Frozen mirror image method

Table of Contents

1. [Introduction](#introduction)
2. [Theoretical background](#theoretical-background)

• 2.1 [Method of images in electromagnetism](#method-of-images-in-electromagnetism)
• 2.2 [Superconductivity basics: type‑I vs. type‑II](#superconductivity-basics-type‑i-vs-type‑ii)
• 2.3 [Magnetic flux pinning and the “hard” superconductor](#magnetic-flux-pinning-and-the‑hard‑superconductor)

1. [Historical development of the frozen mirror image method](#historical-development-of-the-frozen-mirror-image-method)
2. [Core principles of the frozen mirror image method](#core-principles-of-the-frozen-mirror-image-method)

• 4.1 [Geometry of the problem](#geometry-of-the-problem)
• 4.2 [Field‑cooled (FC) condition](#field‑cooled-fc-condition)
• 4.3 [Construction of frozen images](#construction-of-frozen-images)
• 4.4 [Resulting magnetic‑field representation](#resulting-magnetic‑field-representation)

1. [Comparison with the traditional mirror‑image method](#comparison-with-the-traditional-mirror‑image-method)
2. [Illustrative examples](#illustrative-examples)

• 6.1 [Single dipole above a hard superconductor](#single-dipole-above-a-hard-superconductor)
• 6.2 [Array of permanent magnets interacting with a superconducting slab](#array-of-permanent-magnets-interacting-with-a-superconducting-slab)

1. [Why the method matters for modern research](#why-the-method-matters-for-modern-research)
2. [Potential links to the Apiary platform](#potential-links-to-the-apiary-platform)
3. [FAQ](#faq)

10 [Keywords](#keywords)

Introduction

The frozen mirror image method (also called the method of frozen images) is a theoretical tool used to model magnetic‑field distributions in systems that combine permanent magnets with a type‑II superconductor possessing an infinitely hard pinning landscape. First proposed by Alexander Kordyuk in 1998, the method extends the classic method of images—a staple of electrostatics and magnetostatics—so that it can accommodate the magnetic flux pinning phenomenon that dominates the response of hard type‑II superconductors.

In practical terms, the frozen mirror image method provides a simple yet accurate representation of the magnetic field generated by one or several magnets outside an infinitely flat surface of a perfectly hard superconductor when the superconductor has entered its superconducting state under the influence of an external magnetic field (the field‑cooled or FC case). The method is distinct from the traditional mirror‑image approach, which assumes a perfect type‑I superconductor that completely expels magnetic flux (the Meissner effect). Instead, the frozen‑image technique acknowledges that a hard type‑II superconductor does not expel the static field itself; it screens variations of that field while preserving the frozen‑in flux pattern established during cooling.

The following sections unpack the physics that underlies the method, trace its historical origins, detail its construction, and discuss why it remains a valuable analytical instrument for researchers working at the intersection of magnetism and superconductivity.

Theoretical background

Method of images in electromagnetism

The method of images is a mathematical construction that replaces a complex boundary condition with an equivalent set of fictitious (image) sources placed in a region where the governing differential equations are easier to solve. In electrostatics, a classic example is a point charge placed above a grounded conducting plane; the field in the half‑space can be reproduced by introducing an opposite‑sign image charge mirrored across the plane.

In magnetostatics, the same principle applies: a magnetic dipole near a perfectly diamagnetic surface can be represented by an appropriately oriented image dipole. The elegance of the method lies in its ability to convert a boundary‑value problem into a superposition of elementary fields, each of which is analytically tractable.

Superconductivity basics: type‑I vs. type‑II

Superconductors are broadly classified into two families based on their magnetic response:

A perfect type‑I superconductor enforces the Meissner effect: any external magnetic field is expelled from its interior, and the surface currents generated exactly cancel the incident field. Consequently, the traditional mirror‑image method works well for this ideal case because the superconductor behaves as a perfect magnetic mirror.

In contrast, a type‑II superconductor allows magnetic flux to thread the material in the form of quantized vortices once the applied field exceeds the lower critical field \(H_{c1}\). The vortices can become pinned by material defects, preventing them from moving under Lorentz forces. This pinning dramatically alters the magnetic response, especially when the superconductor is hard—i.e., it exhibits an infinite pinning force.

Magnetic flux pinning and the “hard” superconductor

Flux pinning refers to the immobilization of vortices by inhomogeneities such as grain boundaries, dislocations, or artificially introduced nano‑defects. In a perfectly hard superconductor, the pinning force is taken to be infinite, meaning that once a vortex configuration is established (for example, during the cooling process), it cannot be altered by subsequent changes in the external magnetic field.

This extreme pinning regime is an idealization useful for analytical work. It implies that the superconductor screens only the variation of the external field while the static component of the field that was present at the moment of transition to the superconducting state remains frozen inside the material. The frozen‑mirror image method was specifically designed to capture this subtle distinction.

Historical development of the frozen mirror image method

The classical method of images dates back to the 19th century, finding early applications in electrostatics and later in magnetostatics. However, when researchers began to study magnet–superconductor hybrids—systems where permanent magnets interact with superconducting screens—the limitations of the traditional approach became evident.

In 1998, Alexander Kordyuk introduced the frozen mirror image method as an extension of the conventional image technique. The motivation was to incorporate the magnetic flux pinning phenomenon that dominates the behavior of hard type‑II superconductors. By treating the superconductor as a surface that freezes the magnetic field present at the moment of the superconducting transition, Kordyuk’s formulation allowed for a straightforward analytic description of the field‑cooled (FC) configuration, i.e., the situation where the material becomes superconducting while already immersed in an external magnetic field.

Since its inception, the frozen‑image concept has been employed in a range of theoretical studies that require an accurate yet tractable description of magnetic fields near hard superconductors, especially when dealing with arrays of permanent magnets or magnetic levitation configurations.

Core principles of the frozen mirror image method

Geometry of the problem

The method assumes an infinitely flat interface separating two half‑spaces:

• Region I (z > 0) – the space occupied by the magnet system (one or several permanent magnets).
• Region II (z < 0) – the interior of a perfectly hard type‑II superconductor with infinite pinning force.

The coordinate system is chosen such that the plane \(z = 0\) coincides with the superconductor’s surface. All sources (real magnets) are located in Region I, and the magnetic field of interest is evaluated in the same region.

Field‑cooled (FC) condition

In the field‑cooled scenario, the superconductor transitions from the normal to the superconducting state while the external magnetic field is already present. The field distribution at the instant of transition becomes frozen inside the superconductor because of the infinite pinning. Consequently, any subsequent change in the external field (e.g., moving the magnets) is screened by surface currents that prevent the frozen flux pattern from being altered.

The frozen‑image method explicitly incorporates this condition by assigning to the superconductor a set of static image sources that represent the frozen flux, and a dynamic set of image sources that account for the variation of the external field after cooling.

Construction of frozen images

1. Identify the real magnet distribution \(\mathbf{M}(\mathbf{r})\) (or equivalent current/ dipole representation) located at positions \(\mathbf{r}{k}\) with \(z{k}>0\).

1. **Create a static image** for each real source:

• Mirror the source across the plane \(z=0\) (i.e., replace \(z_{k}\) by \(-z_{k}\)).
• Preserve the orientation and magnitude of the source. This image represents the magnetic flux that was present at the moment of cooling and remains frozen inside the superconductor.

1. **Create a dynamic image** that enforces the screening of variations:

• Mirror the source as in step 2, but invert its sign (or, more generally, apply the appropriate transformation that yields a field that cancels the variation of the external field at the surface).
• This dynamic image ensures that any change in the external field beyond the frozen component does not penetrate the superconductor.

1. Superpose the fields of the real sources, the static images, and the dynamic images. The resulting field in Region I satisfies the boundary condition of a perfectly hard superconductor under FC conditions: the normal component of the magnetic induction \(B_{z}\) is continuous across the interface, while the tangential component of the magnetic field \(H_{\parallel}\) adjusts to screen variations.

Resulting magnetic‑field representation

The total magnetic field \(\mathbf{B}(\mathbf{r})\) in the space above the superconductor can be written as

\[ \mathbf{B}(\mathbf{r}) = \sum_{k}\Big[ \mathbf{B}{\text{real}}^{(k)}(\mathbf{r}) + \mathbf{B}{\text{static\;image}}^{(k)}(\mathbf{r}) + \mathbf{B}_{\text{dynamic\;image}}^{(k)}(\mathbf{r})\Big] . \]

Because the static images are identical to the real sources (mirrored but not sign‑reversed), they reproduce the frozen flux pattern inside the superconductor. The dynamic images, being sign‑reversed, cancel any incremental field that would otherwise modify that pattern.

The elegance of the frozen‑image construction lies in its linearity: each real magnet contributes independently, allowing complex magnet assemblies to be treated by simple superposition. Moreover, the method yields closed‑form expressions for the field components when the real sources are elementary (e.g., point dipoles or uniformly magnetized cylinders), facilitating analytical insight and rapid numerical evaluation.

Comparison with the traditional mirror‑image method

The frozen‑image method reduces to the traditional approach only in the limiting case where the pinning force is zero (i.e., the superconductor behaves like a perfect type‑I material). In that limit, the static image disappears and the dynamic image alone reproduces the classic result.

Illustrative examples

Single dipole above a hard superconductor

Consider a magnetic dipole \(\mathbf{m}\) located at \((0,0,d)\) with \(d>0\).

• Static image: A dipole \(\mathbf{m