Assembly of eight spherical magnets into a dotriacontapole configuration

Stefan Hartung, Felix Sommer, Simeon Völkel, Johannes Schönke, and Ingo Rehberg

Phys. Rev. B 98, 214424 – Published 13 December 2018

Abstract

The magnetic field of a cuboidal cluster of eight magnetic spheres is measured. It decays with the inverse seventh power of the distance. This corresponds formally to a multipole named a dotriacontapole. This strong decay is explained on the basis of dipole-dipole interaction and the symmetry of the ensuing ground state of the cuboidal cluster. A method to build such dotriacontapoles is provided.

Received 30 August 2018
Revised 31 October 2018

DOI:https://doi.org/10.1103/PhysRevB.98.214424

Physics Subject Headings (PhySH)

Dipolar interaction Electronic & magnetic properties of clusters Magnetism

Magnetic colloids

Cluster models

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Stefan Hartung^1,*, Felix Sommer¹, Simeon Völkel¹, Johannes Schönke^2,†, and Ingo Rehberg^1,‡

¹Experimentalphysik V, Universität Bayreuth, D-95440 Bayreuth, Germany
²Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan

^*Stefan.Hartung@uni-bayreuth.de
^†Johannes.Schoenke@oist.jp
^‡Ingo.Rehberg@uni-bayreuth.de

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Issue

Vol. 98, Iss. 21 — 1 December 2018

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Images

Figure 1
Magnetic-flux density as measured along a straight path through the center of the cuboid. Only every 30th data point is shown. The solid line corresponds to the numerical superposition of the flux densities of eight accordingly arranged point dipoles with a magnetic moment $m = 3.48 J T^{- 1}$ . The left-hand side inset shows the principal cuboidal arrangement of the eight magnetic spheres, and the right-hand side inset a geometrically similar arrangement, but here with a white Teflon® spacer. The hole in that spacer allows us to take data inside the cuboid by means of the Hall probe, which is visible as the black part above the hole.
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Figure 2
The magnetic-flux density measured along a path starting from the center of the cuboid is represented by the circles. Only every tenth data point is shown at the left-hand side of the maximum, and every 40th data point at the right-hand side. The solid line is the same numerically obtained curve as in Fig. 1. The dash-dotted lines are for comparison with the expected asymptotic slopes. The dashed line depicts the analytical solution (7) for the far field.
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Figure 3
The position $r$ and the position vectors $p_{ℓ}$ of the dipole moments $m_{ℓ}$ are taken from the center of the cluster. The orientations of the dipoles in the continuous ground state are determined by the angle $τ$ . The dipole configuration is sketched here for $τ = 90^{\circ}$ , which corresponds to the largest negative value of $B_{z}$ along the (1,1,0) direction.
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Figure 4
A cluster of 3D-printed dotriacontapoles. The inner part of these spheres contains three perpendicular walls as indicated by the left-hand side inset. The colored magnetic spheres of 5 mm diameter are placed inside these plastic spheres by the eight holes along the space diagonals, as indicated by the right-hand side inset.
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Figure 5
The inset shows $B_{x} (x, y)$ measured in a plane, color coded in red for strong fields to blue for weak ones. The direction of the $x$ coordinate is chosen to be parallel to the dipole moment m and forms a horizontal plane with the perpendicular coordinate $y$ . An estimator for the magnetic moment is obtained from these data with (A2). The result is displayed as a function of $r$ by the circles. The dashed line represents the mean value $m_{1}$ of these data.
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Figure 6
Data obtained from the measurements of the magnetic-flux density $B_{x}$ of two spheres in contact. The raw data are shown in the inset, and the solid line shows the calculated superposition of two dipole fields. $B_{x}$ scaled with $x^{3} 2 π / μ_{0}$ is shown in the larger plot. The horizontal dashed line represents the sum of the magnetic moments of the isolated spheres. The dashed vertical line represents the origin at the contact point of the spheres.
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