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The Stray- and Demagnetizing Field from a Homogeneously Magnetized

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The Stray- and Demagnetizing Field from a Homogeneously Magnetized Published in IEEE Magnetics Letters, Vol. 10, 2110205, 2019 DOI: 10.1109/LMAG.2019.2956895 The stray- and demagnetizing field from a homogeneously magnetized tetrahedron Kaspar K. Nielsen, Andrea R. Insinga, Rasmus Bjørk Abstract The stray- and demagnetization tensor field for a homogeneously magnetized tetrahedron is found analytically. The tetrahedron is a special case of four triangular faces with constant magnetization-charge surface density, for which we also determine the tensor field. The tensor field is implemented in the open source micromagnetic and magnetostatic simulation framework MagTense and compared with the obtained magnetic field from an FEM solution, showing excellent agreement. This result is important for modeling magnetostatics in general and for micromagnetism in particular as the demagnetizing field of an arbitrary body discretized using conventional meshing techniques is significantly simplified with this approach. Department of Energy Conversion and Storage, Technical University of Denmark - DTU, Anker Engelunds Vej 1, DK-2800 Kgs. Lyngby, Denmark *Corresponding author: [email protected] 1. Introduction Here m0 is the vacuum permeability. Ampere’s law is then expressed in terms of H and the free current density J : The stray- and demagnetization field tensor of a uniformly free magnetized body is a fast and possibly analytical approach to ∇ × H = J : (3) calculate the magnetic field generated by a magnetized body free at any point in space. The stray- and demagnetization tensor In absence of free currents, the magnetic field H is an field from uniformly magnetized primitives has been derived irrotational vector field, i.e. its curl is zero. In these conditions for rectangular prisms [1; 2], finite cylinders [4; 3], hollow it is always possible to express H as the gradient of a scalar spheres [5], ellipsoids [6], powder samples [8; 9; 10] and an field, thus guaranteeing that H is irrotational. We can then equilateral triangluar prism [11]. Several such tensor fields introduce the magnetic scalar potential fM: for primitives have also been found through the Fourier space approach [12]. Here, we present the magnetic field produced H = −∇fM: (4) by a general triangular face with uniform magnetization and give, as a special case, the total stray- and demagnetization Combining Eqs.1,2 and4 we obtain the following equation, tensor field for a general tetrahedron, i.e. a polygon defined by which is analogous to Poisson’s equation: four triangles that share four non-coplanar vertices. We note that the stray-field is defined as the magnetic field outside −∇ · (m0∇fM) = −∇ · (m0M) : (5) the magnetized body while the demagnetizing field is the magnetic field inside the body, both produced by the body’s The quantity rM = −∇ · M is interpreted as the magnetic charge volume-density. The formal solution to Eq.5 for a arXiv:2004.11700v1 [cs.CE] 15 Apr 2020 magnetization. The results presented in this paper are valid both inside and outside the homogeneously magnetized body. homogeneously magnetized body with enclosing surface S0 is given by [13]: 2. Model I 0 0 1 nˆ(r ) · M(r ) 0 fM(r) = 0 dS : (6) Magentostatics systems are governed by Gauss’s law for mag- 4p S0 kr − r k netism and Ampere’s law. Gauss’s law for magnetism states that the flux density B is a solenoidal vector field: Note that there are two sets of coordinates. The coordinates marked with a 0 are the coordinates of the face that creates the ∇ · B = 0: (1) magnetic field, whereas the non-marked coordinates are to the point at which the field is evaluated. The quantity sM = nˆ · M The magnetization M is defined as the volume density of is interpreted as the magnetic charge surface-density. The magnetic dipoles. The magnetic field H is defined from B and magnetic field is obtained by combining Eq.4 and Eq.6: M according to the following relation: I 0 0 1 nˆ(r ) · M(r ) 0 H(r) = −∇fM(r) = −∇ 0 dS : (7) B = m0(H + M) (2) 4p S0 kr − r k The stray- and demagnetizing field from a homogeneously magnetized tetrahedron — 2/5 y0 is denoted l. The x0−component of the magnetic field is then 0 Z h Z l(1−y0=h) 0 0 Mz 0 0 A Hx = − dy dx 4p 0 0 ¶ 1 ¶x ((x − x0)2 + (y − y0)2 + z2)1=2 h 0 0 = Nxz;lMz 1 N0 = − (F(x;y;z;h;h;l) − F(x;y;z;0;h;l) D0 xz;l 4p − (G(x;y;z;h) − G(x;y;z;0))) (8) C0 B0 x0 k l with Figure 1. General triangle in the local (primed) coordinate 0 0 0 h f system. The triangle given by A B D is the first triangle F(x;y;z;y0;h;l) = p atanh n (9) considered in the text. Please note that the lengths h, k and l h2 + l2 fd are all > 0. l2 f (x;y;z;y0;h;l) = l2 − lx + hy − hy0 1 + n h2 0 p 2 2 fd(x;y;z;y ;h;l) = h + l 2(l2 − lx + hy)y0 Here it is worth noting that the gradient operator and the × (x − l)2 + y2 − integration are with respect to different sets of coordinates: the h gradient operator is with respect to the unprimed coordinates, l2 1=2 + y02 1 + + z2 while the integration is with respect to the primed coordinates. h2 The order is thus interchangable. ! y − y0 G(x;y;z;y0) = atanh (10) In the following we will first derive the contribution to the px2 + (y − y0)2 + z2 magnetic field from a face shaped as a right triangle. Then we exploit that any triangle can be divided into two right trian- The y−component of the magnetic field is: gles and apply the superposition principle to get the solution for a general triangle. We then provide a transformation nec- 0 Z l Z h(1−x0=l) 0 Mz 0 0 essary for obtaining the field from an arbitrarily positioned Hy = − dx dy 4p 0 0 and rotated triangle from our basic solution. This, in turn, yields the contribution to the components of the stray- and de- ¶ 1 magnetization tensor field from an arbitrarily positioned and ¶y ((x − x0)2 + (y − y0)2 + z2)1=2 0 0 oriented triangle. Finally, we provide the details of building = Nyz;lMz up a tetrahedron consisting of such four triangles and provide 1 N0 = − (K(x;y;z;l;h;l) − K(x;y;z;0;h;l) the total tensor field describing the stray- and demagnetizing yz;l 4p fields from a homogeneously magnetized tetrahedron. This − (L(x;y;z;l) − L(x;y;z;0)) (11) result has been build into the open source micromagnetic and magnetostatic simulation framework MagTense [14]. with l k K(x;y;z;x0;h;l) = p atanh n h2 + l2 kd h2 2.1 The contribution to the field from a right trian- 0 2 0 kn(x;y;z;x ;h;l) = h − hy + lx − lx 1 + 2 gle l 0 p 2 2 kd(x;y;z;x ;h;l) = h + l We consider a right triangle positioned in the x0y0-plane at 2x0(h2 + lx − hy) z0 = 0 with the vertices A0 = (l;0;0), B0 = (0;h;0) and D0 = × (y − h)2 + x2 − (0;0;0) as shown in blue in Fig.1. Only the z0-component l of the magnetization contributes to the magnetic field and the h2 1=2 0 0 + x02 1 + + z2 normal vector is nˆ = zˆ . The primes on the field and magneti- l2 zation components emphasize that the field and magnetization ! x − x0 are evaluated in the local coordinate system of the triangle as L(x;y;z;x0) = atanh (12) defined in Fig.1. The length of the triangle along the x0−axis p(x − x0)2 + y2 + z2 The stray- and demagnetizing field from a homogeneously magnetized tetrahedron — 3/5 The z−component of the field becomes: y B M0 Z l Z h(1−x0=l) H0 = − z dx0 dy0 z 4p 0 0 A ¶ 1 ¶z ((x − x0)2 + (y − y0)2 + z2)1=2 D 0 0 = Nzz;lMz 1 eˆ N0 = − (P(x;y;z;l;h;l) − P(x;y;z;0;h;l) y eˆx zz;l 4p C − (Q(x;y;z;l) − Q(x;y;z;0))) (13) x with Figure 2. General triangle in the global coordinate system. Note that the unit vector in the z−direction points out of the p P(x;y;z;x0;h;l) = atan n plane. pd 2 2 0 0 x h(x + z ) pn(x;y;z;x ;h;l) = x(h − y) − x (h(1 − ) − y) − above cancel out even though the arguments to the atan func- l l tion are infinite since the function arguments approach infinity h2 p (x;y;z;x0;h;l) = z (y − h)2 + x2 + x02 1 + at the same rate. d l2 = 2x0(h2 + lx − hy) 1 2 2.3 Arbitrarily positioned and oriented triangular face − + z2 l We now consider a general triangular face in Euclidian geom- ! etry characterized by three points, A; B and C in the global (x − x0)y Q(x;y;z;x0) = −atan : (14) coordinate system, G. It is a requirement that the three vertices zp(x − x0)2 + y2 + z2 are not collinear. The goal is to find the linear transformation from this Some of these integrals were evaluated using the Rule-based general triangle to a local coordinate system, L, with the integrator [15].
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