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Effective Demagnetization Factors of Diamagnetic Samples of Various Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes R. Prozorov1, 2, ∗ and V. G. Kogan1, y 1Ames Laboratory, Ames, IA 50011 2Department of Physics & Astronomy, Iowa State University, Ames, IA 50011 (Dated: Submitted: 16 December 2017; accepted in Phys. Rev. Applied: 26 June 2018) Effective demagnetizing factors that connect the sample magnetic moment with the applied mag- netic field are calculated numerically for perfectly diamagnetic samples of various non-ellipsoidal shapes. The procedure is based on calculating total magnetic moment by integrating the magnetic induction obtained from a full three dimensional solution of the Maxwell equations using adaptive mesh. The results are relevant for superconductors (and conductors in AC fields) when the London penetration depth (or the skin depth) is much smaller than the sample size. Simple but reasonably accurate approximate formulas are given for practical shapes including rectangular cuboids, finite cylinders in axial and transverse field as well as infinite rectangular and elliptical cross-section strips. I. INTRODUCTION state to compute the approximate N−factors for a 2D sit- uation of infinitely long strips of rectangular cross-section Correcting results of magnetic measurements for the in perpendicular field and he extended these results to fi- distortion of the magnetic field inside and around a finite nite 3D cylinders (also of rectangular cross-section) in sample of arbitrary shape is not trivial, but necessary the axial magnetic field. We find an excellent agreement part of experimental studies in magnetism and super- between our calculations and Brandt's results for these conductivity. The internal magnetic field is uniform only geometries. Also, in our earlier work, the 2D numerical in ellipsoids (see Fig. 2) for which demagnetizing factors solutions of the Maxwell equations obtained using finite can be calculated analytically [1, 2]. In general, however, element method were generalized to 3D cylinders and magnetic field is highly non-uniform inside and outside brought similar results [10]). of finite samples of arbitrary (non-ellipsoidal) shapes and Yet, despite multiple attempts, results published so various approaches were used to handle the problem [3{ far do not describe three dimensional finite samples of 9]. As discussed below, the major obstacle has been that arbitrary non-ellipsoidal shapes to answer an important so far the total magnetic moment of arbitrary shaped practical question: What is the total magnetic mo- samples could not be calculated and approximations and ment of a three dimensional sample of a particular shape assumptions had to be made. As a result, various ap- in a fixed applied magnetic field, H0? We answer this proximate demagnetizing factors were introduced. For question by finding a way to calculate total magnetic mo- example, so-called \magnetometric" demagnetizing fac- ment from the first principles with no assumptions and tor, Nm, is based on equating magnetostatic self-energy introducing the effective demagnetizing factors without to the energy of a fully magnetized ferromagnetic prism referring to the details of the spatial distribution of the or, more generally, considering volume-average magneti- magnetic induction. We will first consider how these ef- zation in magnetized [4, 5, 7] or perfectly diamagnetic fective demagnetizing factors depend on finite magnetic [6] media. Similarly, so-called “fluxmetric” or \ballistic" permeability, µr, which highlights the difference between demagnetizing factor, Nf , is based on the average mag- ellipsoidal and non-ellipsoidal shapes. Complete treat- netization in the sample mid-plane [4, 6]. In these for- ment of finite µr requires separate papers in which we mulations, micromagnetic calculations are used to find will focus on (a) the London-Meissner state in supercon- the distribution of surface magnetic dipoles density that ductors of arbitrary shape with finite London penetration satisfies the boundary conditions and the assumptions depth and (b) demagnetizing corrections in local and lin- made. Then the average magnetization is calculated and ear magnetic media with arbitrary µr. used to compute the N factors using formulas similar In this work we focus on perfectly diamagnetic sam- to those used in this work. One common, but generally ples, the magnetic induction B = 0 inside, which allows arXiv:1712.06037v2 [cond-mat.supr-con] 2 Jul 2018 incorrect, assumption is that the sum of demagnetizing studying pure effects of sample shape. The results can be factors along three principal axes equals to one. This is used for the interpretation of magnetic measurement of true only for ellipsoids. Notably, E. H. Brandt has used a superconductors when London penetration depth is much different approach by numerically calculating the slope, smaller than the sample dimensions (a good approxima- dm=dH , of the magnetic moment m vs. applied mag- 0 tion almost up to 0:95T ) or in conducting samples sub- netic field H in a perfect superconductor in the Meissner c 0 ject to AC magnetic field when the skin depth is small. Our goal is to find simple to use, but accurate enough, approximate formulas suitable for the calculations of the ∗ Corresponding author: [email protected] demagnetizing correction for many shapes that can ap- y [email protected] proximate realistic samples, such as finite cylinders and 2 cuboids (rectangular prisms). 1 1 N = − : (7) II. DEFINITIONS χ0 χ where −1 ≤ χ ≤ 1 and −∞ ≤ χ0 ≤ χ. To eliminate the In local and magnetically linear media without demag- influence of the material, for calculations of N we will netizing effects (infinite slab or cylinder in parallel mag- consider a perfect diamagnet with χ = −1, so that when netic field), χ0 = χ = −1, N = 0 as it should be in case of no de- magnetizing effects (infinite slab or a cylinder in parallel B = µH = µ0 (M + H) ; (1) field) and N ! 1 for infinite plate in perpendicular field B M = − H = χH (2) where χ0 ! −∞, while χ is still equals -1. µ0 The main issue in using Eq. (7) to calculate demagne- −7 2 tizing factors is to calculate the total magnetic moment of where µ0 = 4π ×10 [N/A or H/m] is magnetic perme- ability of free space; µ and χ are linear magnetic perme- a sample of a given (arbitrary) shape. There are two ways ability and susceptibility (in general these quantities are of approaching this. In non-magnetic (super)conductors, second rank tensors, but here we consider the isotropic one first solves Maxwell equations with the help of one of case.) existing numerical software packages (such as COMSOL, It follows then that, [13]), to find the transport current density j(r). Then, the total magnetic moment is given by [1], µ χ = − 1 = µr − 1 (3) 1 Z µ0 m = [r × j(r)] dV (8) 2 where µr = µ/µ0 is relative magnetic permeability; µr = 1 for non-magnetic media and µr = 0 and χ = −1 for a The integral here can be evaluated over the entire space, perfect diamagnet. but the integrand is non-zero only inside the sample For finite samples of ellipsoidal shape, constant demag- where the currents flow. netizing factors N connect the applied magnetic field H0 The second way to calculate the total magnetic mo- along certain principle direction with the internal field, ment, m, is given by, H, Z B(r) 3 m = α − H0 d r; (9) µ0 H = H0 − NM (4) where B(r) is the actual field and H0 is the uniform and in terms of an applied field the magnetization is: applied field. Here the integral must be evaluated in a χ region that includes the sample (can be the entire space). M = H (5) We show in the next section that this integral is not 1 + χN 0 unique, but depends on the way chosen for the integra- In arbitrary shaped samples this simple description tion. This is accounted for by a constant α in Eq. (9), breaks down and we have to introduce similarly struc- α = 3=2 for integration over the large spherical domain tured effective equations, albeit applicable only for inte- that includes the whole sample, whereas α = 1 for inte- gral quantities. Namely, upon application of an exter- gration domain as a large cylinder with the axis parallel to H0. It turns out that for numerical reasons, the cylin- nal field H0, a finite sample of a given shape develops a measurable total magnetic moment m. We now define drical domain is preferable and we used it for our numer- “effective" (or \integral", or \apparent") magnetic sus- ical work. Equation (9) is central to the present work, because it allows calculations without using the current ceptibility, χ0, and corresponding demagnetizing factor, N, by writing relations structurally similar to Eq. (5) distribution. This equation (with α = 3=2) can be found in Jackson's textbook, Ref. [11], Eq. (5.62). A related χH V discussion about the multipole representation of the field m = χ H V = 0 (6) 0 0 1 + χN outside the region where the field sources are localized is given in Ref. [12]. which reduces to conventional equations in the case of For evaluation of B(r), one can use approximation of a a linear magnetic material of ellipsoidal shape. Impor- fully diamagnetic sample imposing \magnetic shielding" tantly, Eq. (6) contains only one property to be deter- boundary conditions available in the COMSOL software. mined - intrinsic susceptibility, χ provided the demag- Employing Eq. (9) with B(r) simplifies numerical pro- netizing factor N can be calculated for given geome- cedure and improves accuracy considerably. However, try. This can be done for model materials with known proving Eq. (9) is not at all trivial and we derive it ana- (assumed) χ and numerically evaluated χ0 by inverting lytically in the next section. We also verified the results Eq. (6) to obtain: by calculating total magnetic moment m utilizing both 3 approaches evaluating current distribution in supercon- Hence, we have ducting samples using London equations and employing Eq.
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