PHYSICAL REVIEW B 88, 125104 (2013) Quantification of the neutron dark-field imaging signal in grating interferometry C. Gr¨ unzweig, 1,* J. Kopecek, 2 B. Betz, 1 A. Kaestner, 1 K. Jefimovs, 3 J. Kohlbrecher, 1 U. Gasser, 1 O. Bunk, 1 C. David, 1 E. Lehmann, 1 T. Donath, 1 and F. Pfeiffer 4 1 Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 2 Institute of Physics ASCR, 182 21 Praha 8, Czech Republic 3 Swiss Federal Laboratories for Material Science and Technology, CH-8600 D¨ ubendorf EMPA, Switzerland 4 Physics Department (E17), Technische Universit¨ at M ¨ unchen, D-85748 Garching, Germany (Received 28 June 2013; revised manuscript received 22 August 2013; published 4 September 2013) Here we report on a mathematical description for the neutron dark-field image (DFI) contrast based on the influence of the thickness-dependent beam broadening caused by scattering interactions and multiple refraction in the sample. We conduct radiography experiments to verify that the DFI signal exponentially decays as a function of thickness for both magnetic and nonmagnetic materials. Here we introduce a material-dependent parameter, the so-called linear diffusion coefficient . This allows us to perform a quantitative DFI-computed tomography. Additionally, we conduct correlative small-angle neutron-scattering experiments and validate the mathematical assumption that the angular broadening of the direct beam is proportional to the square root of the number of discrete layers. DOI: 10.1103/PhysRevB.88.125104 PACS number(s): 61.05.fg, 03.75.Dg I. INTRODUCTION Over the past years the neutron grating interferometry (nGI) technique 1,2 was successfully established as a routine method in neutron imaging and is now used at different facilities. 3–5 By the usage of the nGI technique information complementary to the conventional attenuation-based imaging, such as phase- contrast imaging 1 and dark-field imaging, is simultaneously obtained. Especially, the dark-field image modality gained rapidly in interest. Due to the magnetic moment of the neutron and hence its sensitivity to magnetic materials the dark-field contrast was used, on the one hand, to visualize bulk magnetic domain structures in two dimensions 6,7 and even in three dimensions 8 and, on the other hand, to visualize the magnetization processes. 9,10 Dark-field contrast was also used to investigate structural inhomogeneities which are below the standard detector resolution and varying in the micrometer range. 3,11 For both magnetic and nonmagnetic in- teractions the dark-field image (DFI) signal is related to small- angle/ultrasmall-angle neutron scattering (SANS/USANS) and to multiple refraction in the sample caused by vari- ations of either the nuclear or the magnetic interaction potential. Quantitative neutron attenuation computed tomography (CT) has been known for decades and is based on the Beer-Lambert law, where the intensity decays exponentially as a function of the product of sample thickness and the material-dependent linear attenuation coefficient . Tomo- graphic investigations of USANS signals are reported in Ref. 12 using a double-crystal diffractometer device; how- ever, no quantitative information is reported. Although DFI tomography investigations using the nGI setup are now routinely performed, 3,8 a quantitative description, as for the attenuation-based CT, is still missing. Corresponding activities have been reported for x rays in the literature, 13–16 but for neutrons, both the theoretical description and experimental verification are still missing. Both points are addressed in this paper. II. SETUP AND PHASE-STEPPING ANALYSIS A schematic depiction of the experimental setup used for the quantification of the DFI signal is shown in Fig. 1. In Fig. 1(a) the nGI interferometer setup is schematically shown, and it consists of a source grating G 0 , a phase grating G 1 , and an analyzer attenuation grating G 2 . The wedge-shaped sample is placed as close as possible to the front of the phase grating. The source grating G 0 (periodicity p 0 = 1076 μm) is an aperture mask with transmitting slits, placed in a monochromatic neutron beam with a wavelength of λ = 4.1 ˚ A and a wavelength spread of λ/λ =15 %. It creates an array of periodically repeating line sources and effectively allows the use of relatively large, i.e., centimeter-sized, neutron sources without compromising the coherence requirements for the interferometer formed by G 1 and G 2 . The grating G 1 (p 1 = 7.97 μm), placed at a distance l = 5.23 m behind G 0 , acts as a phase mask and imprints periodic phase modulations onto the incoming wave field. Through the Talbot effect, the phase modulation is transformed into an intensity modulation in the plane of G 2 (p 2 = 4 μm), forming a linear periodic fringe pattern perpendicular to the optical axis and parallel to the lines of G 1 . For our setup, the periodicity of the fringe pattern is equal to 4 μm. The absorption grating G 2 is placed at a distance d T = 19.7 mm behind G 1 . The periodicity and orientation of G 2 is matched to the fringes created by G 1 . G 2 is placed in the detection plane, immediately in front of the detector. The fringe pattern is transformed to an intensity modulation by a linear movement of G 0 . Only the intensity modulation as a function of the position of G 0 is recorded at the detector with a spatial resolution of 100 μm. 17 Both the transmission image (TI) and the DFI were processed out of a series of nine images each taken with an exposure time of 10 s. The series of nine images corresponds to an equidistant stepping of G 0 over one period along the x direction perpendicular to the grating lines. Thereby the intensity signal I (m,n,x g ) in each pixel (m,n) in the detector plane oscillates as shown in Fig. 1(b). To analyze these 125104-1 1098-0121/2013/88(12)/125104(6) ©2013 American Physical Society