Best practices for a proper evaluation and conversion of physical property equations in superconductors: the examples of WHH formulation, Bean model and other cases of interest

In recent years there have been growing concerns about the proper evaluation of physical properties of superconductors, in particular for quantities extracted from magnetic characterizations. Errors can and often do occur due to the following issues: i) several measurement instruments still use Gaussian & cgs-emu units instead of the preferable International System (SI) units; ii) there are decades of valuable publications where, however, equations were expressed in Gaussian & cgs-emu or other unit systems or where constants were normalized to unity, which requires proper understanding and unit conversion in order to correctly evaluate the measured physical quantities; iii) the conversion between unit systems sometimes appears challenging and may not be properly performed. In this paper we will describe how to properly convert physical quantities relevant for the evaluation of magnetic and other properties focusing on the still most used unit systems, SI and Gaussian & cgs-emu. We will provide examples of how to properly verify and understand the physical formulae. We will include examples for the correct method to determine the critical current density Jc of a superconductor from the measurement of its magnetic hysteresis loop through the Bean model, and the correct conversion to SI of the equations for Hc2(T) according to the Werthamer-Helfand-Hohenberg (WHH) formulation. The goals of this paper are to make the readers aware of the unit conversion issue, to provide useful hands-on tools for proper conversion and to strongly encourage future exclusive use of the SI units and formulae.

💡 Research Summary

The manuscript addresses a pervasive problem in superconductivity research: the misuse and mis‑conversion of physical units when extracting key material parameters from magnetic measurements. The authors identify two principal sources of error. First, many commercial magnetometers still output magnetic moments in Gaussian/cgs‑emu units (emu, gauss) while the scientific community increasingly expects results in International System (SI) units (A·m², tesla). Second, decades‑old literature often presents formulas with constants set to unity or without explicitly stating the unit system, leading to ambiguous or incorrect application of those equations today.

To remedy these issues, the paper provides a systematic guide for converting between Gaussian/cgs‑emu and SI, emphasizing that unit conversion is not a simple numerical scaling but a process that must respect the definitions of the physical quantities involved. The authors reproduce and annotate the classic Goldfarb‑Fickett conversion tables, clarifying that magnetization has two common Gaussian definitions—M expressed as emu cm⁻³ and 4πM expressed in gauss—and that each requires a distinct conversion factor (10³ A m⁻¹ for the former, 10 A m⁻¹ / 4π for the latter). They also discuss the updated value of the vacuum permeability μ₀ after the 2019 SI redefinition, noting that μ₀ is now a measured constant (≈1.256 637 061 × 10⁻⁶ H m⁻¹) rather than an exact 4π × 10⁻⁷ H m⁻¹.

The core of the paper consists of three detailed case studies that illustrate how dimensional analysis can uncover hidden errors.

1. McMillan Formula – The original McMillan expression for the superconducting critical temperature assumes k_B = ħ = 1, which masks the underlying dimensions. By reinstating Boltzmann’s constant and Planck’s constant, the authors convert the formula to a fully dimensional SI version, showing how to correctly handle frequencies expressed in kelvin, cm⁻¹, or electron‑volts.

2. Ginzburg‑Levanyuk Number (Gi) – Gi is a dimensionless measure of the fluctuation regime, but its common Gaussian representation mixes magnetic field units (gauss) with length scales (coherence length ξ) without explicit μ₀ or 4π factors. The authors derive the SI expression, demonstrating that the magnetic energy density must be written as B²/(2μ₀) and that the resulting Gi is truly unit‑free.

3. Bean Critical‑State Model – The Bean model is widely used to extract the critical current density J_c from magnetic hysteresis loops. The standard textbook formula J_c = 20 ΔM/