A gentle introduction to magnetism: units, fields,theory, and experimen ( Practice Magnetism )

 development: the greater access to SQUID magnetometers with user-friendly software;

increased collaborations between synthetic and physical chemistry groups; increased

collaborations between chemists and physicists; and the greater ability to form international

collaborations. However, as is frequently the case, many of these scientists are not simply

new to collecting magnetic data; they are also new to the field of magnetism and especially

to understanding the influence of magnetic interactions. Using a room temperature magnetic

moment to determine whether an isolated ion is high spin or low spin can be a straightfor-

ward matter, but the presence of interactions frequently leads to complex temperature- and

field-dependent data. The interpretation of such data is much more challenging to those

without sufficient experience.

This increase in reported magnetic data and magnetic behavior in materials extends

beyond the researchers themselves. As more chemists report magnetic data, it becomes

incumbent upon referees and readers to also be more familiar with routine interpretation of

temperature- and field-dependent magnetic data. This can be especially confusing to the

novice when the literature is taken into account. Historically, different systems of units have

been used for reporting magnetic data, and converting between them is not trivial. Even the

question of the simplest form of the magnetic exchange Hamiltonian can be a daunting task

given the variety of forms used. Is there a factor of two incorporated or not? What is the

meaning of the sign of J?

We present here a “Primer to interpreting magnetic data”that is intended as an introduc-

tion for those new to the field, whether new students or senior scientists, and we hope it

will serve as a stepping-stone to the more in-depth and specific reviews that are already

available in the literature. Please note that there is nothing original in this document;

everything presented here is standard magnetism as described in a long series of books,

monographs, and review articles [1–12].

1.1. Magnetic fields and units in the SI system

The International System of Units [13] (SI for the French name “Système International

d’unités”) is the modern form of the metric system, founded upon seven base units.

For magnetism, it is only necessary to use four of them: meters, kilograms, seconds,

and amperes (MKSA). All other units can be derived from these units plus fundamen-

tal equations of science. For instance, the metric unit for electric charge, the Coulomb

(C), is defined to be the charge transferred by a current of one ampere (A) in one sec-

ond.

The SI system is universally used in education and employs the mechanical and electrical

units with which we are all familiar: joules, watts, newtons, volts, ohms, farads, etc. The

units of magnetism are less familiar but can all be readily derived, as we shall see in the fol-

lowing paragraphs. Unfortunately, much of the research in magnetism is still reported in the

older cgs (centimeters, grams, seconds) system of units and converting from one system to

the other is an endless source of confusion. Following this introduction to SI magnetism,

cgs magnetism will be discussed, along with the appropriate conversion factors, in

Section 1.2.

We begin with the most important field, B, which can be defined in terms of the force on

a moving charge through the equation F=qv×B, where the respective units are Newtons

(N) for force, Coulombs (C) for charge, m s

−1

for velocity, and Tesla (T) for B. One tesla is

thus equivalent to one N A

−1

m

−1

, where the current is expressed in amps. The name of the

2C.P. Landee and M.M. Turnbull

B-field is the magnetic flux density

†

, because it equals the number of magnetic flux lines (in

units of Webers (Wb)) passing through an area of one square meter: 1T = 1 Wb m

−2

.

The B-field is related to the two other magnetic fields, Hand M, by equation (1), in

which the proportionality constant μ

0

is named the permeability of free space and has a

value of exactly 4π×10

−7

TmA

−1

.

B¼l0ðHþMÞ(1)

The H-field, named the magnetic field, arises from electrical currents passing through

wires. (The field created by current in a solenoid is the H-field.) The magnitude of the

magnetic field at a distance rfrom a long straight wire carrying a current Iis given by the

equation

H¼1

2p

I

r(2)

Consequently the units for the H-field are amperes per meter, A m

−1

. As shown later, one

Am

−1

is a very small field. It is also clear from equation (1) that Mand Hmust have the

same unit, A m

−1

. It is essential to realize that the magnetization appearing in equation (1)

is the magnetization per unit volume, and not the more common magnetization per mole

used in experimental work.

The volume magnetization Mequals the vector sum of all the magnetic moments per

cubic meter. Magnetic moments

‡

arise from circulating currents (whether quantum or free

currents) and are equal to the product of the current in the loop times the area of the loop;

moments have units of amp-meter

2

(A m

2

). The moment is normal to the plane of the loop

and in the direction such that the H-field generated by the current passes through the loop

according to the right-hand rule.

M¼X

i

li

!

=unit volume (3)

The units of the volume magnetization are A m

2

per cubic meter, or A m

−1

, the same as

that of the magnetic field H.

Scientists rarely know the volumes of their samples, but they can measure the masses.

For this reason, the magnetization per unit mass M

kg

is more useful. It is obtained from the

volume magnetization by dividing Mby the sample’s density expressed in the SI units of

density, kilograms per cubic meter, equation (4).

MkgðAm

2kg1Þ¼MðAm

1Þ

qðkg m3Þ(4)

One can then obtain the magnetization per mole M

mol

by multiplying the mass

magnetization by the formula weight (FW), equation (5).

MmolðAm

2mol1Þ¼MkgðAm

2kg1ÞFWðkg mol1Þ(5)

†

The B-field is also known as the magnetic induction.

‡

This topic is addressed more completely in Section 2.

Review: magnetism 3

The magnetic susceptibility vis usually the first quantity measured for a new compound.

The volume susceptibility is defined as the ratio of the sample’svolume magnetization to

the applied H-field, in the limit of a vanishingly small field, equation (6):

vvolðdimensionlessÞ¼ lim

H!0

M

H(6)

The volume susceptibility is dimensionless because the units of the volume magnetization

and the magnetic field are the same, equation (1). Nevertheless, in the laboratory, we mea-

sure the magnetization per unit of mass, and calculate the magnetization per mole, so the

corresponding mass and molar susceptibilities are defined as follows:

vkgðm3kg1Þ¼ lim

H!0

Mkg

H(7a)

vmolðm3mol1Þ¼ lim

H!0

Mmol

H(7b)

The units for the mass and molar susceptibilities are those of the inverse density and the

molar volume, respectively.

This section concludes with the important connection between SI magnetic units and

energy. In a B-field, a magnetic moment experiences a torque that tends to align the

moment parallel to the field. The energy Urequired to rotate the moment away from the

field direction is given as

U¼lBðthe Zeeman equationÞ(8)

with the minimum energy configuration occurring when the moment and field are parallel.

As seen from the units of this equation, the SI unit of moment μ(A m

2

) is also equal to the

ratio of energy to field, one joule per tesla: 1 A m

2

=1JT

−1

.

1.2. Magnetic fields and units in the CGS system

The SI is the legal system, but legality is not science. Indeed, this system is particularly

inappropriate in molecular magnetism and, like most researchers involved in this field, we

prefer to use the cgs-emu system. Olivier Kahn [1]

People studied magnetism long before the advent of the SI system and they created unit

systems to suit their own purposes, usually to make a set of fundamental equations as

simple as possible. Length, mass, and time were always in units of centimeters, grams, and

seconds so these unit systems became known as cgs systems. However, multiple cgs

systems were created, each one for the study of one particular branch of physics.

As an example, in the study of electrostatics, the fundamental unit of charge was defined

such that two identical charges, separated by one centimeter, exerted mutual forces of one

dyne (1 g cm s

−2

=10

−5

N). In the cgs-electrostatic unit system (cgs-esu), Coulomb’s Law

has the simple form

Coulomb’s Law ðcgs esu): F¼q1q2

r2(9)

4C.P. Landee and M.M. Turnbull

The charge thus defined is the esu unit of charge, the statcoulomb (1 sC = 3.33 × 10

−10

C).

One statampere is the flow of one statcoulomb per second and equals 3.33 × 10

−10

A.

In contrast, for the study of magnetism, the cgs-electromagnetic system was created

(cgs-emu) based on the definition of one unit of magnetic moment such that two equal

moments separated by one centimeter repel each other with a force equal to one dyne.

Given that the magnetic moment is equal to the product of a current times an area, this

definition of moment defines the cgs-emu current, known as the absolute current or abamp

(1 abA = 10 A) which is different from the stat-amp. Ultimately, the Gaussian system of cgs

units evolved which uses esu units for charges and emu units for magnetic fields and

moments. For the study of magnetism in this Tutorial, the Gaussian and cgs-emu units are

identical. The relationships between the Gaussian and MKSA units are given in Appendix

A. Additional sources for studying this confusing topic are available [14].

In the cgs-emu system, B,H, and Mare related as shown in equation (10). The units of

Band Hare equivalent but are given different names to help identify the field under discus-

sion. The unit of the B-field is the gauss (G, 1 G = 10

−4

T) and those of the H-field are

oersteds (Oe, 1 Oe = 10

3

/4πAm

−1

= 79.6 A m

−1

). The volume magnetization consists of the

vector sum of individual magnetic moments per unit volume, equation (3), and is often said

to have units of emu/cm

3

. This is unfortunate nomenclature for it leads people to believe

that the emu is the cgs-emu unit of magnetic moment; it is not! This use of emu simply

means that the magnetization is given in the emu system, with the cm

3

identifying the mag-

netization as the volume magnetization. Likewise, the mass and molar magnetizations in

cgs units are commonly written as emu g

−1

and emu mol

−1

, respectively (see Appendix A).

B¼Hþ4pM(10)

If not the emu, what is the cgs unit of magnetic moment? It is identified from equation (8),

the Zeeman equation, for the energy (in ergs) of a moment in an applied field (in G). The cgs

moment has units of ergs G

−1

and is smaller than the SI unit of moment, the A m

2

, by exactly

1000.

1erg

G¼107J

104T¼103Am

2(11)

Do we get the same numerical result when we compare definitions of moments (as a

product of currents times area) from the cgs and SI systems? Does 1 A cm

2

=10

−3

Am

2

?

Clearly not, because 1 m

2

=10

4

cm

2

. The inequality arises because the cgs-emu unit of cur-

rent is not the ampere, but the abampere, equal to 10 amperes. This example reveals a

severe disadvantage of the Gaussian system; its units for electrical variables (charge, cur-

rent, voltage, and resistance) are all different from the SI units in everyday use.

The cgs volume magnetization therefore has units of erg G

−1

cm

−3

, usually expressed as

emu cm

−3

. From equation (10), we see that volume magnetization also has the units of

oersteds, so 1 Oe = 1 erg G

−1

cm

−3

= 1 emu cm

−3

. However, it is also seen in equation (10)

that 4πMhas the unit of gauss! In the cgs system, both the free currents in wires and the

bound currents in magnetized matter produce magnetic flux lines but the free currents are

weighted more heavily. This difference does not occur in the SI system, equation (1).

For low fields, and in the absence of any permanent moments or hysteresis, the magneti-

zation is proportional to the H-field, M¼vH, equation (6), with the proportionality

constant vdefined as the magnetic susceptibility. The ratio of M/His dimensionless so

1 emu/cm

3

= 1 Oe = 1 G. Under these conditions, equation (10) can be written as

Review: magnetism 5

c

B¼ð1þ4pvvol ÞH¼lH(12)

where the ratio of B/H is defined as the magnetic permeability μ. Since Band Hhave equal

units, both the permeability and volume susceptibility are dimensionless.

As described in Section 1.1, for experimental work, it is convenient to work with the

mass and molar magnetizations. These are calculated by the same procedure used

previously, with the following results:

Mgðemu g1Þ¼Mðemu cm3Þ=qðgcm

3Þ(13a)

Mgðemu mol1Þ¼Mgðemu g1ÞFWðg mol1Þ(13b)

The mass and molar susceptibilities are defined in terms of the M

g

and M

mol

as before.

vgðcm3g1Þ¼ lim

H!0

Mg

H(14a)

vmolðcm3mol1Þ¼ lim

H!0

Mmol

H(14b)

The units for the mass and molar susceptibilities are once again those of inverse density

and molar volume, respectively, just as found for the equivalent SI susceptibilities in

equation (7).

We conclude this section by evaluating the conversion factors between the cgs and SI

susceptibilities. In addition to powers of 10 that arise from converting base units (1 cm

3

mol

−1

=10

−6

m

3

mol

−1

), a factor of 4πappears, due to the existence of 4πin equation (10)

and its absence in equation (1). For this reason, a cgs molar susceptibility in units of cm

3

mol

−1

equals 4π×10

−6

times the SI molar susceptibility, equation (14). See Appendix A.

vmolðcm3mol1Þ¼4p106vmol ðm3mol1Þ(15)

We have just defined the susceptibilities in the two unit systems, and learned how to con-

vert from one system to the other. However, we are not done. As shown in Section 2, appli-

cation of a magnetic field to a magnetic material will induce a moment that is determined

by the B-field, not the H-field. When calculating the susceptibility, the ratio B/Hwill appear.

This ratio is dimensionless in cgs-emu system [equation (10)], but B/Hhas the units of

(T m A

−1

) in the SI system, equation (1), so the conversion of susceptibility units is more

complex than shown in Appendix A.

2. Paramagnetism –magnetization and susceptibility of a mole of independent spins

In this section, we begin the study of magnetism itself. The connection between angular

momentum and magnetic moments is reviewed in Section 2.1, both for classical currents

and at the quantum level. Section 2.2 explores the effects of temperature and field upon a

collection of non-interacting moments; the case of S= 1/2 is worked out in detail. It is seen

that the net magnetization is due to a competition between the aligning effect of the B-field

6C.P. Landee and M.M. Turnbull

Click here to get instant acces