of these neutrons is determined by the angular divergence due to the finite collimation and the bending of the neutron trajectories in the gravity field. The analytical beam-line analysis method [4] has been employed to calculate the divergence at the sample position. In the original method, the distribution of neutrons is described as a function of the position, angle and corresponding wavelength. However, in our case, there was no crystal monochromator, and a white-beam pulsed-source instrument was considered. The position of any neutron with wavelength l can be described by the distribution I(z,z0 ,l), where z0 is the angle of the wavevector with respect to the horizontal plane. Each element of the reflectometer transforms the distribution I(z,z0 ,l). The transmission functions in each element of the instrument are approximated by Gaussian functions, which make it possible to calculate analytically the final neutron distribution I(z,z0 ,l). It was assumed that the neutron source radiates neutrons isotropically. This assumption is reasonable because the divergence of neutrons before the instrument was large in comparison with the
Let us consider the sample as a vertical slit of width wS ¼ y LS (the corresponding pffiffiffiffiffiffi standard deviation of the transmission function is szS ¼ wS = 12) at distance SD1S ¼SD1D2 +SD2S from the first slit D1 (the corresponding center of the distribution is y SD1S). Then, the distribution after the third slit is given by the following expression: Iðz,zu, lÞp 8 2 2 2 > < 1 z2 z þ zuSD1D2 þ 2g lhm S2D1D2 ySD1D2 6 exp 4 2 þ > s2z2 : 2 sz1 2 39 2 = z þ zuSD1S þ 2g lhm S2D1S ySD1S 7> þ 5 : 2 > szS ;
ð5Þ
The resulting distribution (5) consists of three Gaussian functions and has the following general form: ( " #) 1 ðzAÞ2 ðzBÞ2 ðzCÞ2 exp þ þ : ð6Þ 2 s2A s2B s2C The reduction of expression (6) to a form with one Gaussian function by extracting the perfect square within the exponent gives the following expression: " # " # 1 ðz/zSÞ2 1 ðABÞ2 s2C þ ðACÞ2 s2B þ ðBCÞ2 s2A exp , exp 2 2 s2z s2A s2B þ s2A s2C þ s2B s2C ð7Þ where /zS ¼
Fig. 1. Layout of reflectometer with a horizontal sample plane. The bent solid line demonstrates the real neutron trajectory in the gravity field. The dashed line depicts the axis passing through the centers of both slits and the sample center. The inset shows the ideal reflectivity as a function of neutron wavelengths from monolayer with a critical angle of 5.56 10 4 rad/A˚ and thickness of 1500 A˚ on a substrate with a critical angle of 4.17 10 4 rad/A˚ at a grazing angle of 15 mrad.
As2B s2C þBs2A s2C þ C s2A s2B , s2A s2B þ s2A s2C þ s2B s2C
sA sB sC ffi: sz ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 sA sB þ s2A s2C þ s2B s2C
ð8Þ
Since the second exponent in expression (7) does not depend on z, the integration of expression (7) over z from N to + N gives the constant from the first exponent multiplied by the unchanged second exponent. The resulting angular distribution for the neutrons in the first slit D1 that will eventually be reflected by the sample can be obtained by extracting the perfect square within the second exponent " # " # 1 ðzu/zuSÞ2 1 k/zuS2 Iðzu, lÞpexp exp , ð9Þ 2 2 s2zu s2zu
I. Bodnarchuk et al. / Nuclear Instruments and Methods in Physics Research A 631 (2011) 121–124
123
where /zuS ¼
szu ¼
lm2 h
2 2 SD1D2 S2D1D2 s2zS þ y 2g lhm SD1S S2D1S s2z2 þ y 2g lhm ð2SD1D2 þSD2S Þ S2D2S s2z1 S2D1D2 s2zS þ S2D1S s2z2 þ S2D2S s2z1
,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2z1 s2z2 þ s2z1 s2zS þ s2z2 s2zS , 2 SD1D2 s2zS þ S2D1S s2z2 þ S2D2S s2z1
k¼
y 2g
y 2g
lm2 h
SD1D2
2
ð11Þ
2 2 2 2 S2D1D2 s2zS þ y 2g lhm SD1S S2D1S s2z2 þ y 2g lhm ð2SD1D2 þSD2S Þ S2D2S s2z1 S2D1D2 s2zS þ S2D1S s2z2 þ S2D2S s2z1
The scattering vector Q can be written as: ! 2 4p lm zu þg SD1S , Q¼ h l
ð13Þ
because the angle in the first slit z0 is increased by the factor g(lm/h)2SD1S due to the gravity bending at distance SD1S. Taking into account (13), the distribution (9) can be rewritten as: 2 6 1 IðQ , lÞpexp4 2
2 3 2 " # Q 4lp /zuSþ g lhm SD1S 1 k/zuS2 7 : 5exp 2 2 s2zu 4pszu
ð14Þ
l
The first exponent in Eq. (14) is the Gaussian function responsible for the smearing of the reflected intensity due to the angular divergence with a gravity shift of mean grazing angle /z0 S+ g(lm/h)2SD1S from the angle y. The value of the second exponent is equal to unity at zero gravity. At non-zero gravity, the exponent decreases when wavelength increases. This exponent is responsible for the reduction of the reflected flux due to the neutrons fall before the sample. In TOF experiments, the neutron wavelength can be determined by the neutron time-of-flight t from the source to the detector at distance L th : ð15Þ lt ¼ mL The distribution of wavelengths can be approximated by the following Gaussian function: " # 1 ðllt Þ2 Rl ðl, lt Þp exp 2 , ð16Þ 2 stm þ Dt 2 =12 ðh=mLÞ2 pffiffiffiffiffiffiffiffiffiffiffi where stm ¼ T=ð2 2 ln 2Þ is the uncertainty of the moment of the neutron emission from the source (T is the pulse width), and Dt is the time width of the analyzer channel. Thus, the measured value of /QS can be written as the following expression: /Q S ¼
4p
lt
y:
ð17Þ
In real experiments, y can be determined with a finite precision. In our VITESS simulations, y was determined precisely by the line passing through the centers of all reflectometer elements. The total resolution function of the reflectometer is the integration over all wavelengths l, given as follows: Rð/Q S,Q Þ ¼
Z
ð10Þ
IðQ , lÞRl ðl,4py=/Q SÞdl:
ð18Þ
3. Parameters of simulations in the VITESS package The VITESS software package is a Monte Carlo instrumentation tool for the simulation of neutron scattering at pulsed and steadystate sources [7]. It describes the motion of a neutron as a classical particle in real 3-D space, excluding the sample where neutrons have to be considered as a wave. The choice of instrumental parameters was based on the real beam line characteristics of the reflectometer GRAINS [6]. In all the
:
ð12Þ
simulations, the distance from the source to the first slit D1 was SMD1 ¼10 m, the distance from D1 to the second slit D2 was SD1D2 ¼2 m, the distance from the sample S to the detector was L0 ¼4 m and y ¼15 mrad. The widths of the slits were w1 ¼2 mm and w2 ¼1 mm. The analyzer time channel width was Dt¼64 ms and the neutron pulse width was T¼320 ms. To analyze the role of the sample size along the xaxis, two values that represented a ‘‘big’’ sample with length of 100 mm and a ‘‘small’’ one with length of 20 mm were chosen. A thin monolayer with a critical angle of 5.56 10 4 rad/A˚ and thickness of 1500 A˚ on a substrate with a critical angle of 4.17 10 4 rad/A˚ was used as an idealized sample. The wavelength dependence of reflectivity from such a monolayer consists of a sequence of narrow oscillations whose positions and shapes are very sensitive to the resolution factor (Fig. 1). An incident spectrum of constant intensity for all wavelengths was used to exclude the factor of spectrum shape from consideration.
4. Results and discussion The dependence of the gravity effect on the distance between the second slit and the sample SD2S and on the sample size LS was analyzed. The reflectivities for two sample lengths, LS ¼20 and 100 mm, and three sample positions, SD2S ¼0.033 or 0.05 m (the sample position right after the second slit for two sample lengths correspondingly), SD2S ¼0.5 and 1 m are shown on Fig. 2. It can be seen that the agreement between the theoretical calculations in the frame of proposed approach and the Monte-Carlo simulations by means of the VITESS package is very good. In all cases, the deviations of reflectivity with zero and non-zero gravity can be observed. An increase in the distance between the second slit and the sample leads to higher deviation. Two factors, both of which are due to the bending of neutron trajectories in the gravity field, contribute to the deviation. The first factor is the increase in the grazing incidence angle, which leads to the shift of reflectivity fringes to higher wavelengths. If the angular divergence increases, the fringes broaden and the gravity shift becomes less distinguishable. The second factor is that neutrons fall before the sample, which leads to the flux deficiency for longer wavelengths. This effect is greater for a smaller sample length. In the case of non-zero gravity, each element of the reflectometer acts as a wavelength and angular filter while in the case of zero gravity, each element only selects angles. Positioning the sample as close as possible to the second slit minimizes the effect of gravity on the scattering pattern, but this is not possible to do in particular cases. For example, if it is necessary to install some device for polarization control before the sample (polarizer, flipper, etc.), the influence of gravity on the resolution cannot be avoided.
5. Conclusions The analytical beam-line analysis method for deriving the resolution function of the neutron TOF reflectometer accounting for the effect
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I. Bodnarchuk et al. / Nuclear Instruments and Methods in Physics Research A 631 (2011) 121–124
Fig. 2. Reflectivities for two sample lengths and three sample positions with respect to the second slit. The solid lines are the theoretical calculations, and the triangles and circles are the Monte-Carlo simulations prepared by the VITESS package with zero and non-zero gravity, respectively. All plots are represented from 12 A˚ where all curves are distinguishable.
of gravity was applied. The theoretical calculations are in good agreement with Monte-Carlo simulations, which mimic the real measurements with idealized samples. The shape of the reflectivity curve with non-zero gravity is defined by two factors that depend on the neutron wavelengths: angle deviations due to the neutron trajectory bending and a part of neutrons miss the sample. Positioning the sample right after the second collimating slit minimizes the effect of gravity. However, the effect of gravity is still present even in this case. The proposed theoretical approach makes it possible to take the resolution into account correctly and allows one to carry out real measurements with a broad wavelength band.
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