Neutron tomography using a crystal monochromator

Nuclear Instruments and Methods in Physics Research A 621 (2010) 502–505 Contents lists available at ScienceDirect Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima Neutron tomography using a crystal monochromator W. Treimer a,b,n, S.O. Seidel a,b, O. Ebrahimi a,b a b Beuth Hochschule für Technik, University of Applied Sciences, Department of Mathematics, Physics & Chemistry, Luxemburger Str. 10, D-13353 Berlin, Germany Helmholtz Centre for Materials and Energy, G-G1, 14109 Berlin, Germany a r t i c l e in f o a b s t r a c t Article history: Received 28 April 2010 Received in revised form 12 June 2010 Accepted 14 June 2010 Available online 30 June 2010 For several neutron tomography methods, the use of monochromatic radiation is important. In order to select a wavelength band from a white spectrum commonly used devices are velocity selectors, choppers and crystal monochromators. We show how a crystal monochromator changes the wellknown L/D-ratio in CT instruments and how it determines the spatial resolution. We calculate the L/D-ratio as convolution integral and prove the results by the experimental determination of the modulation transfer function (MTF) of the used tomography system. & 2010 Elsevier B.V. All rights reserved. Keywords: Neutron tomography Energy selective tomography Crystal monochromator Spatial resolution 1. Introduction and theory Several tomography methods such as energy selective tomography, Bragg edge radiography, phase contrast tomography using a grating interferometer, diffraction enhanced radiography, refraction and USANS tomography, or tomography with polarized neutrons use monochromatic radiation due to the wavelength dependency of the interaction with matter [1–10]. Use of monochromatic radiation involves the selection of radiation from a white spectrum, which is done by monochromator crystals or (especially in the case of neutrons) with chopper devices and velocity selectors, respectively. The available intensity of latter techniques is quite high, about 10–20% of the spectrum but the corresponding Dl/l is often too poor for sharp imaging contrast due to wavelength dependent smearing effects that decreases the contrast in an image and thus information of details to be extracted. The width of the wavelength band that monochromator crystals select from the spectrum depends on the mosaic spread, which is usually of the order of 0.51, corresponding to a Dl/l  1%. In the case of tomography with synchrotron radiation the high monochromaticity of radiation is yielded with an (naturally) highly collimated beam, by perfect crystal (Si) Bragg reflections and multilayers suppressing harmonics. The used Si crystals act like perfect mirrors, i.e. the reflected beam divergence is nearly the same as the incident beam due to their narrow ‘‘Darwin n Corresponding author at: Beuth Hochschule für Technik, University of Applied Sciences, Department of Mathematics, Physics & Chemistry, Luxemburger Str. 10, D-13353 Berlin, Germany. Tel.: + 49 30 4504 2213, + 49 30 4504 2428, + 49 30 8062 2221; fax: +49 30 4504 2011. E-mail address: [email protected] (W. Treimer). 0168-9002/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2010.06.256 range’’ of some mrad. This is different for radiography and tomography instruments for neutrons and X-ray tubes. The ray geometry of both is similar to each other, they have a collimating system consisting of a (small) source (slit or hole, diameter D), a distance L to the object under investigation and a distance ld to the detector unit. Due to the size D of the micro-focus of an X-ray source (some mm) L can be kept short ( 1 m), in the case of neutron tomography D is of the order of cm and the length L about 7–18 m. In the case of tomography with monochromatic neutrons, the behavior of crystals as wavelength selecting devices is different to pin hole geometry and the L/D ratio has to be considered in conjunction with the geometry of the tomography experiment. As can be seen from Fig. 1 the beam divergence f is inversely proportionally to L/D that characterizes a tomography instrument. Assuming a pin hole having the diameter D the un-sharpness d (blur) of an image of a point is given by the product ld  f due to L/D¼ld/d (Fig. 1). There are three conditions to decrease the blur: D and ld must be small and L must be large. Considering a standard tomography instrument D is (in the case of neutrons) usually an aperture where the whole area of D contributes to image of a point at the detector. Increasing L decreases f and the blur d and improves the sharpness of the image at the detector position. Calculating the blur b(x, y) of a point in a sample, b(x, y) is the result of the convolution of the beam divergence f with the source function S¼S(x, Z). In the case of a neutron guide the divergence f ¼ fguide (m  0.61/nm)  l [nm]. The parameter m (m ¼1, 1.2, 1.5, 2,y) defines the reflectivity range of the coating of the neutron guide normalized to the critical angle of total reflection of natural Ni. In our case m was 1.2 and l ¼0.524 nm. W. Treimer et al. / Nuclear Instruments and Methods in Physics Research A 621 (2010) 502–505 503 Fig. 1. Definition of the L/D ratio. D is the size of the source, L is the distance between D and a point in the object, ld is the distance of the point in the sample and its image, D/L¼ f is the beam divergence. A point in the sample is blurred to d due to the size of D, length L and distance ld. If the function fincident ¼ f (x, Z)incident in each point (x, Z) of the slit S(x, Z) contains all properties such as the incident beam divergence in front (upstream) of D, the spatial intensity variation and wavelength distribution in the beam) and the L/D ratio, then b(x, y) is given by the 2D convolution integral Z x0 =2 Z Z0 =2 fðx, ZÞincident Sðxx,yZÞ dx dZ: ð1Þ bðx,yÞ ¼ x0 =2 Z0 =2 To calculate the blur b for all points of the detector Eq. (1) must be written as a 2D function of the horizontal and vertical divergences as Z ah =2 Z av =2 Fðyh , yv Þ ¼ fðah , av Þincident fðyh ah , yv av ÞL=D dah dav ah=2 Fig. 2. Two mosaic blocks (or ensembles of mosaic blocks) m1 and m2 are Dm apart from each other, each one reflects neutrons with a certain divergence fmosaic to the sample. Two points in the sample, P1 and P2, are illuminated from different points of the monochromator crystal (mosaic blocks are strongly magnified with respect to the distance L). having an aperture angle proportionally to the mosaic spread S, usually described with a Gaussian function fmosaic(ah, av) S. Assuming mosaic blocks centered at a point Ax,y the divergence can be described by the function fmosaic(ah,av) as ( ) ( ) lnð1=2Þ½av Ay =22 lnð1=2Þ½ah Ax =22 exp : fmosaic ðah , av Þ ¼ exp ðSh =2Þ2 ðSv =2Þ2 ð4Þ av =2 ð2Þ (yh, yv) is the (h ¼horizontal, v¼vertical) divergence incident in the point (x, Z), a is the amount of the divergence due to a given L/D ratio. The integration in Eq. (1) is taken over the total area of the aperture S(x0  Z0), the integration in Eq. (2) over the full angular range of a. In the case of yh ¼ yv and ah ¼ av the integral in Eq. (2) can be simplified to Z FðyÞ ¼ fðaÞincident fL=D ðyaÞ da ð3Þ with Eq. (3) the blur b(x, y) in each detector point can be calculated to be ld  F. A very similar formula for blurring due to scattering is given by the radial PsF-function as was published in [11–13]. These considerations are very likely to Eqs. (1)–(3) and both treatments yield very similar results. Finally, the size of the detector pixel must be taking into account. The size of a detector pixel ( 6 mm) is – up to now – much smaller than the spatial resolution of the instrument (  50 mm–100 mm), so one can simply add the half pixel diameter (3 mm) to the (calculated) blur due to the pure mapping of the image onto the pixel array1. These considerations cannot be applied to radiography and tomography geometries using a crystal monochromator as a source of radiation because it behaves different to an aperture as follows. In the case of an aperture S ¼S(x, Z) each point (x, Z) contributes to the image of a point as shown in Fig. 1. Replacing the aperture by a mosaic crystal, each mosaic block reflects neutrons individually within a cone angle equal the angular mosaic spread. Therefore, a mosaic block cannot illuminate all points in the sample 2 as shown in Fig. 2. The distance of all mosaic blocks to a point in a sample can taken to be the same  L, thus L and the area (size) of the mosaic block (or an ensemble of mosaic blocks SDmosaic) determines the size of illumination around a point P. Realizing the nature of these mosaic blocks the surface of the monochromator crystal must be assumed to be 2D set of individual sources (SDmosaic) that reflects neutrons within a cone 1 The convolution with a square function representing the pixel array yields a very similar result due to 6mmoo 50 mm/100 mm. 2 This holds if and only if L does not exceed a certain Ln (see below). The Sh,v means the full-width at half-maximum (FWHM) and is the deviation from the mean orientation of the mosaic blocks in the horizontal (h) and vertical (v) direction. The assumption Sh ¼ Sv ¼ S to be constant over the whole crystal surface simplifies Eq. (4). Treating a monochromator crystal as source of radiation, which contains in a unit volume mosaic blocks with orientations that cover the angular range of the mosaic spread S, the divergence of reflected radiation is given by convolution integral of the incident divergence with the function fmosaic of Eq. (4). The incident divergence is again represented by the divergence function fincident, so the reflected divergence from the monochromator crystal is given by the convolution integral Fmosaic ðyh , yv Þ ¼ Z ah =2 Z av =2 ah =2 av =2 fðah , av Þincident fðyh ah , yv av Þmosaic dah dav : ð5Þ Using the same assumption as for Eqs. (3),(5) can simplified to Z FðyÞ ¼ fincident ðaÞfmosaic ðyaÞ da: ð6Þ Eq. (6) looks very similar to Eq. (3), however, the amount of fmosaic compared to the D/L ratio changes the characteristics of the CT-instrument. With fincident ¼ 0.61 and fmosaic ¼0.41 we calculate with (6) F(y) ¼0.721. As is seen from Eq. (6) the function fmosaic contains only the angle a, which is independent of L, whereas fL/D contains L (because this divergence is defined as D/L). The consequences and the influence of the mosaic crystal on the L/D are shown in Fig. 3. L is the distance between the mosaic crystal and a sample point and SDmosaic the mean size of an ensemble of mosaic blocks contributing to the image of a point in the sample. For a given L the ratio L/SDmosaic will always be larger than L/D (D¼full size of the crystal) because D4 SDmosaic, only in the case of D¼ SDmosaic and of a perfect crystal where the mosaic block is the whole crystal, L/D¼ L/SDmosaic, perfect. In the case that L becomes large, LZLn, Ln is obtained from the intersection F(y)¼ const (0.721) with the function Dh,v, L/D becomes larger than¼L/SDmosaic, i.e. the corresponding divergence due to D/Ln is smaller than fmosaic and thus F. For LoLn the mosaic crystal converts the D/L ratio into a L-independent divergence  fmosaic which changes the imaging properties as indicated in Fig. 3. 504 W. Treimer et al. / Nuclear Instruments and Methods in Physics Research A 621 (2010) 502–505 2. Experiments Fig. 3. Beam geometry of a mosaic crystal, Dmosaic ¼ SDmosaic means a set of mosaic blocks reflecting radiation only within the angular mosaic spread to a point in the sample and causing a blur d. (D ¼size of the monochromator crystal, L is the distance of the sample from the monochromator crystal, ld ¼ constant distance sample point–detector, d ¼blur), see also Fig. 4. The experiments were performed with the V12c set up as shown in Fig. 5. From images measured earlier with this set up the blur d for ld ¼60 mm was much less than expected, i.e. dh,v was well below 0.8 mm. This behavior was investigated in detail and experimentally verified by measuring the spatial resolution by means of two methods. Firstly slits ld ¼60 mm apart from the detector with different slit openings (200–1000 mm) were moved perpendicular to the neutron beam with a step width twice of the slit opening, creating a fringe pattern. The transmitted intensities showed fringes that contrasts decreased with smaller slit and step width, as displayed in Fig. 6. By means of the contrast function K(n), given by KðnÞ ¼ Imax ðnÞImin ðnÞ Imax ðnÞ þ Imin ðnÞ with Imax(n), Imin(n)¼maximum/minimum intensity for different n (n ¼line pairs/mm), the MTF equivalent contrast function and thus the spatial resolution at 10% K(n) could be determined to be 1.55 line pairs/mm (Fig. 7). This corresponds to a horizontal blur of 320(23) mm or to a L/SDmosaic 187 for ld ¼60 mm. Fig. 4. Plot of the beam divergence vs the distance of the sample from the monochromator crystal. Dh and Dv are the height and width of the monochromator crystal, L¼ 2.08 m is the actual distance monochromator–sample. For a given beam divergence (Eq. (6)) equals 0.721 and for all Lo Ln ¼2.39 m (3.19 m) one yields L/SDmosaic 4L/Dh,v. Depending on SDmosaic L can be diminished much less than 2 m. For L4Lnh,v we get L/D 4Ln/SDmosaic (because D ¼ SDmosaic). So due to the mosaic crystal the incident divergence at each point in a sample is fmosaic becomes independent of L (for all LoLn). This behavior is interesting because moving the sample upstream to the monochromator (decreasing L, keeping ld constant) the incident intensity at a point increases (1/r2 law), whereas the smearing remains constant or even decreases due to the constant divergence fmosaic and ld. This is in contrast to the behavior of conventional CT-instrument where a decreasing L involves an increasing blur d, see also Fig. 4. From these considerations one can conclude that the L/D is smaller than L/SDmosaic, i.e. that a blur d for ld ¼60 mm must horizontally and vertically become smaller. To give some numbers on this behavior we had L¼2080 mm, the size of the crystal D was 30mm  40mm corresponding to ratios (L/D)h 70 and (L/D)v  52. Hence for ld ¼60 mm the horizontal blur dh becomes 0.87 mm and the vertical blur dv ¼1.15 mm. An observation of a blur smaller than these values would prove that the mosaic monochromator crystal as source did not work as a diaphragm. However, to measure the influence of the mosaic spread L must be kept below a certain Ln as follows. One can derive the maximum length Ln from L/DoLn/SDmosaic, i.e. as long as SDmosaic oD holds for the given incident beam divergence F(y). In our case one calculates Ln  2.39 m for Dh ¼30 mm and Ln 3.19 m for Dv ¼40 mm (see Fig. 4). Operating the set up with L¼2.08 m3 one should yield a measurable improvement of the L/SDmosaic due to the mosaicity of monochromator, and keeping in mind that SDmosaic oD should further improve the results. 3 The space between the monochromator crystal and the shutter was limited to approximately 1 m. Fig. 5. Geometry and layout of the V12c set up. C ¼ graphite monochromator, symmetric (0 0 2) reflection, l ¼0.467(2) nm, mosaic spread¼ 0.41, incident beam divergence¼ 0.61. Fig. 6. Intensity modulations for different slit widths (200–1000 mm) derived from their images (radiograms), distance of the slit from the detector ld ¼60 mm. W. Treimer et al. / Nuclear Instruments and Methods in Physics Research A 621 (2010) 502–505 505 be resolved with a L/D 70. These results are important for all tomography instruments using a mosaic crystal or even double monochromator devices [7,8], because in any case the flight paths L can be optimized which makes instruments much shorter than standard ones and due to the 1/r2-law one gets an intensity gain without loss of spatial resolution, which is also a very interesting result of this paper. The spatial resolution of the order 300 mm seems to be rather poor, however, this value was determined for a ld ¼60 mm, so for radiograms or tomography of vast objects or extended samples (diameter  60–120 mm) using environments such as cryostats and magnets, 300 mm spatial resolution is still an acceptable value, for small objects ( o25 mm the spatial resolution can be better than 200 mm). 3. Summary Fig. 7. The contrast function (equivalent to the modulation transfer function (MTF)) derived from images of Fig. 5. For the slit position 60 mm apart from the detector. At 10% contrast one gets  1.55 line pairs/mm corresponding of a blur or horizontal spatial resolution of  320(23) mm. Fig. 8. Spatial resolution (line pairs/mm) as a function of ld (distance from the detector), measured with a 500 mm thick Gd-edge, first point corresponds to a distance Gd-edge to detector  1 mm, difference stems from fh,incident a fv, incident. Unlike the classical L/D behavior of slits or diaphragms as radiation sources for radiography and tomography instruments, mosaic crystals must be considered as a set of individually reflecting sources, which are usually smaller than the whole crystal. In this paper the role of a mosaic crystal as monochromatic source of radiation was calculated and experimentally proved by measuring the spatial resolution with two different methods (slits and edge measurements). Limits are given for which an improvement of spatial resolution can be expected. Due to a shorter flight path one can enhance the neutron flux (1/r2 law), without loss of spatial resolution. This is valid for all LoLn for which L/DoLn/SDmosaic holds, for D¼ SDmosaic the monochromator crystal behaves like a diaphragm, where each point contributes to the image of a sample point. So the length of radiography and tomography instruments using mosaic crystals as source for monochromatic radiation can be planned to be much shorter than e.g. 8 m, e.g.  2 m or even less, gaining intensity and maintaining a spatial resolution of approximately 300 mm or even less. These are interesting features concerning test instruments for more elaborating experiments and for training students and scientist from other disciplines. Acknowledgment The geometrical L/Dh ¼70 would yield a blur¼0.87 mm, so one yields an improvement of spatial resolution of a factor  2.7. The second determination of the MTF and thus the spatial resolution was done by imaging a 500 mm thick Gd-edge recorded at different distances from the detector. The resultant line pairs/mm were deduced from Fourier transform of the first derivative of the particular edge function of the corresponding image (see Fig. 8). Comparing the edge-result with the slit-result we found for both results good agreement (for ld ¼60 mm). For this ld the spatial resolutions agreed within the experimental error, i.e. a horizontal blurring of 320(23) mm was determined with the slit measurements and 350(25) mm) was the results of the edge measurement. Both results are well below the 0.87 mm for the horizontal blur assuming a geometrical L/D. From Fig. 8 one yields for ld ¼60 mm horizontally 1.44 line pairs/mm corresponding to a horizontal blur dh  0. 35 mm and vertically 0.85 line pairs/mm corresponding to dv  0.6 mm. Both results prove the influence of the mosaic monochromator, which decreases the effective dimensions of D. One gets an instrumental (L/Dmosaic)v 180 and (L/Dmosaic)h  100 horizontally more than 2.7 times and vertically 1.9 times better than the geometrical L/D assuming the size of the monochromator crystal as the conventional ‘‘D’’ . This behavior was also verified by the reconstruction of tomograms that showed details that would not This work was supported by the BMBF Project no. 05KN7KF. References [1] M. Arif, R. Gregory (Eds.), Neutron Radiography, Proceedings of the WCNR-8, DEStech Publications, Inc., Downing, 2006. [2] I.S. Anderson, R.L. McGreevy, H.Z. Bilheux (Eds.), Neutron Imaging and Applications, Springer Verlag, 2009. [3] E. Lehmann, PRAMANA—Journal of Physics, Indian Academy of Sciences 71 (4) (2008) 653. [4] W. Treimer, M. Strobl, A. Hilger, Appl. Phys. Lett. 83 (2) (2003) 398. [5] M. Strobl, W. Treimer, A. Hilger, Appl. Phys. Lett. 85 (2) (2004) 488. [6] W. Treimer, N. Kardjilov, U. Feye-Treimer, A. Hilger, I. Manke, M. Strobl, Appl. Solid State Phys. 45 (2005) 407. [7] W. Treimer, M. Strobl, N. Kardjilov, A. Hilger, I. Manke, Appl. Phys. Lett. 89 (2006) 203504. [8] N. Kardjilov, I. Manke, M. Strobl, A. Hilger, W. Treimer, M. Meissner, T. Krist, J. Banhart, Nat. Phys. 4 (2008) 399. [9] M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, F. Pfeiffer, Phys. Rev. Lett. 101 (2008) 123902. [10] F. Pfeiffer, C. Gruenzweig, O. Bunk, G. Frei, E. Lehmann, C. David, Phys. Rev. Lett. 96 (2006) 215505. [11] R. Hassanein, F. de Beer, N. Kardjilov, E. Lehmann, Physica B: Condens. Matter 385–386 (Part 2) (2006) 1194. [12] R. Hassanein, E. Lehmann, P. Vontobel, Nucl. Instr. and Meth. A 542 (2005) 353. [13] R. Hassanein, Doctorate Thesis, Swiss Federal Institute of Technology, Zurich, 2006.