ARTICLE DOI: 10.1038/s41467-018-06593-4 OPEN Tensorial neutron tomography of three- dimensional magnetic vector fields in bulk materials A. Hilger1,2, I. Manke1, N. Kardjilov1, M. Osenberg2, H. Markötter1 & J. Banhart1,2 Knowing the distribution of a magnetic field in bulk materials is important for understanding basic phenomena and developing functional magnetic materials. Microscopic imaging tech- 1234567890():,; niques employing X-rays, light, electrons, or scanning probe methods have been used to quantify magnetic fields within planar thin magnetic films in 2D or magnetic vector fields within comparable thin volumes in 3D. Some years ago, neutron imaging has been demon- strated to be a unique tool to detect magnetic fields and magnetic domain structures within bulk materials. Here, we show how arbitrary magnetic vector fields within bulk materials can be visualized and quantified in 3D using a set of nine spin-polarized neutron imaging mea- surements and a novel tensorial multiplicative algebraic reconstruction technique (TMART). We first verify the method by measuring the known magnetic field of an electric coil and then investigate the unknown trapped magnetic flux within the type-I superconductor lead. 1 Helmholtz Centre Berlin for Materials and Energy (HZB), Institute of Applied Materials, Hahn-Meitner-Platz 1, 14109 Berlin, Germany. 2 Department of Materials Science and Technology, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany. Correspondence and requests for materials should be addressed to I.M. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:4023 | DOI: 10.1038/s41467-018-06593-4 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06593-4 urveying magnetic fields inside solid matter is a difficult down |↓〉 so that only spin up |↑〉 neutrons can pass. Ignoring for fl Schallenge but new methods are developing that allow one to a moment the ippers (F1 to F4) the now spin-polarized neutron fi visualize and quantify magnetic structures: Techniques for beam, intensity pro le I0(x, y), passes through the sample, during surface measurements such as magneto-optical Kerr microscopy1, which in each ray of the beam the spin orientation may change electron magnetic exchange force microscopy2,3 and spin- depending on the magnetic vector field distribution along the ray. 3 polarized scanning tunneling microscopy on the one hand, or Finally, another spin polarizer P2 called the spin analyzer and a transmission techniques such as soft X-ray holography/dichroism 2D detector are used to measure the transmitted intensity I(x, y) microscopy4–6 and Lorentz transmission electron microscopy/ given by: 7–9 holography on the other have been used to probe the internal Z ! magnetic structure of thin samples (up to some 100 nm thickness) ðÞ; ¼ ðÞ; Á À μ ðÞ Á 1 ½þ ðÞϑðÞ; ; even in three dimensions10. Ixy I0 x y exp att s ds |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}1 cos x y path 2 However, there is still a fundamental gap: The complete 3D |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ≤ 1ðspin rotationÞ structure of a magnetic vector field in (thick) bulk samples is ≤ 1ðattenuation by sampleÞ inaccessible by any of these techniques. A first approach in this ð1Þ direction provided maps of the magnetic domains within a thin film rolled into a cylinder11. More recently, an approach based on ptychographic X-ray nano-tomography has been demonstrated to μ fi provide magnetic contrast12,13. This technique has been further where att(s) is the linear attenuation coef cient in the sample extended to investigate the magnetic vector field in 3D in a along path s and defines the first term. The second term occurs 14 because neutrons pass the polarization analyzer P with an angle GdCo2 sample . The contrast used is based on the X-ray mag- ~ 2 netic circular dichroism (XMCD) at X-ray absorption edges of the ϑ(x, y) between the beam polarization vector P and the ~y axis corresponding material and restricts the usable X-ray energies to (see coordinate system in Fig. 1) at location (x, y). Therefore, from the measurement of transmission I(x,y) alone, only the polar that of the X-ray absorption edges of the corresponding materials, ~ which affects the maximum achievable penetration depths. angle between P and ~y is obtained. The azimuthal angle remains The fundamental problem of imaging and measuring magnetic unknown and requires additional measurements using two pairs fl vector fields inside solid matter is the necessity to use a probe that of spin ippers F1-F4. fi fl is sensitive to magnetic fields on the one hand, but penetrative The rst pair of spin- ippers F1/F2 is used to rotate the initial enough to reach the region of interest inside a material on the spin polarization vector of the neutron beam (parallel to ~y after ~ ~ ~ other. Light and X-rays interact mainly with atomic electron polarization by P1) into one of three possible axes x, y, and zby fi π shells and only weakly with a magnetic eld directly. Due to their an angle of 2. After passing the sample, the spin-polarization of spin, electrons are very sensitive to magnetic fields, but they have the neutrons can be in any direction. A second pair of spin- fl only limited penetration depths. In contrast, neutrons satisfy both ippers F3/F4 (see Fig. 1a) is used in combination with the spin conditions as their spin is affected by magnetic fields due to their analyzer to probe the spin polarization changes of the neutron intrinsic magnetic moment while they penetrate deeply into many beam in any of the three directions ~x, ~y, and ~z separately by π fl materials, especially metals, due to their zero electric charge and applying 2 spin ips again. Altogether 3 × 3 different measure- weak interaction with the atomic shell. It has been shown that ments are performed for different combinations of neutron beam neutron phase grating interferometry15,16 can be used to inves- spin polarization before and after the sample. In this way, all tigate magnetic domains17 and other magnetic structures such as components of the magnetic field in the sample can be probed. vortex-lattice domains18 two-dimensionally and three- Further details are explained in Methods. dimensionally19 in bulk samples, but only the domain walls Beside the difficulty to precisely set-up and adjust the various could be visualized and no information on the magnetic field neutron-optical elements, a further major challenge is to find a direction was obtained. On the other hand, spin-polarized neu- mathematical reconstruction procedure that allows one to extract tron imaging provides access to the magnetic field direction due information from the nine 3D data sets and eventually obtain to the precession of neutrons when passing a magnetic field and three sets of 3D data representing each magnetic vector field the measurable change of the spin-polarization direction of a component. Here, we develop a technique that resembles the beam. Several approaches have been suggested to make use of the well-known, although due its complexity barely used, multi- potential of spin-polarized neutron imaging for 3D magnetic plicative algebraic reconstruction technique (MART) algorithm vector field measurements in bulk materials20–23, but quantitative that was only made for scalar values, but can be extended to use tomography has only been demonstrated for simple, one- tensors, then called tensorial MART (TMART), see Methods. dimensional magnetic fields, where the field vectors vary only in strength but not in direction24,25. Magnetic vector field of an electric coil. We apply a three-stage Here we show a way to overcome the current limits by procedure to verify and apply the new method: First, the new designing a neutron imaging setup equipped with four spin- TMART technique is tested by letting it reconstruct a simulated flippers and two spin-polarizers and obtaining a set of nine tomography experiment on a calculated magnetic field of an individual tomography measurements instead of just one. The electric coil. Second, a real coil producing a known field is probed data obtained serve as an input for a specially developed recon- by neutrons and TMART applied. Comparison with the all- struction algorithm used to extract the full three-dimensional simulated experiment validates the experimental approach. magnetic vector field distribution. Finally, an unknown complex magnetic field inside a real sample is measured. Accompanying simulations based on simplified current patterns help to understand the results. Results First, we prove the reliability of the TMART algorithm by Basic principles. A sketch of the imaging setup is shown in Fig. 1. reconstructing the well-known magnetic field of an electric coil After monochromatization the neutrons pass the first spin that is calculated using the Biot-Savart law (Fig. 2a–c). Figure 1b fi fi polarizer P1. In a magnetic eld a neutron spin can take only one and Supplementary Movie 1 show some selected magnetic eld of two possible eigenstates, namely spin up |↑〉 or spin down |↓〉. lines taken from the calculated magnetic vector field to The spin-polarizer shown in Fig. 1 absorbs neutrons with spin demonstrate the complexity of the field. Based on the calculated 2 NATURE COMMUNICATIONS | (2018) 9:4023 | DOI: 10.1038/s41467-018-06593-4 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06593-4 ARTICLE a Spin-flipper Detector b Polarizer D P2 F4 F3 F2 y F 1 z P1 x Fig. 1 Tensor tomography. a Schematic drawing of the setup used for tensor tomography with spin-polarized neutrons, comprising spin polarizers (P), spin flippers (F) and a detector (D). b Selected magnetic field lines around an electric coil (calculation, see text and Methods) magnetic vector field we simulated an ideal neutron tensor Simulation and discussion. These observations suggest how tomographic measurement and used the virtually created nine 3D currents might flow in the superconductor and lead to a simple data sets as an input for our algorithm.