PHYSICAL REVIEW B 98, 214424 (2018)

Assembly of eight spherical magnets into a dotriacontapole configuration
Stefan Hartung,1,* Felix Sommer,1 Simeon Völkel,1 Johannes Schönke,2,† and Ingo Rehberg1,‡
2

1
Experimentalphysik V, Universität Bayreuth, D-95440 Bayreuth, Germany
Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan(Received 30 August 2018; revised manuscript received 31 October 2018; published 13 December 2018)
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.
DOI: 10.1103/PhysRevB.98.214424

I. INTRODUCTION

Within the forces determining the interplay of condensed
matter, the dipole-dipole interaction can be considered as
the most important one, because monopoles do not exist forneutral matter, and pure quadrupole, octopole, or hexadecapolinteraction tends to be masked by induced dipole moments.
While the interaction of quadrupoles is not too exotic [1]and includes examples from continuum mechanics [2], pureoctopole or even higher order interaction is different. Here wedemonstrate that the combination of eight dipoles in a simple
cubic arrangement leads to a 32-pole or dotriacontapole.
The exploration of the cuboidal dipole arrangement discussed here is triggered by the investigation of magnetic
nanoparticles, which have been reported to self-assemble intosuch configurations [3,4]. The most elementary cluster ofthis type contains only eight particles. It can also be assembled macroscopically as a cubic cluster from eight magnetic
spheres, as indicated by the left-hand side inset of Fig. 1, anddescribed previously [5,6]. The ground state of this arrangement is stable, and an interesting continuum [5,7]. In this state,
the spheres attract each other by the magnetic interaction. The
cuboidal arrangement is an attractor, provided that the spheresare brought sufficiently close to that configuration and areallowed to adjust their orientation towards the ground state,i.e., their mutual friction must not be too large. That is thereason why the arrangement shown by the left-hand side inset
of Fig. 1 can be assembled without needing a tremendous
amount of dexterity, and in that sense the arrangement canbe considered as almost self-assembled.
While the hallmark of a dipole is its field decay with thethird power of the distance, the combination of eight dipoles
could be expected to form a 16-pole or hexadecapole witha decay according to the sixth power. Amazingly enough, itturns out that the ground state of a cuboidal cluster of eight
dipoles shows a field decay with the seventh power. This is*

Stefan.Hartung@uni-bayreuth.de
Johannes.Schoenke@oist.jp‡
Ingo.Rehberg@uni-bayreuth.de†2469-9950/2018/98(21)/214424(6)explained by the symmetry of the ensuing ground state whichmake all lower-order terms vanish.
II. EXPERIMENTAL RESULTS

For reaching the ground state of the cluster, the eightspheres should be allowed to rotate freely. For that purpose
it is useful to provide a Teflon® spacer to reduce the frictionof the spheres, as shown in the right-hand side inset of Fig. 1.
Here, the eight neodymium magnets of diameter d = (19 ±0.05) mm are arranged in a cuboidal configuration by theholes at the corners of the white Teflon® cube, and kept
at an edge length L = (39.5 ± 0.05) mm by means of thenonmagnetic Teflon® spacer. A hole is drilled into that spacer
along the face diagonal, the (1,1,0) direction of the cube. This
allows us to move the Hall probe (the black tip) into the
cuboid, down to its center, by means of a stepper motor, using
0.1-mm steps. We adjust the spheres within their continuous
ground state to maximize the measured magnetic-flux density.
This is achieved by manually turning just one sphere aroundthe space diagonal as rotation axis; the other ones follow
accordingly due to the magnetic interaction.
The measured magnetic-flux density along the (1,1,0)
direction is shown in Fig. 1. It has a maximum at aboutr = 28 mm—where the Hall probe is closest to the spheres—and decays to zero both when approaching the center, andwhen increasing the distance from the cube. The solid line
corresponds to a fit of the numerical superposition of the flux
densities of eight accordingly arranged point dipoles, as givenby (1) discussed below.
The most important feature of this cuboidal arrangementof dipoles is the unusually steep decrease of the magneticflux density outside the cube. To quantify this decrease, Fig. 2
shows the data from Fig. 1 in a logarithmic plot. It becomes
obvious that the magnetic-flux density decays with the inverse
seventh power of the distance. To characterize this magnetic
cluster with an appropriate name, it must be recalled that thefield of dipoles decays with the third power, quadrupoles withthe fourth power, and so on. In that sense, the seventh power
corresponds to a dotriacontapole.
The fact that the field is expected to be zero at that
center of the arrangement is caused by the symmetry of the214424-1©2018 American Physical Society

HARTUNG, SOMMER, VÖLKEL, SCHÖNKE, AND REHBERG

FIG. 1. Magnetic-flux density as measured along a straight paththrough 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 righthand side inset a geometrically similar arrangement, but here with awhite 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 theblack part above the hole.

ground state. The measured deviations from that value can be
attributed to geometrical and experimental imperfections: The
dipole moments are not mathematically identical; they might
not have reached their ground state due to the finite amountof friction, and the Hall probe can reach the center of thearrangement only with a mm precision.
The increase of the flux density with the fourth power is inagreement with the numerical evaluation of the ground state.
According to that simulation, it even seems to be universal,
i.e., independent of the direction along which the field is
calculated. Compared to the seventh power of the decay thisfourth power seems less exotic. It is somehow reminiscent of
the field near the center of a Helmholtz pair of coils, where

PHYSICAL REVIEW B 98, 214424 (2018)

FIG. 3. The position r and the position vectors p of the dipole
moments m are taken from the center of the cluster. The orientationsof the dipoles in the continuous ground state are determined by theangle τ . The dipole configuration is sketched here for τ = 90◦ , whichcorresponds to the largest negative value of Bz along the (1,1,0)
direction.

the second-order terms vanish at a singular value of the coildistance.
III. THEORY

The scalar potential φ at position r for a distribution of N
dipoles with position vectors p and dipole moments m (seeFig. 3) is given byNm · (r − p )φ=.
(1)
4π |r − p |3
=1This equation is expected to describe the cluster field, because
the individual spheres have a pure dipole field, in agreementwith the theoretical expectation for homogeneously magnetized spheres [8] and our measurements presented in Appendix A. The numerical results of this equation correspondto the gray lines in Figs. 1 and 2.
To explain the behavior of the magnetic-flux density B in
the far field, we perform a multidipole expansion, where thepotential is expanded in a series for |p |  |r|,∞N
1  ∂ ν φ 
φ =· (p ⊗ · · · ⊗ p ).
ν! =1 ∂pν p =0 
ν=0ν times

FIG. 2. The magnetic-flux density measured along a path startingfrom 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, andevery 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 forcomparison with the expected asymptotic slopes. The dashed line
depicts the analytical solution (7) for the far field.

As an example, the quadrupole (second term in the expansion)
reads
N
1 
φ (2) =
[3(m · r)r − |r|2 m ] · p
4π |r|5 =1214424-2

1 [3 m ⊗ p − (m · p )I] · (r ⊗ r),
4π |r|5 =1 
N

=

M2

ASSEMBLY OF EIGHT SPHERICAL MAGNETS INTO A …

PHYSICAL REVIEW B 98, 214424 (2018)with the second-order unit tensor I. The second-order tensor
M2 is the quadrupole moment. Using Cartesian coordinatesyyp = (px , p , pz ), m = (mx , m , mz ) and r = (x, y, z), we
obtain
⎡
⎢ N
⎢
1
⎢
y y
φ (2) =
2px mx − p m − pz mz x 2⎢
5
4π |r| ⎢ =1
⎣


M2200

⎤

⎥
⎥
⎥
y
y
+3
px m + p mx xy + · · · ⎥.
⎥
=1
⎦



N


M2110

M2ij k are the Cartesian components of the moment M2
with i + j + k = 2. Using the moments, the potential can be
written as
φ=

∞

α=1

φ (α) =∞
1Mαij k x i y j zk .
2α+1
4π
|r|
α=1
i+j +k=α(2)

The cube ground state [5,9] is a highly shielded structure. Fora cube with edge length L and dipole moment magnitudes|m | = m we have(dipole)M1ij k = 0,(quadrupole)

M2ij k = 0,(octopole)

M3ij k = 0,

(hexadecapole)

M4ij k = 0,

(dotriacontapole)

The next order decays with |B| ∝ 1/s 9 because all momentswith even α are zero.
Equation (7) is displayed in Fig. 2 by the dashed line. The
solid lines in Figs. 1 and 2 are obtained numerically fromthe exact (1), with τ = 90◦ taken as the phase angle of thecontinuous ground state (see Appendix C). It is amazing that
this asymptotic prediction reaches the exact solution alreadyat a distance of about 50 mm, which can be considered as
sheer luck from an experimental point of view, because thefield is hardly detectable for our equipment at distances larger
than 100 mm. For the measurements shown there, the angle
τ was adjusted manually to obtain the largest signal of theHall probe to achieve an optimal signal-to-noise ratio. This
corresponds either to τ = 90◦ or to τ = 270◦ .
Note that the shape of the B(r ) curve shown in Figs. 1
and 2 is not universal, it rather depends on the direction of theline along which the flux density is measured. The 1/r 7 decay,
however, is a universal feature for all directions in the far-field
limit, |p |  |r|.
IV. CONCLUSION AND OUTLOOK(3)

M5311 = C sin(τ + π/3),

M5131 = C sin(τ + 5π/3),(4)M5113 = C sin(τ + 9π/3),(5)where τ = 0 . . . 2π is the current phase angle [5,9] ofthe continuous ground state√ as indicated in Fig. 3 andAppendix B, and C = 105 3/2 L4 m. There are restrictions
for the cube moments following from the symmetries of theground state [5]. The potential φ has to be zero in the threeplanes, x = 0 , y = 0 , z = 0, as well as on the four volume
diagonals, |x| = |y| = |z|. Together with (2) this leads toconditions for the nonzero moments Mαij k :Mαij k = 0.
i, j, k positive, odd ⇒ α odd, andi+j +k=αThis explains why the first nonzero moments appear in the
dotriacontapole,φ (5) =√
(x, y, z) = (s, s, 0)/ 2 and obtain the following expression
for the z component of the magnetic-flux density from (3)–(6):∂φ Bz (s, τ ) = −μ0 ∂z x=y=s/√2, z=0√ 
105 3/2 μ0 L4 m sin τ1=−+ O 9 . (7)
7
16π s
s

M5311 x 3 yz + M5131 xy 3 z + M5113 xyz3.
4π |r|11(6)

The magnetic-flux density is related to the potential throughB = −μ0 ∂φ/∂r. We parametrize the measurement along
the direction (1,1,0) with the radius parameter s throughIn summary, we have demonstrated that eight sphericalpermanent magnets assemble into a configuration which behaves like a dotriacontapole. This can be explained by a model
based on pure dipole-dipole interaction. This model is basedon symmetry considerations which are an idealization of theexperimental situation. The measurements make it clear thatthe conclusions drawn from the idealization are robust against(small) distortions, in particular the decay of the magneticflux density with 1/r 7 —a hallmark for a highly shielded
structure—survives.
This finding implies that storing strong magnets in a cubic
packing might be the optimal way for suppressing their field in
the outer surrounding. Moreover, the extremely steep field decay has remarkable consequences for the clustering dynamics:
If two dipole spheres, initially separated by say ten diameters,
needed one second to collide due to their attractive force, fordotriacontapoles of comparable strength, this process would
take more than one year (see Appendix B). Thus, dipoles
which manage to arrange themselves in this configuration are
fairly robust against further clustering. This argument is scale
invariant. It applies to macroscopic granules in the early stagesof planet formation [10], but could also shed some light on the
self-assembly dynamics of colloidal nanomagnets [3,4] used
for medical applications [11].
The plastic spheres shown in Fig. 4 demonstrate an attempt
to build a handful of such dotriacontapoles with the help ofa three-dimensional (3D) printer. Each sphere contains eightmagnetic dipoles in a cubic arrangement. This is provided
by three perpendicular walls inside these spheres, indicatedin the left-hand side inset, and eight holes along the space

214424-3

HARTUNG, SOMMER, VÖLKEL, SCHÖNKE, AND REHBERG

PHYSICAL REVIEW B 98, 214424 (2018)

FIG. 4. A cluster of 3D-printed dotriacontapoles. The inner partof these spheres contains three perpendicular walls as indicated by
the left-hand side inset. The colored magnetic spheres of 5 mmdiameter are placed inside these plastic spheres by the eight holes
along the space diagonals, as indicated by the right-hand side inset.

FIG. 5. The inset shows Bx (x, y ) measured in a plane, color
coded in red for strong fields to blue for weak ones. The directionof the x coordinate is chosen to be parallel to the dipole moment mand 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 m1 of these data.

diagonals, as indicated by the right-hand side inset. These
plastic spheres should thus interact with an extremely shortranged interaction force, which should asymptotically decay with the inverse 12th power of the mutual distance—
provided that the magnetic dipoles inside a sphere are in
their ground state. Measuring such a short-range interaction
between dipole clusters provides a challenge left to be facedin future work.

dipole [8],μ0 m (3 cos2  − 1)
,
(A1)
4π
r3

with  = arctan(y/x), r = x 2 + y 2 measuring the angle between the dipole moment and the position vector,
and the magnetic constant μ0 . With the shorthand notation
4π= f , this provides the magnitude of the magμ0 (3 cos2 −1)
netic moment,
Bx =ACKNOWLEDGMENTS

It is a pleasure to thank F. Braun, K. Huang, R. Richter,
W. Schöpf, and A. Weber for valuable hints and discussions.
This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) - Project No. 389197450.

m = Bx r 3 f .

(A2)

The resulting m as a function of the measured value ofBx (x, y) is plotted in Fig. 5 as a function of the distance ofthe Hall probe from the center of the sphere. The increasing

APPENDIX A: MAGNETIC SPHERES AS DIPOLES

The magnetic spheres (MK-19-C from magnets4youGmbH) have a diameter of d = (19 ± 0.05) mm. For explaining the experimental findings with a theoretical model basedon pure dipole-dipole interaction, it is crucial to demonstratethat these spheres can be described as magnetically hardpoint dipoles. Thus, we have measured the axial componentof the magnetic-flux density Bx of a single sphere along
the x direction in a 170-mm × 20-mm xy plane, as shownin the inset of Fig. 5. The flux density is measured by aHall probe (HU-ST1-184605, MAGNET-PHYSIK Dr. Steingroever GmbH). The 3D positioning of this probe is donewith a stepper motor (High-Z S-400T, with Zero-3 controller
from CNC-STEP), the interface (CNCPod) is programmablein G-Code, DIN/ISO 66025. A single-board microcontroller(Leonardo, Arduino) is additionally used for interfacing it toa PC.
To emphasize deviations from the point dipole approximation, and to extract the underlying magnetic moment, we
make use of the theoretically expected flux density of a point

FIG. 6. Data obtained from the measurements of the magneticflux density Bx 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. Bx scaled with x 3 2π/μ0 is shown in the larger plot. The
horizontal dashed line represents the sum of the magnetic momentsof the isolated spheres. The dashed vertical line represents the origin
at the contact point of the spheres.

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ASSEMBLY OF EIGHT SPHERICAL MAGNETS INTO A …

PHYSICAL REVIEW B 98, 214424 (2018)

scatter at larger distances r is caused by the fast decay ofthe magnetic-flux density. Based on these data, it seems
safe to conclude that the point dipole approximation for themagnetic-flux density of the sphere is reliable within ±2 %.
The mean value is (3.51 ± 0.18) J T−1 , which is well withinthe (3.54 ± 0.11) J T−1 claimed by the manufacturer. We havemeasured all eight dipoles used in the experiments describedhere in a similar way, they differ by an amount of ±3%.
To measure the mutual influence of such magnetic spheres,
we brought them in direct contact as shown in the left-handside inset of Fig. 6. The measured flux density along theaxis of the resulting two-dipole cluster is shown as the righthand side inset in Fig. 6. The position of the Hall probe
is measured as the distance from the center between thespheres. The data reveal roughly the typical 1/x 3 descent of adipole, but deviations from that scaling are hard to judge fromthis inset plot. To get a better resolution for the deviations
from the overall 1/x 3 decay, the data were multiplied withx 3 . After scaling with 2π/μ0 one gets an estimate for themagnetic moment, which is displayed on the vertical axisof Fig. 6. These scaled data decay monotonically with theposition x and reach the value of the sum of the two magnetic
moments asymptotically, which is indicated by the dashedline. The solid line is the theoretical estimation, based on
the superposition of the fields of the individually measured
moments m1 = 3.51 J T−1 and m2 = 3.50 J T−1 , with theirmutual distance given by the diameter of the spheres. The
good agreement between this curve and the data indicatesthat the magnets are hard ones: Their magnetic moment stays
constant even under the influence of the immediately adjacent
other magnet, at least within the experimental resolution on apercentage level.
APPENDIX B: DIPOLE ORIENTATION IN THE CUBEGROUND STATE

The dipole orientation within the ground state of the
cuboidal cluster can be parametrized by a single parameter,
namely the phase angle τ = 0◦ . . . 360◦ [5,9]. The configuration for the angle τ = 90◦ is illustrated in Fig. 3. The sourcecode of a Python script animating this state together with thecorresponding fields for adjustable values of τ is available[12]. The exact positions and orientations of the dipoles withrespect to τ are provided by Table I.

TABLE I. Description of pi and mi in the cube ground state.
i
⎛

0.5

mi ·
⎛

⎞

⎛

0.5
0.5

⎛

−0.5
0.5

⎞

⎛

0.5
0.5

⎛

−0.5
−0.5

⎝+sin(τ − 120◦ )⎠

⎞

⎛

0.5
−0.5

⎛

−0.5
−0.5

⎝−sin(τ − 120◦ )⎠

⎞

⎛
8

0.5
−0.5

+sin(τ )⎛⎞−sin(τ − 240◦ )⎝+sin(τ − 120◦ )⎠⎞+sin(τ )⎛⎞−sin(τ − 240◦ )⎝+sin(τ − 120◦ )⎠⎞+sin(τ )⎛⎞+sin(τ − 240◦ )⎝−sin(τ − 120◦ )⎠⎞+sin(τ )⎛⎞+sin(τ − 240◦ )⎝−0.5 ⎠

7⎞

+sin(τ − 240◦ )⎝ 0.5 ⎠

6

−sin(τ )⎛⎝ 0.5 ⎠

5⎞+sin(τ − 240◦ )⎝−0.5 ⎠

4−sin(τ )⎛⎝−0.5 ⎠

3

3/2/m
⎞
◦

⎝−sin(τ − 120◦ )⎠⎝ 0.5 ⎠

2√−sin(τ − 240 )⎝ 0.5 ⎠

1⎝+sin(τ − 120 )⎠◦⎞⎛−sin(τ )⎞−sin(τ − 240◦ )⎝−0.5 ⎠⎝−sin(τ − 120 )⎠−0.5−sin(τ )◦their mutual attraction—a characteristic time for the dynamics
of the self-assembly of magnetic clusters [3,4]—is obtainedby integrating over their inverse velocity. When assuming thatthese particles are suspended in a viscous fluid, that velocity is
proportional to the attractive force (Stokes’s law). T2 denotes
the pair of dipoles, T32 denotes the pair of dotriacontapoles.
The attracting force of these multipole pairs is assumed to be
the same when they are in contact at the distance of 1d,
 d/2 1
 d/2
2 r 12dr
T32
5 d v32 dr v∝F 5 d − d=  d/2=  d/2
4
1
2
r
T2
dr
−
dr
5d

=
APPENDIX C: ASSEMBLY TIME FOR DIPOLES VERSUS
THAT FOR DOTRIACONTAPOLES

pi

v2

5d

d

5 10 − 1≈ 0.4 × 108 .
13 105 − 1
13The time Tm for two multipoles of diameter d starting at
a distance of 10d to come into contact under the influence ofThis ratio turns, e.g., 1s for a dipole pair into 1a for thecorresponding pair of dotriacontapoles: They are fairly robust
against further clustering.

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PHYSICAL REVIEW B 98, 214424 (2018)

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