## Abstract

We provide an exhaustive description of the magnetostatics of the uniformly polarized torus and its derivative self-intersecting (spindle) shapes. In the process, two complementary approaches have been implemented, position-space analysis of the Laplace equation with inhomogeneous boundary conditions and a Fourier-space analysis, starting from the determination of the shape amplitude of this topologically non-trivial body. The stray field and the demagnetization tensor have been determined as rapidly converging series of toroidal functions. The single independent demagnetization-tensor eigenvalue has been determined as a function of the unique aspect ratio *α* of the torus. Throughout the range of values of the ratio, corresponding to a multiply connected torus proper, the axial demagnetization factor *N*_{z} remains close to one half. There is no breach of smoothness of *N*_{z}(*α*) at the topological crossover to a simply connected tight torus (*α*=1). However, *N*_{z} is a non-monotonic function of the aspect ratio, such that substantially different pairs of corresponding tori would still have the same demagnetization factor. This property does not occur in a simply connected body of the same continuous axial symmetry. Several self-suggesting practical applications are outlined, deriving from the acquired quantitative insight.

## 1. Introduction

Magnetostatics of macroscopically polarized bodies is a venerable subject and a well-established part of the electrodynamics of continua (Maxwell 1873; Brown 1969). Very often, it is developed mostly by analogy with electrostatics, and especially so in deductive expositions (Landau & Lifshitz 1960). The problem at hand is sufficiently complicated in itself, which is why we will conduct the discussion within a strictly ‘magnetic’ context, even though most of the rigorous results below are immediately transferrable to particles or media exhibiting permanent electrical polarization.

A finite body whose polarization is a constant vector is bound to give rise to a stray field in its surrounding space. The body itself, being immersed in its own stray field, interacts with it. This is the archetypal schematic of self-interaction. Within the body, the stray field superposes onto the internal magnetization field. As a rule, this leads to deviations from field homogeneity inside the body, which explains why the stray field is often referred to as the demagnetizing field. Since the times of Maxwell, it has been parameterized by means of a second-rank demagnetization tensor (Maxwell 1873; Osborn 1945; Stoner 1945; Stoner & Wohlfarth 1948; Brown 1963). Accordingly, the energy of self-interaction of the body with its own stray field is known as the demagnetizing energy.

In principle, then, the magnetostatics of a permanently polarized body should be considered solved if and when the demagnetization tensor for the given body has been determined. Once this is done, the practically important case of placing the magnetized body in an external field can be solved by simple superposition of the fields. The tensor on its part is fully specified if its eigenvalues have been determined. There is an additional general circumstance of great value, namely that the trace of the demagnetization tensor is unity in suitable units (Brown 1960; Moskowitz & Della Torre 1966). Thus, without any loss of generality, one needs to calculate only two eigenvalues of the demagnetization tensor.

For a whole class of bodies possessing axial symmetry, two of the eigenvalues are equal and the problem reduces to determining just one independent eigenvalue. The present magnetostatic problem is of this variety, as the torus does exhibit the requisite high symmetry. Indeed, it is the body of revolution, obtained by rotating a plane disc of radius *r* about an axis that is a distance away from the disc’s centre (cf. figure 1*a*). The various possibilities that arise depending on the ratio will be treated exhaustively below. However, the torus proper, also known as 1-torus or torus of genus one, arises when . Such a body is compact, yet multiply connected and thus fundamentally different from all simply connected bodies, which are topologically equivalent to a spherical ball. At the same time and despite its higher degree of topological complexity—it suffices to mention that the number of colours sufficient for map colouring on its surface, the so-called *chromatic number*, is seven while the chromatic number of a plane or a sphere is four—the shape of the torus is characterized by a single geometric ratio, , which will be referred to simply as the *aspect ratio* or *shape parameter*. (The only other body that is of genus one and is characterized by a unique aspect ratio is the square ring, obtained by rotating a square rather than a circle; the arising edges are, of course, of no topological consequence, yet generate significant effective magnetic charge, which is bound to make the assumption of magnetic uniformity of much more limited validity.) It is so much the more remarkable that such a shape, featuring both single-parameter axial symmetry and fundamental topological interest, has remained mostly unquantified, magnetostatically speaking.

The non-triviality of the problem whose solution we report in the present study may be appreciated better when placed in the perspective of what other relevant shapes have actually been satisfactorily described. Very few exact analytical results for the demagnetization tensors and the concomitant magnetostatics have been derived by means of direct real-space calculations. Maxwell solved the case involving ellipsoids of revolution by exploiting the celebrated result, going back to Poisson, that the only bodies that can sustain strictly uniform magnetization are those enclosed by quadratic surfaces. Of those, only the ellipsoidal ones can be finite, emphasizing the great practical importance of ellipsoidal particles/bodies including spherical ones as a simplest limiting case. Stoner (1945) and, independently, Osborn (1945) resolved the triaxial ellipsoid. Rhodes & Rowlands (1954) and Rowlands (1956) presented an exhaustive and highly influential study of the general (rectangular) prism, including the exact analysis of *interacting* magnetized prisms in regular arrays. This calculation has apparently been repeated independently by a number of authors, e.g. Schabes & Aharoni (1987) and others. The more demanding case of triangular prisms was outlined by Rowlands (1956). Magnetized right circular cylinders were examined independently by Rowlands (1956), Brown (1963), Joseph (1966), Kraus (1973), and, more recently, by Millev *et al*. (2003) in the context of ultrathin discs. The analytical results, and particularly those for ellipsoidal and rectangular-shaped bodies, have been used extensively to elucidate the quantitative aspects of the geometry of domain structure. For an overview of this subject, one should consult the authoritative book by Hubert & Shäfer (1998).

Calculations under the assumption of uniform magnetization are applicable beyond the quadratic-boundary case in practical situations when strong applied fields would ensure nearly perfect saturation, as well as when the size of the particles involved is below the critical one for the single-domain state to be most stable among competing micromagnetic configurations. Additionally, very recently, highly non-trivial classes of inclusion shapes have been discovered (Liu *et al*. 2007; Liu 2008) within the context of Eshelby’s elasticity problem (Eshelby 1957, 1959), which would exhibit homogeneous strain states for homogeneous stresses. By analogy with the magnetostatic problem, captured already by Khachaturian (1983), the same exotic shapes would support uniform magnetization.

A fundamental reason why shape plays such a crucial role in magnetostatics lies with the long-range character of the interactions involved. These interactions sense the finite extent of the magnetized body in a non-negligible way and cannot be reduced to an effective additive or multiplicative energy rescaling. From the perspective of our study, shape as an epitomy of finiteness in a specific guise takes the centre stage of the analysis. Accordingly, it is rather surprising that shape in magnetostatic analyses was tackled head on only very recently (Beleggia & De Graef 2003) by means of a Fourier-space alternative to the traditional position-space approach. With the development of this technique, where shape is treated with priority before anything else is, the class of exact magnetostatic solutions has become much broader, while most of the significant classical (direct space) results have been rederived in a rather satisfactory and completely self-contained fashion (Beleggia *et al*. 2006). In the present study, both real-space and position-space angles of attack have been implemented to the purpose of an exhaustive solution in a situation where none of the complementary approaches turns out to be ultimately advantageous.

The bulk of this paper is organized as follows. After defining the torus geometry in §2*a*, in §2*b* we present the determination of the magnetic scalar potential for the case of uniform magnetization parallel to the axis of revolution. The advance involves solving Poisson’s equation with the particular inhomogeneous Dirichlet-type boundary conditions, dictated by the induced surface magnetic charges. A most straightforward way for applying the general integral form of solution turns out to be the use of the required Green’s function in terms of *toroidal* functions (Cohl *et al*. 2000); recently, Selvaggi (2005, 2008) has revived the interest in these functions by their successful and systematic implementation to the solution of magnetic, electrostatic and gravitational problems of potential theory. The stray magnetic field then follows in §2*c* through, essentially, straightforward differentiation in toroidal coordinates. Section 2*d* builds on the complete description of the magnetic field and culminates in the expression for the axial demagnetization eigenvalue in terms of fast converging series, depending on the unique aspect ratio, *α*, of the torus.

Section 3*a* is devoted to the development of the Fourier-space approach, based on the calculation of the shape amplitude of the torus, as well as of its self-intersecting variations, known as *apple* and *lemon* tori. The shape amplitude is then used to calculate the point demagnetization tensor, allowing for the determination of the stray field in a way alternative to the Poisson equation approach. Following the same path, alternative yet equivalent expressions for the axial demagnetization eigenvalue are derived.

Section 4 summarizes our main findings and outlines some immediate promising applications.

## 2. Demagnetization field of the uniformly magnetized torus

### (a) Geometrical

Consider a torus, also known as an anchor ring, of inner radius *R*_{1} and outer radius *R*_{2}, as shown in figure 1*a*. The radius of the generating disc is *r*=(*R*_{2}−*R*_{1})/2, while the radius of the circular axis is . The parametric equations of the torus can then be written as
2.1
where 0≤*ρ*≤*r*, 0≤*θ*,*ϕ*≤2*π*. We choose *r* and as independent parameters and define the dimensionless shape parameter (aspect ratio) as . The limiting case of *α*→0 corresponds physically to a very thin wire whose demagnetization tensor is trivial and will provide a meaningful check on the calculations that follow. Over the range of values between zero and unity, 0≤*α*<1, the body exhibits the multiply connected topology of the 1-torus. At *α*=1, there is a dramatic crossover to a simply connected body without a hole in the centre; this body has been denoted as a *horn* or *tight* torus in the literature. It is of immediate interest to investigate if the change of topology brings about substantial peculiarities in the magnetostatic behaviour. The consequences of this topological change will emerge below. As *α* grows past unity, the body becomes self-intersecting, yet simply connected. The usual designation for this shape has been *spindle torus*. It features both an internal and an external surface. For that case (*α*>1), it is advantageous to work with the reciprocal of *α*, with 0≤*ω*<1. The external surface is known as an apple shape, and the top and bottom surfaces are given by
2.2
in normalized cylindrical coordinates . For the internal surface, known as the lemon shape, we have
2.3
The rationale behind the self-suggesting names can be recognized in figure 1*b,c* for the apple and the lemon, respectively, as rendered at *ω*=0.75. In the limit of , both the apple and the lemon tend to a sphere. For , the apple approaches the horn torus, while the lemon tends to a cylinder of infinitesimal radius.

### (b) General solution for the magnetic scalar potential

The magnetic scalar potential of a magnetized object is, in general, given by (Brown 1962; Jackson 1999)
2.4
where *ρ*_{M}(**r**′)=∇⋅**M**(**r**′) is the bulk magnetic charge density and is the surface magnetic charge density, with the outward unit surface normal. This form is rather telling, even if too general for a detailed description. Mathematically, the expression solves the Dirichlet problem with inhomogeneous boundary conditions, as dictated by *σ*_{M}. Physically, the additive form of the solution separates volume and surface contributions in an intuitively satisfactory way. Reduction of either type of magnetic charge would lead to a reduced potential and, eventually, to a smaller contribution of the self-energy to the overall micromagnetic energy that, in a detailed analysis, would encompass exchange, anisotropy and magnetoelastic contributions (Brown 1962). The *pole avoidance* principle, which is an indispensable guiding tool in heuristic analyses of domain structures, has its roots in the expression for the magnetic scalar potential (Brown 1963). In fact, the magnetostatic self-energy of the torus can become zero, which is the absolute mininum possible (Arrott *et al*. 1974; Aharoni 1996). That state corresponds to the vortex state of zero divergence in the bulk whose magnetization is at the same time everywhere tangential to the surface, leading to zero surface charge.

For the case of constant magnetization, which is the one we are preoccupied with, the volume charge density vanishes—and with it its corresponding integral contribution—but the second integral survives and needs to be considered carefully.

It is self-suggesting that one should employ toroidal coordinates, first introduced by Riemann (1876) and defined, in modern notation, as
2.5
where () is a strictly positive constant, and , −*π*<*θ*′≤+*π* and 0≤ψ′<2*π* (Moon & Spencer 1961; Lebedev 1965). The coordinate surfaces *η*= constant are toroidal surfaces. In particular, *η*=arcsech *α* corresponds to the surface of the torus. Furthermore, in toroidal coordinates, the integration kernel can be cast into the form (Morse & Feshbach 1953; Cohl *et al*. 2000)
2.6
where *η*_{<}(*η*_{>}) is the smaller (larger) of *η* and *η*′. *P* and *Q* are the Legendre functions of the first and the second kinds, respectively, and of half-integer degree *n*−1/2 and order *m*. They were first studied in remarkable clarity and depth by Hicks (1881), and are also known as toroidal or *ring* functions (Erdelyi *et al*. 1953; Moon & Spencer 1961; Lebedev 1965).

The other required ingredients for an explicit evaluation of the surface charge contribution to the potential are determined as follows. The Jacobian of the coordinate transformation from Cartesian to toroidal coordinates is given by
2.7
To remain on the surface of integration, one has to keep the *η* variable constant, whereby the relevant surface element becomes
2.8
The inhomogeneous surface charge density for an axially magnetized torus with is
2.9
where is the unit vector in the axial direction, while *r* is, as before, the minor radius of the torus.

To find the scalar potential inside the torus, we put *η*_{<}=*η*′=arcsech *α* and *η*_{>}=*η*; outside, the opposite is true. The integral over *ψ*′ equals 2*π**δ*_{m0}, so that the scalar potential is reduced to a single sum,
2.10
Here, the top function between curly braces refers to the space inside the torus, corresponding to , while the bottom function has to be used for the outside region. Using the symmetry of the Legendre functions with respect to its order, *P*_{−n−1/2}(*z*)=*P*_{n−1/2}(*z*) and *Q*_{−n−1/2}(*z*)=*Q*_{n−1/2}(*z*), and expanding the angular term in *θ*, the integral in the expression for the potential can be written in terms of a hypergeometric function or, even more uniformly, as a Legendre function of the second kind of half-integer degree and order 1 (Gradshteyn & Ryzhik 2000, 8.751.3),
2.11
2.12

The scalar magnetic potential for the torus can finally be expressed as 2.13 with 2.14

### (c) The stray field of the torus

At this point, we have sufficient information to compute the demagnetization field in an arbitrary *ψ*-plane, say *ψ*=*π*/2 (i.e. the *x*=0 plane). The demagnetization field is equal to the negative gradient of the scalar potential. In toroidal coordinates, the field components are
2.15
We have omitted the *H*_{ψ} component, which vanishes due to the rotational symmetry. The derivatives are computed readily, using . The final result is given by
2.16
and
2.17
Both series converge reasonably fast, and especially so for small *α* where five terms already produce sufficient accuracy. For *α* close to 1, that is, close to the topologically significant crossover to simple connectedness, one needs some 30 terms of the series, which presents no computational difficulty whatsoever.

Note that the conversion from toroidal to Cartesian components requires two steps. First, the Cartesian components of the demagnetization field are related to its toroidal components via the following relation:
2.18
Furthermore, the Cartesian coordinates (*x*,*y*,*z*), which are henceforth normalized with respect to , must be converted into the toroidal coordinates (*β*,*θ*) using the relations
2.19
and
2.20
with *ρ*^{2}=*x*^{2}+*y*^{2}. Figure 2 shows the demagnetization field component *H*_{z}(0,*y*,0) for three different values of *α*: 0.1, 0.5 and 0.9.

The complete demagnetizing field for *α*=0.5 is shown in figure 3 as a two-dimensional vector field. Inside the torus, the magnetization has been added to the demagnetizing field, so that the figure represents the magnetic induction . Here, is the unit vector along the *z*-axis, *D*(**r**) is the indicator function of the torus, which, by definition, equals unity within the body and is zero without. Its Fourier transform will play a central part in the discussion of the alternative Fourier-space approach in the following section. Examining the figure, we note that the *H*_{z}(0,0,*z*) component of the demagnetization field changes sign twice along the *z*-axis: it is negative in between the two arrow-marked points on the *z*-axis and positive elsewhere. In the limit of , the points correspond to zero stray field. When *α* increases, these points move away from the origin. When *α*=0.5, they are located at , as indicated in figure 3. For *α*=1 (horn torus), the zero-field points are located at .

### (d) The demagnetization tensor of a torus

The symmetric second rank demagnetization tensor of a uniformly magnetized object relates the magnetization vector, **M**, to the demagnetizing field, **H**(**r**), as
2.21
as usual, summation over repeated subscripts is implied. The trace of this tensor can be shown to be equal to the indicator function, *D*(**r**), of the object (Beleggia & De Graef 2003). In particular, the trace equals unity inside the object and vanishes everywhere outside it. The demagnetization factors, *N*_{i}, with *i*=*x*,*y*,*z*, are defined as the average of the corresponding diagonal elements over the volume of the object, e.g. *N*_{z}=〈*N*_{zz}〉. As discussed in the introduction, one only needs to determine the axial factor for our highly symmetric case of a body of revolution.

In Cartesian coordinates, *H*_{z}=−*N*_{zz}*M*_{z} holds for uniform axial magnetization. Since, in equation (2.18), one already has an analytical expression for *H*_{z}, one can deduce, essentially by inspection, that
2.22
This expression is now averaged over the volume of the torus, using once again the Jacobian from equation (2.7), and one is led to consider the axial demagnetization factor *N*_{z}(*α*) in the following form:
2.23
The first of these integrals can be simplified by means of integration by parts with respect to *β*
2.24
The second term in this expression, when integrated over *θ* and combined with the integration over the second term in equation (2.23), evaluates to zero. Therefore, one is left with
2.25
Evidently, the integral over *θ* is identical to that in equation (2.12). Putting all pieces together, one finds that the axial demagnetization factor of the torus is given by the following expression, which explicitly befits the symmetry of the problem and is of no insignificant compactness given the physical, mathematical and geometric intricacies of the situation:
2.26

This is one of the central results of this work. Here, it is helpful to remind the reader that the aspect ratio *α* is defined as , where *r* and are the radii specified in figure 1*a* and in the introduction. The functions *P* and *Q* in the formula are the Legendre functions of the first and the second kinds, respectively, of order *m*=0 and of degree *n*−1/2. The demagnetization factor is shown as a function of the aspect ratio *α* in figure 4 (see also figure 5 which extends into the self-intersecting varieties of the torus that will be characterized in the next section by the alternative Fourier approach). There are a few remarkable properties of *N*_{z} that need to be discussed. They are encoded in the exact mathematical expression equation (2.26), but it is much easier to grasp them from the graphical representations in figures 4 and 5.

First, in the limit of *α*→0, the toroidal axial factor goes to 1/2. This limiting value makes a lot of sense, since now one has a transversely magnetized wire of radius that is much thinner than the distance to its centre, , implying an infinitely small curvature of the wire and, hence, its equivalence to a straight wire. But for this last case, which is the same as that of the infinitely long cylinder, the transverse demagnetization factors are equal and add up to one; hence each of them is precisely equal to 1/2. The formal limiting process from toroidal to cylindrical coordinates, as dictated by the change of symmetry in the process of *α*→0, has been described in detail, e.g. by Arrott *et al.* (1974).

Second, as *α*→1 the 1-torus tends to a tight torus that, unlike the torus proper, is a body of simple connectedness. Past this critical value of *α*, one has to deal with the self-intersecting varieties, the apple and the lemon, which are simply connected. Thus, at *α*=1 the body undergoes a topological crossover. From a micromagnetic point of view, however, there is nothing spectacular happening at this point, since, apart from the precise value of *N*_{z}=0.486 that we are able to determine, the axial demagnetization factor is continuous and *smooth* through the crossover, as is easily recognizable from figure 5. One should not forget, however, that our conclusions originate from the assumption of uniform magnetization. The topological crossover might still make itself notable in a detailed investigation of competing micromagnetic textures in appropriately designed remagnetization processes (Hertel & Kronmueller 2002).

Another limiting point of interest is at the crossover between the apple and the lemon, and this is precisely the sphere. Here, one recovers nicely the reference value of *N*_{z}=1/3 (see figure 5 and the analysis in the next section).

More importantly, the axial magnetization factor as a function of a unique aspect ratio, *N*_{z}(*α*), exhibits non-monotonic behaviour over a wide range of values of . Looking again at figure 4, one realizes that this unusual behaviour holds for 0≤*α*≤0.82, with the axial factor never deviating by much from the limiting value of 1/2; indeed, over the domain of non-monotonicity *N*_{z} stays between the minimum of 1/2 and the maximum of 0.51, which is attained at *α*=0.51. (Incidentally, at this level of accuracy, one gets an easy-to-remember reference point: .) Depending on the degree of sophistication of possible applications of the explicit result for *N*_{z}(*α*) as given in equation (2.26), one might settle for an approximation of *N*_{z}≈0.5 with in-plane components of the demagnetization tensor both equal to each other and approximately equal to 0.25, *N*_{x}=*N*_{y}≈0.25, throughout the region of the torus proper all the way out to the topological crossover to a tight torus (*α*=1), or one might use the exact results as given in analytical and graphical form here. Along this second route of application, one may establish the exact micromagnetic equivalence of uniformly magnetized pairs of tori with rather different aspect ratios with the range of 0≤*α*≤1. Indeed, the possibility for such an equivalence is an immediate consequence of the non-monotonicity of *N*_{z}(*α*), as described above. The pairs of equivalent tori of different aspect ratios can be determined graphically by drawing a horizontal line at any desired value of *N*_{z}>1/2 and examining its cross points with the graph of *N*_{z}(*α*). Their *α*-abscissae determine completely the equivalent tori. Thus, it might come as a surprise that the rather robust torus of aspect ratio *α*=0.82 is equivalent to a very thin transversely magnetized wire.

It is also physically satisfying that the result for the ring torus appears similar to the axial demagnetization factor for the ring of square cross section (see Beleggia *et al*. 2009), the latter being the only other non-simply connected body characterized by a single aspect ratio. Note that the relationship for establishing the correspondence between the two bodies is given by a simple formula between their unique aspect ratios, *α* for the torus and *σ* for the square ring, with *σ* being the ratio of the inner and outer radii of the square ring: *α*=(1−*σ*)/(1+*σ*), or conversely, *σ*=(1−*α*)/(1+*α*). The demagnetization factors show striking similarities except, of course, for the different values at *α*=1 (*σ*=0) because, at that value, one compares the tight torus (*α*=1) with the rather different right circular cylinder. Nonetheless, the actual numbers for the two cases in question are quite close, 0.486 for the tight torus versus 0.474 for the cylinder. The maximum demagnetization factor for the torus was given above, while for the square ring, the maximum is 0.510 at *α*=0.444 (corresponding to *σ*=0.385).

## 3. Fourier-space approach

As mentioned in §1, in the magnetostatics of uniformly magnetized bodies much, if not all, depends on *shape*. In position space, one can approach the problem of shape by looking at the indicator function *D*(**r**), introduced above. Such a definition appears almost tautological by, more or less, stating the obvious. In any case, one does not get any wiser before one attempts to find its Fourier transform, which is known as the *shape amplitude* in the context relevant to the present discussion. Finding the shape amplitude immediately creates the practical problem of using appropriate parameters to describe the surface of the body as a particular two-parameter manifold, which is indeed the first step in getting more specific about quantifying the shape. The other crucial step from the point of view of shape determination is in defining the appropriate limits for the Fourier integral. Once this is done, there remains the problem of solving the integral to find *D*(**k**). Note that we will not use a hat to denote the Fourier transform of the indicator function or other transforms, as no misunderstanding is likely to occur. With the knowledge of the latter, one can determine the demagnetization tensor for the body of the given shape and all further details needed for a complete micromagnetic description (Beleggia & De Graef 2003). With the advent of this approach, the number of shapes whose magnetostatics can be completely specified has increased substantially. Even when a simple closed-form solution cannot be found, the Fourier-space method with its paradigm of ‘shape first’ quantification allows for straightforward, fast-converging numerics to be unleashed in an essentially algorithmic procedure.

### (a) The shape amplitude of the 1-torus and its derivative self-intersecting tori

One sets out to execute the programme outlined above by looking into the shape amplitude of the torus *D*(**k**), defined as the Fourier transform of the indicator function, *D*(**r**), of the shape
3.1
where the coordinate change from equation (2.1) with the Jacobian has been used for the second equality in the chain. First, we verify that *D*(**0**) yields the correct volume, *V*_{t}(*α*), of the ring torus,
3.2
Then, with , we evaluate the shape amplitude according to
3.3
Reverting to a Cartesian frame of reference with and , one obtains
3.4
It is convenient, at this point, to rescale the reciprocal coordinates , and to use the aspect ratio . The shape amplitude may then be written as
3.5
3.6
where, in equation (3.5), we have defined as a shorthand notation. These integral expressions can be expanded in a number of different ways to generate convenient expressions for the numerical calculation of the shape amplitude.

The shape amplitudes of the apple and the lemon, the self-intersecting varieties of the torus that were mentioned earlier, can be derived more conveniently in cylindrical coordinates. The derivation is straightforward (the lower symbol in ± and ∓ refers to the apple, the upper symbol to the lemon),
3.7
where *q*=*k*_{⊥}*r*, *q*_{z}=*k*_{z}*r*, *u*=*ρ*/*r* and . The volume of these shapes can be obtained in the limit of , and results in
3.8
and
3.9
where the subscripts ‘a’ and ‘l’ refer to the apple and the lemon, respectively. Note the similarity between the integrals (3.7) and those for the torus shape amplitude in equation (3.5).

To conclude this section, we render the shape amplitudes of the torus, apple and lemon graphically as surfaces of a given amplitude (iso-amplitudinal surface) in figure 6. The amplitude level for this plot was chosen as 1 per cent of the volume or 0.01*V* (recall that *D*(**0**)=*V*). For the 1-torus, the shape amplitude surface consists of a series of coaxial cylinder-like objects with amplitude oscillations along the *k*_{z} direction. As *α* increases above 1 (i.e. *ω*=0.75=1/*α*), the central cylinder becomes more ovoid-shaped and the oscillations along the *k*_{z} axis move closer to the origin (figure 6*b*). For the lemon, the shape amplitude flattens out and the oscillations along *k*_{z} develop a central hole (figure 6*c*).

### (b) Alternative calculation of the demagnetization field based on the knowledge of the shape amplitude

The nine components of the so-called point-function demagnetization tensor associated with a body, characterized by its shape amplitude *D*(**k**), are given by (Beleggia & De Graef 2003)
3.10
Working in (dimensionless) cylindrical coordinates, the tensor field is given explicitly by
3.11
where the integrations are carried out over the full coordinate ranges.

The simplest application of this relation is the computation of the *z*-component of the reduced demagnetization field *h*_{z}=*H*_{z}/*M* associated with an axially magnetized torus. The demagnetization field in an arbitrary (*ρ*,*z*) cross section, where we keep *z*>0 for simplicity and work with dimensionless coordinates *z* normalized to *r* and normalized to , is
3.12
where we have already integrated over the polar angle. Choosing the form (3.6) for the shape amplitude with the substitution and defining , one obtains
3.13
The *q*_{z}-integral is carried out first,
3.14
where *S*(*z*) is the sign function. For the *q*-integral, now featuring the product of two Bessel functions and an exponential, we rely on a Lipschitz–Hankel integral (Watson 1966, ch. 13), with the appropriate identification of the parameters involved (see also eqn 2.12.38 in Prudnikov *et al*. 1983). In particular,
3.15
with
3.16
Here and below, *E* and *K* are the complete elliptic integrals (Abramowitz & Stegun 1964). The demagnetization field component *h*_{z} then becomes
3.17
which can be evaluated numerically without any problem. The reduced field component *h*_{z}(*ρ*,*z*;*α*) is in complete agreement with the expression derived in §2*c* using the scalar magnetic potential approach in toroidal coordinates. The advantage of the shape-amplitude approach is that the field component is represented by a single, numerically well-behaved integral instead of an infinite summation.

The field at the origin is given by
3.18
and is shown in figure 7*a*. The reduced field in units of *M* at the centre of the torus (i.e. at ) is shown in figure 7*b* and varies between −0.5 for *α*=0 and −0.516 for *α*=0.635; for *α*=1, the field at the centre equals −0.501. Finally, the field along the torus axis (*ρ*=0) is shown for a number of different *α* values in figure 7*c*; as derived in the previous section, the field *h*_{z}(0,*z*;*α*) changes sign between for *α*=0 and for *α*=1.

### (c) Determination of the axial demagnetization factor via the shape-amplitude approach

The demagnetization factors can be computed as the volume average of the demagnetization tensor field in equation (3.11). Noting the similarities between the shape-amplitude expressions (3.5) and (3.7) for torus, apple and lemon, we can compute all three demagnetization factors using the approach described below. The various factors (*Ω*, *s*, *a*, *b* and *c*(*t*)) used in this derivation are shown in table 1. One proceeds to evaluate the axial demagnetization factor as
3.19
where *D*^{2}(*q*,*q*_{z}) can be written in a double-integral form (see equations (3.5) and (3.7)),
3.20
One can reduce the complexity down to two integrals by carrying out the *q*_{z} integration first,
3.21
and, subsequently, the *q* integration
3.22
with
3.23
3.24
so that
3.25

As this last equation can be used as an alternative to equation (2.26), we spell out the meaning of the symbols involved. Namely, *A*_{1} and *A*_{2} are defined as functions of the integration variables *u* and *v* immediately before the last equation; there, one also finds the definition of the respective arguments *k*_{1} and *k*_{2} of the complete elliptic integral of the first kind *K*. The quantities *s* and *ω*, appearing in the prefactor to the integral in the preceding equation, and the corresponding limits of integration *a* and *b* are collected in table 1. Resorting to the appropriate column in this same table 1 allows for the numerical computation of the axial demagnetization factor of the torus proper, the apple and the lemon. In this form, the demagnetization factor can be handled, albeit rather slowly, by symbolic software (e.g. Mathematica; Wolfram 2003). For practical convenience, we computed the demagnetization factor at a set of points in the range [0,1], and interpolated the results with a sixth-degree polynomial,
3.26
The coefficients *a*_{i} are given in table 2. These expansions provide the demagnetization factors with a maximum error of 10^{−4} over the entire parameter range. The results are in complete agreement with the analytical solution derived in the preceding section using the magnetic scalar potential in toroidal coordinates. The demagnetization factors for the apple and the lemon are shown graphically in figure 5.

Let us compare briefly the expressions for the computation of the demagnetization factor that result from the two different approaches, as given in equations (2.26) and (3.25). Each of them offers certain advantages and features some shortcomings in comparison with its ‘dual-space’ counterpart. Thus, the series in equation (2.26) is converging fast, yet only applicable to the case of the torus proper (0≤*α*<1), while the alternative expression in equation (3.25) is clumsier and slow-converging, but can be implemented for the self-intersecting varieties of the torus as well. For the case of the torus proper, one could always use the truncated polynomial interpolation above if expedience is called for. Finally, the graphical representations we have included may also serve as a source for quick, if crude, determination of the demagnetization tensor.

## 4. Discussion

In this paper, we presented a complete magnetostatic analysis of the uniformly magnetized torus and its derivative self-intersecting shapes. The results are immediately transferrable to the analogous electrostatic case. Two approaches have been implemented, the one involving a position-space analysis and the other rooted in the Fourier-space description of the shape of the torus. The two lines of attack turn out to be complementary in a substantial way, since none of them can deliver, all by itself, a complete solution to the problem.

The first (position-space) approach involves the exact determination of the magnetic scalar potential from the Poisson equation for the exterior and interior of the torus with inhomogeneous Dirichlet boundary conditions. The same solution also allows for a complete quantitative description of the stray field of the uniformly magnetized torus. From a practical point of view, the most important achievement probably lies with the determination of the demagnetization tensor of the torus in terms of the unique aspect ratio . We have presented in analytical and graphical form the results for the *N*_{z} component, but it is a trivial consequence of the symmetry of the problem that the remaining two in-plane components are equal to each other and to (1−*N*_{z})/2. We have ascertained that the axial demagnetization factor, *N*_{z}, of a torus varies little, and remains numerically close to the value of 1/2, ranging over the interval [0.486,0.510] as the aspect ratio sweeps through the interval [0,1]. A remarkable non-monotonicity over most of the aspect-ratio range has been established. It implies, among other things, that the pairs of ferromagnetic tori of fairly distinct aspect ratios will still have the same magnetostatic behaviour when uniformly polarized.

Along the Fourier-space line of analysis of the problem, we have not been able to determine a closed-form expression for the shape amplitude, which is a quantity of foremost importance. However, we were still able to go ahead and make use of the nearly algorithmic strength of the approach, working with suitable series representations of the shape amplitude, the demagnetization factor and the stray field. Thus, we have been able to examine by this method the self-intersecting derivative shapes of the torus—the apple and the lemon—and to establish along the way the correct relevant limiting cases. More importantly, there are no peculiarities in the magnetostatic behaviour throughout the topologically significant crossover between the multiply connected torus proper and the simply connected apple. This does not preclude that some peculiarities be discovered in a detailed micromagnetic study of spin textures.

We would like to conclude by outlining some immediate and exciting types of application. In the order given below, the first group would most certainly rest on the direct-space advance reported here, while the second group would be most expediently carried out within the alternative Fourier approach.

Thus, the complete position-space characterization of the magnetic field of the uniformly magnetized torus could involve the extension of the analysis by Berry (1996) of the static equilibrium of a spinning-top magnet in the field of a massive magnetized substrate. There, two configurations have been considered for a spin hovering above a magnetized base—a magnetized disc and a uniformly magnetized square slab with a non-magnetized central hole. In each case, a detailed knowledge of the potential energy has been necessary and contour maps of axial sections through the axis of rotational symmetry have been provided. Our detailed knowledge of the corresponding maps for the torus, based on the complete determination of the magnetic scalar potential, implies that the characterization of yet another adiabatic trap for spins (in its two incarnations, pointed out by Berry (1996)—the levitron and the microscopic-particle trap) can be calculated in all the detail that one or the other application might require. A very closely related problem is that of diamagnetic levitation, where a critical parameter is the force of repulsion felt by a diamagnetic body in an optimally chosen, and typically strong, magnetic field (Berry & Geim 1997). The overall repulsive force depends on a number of parameters, and even if the diamagnetic material is chosen, there still remain the field strength and the field gradient that could be optimized (Profijt *et al*. 2009). But the latter two have been completely characterized above for the uniformly magnetized torus, and this paves the way for the exhaustive examination of this type of diamagnetic levitation setup.

Recently, the knowledge of the shape amplitude for a given body has greatly facilitated the analysis of the *interaction* between particles of that shape (Beleggia & De Graef 2005). There is much interest in studying down to the nanoscale the interactions between uniformly polarized particles, arranged in chains, two-dimensional arrays or three-dimensional lattices. The first step, however, has always been the interaction between a pair of such particles, as has been masterfully demonstrated by Rhodes & Rowlands (1954) and Rowlands (1956) for interacting prismatic particles and by Wohlfarth (1955) for ellipsoidal ones. Having now established that the demagnetization factor of a torus is rather robust with respect to variation of its aspect ratio and that it remains close to one half, there is definite value, from the point of view of information storage, in establishing, for instance, the influence of interaction on the stability of that storage. Whether or not there would be some ultimate advantage in using this shape, rather than the traditionally better studied ones mentioned above, is something that can now be settled for each particular case of prospective application on the basis of the complete quantitative understanding provided here. In a different type of application based on the Fourier-space technique, our analysis makes possible the immediate and exhaustive determination of the electron-optical phase shift distribution for transmission electron microscopy applications, set up for a torus with in-plane polarization. Lorentz images can also be computed rather straighforwardly. These and other related applications of practical interest are currently under examination and will be worked out separately.

## Acknowledgements

M.B. acknowledges, with gratitude, the awarding by the Royal Society of a generous Royal Society Relocation Fellowship for his work at the University of Leeds on this and related projects. M.D.G. acknowledges support from the US Department of Energy, Basic Energy Sciences, contract number DE-FG02-01ER45893.

## Footnotes

- Received July 8, 2009.
- Accepted August 18, 2009.