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Analysis of water scattering used for calibration of small-angle neutron scattering (SANS) measurements
Physica B 291 (2000) 152}158
Analysis of water scattering used for calibration of small-angle neutron scattering (SANS) measurements P. Lindner *, F. Leclercq, P. Damay Institut Laue Langevin, BP 156, F-38042 Grenoble cedex 9, France LASIR-CNRS, 13, rue de Toul, F-59046, Lille, France Received 27 October 1999
Abstract In a SANS study of critical scattering on CO , we were faced with the problem of achieving highest resolution: a deviation from the Q\ power law of a few percent had to be characterized and critical #uctuations in the size range between 10 and 1000 As were measured. Experiments were done in the broad Q-range available at the instrument D11 at ILL, using several wavelengths (4.5)j (As ))10) and sample-to-detector positions (1.7)¸ (m))35.7). In this paper we discuss the problem of calibration using a water sample at di!erent experimental con"gurations, i.e. sample-to-detector distances and wavelengths. We consider geometric e!ects and the e$ciency of the two-dimensional BF -multidetector, as well as inelasticity- and multiscattering e!ects as a function of incident wavelength and water thickness. 2000 Elsevier Science B.V. All rights reserved. Keywords: Small-angle neutron scattering; Calibration; Critical scattering; Water thermalization
1. Introduction Calibration is an important problem in smallangle neutron scattering (SANS). Several aspects must be taken into account, the relative importance of each depending on the speci"c aim of the experiment. One of the main purposes of SANS is to determine the size of large objects and thus the relative angular dependence of the scattered intensity has to be determined with the best possible accuracy. In many cases, the absolute intensity (differential scattering cross section) is needed: absolute calibration is a very di$cult task, which has
* Corresponding author. Fax: #33-4-76-20-71-20. E-mail address: [email protected] (P. Lindner).
been considered in several papers [1], the work of Wignall and Bates [2] being certainly the reference for SANS. In few cases, a high precision is also required, but not necessarily on an absolute scale. In a study of critical scattering of CO [3], we were faced with the problem of achieving highest resolution: a deviation from the Q\-power law of a few percent had to be characterized and critical density #uctuations of size m, ranging from 10 to 1000 As were measured. Experiments were done in the broad Q-range available at D11 (ILL, Grenoble), using several wavelengths (4.5)j (As ))10) and sample-to-detector positions (1.7)¸ (m))35.7). At a given value of the correlation length m, all data recorded at various j and ¸ had to be regrouped to yield a single scattering curve in the whole accessible
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 3 9 7 - 6
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momentum transfer range (10\)Q (As \))3; 10\); the required relative precision is of the order of 0.3% which is not usual in SANS. The instrument D11 is equipped with a twodimensional multidetector with +3800 active `cellsa of size 1 cm. Data treatment is achieved in two steps, the critical scattering being expected to be azimuthally isotropic: (i) intensities I"f (x, y) are "rst regrouped for each con"guration (j, ¸) as a function of the radius r (cm) around the actual centre of di!usion, yielding I"f (r). (ii) Calibration with a reference scatterer, measured with the same con"guration allows to regroup, as a function of momentum transfer Q, all data obtained from di!erent experimental con"gurations at a given m. The calibration procedure has to meet three points, given the required precision: corrections for di!erent e$ciencies of individual `cellsa, correction for a detector intrinsic pro"le as a function of r and relative normalization of the scattering level obtained at di!erent values of ¸ and j.
153
Fig. 1. Experimental patterns obtained with the instrument D11 (ILL, Grenoble) for a light water sample (0.1 cm thick, temperature ¹"253C) for j"10 As as a function of r (r"distance from the centre of di!usion). The normalization of the intensity by the scattering level at r"0 allows to compare the parabolic shapes displayed by water in terms of the slope B as de"ned in Eq. (1). ¸"10 m (triangles): B"9.75;10\; ¸"2.5 m (squares): B"7.28;10\; ¸"1.1 m (circles): B"!2.72; 10\.
2. Water as a SANS calibrator radius r with 2.1. Angular dependence of the observed water scattering pattern
I(r)"I(r"0)[1#Br].
The calibration of the D11-detector for variations in e$ciency of individual `cellsa and scattering levels for each wavelength j and shorter sample-to-detector distances ¸ ()10 m) was performed with a standard sample of water (H O) of thickness 0.1 cm at ambient temperature. Such a calibration is based on the assumption that intrinsic water scattering is #at in the low-Q range, i.e. predominantly incoherent (multiple scattering processes are mainly incoherent-incoherent) and the intensity decrease as a function of Q due to inelasticity is not detectable. Inelasticity results also in a stronger scattering in forward direction and this e!ect has to be taken into account only for absolute calibration of the scattering power [2,4]. Instead of an expected #at pattern for the water sample, the experimental signals display a parabolic shape as shown in Fig. 1 for three sample-to-detector distances. The intensity can be "tted as a function of the
This parabolic shape of the water scattering is observed for all sample-to-detector distances ¸, investigated in the range 1.1)L (m))13. The value of the parameter B increases with ¸ for a given wavelength. Furthermore, B depends also on the wavelength j, for a given sample-to-detector distance ¸. At larger distances ¸'10 m, however, the water counting rate is too low for obtaining the slope B precisely enough to be useful for calibration. Water measurements at larger ¸ are only used to determine the average counting level and the calibration spectrum has to be reconstructed from measurements obtained with a good statistics at small ¸ [3]. In order to be able to calibrate sample runs measured at large sample-to-detector distances ¸, and to regroup data obtained at several wavelengths, it is thus necessary to analyse the behaviour B"f (¸, j).
(1)
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2.2. Analysis of the apparent curvature of the water scattering pattern In order to perform a correct normalization of the data, collected at di!erent experimental con"gurations, it is necessary to identify the causes of the shape of the water scattering curve. The SANS-instrument D11 (ILL, Grenoble) had been operated at the time of the experiment with a LETI position-sensitive multidetector of size 64;64 cm (see schematic view in Fig. 2). Its spatial resolution is 1 cm in the x- and in the y-direction. The central part of the detector consists of two glass plates at a distance of 2 cm with electrically conducting metallic tracks (cathodes) and anode wires placed in the middle. The detection volume is "lled with BF gas. Following the nuclear n-a detection reaction between neutrons and the BF, the electrical charges of the a-particles are registered. Towards the outside part of the detector, the `activea detection volume is separated by a thin aluminium membrane against a second pressure compensation volume "lled with CO gas. The volume between the front glass plate and the (theoretically #at) membrane contains also BF -gas;
this `deada (non-counting) volume has a path length of the order of 0.1 cm. To a "rst-order approximation, the parabolic shape of the water scattering can be associated with three e!ects: (1) A parallax ewect: the planar detector is tangent to the scattering sphere and the 1 cm detector `cellsa intercept decreasing scattering solid angles d) as the scattering angle 2h increases. An isotropic scattering will then give rise to a decreasing signal when going from the centre to the edges of the detector. The overall intensity of the scattering cone de"ned by the angle 2h, which intercepts the detector plane on a circular annulus of mean radius r, can be written as I(r) "I(r) cos(2h)"I(r) cos[arctan(r/¸)]. (2) (2) A specixc D11 **parabola++ ewect: the D11 BF -detector has the particularity that the mem brane, separating the BF -detection gas volume from the CO -compensation gas volume is de formed towards the outside, due to a pressure di!erence between both volumes [3]. As a consequence, the detection e$ciency is not homogeneous
Fig. 2. Schematic view of the LETI-type BF detector at D11 (in use until May 1999). Due to a pressure di!erence between the CO and BF volumes the distance l is a function of the radius from the detector centre: l "l [1!er], assuming a parabolic deformation of the P separating membrane (curvature e) [4,7].
P. Lindner et al. / Physica B 291 (2000) 152}158
over the surface of the detector: the deformed membrane induces a radius-dependent absorption of the scattered neutrons in the non-active (`deada) BF volume, just in front of the active volume, between the electrodes. Assuming that the deformation of the separation membrane is parabolic, the absorption path in the BF -`deada volume is written as l(r)"l [1!er], (3) where l is the thickness of the dead volume at the detector centre (r"0) and e is the curvature of the deformed separation membrane. (3) An angular ewect inside the detection chamber: for a scattering angle 2h, the path of neutrons falling on the detection volume (separation distance of the electrodes l "2 cm) is lengthened by ? 1/cos(2h). Thus, the radius dependence of the water signal, considering these three e!ects, can be completely described by I(r) "cos(2h) I(r"0) exp(!ajl (1!er)/cos2h) ; exp(!ajl ) 1!exp(!ajl cos2h) ? ; . (4) 1!exp(!ajl ) ? Eq. (4), divided by the heading factor cos(2h), may be expanded to the "rst order in r, using the cosine expansion cos(2h)"1!r/(2¸)#O(r):
I(r) 1 1 "1#ajr l e! I(r"0) cos(2h) 2¸
l exp(!ajl ) ? # ? 2¸ 1!exp(!ajl ) ? #O(r)+1#B r,
(5)
where B +B#3/(2¸), l (cm) is the path length in the `deada volume for r"0, l (cm) is the `activea ? path length in the detection volume, l "2 cm, and ? p (j ) a"N j with N the number of absorbing B atoms per cm, and p "3835 barn is the absorption
155
Fig. 3. Relative contributions to the apparent parabolic shape of the normalized water scattering intensity as a function of r (squares: j"10 As , ¸"2.5 m). The resulting curvature is a combination of three e!ects: the projection of the di!usion sphere on the planar detector (a cos(2h)-term, line 1), e!ective path in the detection chamber (line 2) and absorption in the dead volume whose thickness is r-dependent [line 3, see Eq. (3)].
cross section of B at a wavelength j "1.798 As ; the absorption coe$cient k(cm\) is k"aj. Eq. (5) describes the apparent slope B for the water signal as a function of r, depending on the detector geometry, the wavelength j and the sample-to-detector distance ¸. Fig. 3 shows the respective contribution of these three e!ects to the shape of a water scattering pattern, recorded at ¸"2.50 m and j"10 As . Assuming known values of the detector parameters k and l from e$ciency measurements, the deformation parameter e can be evaluated from a set of B -values, obtained by "tting measured water scattering patterns at di!erent ¸ and j with I "A[1#B (¸, j)r]. 3. Results and analysis of inelasticity e4ects on the water scattering 3.1. A direct measure of a and l in the forward direction (r"0) The overall detector-e$ciency in the forward direction, including the wavelength dependent
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Fig. 4. Wavelength-dependent e$ciency E(j) of the BF de tector [4,7]. The full line is a "t according to Eq. (6). The e$ciency-decrease at larger wavelengths is due to neutron absorption in the `deada volume before detection. With l "2 cm ? we obtain for the detector constant k"kl /j"aj"0.153 As \ ? and for l "0.41 cm.
Fig. 5. Experimental (symbols, same as in Fig. 1) and calculated values using Eq. (5) (lines) of the slope B as a function of the incident wavelength j . Values for ¸"1.1 m are shown as a dashed line and the limiting behaviour for ¸'5 m is reported as a solid line (see also Eq. (7)).
e$ciency of the LETI BF gas detector and the absorption e!ect in the `deada volume, can be written as
At small ¸ distances, however, a change of slope B and a slight departure from linearity should be observed when 1/¸ is no more negligible, compared to l e. We have measured the curvature of the standard water sample (thickness d"0.1 cm, temperature ¹"253C) for many con"gurations as a function of ¸ (from 1.1 to 13 m) and j (ranging from 4.5 to 20 As ). The resulting B at distances ¸"1.10 and 5 m are reported in Fig. 5. The experimental results show a strong departure from the expected behaviour at long wavelengths. If one is con"dent that Eq. (5) takes into account the main geometric parameters of the detector, one is obliged to question the value of the actual wavelength of the scattered neutrons and to consider inelasticity e!ects on the scattering by light water. The thermal energy of water molecules at room temperature is much larger than the kinetic energy of a long wavelength neutron beam and there is no doubt that thermalisation occurs; this e!ect is increased by multiscattering. One may "rst assume that inelasticity could be neglected for the shorter wavelengths. Then the
E(j)"[1!exp(!ajl )] exp(!ajl ). ?
(6)
From a direct measurement of the overall e$ciency at several wavelengths, using a calibrated monitor, it was possible to determine with accuracy the detector constant k"al as well as the absorption ? path length in the `deada volume l . The result of a least-squares "t using Eq. (6) yields k"0.153 As \ and l "0.41 cm for the D11 BF detector (see Fig. 4) [4,7]. 3.2. Data analysis and inelasticity ewects In Eq. (5), at large values of sample-to-detector distances ¸, the parameter B is expected to vary almost linearly with wavelength; indeed, for the detector under study, the leading term inside the bracket is l e which is an order of magnitude larger than the other parameters for ¸'2.5 m; then B +al ej.
(7)
P. Lindner et al. / Physica B 291 (2000) 152}158
157
curvature e of the separation membrane may be calculated from Eq. (7) at large enough sample-todetector distances. For j)6 As one "nds e+ 0.00038 cm\. The simplest model is to assume that the energy exchange is proportional to the energy di!erence between the water sample and the neutron beam, giving rise to an average e!ective scattered wavelength j : j . (8) j " (1#gj This zero-order correction may be used in Eq. (5) with j in place of the incident wavelength j . For the whole set of data (110 values of B at all distances and wavelengths), one "nds, with a good account of all data, e"0.000397 cm\ and a thermalization factor g"0.0057 As \, that is to say for incident wavelengths j of 5, 10 and 15 As , j are found to be 4.8, 8.0 and 10.0 As , respectively. 3.3. Measurement of the slope B as a function of ¸, j and water thickness d A simplistic model as developed in Section 3.2 is probably su$cient to perform a relative calibration, since the wavelength problem is a secondary correction e!ect. A precise characterization of inelasticity and multiscattering e!ects on the water sample is necessary, however, in order to perform a correct absolute calibration. More insight on the e!ect is obtained by changing the thickness of the water sample. Fig. 6 reports the values of B as a function of j obtained for 5 water cell thicknesses ranging from 0.03 to 0.5 cm [4]; the experiments have been performed at a sample-to-detector distance of 2.5 m and the almost straight line corresponds to the values of B obtained with Eq. (5) and the corresponding incident wavelength j . At a given wavelength, the value of B extrapolated at thickness zero should be placed on this straight line. It appears, as suggested in the previous section, that thermalization is weak at short wavelengths (4.5}6 As ). A detailed analysis of these data will help to develop a precise model of the wavelength dependence [5]. This analysis will be carried out together with that of a direct
Fig. 6. Values of the slope B obtained for "ve values of water thickness ranging from 0.03 to 0.5 cm and at a sample-todetector distance of ¸"2.5 m, as a function of the incident wavelength [4]. The solid line is the behaviour described by Eq. (5) with ¸"2.5 m. The deviation of the experimental points from this line, at wavelength values depending upon the thickness of the water sample, indicates thermalization of incident neutrons by light water.
measurement of the energy exchange obtained by time-of-#ight on the same sample [6].
4. Discussion In Fig. 7, are reported the results obtained for critical scattering of CO at a temperature corre sponding to a value of the correlation length m"226 As , at several sample-to-detector distances (3.5, 7.5, 16 and 35 m) and a wavelength of j "6 As [3]. The four experimental con"gura tions gave a large overlap in Q. All the data agree within the experimental uncertainty (0.3% for ¸"3.5 m, 0.5% for ¸"7.5 m, 1% for ¸"16 m and 3% for ¸"35 m). Without applying the above corrections, the di!erence between the overlapping parts of the spectra recorded at 3.5 and 7.5 m was larger than 3%. This shows that a careful calibration does improve considerably the precision of the results. The work reported in this paper mainly refers to a relative calibration of the D11-BF detector, us ing di!erent experimental con"gurations (¸, j); in
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standard sample, using the empirical correction factor g(j) for calculation of the e!ective di!erential scattering cross section of water [1,2,7], a!ects at longer wavelengths (j *12 As ) the shape of the scattering curve by a combination of the geometric and inelastic e!ects described above [7]. Indeed, the precise determination of the inelastic e!ect in the water sample which is in progress [5], will allow a better determination of the empirical correction factor g(j), separating a general term depending on the scattering of water (incoherent scattering and inelasticity e!ect) from a speci"c term depending on each particular instrument (e$ciency, detector geometry).
Fig. 7. Normalized data for critical scattering of CO with j"6 As , using four sample-to-detector distances and covering an overall Q-range from 1.2;10\ to 1.1;10\ As \. The experimental con"gurations were chosen in such a way that they overlap in a large range. The quality of the regrouping is a good test of the validity of Eqs. (4) and (5).
other words, spectra recorded at di!erent sampleto-detector distances or di!erent wavelength may be regrouped with a good con"dence. But these results are also important with respect to the conditions of absolute calibration. For instance, the absolute calibration of SANS data with a H O
References [1] J.P. Cotton, in: P. Lindner, Th. Zemb (Eds.), Neutron, X-ray and Light Scattering } Introduction to an Investigative Tool for Colloidal and Polymeric Systems, North-Holland, Amsterdam, 1991, pp. 19}31. [2] B.F. Wignall, F. Bates, J. Appl. Crystallogr 20 (1987) 28. [3] P. Damay, F. Leclercq, R. Magli, P. Lindner, Phys. Rev. B 58 (18) (1998) 12038. [4] P. Lindner, Technical Report ILL98 LI 10 T, 1998. [5] P. Lindner, F. Leclercq, P. Damay, in preparation, 2000. [6] A.R. Rennie, R. Ghosh, J. Appl. Crystallogr. SAS Proceedings, in press. [7] P. Lindner, J. Appl. Crystallogr. SAS Proceedings, in press.
Analysis of water scattering used for calibration of small-angle neutron scattering (SANS) measurements P. Lindner *, F. Leclercq, P. Damay Institut Laue Langevin, BP 156, F-38042 Grenoble cedex 9, France LASIR-CNRS, 13, rue de Toul, F-59046, Lille, France Received 27 October 1999
Abstract In a SANS study of critical scattering on CO , we were faced with the problem of achieving highest resolution: a deviation from the Q\ power law of a few percent had to be characterized and critical #uctuations in the size range between 10 and 1000 As were measured. Experiments were done in the broad Q-range available at the instrument D11 at ILL, using several wavelengths (4.5)j (As ))10) and sample-to-detector positions (1.7)¸ (m))35.7). In this paper we discuss the problem of calibration using a water sample at di!erent experimental con"gurations, i.e. sample-to-detector distances and wavelengths. We consider geometric e!ects and the e$ciency of the two-dimensional BF -multidetector, as well as inelasticity- and multiscattering e!ects as a function of incident wavelength and water thickness. 2000 Elsevier Science B.V. All rights reserved. Keywords: Small-angle neutron scattering; Calibration; Critical scattering; Water thermalization
1. Introduction Calibration is an important problem in smallangle neutron scattering (SANS). Several aspects must be taken into account, the relative importance of each depending on the speci"c aim of the experiment. One of the main purposes of SANS is to determine the size of large objects and thus the relative angular dependence of the scattered intensity has to be determined with the best possible accuracy. In many cases, the absolute intensity (differential scattering cross section) is needed: absolute calibration is a very di$cult task, which has
* Corresponding author. Fax: #33-4-76-20-71-20. E-mail address: [email protected] (P. Lindner).
been considered in several papers [1], the work of Wignall and Bates [2] being certainly the reference for SANS. In few cases, a high precision is also required, but not necessarily on an absolute scale. In a study of critical scattering of CO [3], we were faced with the problem of achieving highest resolution: a deviation from the Q\-power law of a few percent had to be characterized and critical density #uctuations of size m, ranging from 10 to 1000 As were measured. Experiments were done in the broad Q-range available at D11 (ILL, Grenoble), using several wavelengths (4.5)j (As ))10) and sample-to-detector positions (1.7)¸ (m))35.7). At a given value of the correlation length m, all data recorded at various j and ¸ had to be regrouped to yield a single scattering curve in the whole accessible
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 3 9 7 - 6
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momentum transfer range (10\)Q (As \))3; 10\); the required relative precision is of the order of 0.3% which is not usual in SANS. The instrument D11 is equipped with a twodimensional multidetector with +3800 active `cellsa of size 1 cm. Data treatment is achieved in two steps, the critical scattering being expected to be azimuthally isotropic: (i) intensities I"f (x, y) are "rst regrouped for each con"guration (j, ¸) as a function of the radius r (cm) around the actual centre of di!usion, yielding I"f (r). (ii) Calibration with a reference scatterer, measured with the same con"guration allows to regroup, as a function of momentum transfer Q, all data obtained from di!erent experimental con"gurations at a given m. The calibration procedure has to meet three points, given the required precision: corrections for di!erent e$ciencies of individual `cellsa, correction for a detector intrinsic pro"le as a function of r and relative normalization of the scattering level obtained at di!erent values of ¸ and j.
153
Fig. 1. Experimental patterns obtained with the instrument D11 (ILL, Grenoble) for a light water sample (0.1 cm thick, temperature ¹"253C) for j"10 As as a function of r (r"distance from the centre of di!usion). The normalization of the intensity by the scattering level at r"0 allows to compare the parabolic shapes displayed by water in terms of the slope B as de"ned in Eq. (1). ¸"10 m (triangles): B"9.75;10\; ¸"2.5 m (squares): B"7.28;10\; ¸"1.1 m (circles): B"!2.72; 10\.
2. Water as a SANS calibrator radius r with 2.1. Angular dependence of the observed water scattering pattern
I(r)"I(r"0)[1#Br].
The calibration of the D11-detector for variations in e$ciency of individual `cellsa and scattering levels for each wavelength j and shorter sample-to-detector distances ¸ ()10 m) was performed with a standard sample of water (H O) of thickness 0.1 cm at ambient temperature. Such a calibration is based on the assumption that intrinsic water scattering is #at in the low-Q range, i.e. predominantly incoherent (multiple scattering processes are mainly incoherent-incoherent) and the intensity decrease as a function of Q due to inelasticity is not detectable. Inelasticity results also in a stronger scattering in forward direction and this e!ect has to be taken into account only for absolute calibration of the scattering power [2,4]. Instead of an expected #at pattern for the water sample, the experimental signals display a parabolic shape as shown in Fig. 1 for three sample-to-detector distances. The intensity can be "tted as a function of the
This parabolic shape of the water scattering is observed for all sample-to-detector distances ¸, investigated in the range 1.1)L (m))13. The value of the parameter B increases with ¸ for a given wavelength. Furthermore, B depends also on the wavelength j, for a given sample-to-detector distance ¸. At larger distances ¸'10 m, however, the water counting rate is too low for obtaining the slope B precisely enough to be useful for calibration. Water measurements at larger ¸ are only used to determine the average counting level and the calibration spectrum has to be reconstructed from measurements obtained with a good statistics at small ¸ [3]. In order to be able to calibrate sample runs measured at large sample-to-detector distances ¸, and to regroup data obtained at several wavelengths, it is thus necessary to analyse the behaviour B"f (¸, j).
(1)
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2.2. Analysis of the apparent curvature of the water scattering pattern In order to perform a correct normalization of the data, collected at di!erent experimental con"gurations, it is necessary to identify the causes of the shape of the water scattering curve. The SANS-instrument D11 (ILL, Grenoble) had been operated at the time of the experiment with a LETI position-sensitive multidetector of size 64;64 cm (see schematic view in Fig. 2). Its spatial resolution is 1 cm in the x- and in the y-direction. The central part of the detector consists of two glass plates at a distance of 2 cm with electrically conducting metallic tracks (cathodes) and anode wires placed in the middle. The detection volume is "lled with BF gas. Following the nuclear n-a detection reaction between neutrons and the BF, the electrical charges of the a-particles are registered. Towards the outside part of the detector, the `activea detection volume is separated by a thin aluminium membrane against a second pressure compensation volume "lled with CO gas. The volume between the front glass plate and the (theoretically #at) membrane contains also BF -gas;
this `deada (non-counting) volume has a path length of the order of 0.1 cm. To a "rst-order approximation, the parabolic shape of the water scattering can be associated with three e!ects: (1) A parallax ewect: the planar detector is tangent to the scattering sphere and the 1 cm detector `cellsa intercept decreasing scattering solid angles d) as the scattering angle 2h increases. An isotropic scattering will then give rise to a decreasing signal when going from the centre to the edges of the detector. The overall intensity of the scattering cone de"ned by the angle 2h, which intercepts the detector plane on a circular annulus of mean radius r, can be written as I(r) "I(r) cos(2h)"I(r) cos[arctan(r/¸)]. (2) (2) A specixc D11 **parabola++ ewect: the D11 BF -detector has the particularity that the mem brane, separating the BF -detection gas volume from the CO -compensation gas volume is de formed towards the outside, due to a pressure di!erence between both volumes [3]. As a consequence, the detection e$ciency is not homogeneous
Fig. 2. Schematic view of the LETI-type BF detector at D11 (in use until May 1999). Due to a pressure di!erence between the CO and BF volumes the distance l is a function of the radius from the detector centre: l "l [1!er], assuming a parabolic deformation of the P separating membrane (curvature e) [4,7].
P. Lindner et al. / Physica B 291 (2000) 152}158
over the surface of the detector: the deformed membrane induces a radius-dependent absorption of the scattered neutrons in the non-active (`deada) BF volume, just in front of the active volume, between the electrodes. Assuming that the deformation of the separation membrane is parabolic, the absorption path in the BF -`deada volume is written as l(r)"l [1!er], (3) where l is the thickness of the dead volume at the detector centre (r"0) and e is the curvature of the deformed separation membrane. (3) An angular ewect inside the detection chamber: for a scattering angle 2h, the path of neutrons falling on the detection volume (separation distance of the electrodes l "2 cm) is lengthened by ? 1/cos(2h). Thus, the radius dependence of the water signal, considering these three e!ects, can be completely described by I(r) "cos(2h) I(r"0) exp(!ajl (1!er)/cos2h) ; exp(!ajl ) 1!exp(!ajl cos2h) ? ; . (4) 1!exp(!ajl ) ? Eq. (4), divided by the heading factor cos(2h), may be expanded to the "rst order in r, using the cosine expansion cos(2h)"1!r/(2¸)#O(r):
I(r) 1 1 "1#ajr l e! I(r"0) cos(2h) 2¸
l exp(!ajl ) ? # ? 2¸ 1!exp(!ajl ) ? #O(r)+1#B r,
(5)
where B +B#3/(2¸), l (cm) is the path length in the `deada volume for r"0, l (cm) is the `activea ? path length in the detection volume, l "2 cm, and ? p (j ) a"N j with N the number of absorbing B atoms per cm, and p "3835 barn is the absorption
155
Fig. 3. Relative contributions to the apparent parabolic shape of the normalized water scattering intensity as a function of r (squares: j"10 As , ¸"2.5 m). The resulting curvature is a combination of three e!ects: the projection of the di!usion sphere on the planar detector (a cos(2h)-term, line 1), e!ective path in the detection chamber (line 2) and absorption in the dead volume whose thickness is r-dependent [line 3, see Eq. (3)].
cross section of B at a wavelength j "1.798 As ; the absorption coe$cient k(cm\) is k"aj. Eq. (5) describes the apparent slope B for the water signal as a function of r, depending on the detector geometry, the wavelength j and the sample-to-detector distance ¸. Fig. 3 shows the respective contribution of these three e!ects to the shape of a water scattering pattern, recorded at ¸"2.50 m and j"10 As . Assuming known values of the detector parameters k and l from e$ciency measurements, the deformation parameter e can be evaluated from a set of B -values, obtained by "tting measured water scattering patterns at di!erent ¸ and j with I "A[1#B (¸, j)r]. 3. Results and analysis of inelasticity e4ects on the water scattering 3.1. A direct measure of a and l in the forward direction (r"0) The overall detector-e$ciency in the forward direction, including the wavelength dependent
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Fig. 4. Wavelength-dependent e$ciency E(j) of the BF de tector [4,7]. The full line is a "t according to Eq. (6). The e$ciency-decrease at larger wavelengths is due to neutron absorption in the `deada volume before detection. With l "2 cm ? we obtain for the detector constant k"kl /j"aj"0.153 As \ ? and for l "0.41 cm.
Fig. 5. Experimental (symbols, same as in Fig. 1) and calculated values using Eq. (5) (lines) of the slope B as a function of the incident wavelength j . Values for ¸"1.1 m are shown as a dashed line and the limiting behaviour for ¸'5 m is reported as a solid line (see also Eq. (7)).
e$ciency of the LETI BF gas detector and the absorption e!ect in the `deada volume, can be written as
At small ¸ distances, however, a change of slope B and a slight departure from linearity should be observed when 1/¸ is no more negligible, compared to l e. We have measured the curvature of the standard water sample (thickness d"0.1 cm, temperature ¹"253C) for many con"gurations as a function of ¸ (from 1.1 to 13 m) and j (ranging from 4.5 to 20 As ). The resulting B at distances ¸"1.10 and 5 m are reported in Fig. 5. The experimental results show a strong departure from the expected behaviour at long wavelengths. If one is con"dent that Eq. (5) takes into account the main geometric parameters of the detector, one is obliged to question the value of the actual wavelength of the scattered neutrons and to consider inelasticity e!ects on the scattering by light water. The thermal energy of water molecules at room temperature is much larger than the kinetic energy of a long wavelength neutron beam and there is no doubt that thermalisation occurs; this e!ect is increased by multiscattering. One may "rst assume that inelasticity could be neglected for the shorter wavelengths. Then the
E(j)"[1!exp(!ajl )] exp(!ajl ). ?
(6)
From a direct measurement of the overall e$ciency at several wavelengths, using a calibrated monitor, it was possible to determine with accuracy the detector constant k"al as well as the absorption ? path length in the `deada volume l . The result of a least-squares "t using Eq. (6) yields k"0.153 As \ and l "0.41 cm for the D11 BF detector (see Fig. 4) [4,7]. 3.2. Data analysis and inelasticity ewects In Eq. (5), at large values of sample-to-detector distances ¸, the parameter B is expected to vary almost linearly with wavelength; indeed, for the detector under study, the leading term inside the bracket is l e which is an order of magnitude larger than the other parameters for ¸'2.5 m; then B +al ej.
(7)
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curvature e of the separation membrane may be calculated from Eq. (7) at large enough sample-todetector distances. For j)6 As one "nds e+ 0.00038 cm\. The simplest model is to assume that the energy exchange is proportional to the energy di!erence between the water sample and the neutron beam, giving rise to an average e!ective scattered wavelength j : j . (8) j " (1#gj This zero-order correction may be used in Eq. (5) with j in place of the incident wavelength j . For the whole set of data (110 values of B at all distances and wavelengths), one "nds, with a good account of all data, e"0.000397 cm\ and a thermalization factor g"0.0057 As \, that is to say for incident wavelengths j of 5, 10 and 15 As , j are found to be 4.8, 8.0 and 10.0 As , respectively. 3.3. Measurement of the slope B as a function of ¸, j and water thickness d A simplistic model as developed in Section 3.2 is probably su$cient to perform a relative calibration, since the wavelength problem is a secondary correction e!ect. A precise characterization of inelasticity and multiscattering e!ects on the water sample is necessary, however, in order to perform a correct absolute calibration. More insight on the e!ect is obtained by changing the thickness of the water sample. Fig. 6 reports the values of B as a function of j obtained for 5 water cell thicknesses ranging from 0.03 to 0.5 cm [4]; the experiments have been performed at a sample-to-detector distance of 2.5 m and the almost straight line corresponds to the values of B obtained with Eq. (5) and the corresponding incident wavelength j . At a given wavelength, the value of B extrapolated at thickness zero should be placed on this straight line. It appears, as suggested in the previous section, that thermalization is weak at short wavelengths (4.5}6 As ). A detailed analysis of these data will help to develop a precise model of the wavelength dependence [5]. This analysis will be carried out together with that of a direct
Fig. 6. Values of the slope B obtained for "ve values of water thickness ranging from 0.03 to 0.5 cm and at a sample-todetector distance of ¸"2.5 m, as a function of the incident wavelength [4]. The solid line is the behaviour described by Eq. (5) with ¸"2.5 m. The deviation of the experimental points from this line, at wavelength values depending upon the thickness of the water sample, indicates thermalization of incident neutrons by light water.
measurement of the energy exchange obtained by time-of-#ight on the same sample [6].
4. Discussion In Fig. 7, are reported the results obtained for critical scattering of CO at a temperature corre sponding to a value of the correlation length m"226 As , at several sample-to-detector distances (3.5, 7.5, 16 and 35 m) and a wavelength of j "6 As [3]. The four experimental con"gura tions gave a large overlap in Q. All the data agree within the experimental uncertainty (0.3% for ¸"3.5 m, 0.5% for ¸"7.5 m, 1% for ¸"16 m and 3% for ¸"35 m). Without applying the above corrections, the di!erence between the overlapping parts of the spectra recorded at 3.5 and 7.5 m was larger than 3%. This shows that a careful calibration does improve considerably the precision of the results. The work reported in this paper mainly refers to a relative calibration of the D11-BF detector, us ing di!erent experimental con"gurations (¸, j); in
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standard sample, using the empirical correction factor g(j) for calculation of the e!ective di!erential scattering cross section of water [1,2,7], a!ects at longer wavelengths (j *12 As ) the shape of the scattering curve by a combination of the geometric and inelastic e!ects described above [7]. Indeed, the precise determination of the inelastic e!ect in the water sample which is in progress [5], will allow a better determination of the empirical correction factor g(j), separating a general term depending on the scattering of water (incoherent scattering and inelasticity e!ect) from a speci"c term depending on each particular instrument (e$ciency, detector geometry).
Fig. 7. Normalized data for critical scattering of CO with j"6 As , using four sample-to-detector distances and covering an overall Q-range from 1.2;10\ to 1.1;10\ As \. The experimental con"gurations were chosen in such a way that they overlap in a large range. The quality of the regrouping is a good test of the validity of Eqs. (4) and (5).
other words, spectra recorded at di!erent sampleto-detector distances or di!erent wavelength may be regrouped with a good con"dence. But these results are also important with respect to the conditions of absolute calibration. For instance, the absolute calibration of SANS data with a H O
References [1] J.P. Cotton, in: P. Lindner, Th. Zemb (Eds.), Neutron, X-ray and Light Scattering } Introduction to an Investigative Tool for Colloidal and Polymeric Systems, North-Holland, Amsterdam, 1991, pp. 19}31. [2] B.F. Wignall, F. Bates, J. Appl. Crystallogr 20 (1987) 28. [3] P. Damay, F. Leclercq, R. Magli, P. Lindner, Phys. Rev. B 58 (18) (1998) 12038. [4] P. Lindner, Technical Report ILL98 LI 10 T, 1998. [5] P. Lindner, F. Leclercq, P. Damay, in preparation, 2000. [6] A.R. Rennie, R. Ghosh, J. Appl. Crystallogr. SAS Proceedings, in press. [7] P. Lindner, J. Appl. Crystallogr. SAS Proceedings, in press.