Experimental observation of magnetic poles inside bulk magnets via q ≠ 0 ?> Fourier modes of magnetostatic field

The pole-avoidance principle of magnetostatics results in an angular anisotropy of the magnetic neutron scattering cross section d &Sgr; M / d &OHgr; ?> . For the case of a sintered Nd–Fe–B permanent magnet, we report the experimental observation of a ‘spike’ in d &Sgr; M / d &OHgr; ?> along the forward direction. The spike implies the presence of long-wavelength magnetization fluctuations on a length scale of at least 60 nm. Using micromagnetic theory, it is shown that this type of angular anisotropy is the result of the presence of unavoidable magnetic poles in the bulk of the magnet and is related to the q ≠ 0 ?> Fourier modes of the magnetostatic field. Thus, our observation proves the existence of such modes.


Introduction
The dipole-dipole interaction is one of the most fundamental interactions in condensed-matter physics. For instance, the scattering of thermal neutrons by magnetic materials, the Overhauser effect in nuclear magnetic resonance and the related phenomenon of dynamic nuclear polarization [1], or the appearance of van-der-Waals forces can all be described using the concept of the interaction between (electrical and magnetic) dipoles.
In recent years, the magnetodipolar interaction moved into the focus of attention within the framework of the study of so-called spin-ice compounds (Dy and Ho-based pyrochlores) (e.g. [2][3][4][5][6][7][8][9][10][11]). In these systems, it is possible to achieve a substantial magnetic-pole separation such that individual poles behave like magnetic monopoles, in this way realizing Diracʼs hypothesis in practice. Magnetic monopoles in spin-ice compounds give rise to frustration, dipolar ferromagnetic coupling, exotic field-induced phase transitions and unusual glassiness (see, e.g. [12] for a recent review).
Besides their long-range nature, dipole-dipole interactions are anisotropic in the sense that the energy of a given arrangement of dipoles depends not only on the distance between them but also on their orientation. A textbook example is the energy of two parallel-aligned magnetic moments m 1 and m 2 , which takes on values of Unlike spin-ice compounds, for macroscopic bulk magnets with ∼10 23 atomic spins per cm 3 , the picture of discrete magnetic moments becomes a burden. It is more convenient to describe their magnetization state by a continuous magnetization vector field M r ( ), and to associate the interaction of the discrete atomic magnetic moments-within the Lorentz continuum approximation-with the magnetostatic self-interaction energy where H r ( ) d denotes the magnetostatic field and the integral is taken over all space. Since the sources of H d are inhomogeneities of M (either in orientation and/or in magnitude), positiveness of E m (or, strictly speaking, the existence of its lower bound, achieved only in the absence of sources) implies that the magnetodipolar interaction tries to avoid any sort of magnetic volume (− M · ) or surface (n M · ) poles; this is the content of the so-called poleavoidance principle.
Similarly to the poles themselves, the field H r ( )  [14]. For uniformly magnetized bodies of ellipsoidal shape, H d s is uniform inside the material; if furthermore M is aligned along a principal axis of the sample (assumed in the following to be the z-direction of a . Under these assumptions and with reference to the title of the paper, the field H d s may then be considered as the macroscopic , is obtained by solving Maxwellʼs magnetostatic equations (no currents) where  M q ( ) represents the Fourier transform of M r ( ). Equation (2) implies that pole avoidance prefers magnetic structures with Fourier modes  M normal to the wave (or scattering) vector q. Moreover, the anisotropic character of the dipole-dipole interaction, which is embodied in equation (2), can also be expected to induce anisotropic features in the magnetic neutron scattering cross section, which is a function of  M q ( ). Here, we report the results of neutron-scattering experiments on a Nd-Fe-B magnet which reveal a 'spike' in the magnetic neutron scattering cross section. We will demonstrate that the origin of the spike is related to the Fourier coefficient h q of the magnetostatic field and that it is a direct manifestation of pole avoidance (flux closure) in the bulk of magnetic materials.

Experiment
The neutron-scattering experiments were carried out at 295 K at the instrument Quokka [17] at the Australian Nuclear Science and Technology Organisation. Figure 1 depicts a sketch of the experimental setup. Incident unpolarized neutrons with a mean wavelength of λ = Å 4.8 (Δλ λ = 10 % (FWHM)) were selected by means of a velocity selector. The magnetic field H 0 at the sample position was applied perpendicular to the incoming neutron beam (wave vector k 0 ). Scattered neutrons were counted on a two-dimensional position-sensitive detector. Neutron data were corrected for background (empty sample holder) scattering, transmission, and detector efficiency and set to absolute units using a calibrated attenuated direct-beam measurement. The sample under study was a sintered isotropic (i.e. untextured) Nd-Fe-B permanent magnet (grade: N42); the neutron sample was prepared in the form of a circular disc with a diameter of 22.0 mm and a thickness of 512 μm ≅ N ( 0.0178). On reducing the field starting from 10 T, where a θ sin 2 -type anisotropy with maxima perpendicular to H 0 is visible ( figure 2(a)), one clearly observes the emergence of a spike in Σ Ω d d along the field direction (θ =°0 and θ =°180 ) at fields below about 4 T and extending even to negative values (figures 2(b)-(d)). In the following, we provide a more detailed discussion of the origin of this angular spike anisotropy in Σ Ω d d in terms of the continuum theory of micromagnetics.

Results and discussion
The static equations of micromagnetics for the bulk of magnetic media can be conveniently expressed as a balance-of-torques equation [13][14][15],     (equation (2)). As we will see below, these contributions decisively determine the angular anisotropy of the elastic (small-angle) neutron scattering (SANS) cross section Σ Ω d d , which, for unpolarized neutrons and the above scattering geometry ( ⊥ k H 0 0 ), reads [19,20] Σ Ω π θ θ θ θ V is the scattering volume, b H = 2.9 × 10 8 A −1 m −1 , and ͠ N q ( ) is the Fourier coefficient of the nuclear scattering-length density; the asterisks '*' mark the complex-conjugated quantity. When equations (4) and (5) are inserted into equation (6), Σ Ω d d can be expressed as [18] Σ Ω represents the (nuclear and magnetic) residual SANS cross section, which is measured at complete magnetic saturation (infinite field), and