Comments on Units in Magnetism

1. Introduction

Perusing the 1974 M³ Conference Proceedings indicates that, at the present time, Systeme Internationale (SI) units are avoided by most leading scientists and engineers in the field of magnetism. Throughout the Proceedings, almost universal preference is displayed for the cgs electromagnetic system (or for the Gaussian system, which gives an equivalent description of magnetic quantities). However, usage of SI units in the field of magnetism will undoubtedly increase with time. One barrier to increased usage is the present lack of standardized and agreed upon relationships between magnetic quantities within the SI. In this paper we will tentatively propose notation and definitions for those relationships most frequently used by experimentalists, with the hope that this will help stimulate the magnetism community to make their views known on preferred definitions.

2. Some Considerations on the Two Systems

One major property of the Gaussian (and the cgs emu) system, considered an advantage by some and a disadvantage by others, is that B and H have the same numerical value in empty space. Changing to the SI, where not only do B and H have different units in empty space, but also different numerical magnitudes, puts one somewhat in the position of Casimir’s [1]¹ mythical tangenometrists who decided that, “The volumetric displacement of empty space – although equal to unity – had the dimension Archimedes per Euclid.”

The SI is a “rationalized” system, whereas the Gaussian is unrationalized. Thus, when magnetic susceptibilities are converted between the two systems a factor of 4π is involved. Further factors of 10 are involved depending on whether volume, mass, or molar susceptibility is in question. This gives considerable latitude for errors and ambiguities in data compilations, handbooks, and treatises which attempt to convert existing numerical values to SI units, and numerous examples of such errors can be found. For example, in the recent treatise on magnetic materials by Heck [2], who endeavors to use SI units as much as possible, a table of paramagnetic susceptibilties apparently gives the rationalized mass susceptibility for Pt in cm³/g, the unrationalized mass susceptibility for λ-Fe in cm³/g, and the rationalized volume susceptibility for Li (dimensionless). Since these differences in units are not listed in the table, an unsuspecting user could easily be misled. As most commonly used with SI, the relation between B, H, and M is defined as B = μ₀ (H + M), χ = M/H. Some authors [3] exhibit the μ₀ associated with the SI explicitly by replacing, H by B/μ₀, giving χ = μ₀M/B. This is, of course, approximately correct for the small susceptibilities found in most diamagnetic and paramagnetic materials, but could be misapplied to super-paramagnetic or ferromagnetic materials.

3. Recommendations

In order to ease conversion from Gaussian (and cgs emu) to SI units, the names, definitions, and symbols for magnetic quantities should be standardized. This requires agreement within the magnetism community. Our current recommendations are summarized in the tables.

Table 1 lists recommended symbols and names for magnetic quantities in SI and cgs emu. When using SI units to express susceptibility, we believe it would be useful to label it ‘rationalized’ and give it the symbol κ, reserving χ for the non-rationalized cgs emu system. What we have labeled the “volume susceptibility” in table 1 is often referred to simply as just “susceptibility.” The introduction of the symbol J (where J = μ₀M) in the SI is useful due to the controversy [4] over whether one should define B = μ₀ (H + M) or B = μ₀ H + M. Further, the symbol J and the associated name ‘magnetic polarization’, are in current use [5].

Table 1.

Symbols and names for magnetic quantities in SI and cgs, Gaussian (or cgs emu).

Symbol	Name
	cgs emu	SI
B	flux density magnetic induction	flux density (magnetic induction)
H	magnetic field strength	magnetic field strength
M	magnetization	magnetization
J	—	magnetic polarization
χ	volume susceptibility	—
κ	—	rationalized volume susceptibility
χρ	mass susceptibility	—
κρ	—	rationalized mass susceptibility
χmole	molar susceptibility	—
κmole	—	rationalized molar susceptibility
m	magnetic moment	magnetic moment
μB	Bohr magneton	Bohr magneton

Table 2 compares several of the more important equations in the field of magnetism. Equations (1) and (2) define the recommended usage of the symbols M and J in SI, as mentioned above. In both Gaussian and SI units, the volume susceptibility, defined by eq (3), is dimensionless and is the ratio of M to H, (both with magnitudes which will change by a factor of 4π upon rationalization). Equation (4) gives the force on a material placed in a magnetic field gradient. (This equation involves certain assumptions and is most useful for small samples with small susceptibilties.) Equation (5) gives the energy of a (point) magnetic moment in a magnetic field, and eq (6) gives the volume energy density associated with a magnetostatic field.

Table 2.

Corresponding equations in SI and cgs Gaussian (or cgs emu). In this table, F refers to force, W refers to the energy of a magnetic dipole in a filed, w refers to the volume energy density. Other symbols are defined in table 1.

Gaussian (or cgs emu)	SI
B = H + 4πM	B = μ0 (H + M) (1)
	B = μ0 H + J (2)
χ= M/H	κ= M/H (3)
$\text{F}=\chi \text{VH}\frac{\partial \text{H}}{\partial x}$	F = μ0 κVH ∂H/∂x (4)
W = −mBcosθ	W = − mBcosθ (5)
$\text{w}=\frac{\text{BH}}{8\pi }$	$\text{w}=\frac{1}{2}\text{BH}$ (6)

Table 3 gives numerical factors for converting between the two unit systems. The conversions for flux density, B, and susceptibility, χ and κ, are independent of the conventions adopted, i.e. whether B = H + M, B = μ₀ H + M, etc. Other conversions will depend on these conventions. One problem for those not thoroughly familar with current magnetic unit usage is that ‘emu’ is not really a unit but rather a flag to describe the unit system being used. Often, though not always, a dimensional anaylsis on susceptibility units may be performed if ‘emu’ is replaced by cm³. Another problem which undoubtedly gives further difficulty to the uninitiated is the variety of units used for the same quantity in the Gaussian system. For example, in the 1974 M³ conference proceedings we find the following units used for “magnetization”: G, Oe, emu/g, μ_B/atom, B.M./FORMULA UNIT, μ_B/impurity, G cm³/g, emu/cm³, and emu: and for “susceptibility” we find the following variety of units: emu/g, emu/cm³, emu/mole, emu/g KOe, emu/gm-At. V, and emu/Oe mole.

Table 3.

Conversion from Gaussian to SI Units

Multiply the Number for		by	To Obtain the Number for
Gaussian Quantity	Unit		SI Quantity	Unit
flux density, B	G	10−4	flux density, B	T(≡Wb/m2 ≡Vs/m2)
magnetic field strength, H	Oe	103/4π	magnetic field strength, H	A/m
volume susceptibility, χ	emu/cm3 (dimensionless)	4π	rationalized volume susceptibility κ	dimensionless
mass susceptibility, χρ	emu/g (≡cm3/g)	4π · 10−3	rationalized mass susceptibility, κρ	m3/kg
molar susceptibility,* χmole	emu/mol (≡cm3/mol)	4π · 10−6	rationalized molar susceptibility, κmole	m3/mol
magnetization, M	G or Oe	103 4π · 10−4	magnetization, M magnetic polarization, J	A/m T
magnetization, 4πM	G or Oe	103/4π 10−4	magnetization, M magnetic polarization, J	A/m T
magnetization, M	μB/atom or μB/form. unit, etc.**	1	magnetization, M	μB/atom or μB/form. unit, etc.**
magnetic moment of a dipole, m	erg/G	10−3	magnetic moment of a dipole, m	J/T (≡Am2)
demagnetizing factor, N	dimensionless	l/4π	rationalized demagnetizing factor, N	dimensionless

^aAlso called atomic susceptibility. Molar susceptibility is preferred since atomic susceptibility has also been used to refer to the susceptibility per atom.

^**“Natural” units, independent of unit system. However, the numerical value of the Bohr magneton does depend on the unit system.

To convert an equation given in the Gaussian system to the corresponding equations in the SI, table 4 can often be useful. For example, in the Gaussian system the magnetization can be considered as the magnetic moment per unit volume,

$$M=\frac{m}{V}$$ (1)

where M is the magnetization in G, m is an appropriate magnetic moment in erg/G, and V is an appropriate volume in cm³. Using the substitutions of table 4 we have

$$\sqrt{\frac{{\mu }_0}{4\pi }}M=\frac{\sqrt{{\mu }_0/4\pi }m}{V}$$ (2)

which reduces to

$$M=\frac{m}{V}.$$ (3)

Thus the magnetization in our suggested SI system can also be considered as the magnetic moment per unit volume, with magnetization in A/m, dipole moment in J/T, and volume in m³. Table 5 gives the numerical value of three important fundamental magnetic constants in the two unit systems, and table 6 compares demagnetizing coefficients, N, for several familiar shapes, where the defining equation for N for both systems is

$$H=H_0-NM$$ (4)

with H the magnetic field strength within the magnetized body and H₀ the applied magnetic field strength.

Table 4. Substitutional symbols for equations.

To convert an equation in Gaussian units to a corresponding equation in SI, replace the symbols in the column labeled Gaussian by the combination of symbols in the column labeled SI. Symbols representing quantities with units involving only volume, force, energy, and length transform directly.

Gaussian Quality	Gaussian symbol	SI symbol
flux density	B	$\sqrt{4\pi /{\mu }_0}\text{B}$
magnetic field	H	$\sqrt{4\pi {\mu }_0}\text{H}$
magnetization	M	$\begin{array}{l}\sqrt{{\mu }_0/4\pi }\text{M},\text{ or}\\ \sqrt{1/4\pi {\mu }_0}\text{J}\end{array}$
volume susceptibility	χ	(1/4π)κ
magnetic moment	m	$\sqrt{{\mu }_0/4\pi }\text{\hspace{0.17em}m}$

Table 5.

Important fundamental constants

Quantity	Gaussian	SI
μ0, permeability of free space	1 (dimensionless)	$4\pi \times {10}^{-7}\text{H}/\text{m}\left(\equiv \frac{\text{Tm}}{\text{A}}\equiv \frac{\text{Vs}}{\text{Am}}\right)$
μB, Bohr magneton	$9.274078(36)\times {10}^{-21}\frac{\text{erg}}{\text{G}}$	$9.274078(36)\times {10}^{-24}\frac{\text{J}}{\text{T}}\left(\equiv {\text{Am}}^2\right)$
μN, Nuclear magneton	5.050824(20) × 10−24 erg/G	5.050824(20) × 10−27 J/T

Table 6.

Demagnetizing Coefficients, N, for homogeneous isotropic bodies of various shapes.

Shape	N Gaussian (unrationalized)	N SI (rationalized)
‖ to axis of long needle	0	0
⊥ to axis of long needle	2π	1/2
sphere	4π/3	1/3
⊥ to plane of a thin disc	4π	1

4. Discussion

There are currently several systems of electromagnetic equations that may be used with SI units [4, 6] In order to apply SI units in the field of magnetism with a minimum of confusion, agreement and uniformity in symbols and definitions would be extremely helpful. Here we have suggested such a set of symbols and definitions which covers most of the quantities of current interest to those who publish in the M³ proceedings. We would emphasize that this set is possibly not the one most desirable to a majority of magneticians. It was selected as one which appeared to us to be most in conformity with current international usage. An example of an alternative system would be the SI analog of a rationalized ‘Gaussian’ system. In such a system B, H, and M would be given the relation B = H + M, and H and M would also have units of ‘tesla’. This would overcome the problem, troublesome to some, of giving B and H different numerical values in a vacuum. Another possibility, favored by Coleman [7], is the “SI electric” in which one defines B = H + μ₀M as the general relationship between B. H and M. In this system the unit for B and H is tesla and the unit for M is Am⁻¹, again giving B and H the same numerical value in empty space. However, both of these systems have the advantage (or disadvantage) found in the Gaussian system that B and H have the same numerical value in empty space.

Many of the details listed in the tables given here depend on the particular SI relationship adopted for magnetic quantities. However, whichever relationships are adopted, the conversions for magnetic induction and susceptibility listed in table 3 will remain valid, and the use of the proper unit and of the term ‘rationalized’ whenever susceptibility values are given in SI units would do much to reduce the possibility for errors and misinterpretation.