Table 5

Important Fundamental Constants

Quantity Gaussian
ug» permeability of

free space 1 (dimensionless)
ug» Bohr magneton

Uy, Nuclear magneton

9,274078(36) x 10°? oe

5.050824(20) x 10°°* ere/G

9.274078(36) x 1077" 4 ( An? )

5.050824 (20) x 107°? g/t

M=— : es)
v won

where M is the magnetization in @% m is an appro-

priate magnetic moment in erg/G, and V is an appro-

priate volume in cm?, Using the substitutions of

Table 4 we have

fe Moe ye,/* m Q
a

which reduces to

ee? (3)
v
Thus the magnetization in our suggested $I system
can also he considered as the magnetic moment per
unit volume, with magnetization in A/m, dipole

-:)
moment in J/T, and volume in m?. Table 5S gives the
numerical value of three important fundamental mag-
netic constants in the two unit systems, and Table
6 compares demagnetizing coefficients, N, for several
familar shapes, where the defining equation for N
for both systems is

Hoe Wt, -NM @)

with H the magnetic field strength within the mag-
netized body and Ng the applied magnetic field
strength,

DISCUSSION

There are currently several systems of electro
magnetic equations that may be used with SI units’ ?®.
In order to apply ST units in the field of mag-
netism with a minimum of confusion, agreement and
uniformity in symbols and definitions would he e
tremely helpful. Here we have suggested such a set
of synbols and definitions which covers nost of the
quantities of current interest to those who publish
in the M? proceedings. We would emphasize that this
set is possibly not the ono most desirable to a
majority of magneticians, It was selected as one
which appeared to us to he most in conformity with
current international usage, An example of an al-
ternative system would be the SI analog of a ra-
tionalized 'Gaussian’ system, In such a system B,
H, and M would be given the relation B-t+M, and I!
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’,
is the "SI electric" in which one defines BeH+gM
as the general relationship between 8, H and M. In

TABLE 6

Demagnetizing Covfficients, N, for homogeneous iso—
tropic bodies of various shapes.

N

Gaussian SI
Shape (unratfonalized) | (rationalized)
Il to axis
lof lon)
Incedle 0 0
long needle aa 1/2
sphere 47/3 U3
Lto plane
of a thin
disc a5 L

this system the upit for B and H is tesla and the
unit for Mis An’, again giving 8 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 iisted in the Tables given
here depend on the particular $I relationship
adopted for magnetic quantities. Nowever, which-
ever relationships are adopted, the conversions for
magnetic induction and-susceptibility listed in
Yable 3 will renain 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,

RE

CES

I. HG, 8. Casimir, Helv, Physica Acta 41, 741
(1968). ~

2. C, Heck, "Magnetic Materials and Their Applica-
tions", (Crane, Russak and Company, Inc,, New
York, 1974), p. 16.

3. For example, C. Kittel, “Introduction to Solid
State Physics", Fourth Edition, (J. Wiley and
Sons, Inc., New York, 1971), p. 499.

4, L, J. Giacoletto, TEEE Trans. on Magnetics 4,
1134 (1974, ~

5. For example, I. Zijlstra, Philips Tech, Rev. 34,
193 (1974). ~~

6, See, for example, C,H. Page, Aner. J. of Phys.
38, 421 (1970); H. V. Stopes-Roe, Nature 222,
S00 (1969); 224, 579 (1969); R. Green, Geophys,
Prosp. 16, 1 (1968).

7. J, E. Coleman, Amer. J. Phys. 41, 221 (1973).



AIP Conference Proccedings
Series Editor: Hugh C, Wolfe
Number 29

Magnetism and Magnetic Materials—1975

(2st Annual Conference —Philadelphia)

Editors

J. J. Becker
General Electric

G. Lander
Argonne National Laboratory

J.J. Rhyne
National Bureau of Standards

American Institute of Physics
New York 1976


COMMENTS ON UNITS IN 5

AGNETISM

L, Il, Bennett, C. MH, Page, and L, J. Swartzendruber
National Bureau of Standards

ABSTRACT

Suggestions are given on how to express mag-
netic quantities in ST units.

INTROOUCTION

Perusing the 1974 M® Conference Proceedings
indicates that, at the present time, Systeme
Internationale (SI) units are avoided by most
leading scientists and engincers in the field of
magnetism, Throughout the Proceedings, almost
universal preference is displayed for the crs
electromagnetic system (or €or 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 defi-
nitions 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 definition:

EMS

SOME CONSIDERATIONS ON THE TWO SYST

One major property of the Gaussian (and the
cas emu) system, considered an advantage by sone
and a disadvantage hy others, is that B and H have
the same numerical value in empty space. Changing
to the SI, where not only do B and I! have different
units in empty space, but also different numerical

Table 1

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

Name

1
symbol css emu ST

B flux density
magnetic induction

flux density
(magnetic induction)

a magnetic field magnetic field
strength strength

M magnet ization vagnetization

J --- magnetic polarization

x volume susceptibility

_ _ rationalized volune

susceptibility

x mass susceptibility —

% _ rationalized mass

° susceptibility

Xpole | Molar susceptibilicy

rationalized colar
mole susceptibility

nm magnetic moment magnetic moment

My Bohr magneton Bohr magneton

Gaithersburg, Maryland 20234

magnitudes, puts one somewhat in the position of
Casimir's! mythical tangenometrists who decided
that, "The volumetric displacement of empty space -
although equal to unity - had the dimension
Archimedes per FuclidTM.

The ST is a “rationalized” system, whereas the
Gaussian is unrationalized, Thus, when magnetic
susceptibilities are converted between the two
systems a factor of 47 is involved, Further factors
of 10 are involved depending on whether volunc,
mass, or molar susceptibility is in question, This
gives considerable latitude for errors and
arbiguities in data compilations, handbooks, and
treatises which attempt to convert existing nu-
merical values to ST units, and numerous examples
of such errors can be found, For example, in the
recent treatise on magnetic materials by Heck?, who
endeavors to use SI units as much as possible, a
table of paramagnetic susceptibilities apparently
gives the rationalized mass susceptibility for Pt
in cm3/g, the unrationalized mass susceptibility
for y-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 uscr could easily be misled. As
rost commonly used with ST, the relation between 3,
H, and M is defined as B=u,(li#M), y=4/H, Some
authors? exhibit the ug associated with the SI
explicitly by replacing, H by B/ugs giving x-voM/3.
‘This is, of course, approximately correct for the
small susceptibilities found in most dianagnetic
and paramagnetic materials, but could he misapplied
to superparamagnetic or ferromagnetic materials,

RECOMMENDATIONS,

In order to case conversion from Gaussian (and
cgs emu) to SI units, the names, definitions, and
symbols for magnetic quantities should he standardized.
This requires agreeement within the magnetism
community, Our current recommendations arc sum-
marized in the Tahles,

Table 1 lists recommended symbols and names
for magnetic quantities in SI and cgs emu. When
using ST units to express susceptibility, we be-
lieve it would be useful to lahel it ‘rationalized!
and give it the synbol «, reserving x for the non-
rationalized cgs em system, What we have Labeled
the "volume susceptibility" in Table 1 is often
referred to simply as just “susceptihility". The

TABLE 2

Corresponding equations in SI and cga Gaussian (or
ces emu). In this table, F refers to force, W refers
to the energy of a magnetic dipole in a field, w re-
fers to the volume energy density. Otber symbols

are defined in Table 1,

Gaussian (or egs emu) SI

Buy (H+TM) a)
ROO + 45M

Bey, hed (2)
x © M/A oe M/H Qa)
Fs xvi F © uge¥l aH/3x (4)
Wo= -aBeost W = -mBcos9 oO)

BH .
= we 6.
an 38H (6)



Table 3, Conversion from Gaussian to $,1. Units

| sotetey the Number for To Obtain the Number for
by
ussian Quantity Unit ST Quantity Unit
flux density, B G tor flux density, B
magnetic Field 3 magnetic field
strength, H Ge 10"/4e strength, H ala
: rationalized vol
volume susceptibility, X emu/cm3 (dimensionless) An piscsncinteee, d@imenstonises |
€ 3 +1072 || rationalized mass 3
mass susceptibility, x, emu/g. /8) 45°10 aecapeinaliey, % m/kg
molar svaceptibility;* ery s1o7® || Fatdonalized molar 3
ar suscep y emu/mol (=cm?/mol) 410 dincoreibiliceey. m3/mol
ap Tia _ | mole
19? magnetization, M Alm 7]
magnetization, G or Oe = =e
4x*10 magnetic polarization, J Tr
10° fax magnetization, M Alo
magnetization, 47TM G or Oe 73 —
10” magnetic polarization, J 13
up/atom or p/atom or *e
magnetization, M up/form. unit, etc. ** 1 magnetization, M up/ form. unit, etc.
magnetic moment of a -3 magnetic moment of a ene |
dipole, erg/G au) dipole, SE Comey
demagnetizing factor, N dinensionless Van rationalized demag- dimensionless
netizing factor, N
*

Also 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
does depend on the unit system.

» the numerical vaiue of the Bohr magneton

introduction of the symbol J (where J = ugM) in the used for "magnetization": G, Oe, emu/g, np/atom
Sl is useful due to the controversy“ over whether BM./PORMULA UNIT, ug/impurity, G cm?/g, enu/end,
one should define B=ug(H*M) or B=uoH+M. Further, and emu: and for "susceptibility" we find the fol-
the symbol J and the associated name ‘magnetic lowing variety of units: emu/g, emu/em}, emu/mole,
polarization", are in current usa’. 2 enu/g kOe, emu/gn-At. V, and emu/Oe mole.

Table 2 compares several of the more important To convert an equation given in the Gaussian
equations in the field of magnetism. Eqs. (1) and system to the corresponding equations in the SI,
(2) define the recommended usage of the synhols M Table 4 can often be useful, For example, in the

and J in SI, as mentioned above, In both Gaussian
and SI units, the volume susceptibility, defined by a
Eq. (3), is dimensionless and is the ratio of M to

H, (both with magnitudes which will change by a

factor of 4: upon rationalization), Eq. (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 susceptibilities.) Fa. (S) 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 3 gives numerical factors for converting
between the two unit systems, The conversions for
flux density, B, and susceptibility, -y ands; are
independent of the conventions adopted, i.e, whether
Beit+M, Beygh+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 alw: » a dimensional anaylsis on sus-
ceptibility units may be performed if ‘emut is replaced
by cm’, Anather 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 M* con-
ference proceedings we find the following units

Gaussian system the magnetization can be considered

s the magnetic moment per unit volume,

TABLE 4
Substitutional Symbols for Equations

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

[eauieian Ginitey Gaussian gyabol| Si aywbo1 |
flux density B Vila, 8
magnetic field 4 Yang, H
magnetization M Yp,74e M, or
Witm 3
volume susceptibility dL Gare
magnetic moment m “ielhe 0