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High Homogeneity Permanent Magnet for Diamond Magnetometry

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High Homogeneity Permanent Magnet for Diamond Magnetometry High homogeneity permanent magnet for diamond magnetometry Arne Wickenbrock,1, 2 Huijie Zheng,1, 2 Georgios Chatzidrosos,1, 2 Joseph Shaji Rebeirro,1, 2 Tim Schneemann,1, 2 and Peter Blmler2 1Helmholtz-Institut, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany 2Johannes Gutenberg-Universit¨atMainz, 55128 Mainz, Germany (Dated: August 18, 2020) Halbach magnets are a source of homogeneous magnetic field in an enclosed volume while keeping stray fields at a minimum. Here, we present the design, construction, and characterization for a 10 cm inner diameter double Halbach ring providing a homogeneous (< 100 ppm over 1:0 × 1:0 × 0:5 cm3) magnetic field of around 105 mT, which will be used for a diamond based microwave-free widefield imaging setup. The final characterization is performed with a novel fiberized diamond-based sensor on a 3D translation stage documenting the high homogeneity of the constructed Halbach array and its suitability for the proposed use. Keywords: Halbach, dipole, Mandhala, magnetic resonance, NMR, MRI, NV-center, GSLAC, wide-field, microwave-free I. INTRODUCTION the proposed experiment can be seen in Figure 1. A large area diamond containing NV centers is placed in a back- ground field of 102.4 mT. A camera can then be used to Cylindrical Halbach magnets [1] have become a stan- directly read-off magnetic field variation originating from dard design for nuclear magnetic resonance (NMR) at samples placed on the top of the diamond surface. De- field strength typically below 2 tesla [2]. Depending on ploying the protocols developed in [6] would allow full 3D the ratio of inner and outer diameter, they can gener- vector magnetic field images. The major component of ate very homogeneous magnetic flux densities. They are the magnetic bias field is supposed to be supplied by a particularly attractive because their design has the high- Halbach magnet. These magnets provide very stable and est flux to mass ratio [3] and they essentially produce highly homogeneous fields and avoid strong electrical cur- no stray fields. Halbach magnets can also be designed rents and associated cooling problems. The fine tuning to allow a high degree of optical or mechanical access can then be done by small resistive coil arrangements (cf. (e.g. for lasers, optics, mechanical parts or very long Fig. 1). samples), which is a requirement for certain experiments. In conjunction with magnetometry based on nitrogen- From the perspective of magnet design, the request vacancy (NV) centers in diamond a background field of of a bias-field of defined strength and high homogeneity 102.4 mT allows for magnetic sensitivity without the need is somewhat different from the typical NMR-approach, for microwaves [4, 5], which are required at most other where the field has to be as strong as possible for given magnetic background fields. A homogeneous magnetic constraints. Therefore, we do review the calculation of field of this magnitude over a certain volume allows for the resulting magnetic flux in Halbach multipoles and microwave-free imaging of magnetic structures on top of compare the homogeneity of different discrete designs. an NV center hosting diamond sample. As suggested in [11], the best homogeneity can be ob- Recent magnetometry-protocol developments even al- tained by avoiding pronounced edges in the magnetic low for microwave-free vector-magnetic field measure- material. For this reason, the homogeneity of a mag- ments using just a single axis NV center ensemble [6]. net assembled from pieces improves with the order of the Wide-field imaging of magnetic phenomena is an estab- polygons used. As the limiting case deploying cylinders, lished technique in the field of diamond magnetometry spherocylinders or spheres result in even better perfor- and requires applied oscillating magnetic fields at around mance. The latter have the disadvantage that their mag- 2.8 GHz. The range of application goes from imaging netization direction can not be aligned with a geometrical magnetic bacteria in a biology context [7] to probing re- feature. Hence, it is very likely that alignment errors are arXiv:2008.07339v1 [physics.ins-det] 17 Aug 2020 manent magnetization of small rock samples to study the introduced during magnet construction. In [12] this prob- history of geophysical magnetic fields [8]. Microwaves lem is solved by aligning cylindrical magnets in a Halbach can have a detrimental effect on living samples and get configuration by an anchor magnet which represents the severely altered in the presence of conductive materials. line dipole whose stray field defines the orientation of the However, oscillating fields can also be used to probe con- magnetization in an ideal Halbach [13]. However, the ductivity, which is commonly done with coils, but has mutual interaction between the cylindrical magnets sets recently become more popular in a low frequency regime a principal limit to the achievable homogeneity by this with atomic magnetometers [9] as well as NV center approach. Nevertheless, we used the idea of orienting based sensors [10]. With a microwave-free magnetic wide- magnetic cylinders in the field of another (orientation) field imaging setup, conductivity maps can be produced magnet, and then fixate them with well-defined orienta- conveniently with high spatial resolution. A schematic of tion inside a suitable support structure. These composite 2 FIG. 1. Schematic of a proposed experiment using the described Halbach magnet: a large diamond with a thin layer of nitrogen vacancy centers is placed in a homogeneous 102.4 mT magnetic field. At this field, the so-called ground-state level anticrossing (GSLAC) occurs, and the red photoluminescence (PL) of the diamond under illumination with a green 532 nm pump laser beam depends on the local magnetic field. Small changes in the local magnetic field from a sample placed on top of the diamond can now be directly imaged with an objective and a camera. Providing millions of simultaneous magnetic field measurements with potential diffraction limited resolution. Additional field coils can be used to shim the field (e.g. from the 105 mT to the working point at 102.4 mT i.e. the field corresponding to the GSLAC), apply gradients or oscillating fields to probe different aspect of the magnetic properties of the sample and/or allow demodulation with a lock-in amplifier. elements are then used to assemble the final magnet with II. DESIGN CONCEPT high precision. A. Halbach magnets The manuscript is structured as follows: at first we present in detail the design ideas behind this Halbach In order to avoid confusion we first want to define a magnet and how the flux can be calculated for realistic nomenclature. The final assembly is called \Magnet". assemblies. The problem to predict the target field of a It consists of several smaller, permanent magnets, refer- given magnet assembly is reviewed and solved in detail enced as \PM". If the PMs have an identical shape, mag- because we are not aware of a comprehensive summary in netization strength and direction the resulting magnet the literature. Furthermore, the analytical treatment al- is called \Mandhala" (Magnetic Arrangement for Novel lows to create seed designs, that then can be refined in nu- Discrete Halbach Layout [14]). merical simulations. We then compare different magnet In this section we review the equations to calculate geometries and designs in terms of homogeneity and field the magnetic field of Halbach cylinders and particularly strength. In the following section, we document the con- Mandhalas and check their validity by comparison with struction process from the individual permanent magnet finite element (FEM)-simulations. pieces to the final assembled double Halbach ring mag- Cylindrical Halbach magnets consist of magnetization net. Finally, the magnet is characterized with a fiberized which is arranged around a center at a position angle, α, diamond vector magnetometer moved via a 3D transla- while the magnetization points at a magnetization angle tion stage in the volume of the assembly. The resulting (see Fig. 2) , central homogeneity of 7µT (67 ppm) at a central field β = (1 + k)α with k 2 Z n f0g (1) around 104.5 mT is confirmed with NMR measurements and suitable for the suggested widefield imaging experi- which is an integer multiple of α. The constant k deter- ment. mines the polarity p = 2k of the resulting magnetic field. 3 The sign of k determines if the entire flux will be inside (k > 0) or outside (k < 0) the magnetic structure [14]. The most interesting settings for NMR-magnets are first of all k = +1, resulting in an inner dipole (p = 2) with a homogeneous field inside and no outer stray field. How- ever, for spatially encoding experiments (~k- or ~q-space) k = +2, resulting in a quadrupolar field (p = 4), is very interesting since very strong magnetic field gradients can be generated this way [15]. In ideal Halbach magnets [1], the direction of the magnetization ~m rotates continuously in the magnetic material at a fixed radius R, ~iβ ~i(1+k)α ~iα m~ (~r) = BRe = BRe withr ~ = Re (2) FIG. 2. The magnets are positioned on a circle of radius R at an angle α with respect to the x-axis. Their magnetization where the tilde indicates complex variables and B is the (red arrow) is oriented at an angle β which is a multiple of α R (cf. eq. (1). remanence of the magnetic material. As long as cylinders cannot be magnetized with such properties, real Halbach systems have to be made from discrete PM pieces and The following abbreviations are used in table I: many different designs have been discussed to do so [2]. They consist either from cylindrical or wedge-shaped seg- p cos 2π − sin 2π − 2 sin π − 4π ments with individual magnetization directions or from Ξ(N) := N N p 4 N (4) π 4π identical pieces which are mounted in a supporting struc- 2 cos 4 − N + 2 ture at individual angles, the latter is also known as a Mandhala [14].
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