Ames Laboratory Accepted Manuscripts Ames Laboratory 7-27-2018 Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes Ruslan Prozorov Iowa State University and Ames Laboratory, [email protected] Vladimir G. Kogan Ames Laboratory, [email protected] Follow this and additional works at: https://lib.dr.iastate.edu/ameslab_manuscripts Part of the Condensed Matter Physics Commons, and the Metallurgy Commons Recommended Citation Prozorov, Ruslan and Kogan, Vladimir G., "Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes" (2018). Ames Laboratory Accepted Manuscripts. 216. https://lib.dr.iastate.edu/ameslab_manuscripts/216 This Article is brought to you for free and open access by the Ames Laboratory at Iowa State University Digital Repository. It has been accepted for inclusion in Ames Laboratory Accepted Manuscripts by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes Abstract Effective demagnetizing factors that connect the sample magnetic moment with the applied magnetic field are calculated numerically for perfectly diamagnetic samples of various nonellipsoidal shapes. The procedure is based on calculating the total magnetic moment by integrating the magnetic induction obtained from a full three-dimensional (3D) solution of the Maxwell equations using an adaptive mesh. The er sults 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 fields, as well as infinite rectangular and elliptical cross-section strips. Disciplines Condensed Matter Physics | Metallurgy | Physics This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/ameslab_manuscripts/216 PHYSICAL REVIEW APPLIED 10, 014030 (2018) Effective Demagnetizing Factors of Diamagnetic Samples of Various Shapes R. Prozorov1,2,∗ and V. G. Kogan1 1Ames Laboratory, Ames, Iowa 50011, USA 2Department of Physics & Astronomy, Iowa State University, Ames, Iowa 50011, USA (Received 16 December 2017; revised manuscript received 18 March 2018; published 27 July 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 nonellipsoidal shapes. The procedure is based on calculating the total magnetic moment by integrating the magnetic induction obtained from a full three-dimensional (3D) solution of the Maxwell equations using an 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 approxi- mate formulas are given for practical shapes including rectangular cuboids, finite cylinders in axial and transverse fields, as well as infinite rectangular and elliptical cross-section strips. DOI: 10.1103/PhysRevApplied.10.014030 I. INTRODUCTION formulas similar to those used in this work. One com- mon, but generally incorrect, assumption is that the sum The correction of results of magnetic measurements for of the demagnetizing factors along the three principal axes the distortion of the magnetic field inside and around a equals one. This is true only for ellipsoids. Notably, Brandt finite sample of arbitrary shape is not trivial, but is a [9] has used a different approach by numerically calcu- necessary part of experimental studies in magnetism and lating the slope, dm/dH , of the magnetic moment m vs superconductivity. The internal magnetic field is uniform 0 the applied magnetic field H in a perfect superconductor only in ellipsoids (see Fig. 2), for which demagnetizing 0 in the Meissner state to compute the approximate N fac- factors can be calculated analytically [1,2]. In general, tors for a two-dimensional (2D) situation of infinitely long however, the magnetic field is highly nonuniform inside strips of rectangular cross section in the perpendicular field and outside of finite samples of arbitrary (nonellipsoidal) and he extended these results to finite 3D cylinders (also shape and various approaches have been used to handle of rectangular cross section) in the axial magnetic field. the problem [3–9]. As discussed below, the major obstacle We find excellent agreement between our calculations and has been that, so far, it has not been possible to calculate Brandt’s results for these geometries. Also, in our earlier the total magnetic moment of arbitrary-shaped samples and work, the 2D numerical solutions of the Maxwell equations approximations and assumptions have had to be made. As obtained using the finite element method were generalized a result, various approximate demagnetizing factors have to 3D cylinders and yielded similar results [10]). been introduced. For example, the so-called “magnetomet- Yet, despite multiple attempts, the results published so ric” demagnetizing factor, N , is based on equating the m far do not describe three-dimensional finite samples of magnetostatic self-energy to the energy of a fully magne- arbitrary nonellipsoidal shapes to answer an important tized ferromagnetic prism or, more generally, considering practical question: What is the total magnetic moment of volume-average magnetization in magnetized [4,5,7]or a three-dimensional sample of a particular shape in a fixed perfectly diamagnetic [6] media. Similarly, the so-called applied magnetic field, H ? We answer this question by “fluxmetric” or “ballistic” demagnetizing factor, N ,is 0 f finding a way to calculate the total magnetic moment from based on the average magnetization in the sample mid- first principles with no assumptions and introducing the plane [4,6]. In these formulations, micromagnetic calcu- effective demagnetizing factors without referring to the lations are used to find the distribution of the surface mag- details of the spatial distribution of the magnetic induction. netic dipole density that satisfies the boundary conditions We will first consider how these effective demagnetiz- and the assumptions made. Then the average magnetiza- ing factors depend on the finite magnetic permeability, tion is calculated and used to compute the N factors, using μr, which highlights the difference between ellipsoidal and nonellipsoidal shapes. Complete treatment of finite μr will require separate papers, in which we will focus on (a) the London-Meissner state in superconductors of ∗ Corresponding author. [email protected] arbitrary shape with a finite London penetration depth and 2331-7019/18/10(1)/014030(12) 014030-1 © 2018 American Physical Society R. PROZOROV and V. G. KOGAN PHYS. REV. APPLIED 10, 014030 (2018) (b) demagnetizing corrections in local and linear magnetic total magnetic moment m. We now define the “effective” media with arbitrary μr. (or “integral,” or “apparent”) magnetic susceptibility, χ0, In this work, we focus on perfectly diamagnetic sam- and the corresponding demagnetizing factor, N, by writing ples, inside which the magnetic induction B = 0, which relations that are structurally similar to Eq. (5): allows the study of pure effects of the sample shape. χH V The results can be used for the interpretation of magnetic m = χ H V = 0 ,(6) measurements of superconductors when the London pene- 0 0 1 + χN tration depth is much smaller than the sample dimensions (usually, a good approximation almost up to 0.95Tc)orin which reduces to conventional equations in the case of a conducting samples subject to an ac magnetic field when linear magnetic material of ellipsoidal shape. Importantly, the skin depth is small. Our goal is to find simple-to-use, Eq. (6) contains only one property to be determined—the but sufficiently accurate, approximate formulas suitable for intrinsic susceptibility, χ—provided that the demagnetiz- the calculation of the demagnetizing correction for many ing factor N can be calculated for a given geometry. This shapes that can approximate realistic samples, such as can be done for model materials with known (assumed) χ χ finite cylinders and cuboids (rectangular prisms). and numerically evaluated 0 by inverting Eq. (6) to obtain II. DEFINITIONS = 1 − 1 N χ χ ,(7) In local and magnetically linear media without demag- 0 − ≤ χ ≤∞ −∞ ≤ χ ≤ χ netizing effects (an infinite slab or cylinder in a parallel where 1 and 0 . To eliminate the magnetic field), influence of the material, for calculations of N we will consider a perfect diamagnet with χ =−1, so that when χ = χ =−1, N = 0, as it should in the case of no demag- B = μH = μ0(M + H),(1)0 netizing effects (an infinite slab or a cylinder in a parallel field), and N → 1 for an infinite plate in a perpendicular field where χ →−∞, while χ still equals −1. = B − = χ 0 M H H,(2)The main issue in using Eq. (7) to calculate demagnetiz- μ0 ing factors is to calculate the total magnetic moment of a −7 2 sample of a given (arbitrary) shape. There are two ways where μ0 = 4π × 10 (N/A or H/m) is the magnetic per- meability of free space; μ and χ are the linear magnetic of approaching this. In nonmagnetic (super)conductors, permeability and susceptibility (in general, these quantities one first solves Maxwell equations with the help of one are second-rank tensors, but here we consider the isotropic of the existing numerical