Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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10 CHAPTER 1. PRINCIPLES OF MAGNETIC TREATMENT

1.3.2 Magnetic susceptibility and permeability

The magnetization of a material, in general, depends on the magnetic ﬁeld

acting on it. For many materials, P is proportional to K (at least when K is

not too large) and we may write:



P = 



K (1.22)

where , the volume magnetic susceptibility, is a property of the material. Since

P and K have the same dimensions,  is dimensionless. We can combine eq.

(1.22) with eq. (1.19) to get



E = 

(1 + )



K = 





K = 



K (1.23)

where



=1+ and  = 

(1 + ) (1.24)

The quantity 

is called relative magnetic permeability and is dimensionless,

while  is called magnetic permeability and has a unit of H/m.

While eq. (1.19) is general, eqs. (1.23) and (1.24) are based on assumption

that the material is both isotropic and linear. In other words, that P is pro-

portional to H and in the same direction. This assumption is never completely

true for ferromagnetic materials.

Either 

or  may be used to characterize a material. Volume magnetic

susceptibility  ranges from values close to 0, both positive and negative, to

positive values greater than 1, for dierent materials. For materials that have

very small susceptibilities, it is much more convenient to use  than 

.For

instance for  =10

6

, 

= 1.000001, and for  = -10

6

, 

= 0.999999.

Magnetic susceptibility can also be expressed with respect to the unit mass

of material of density ,andthen

" =





(1.25)

where " is the mass, or speciﬁc, magnetic susceptibility. Since a sample mass

p is usually better known than its volume, mass magnetic susceptibility is very

commonly employed. For the same reason the mass magnetization  rather

than magnetization P is often used to characterize materials. The mass mag-

netization is deﬁned as

 =



thus P =  (1.26)

1.3.3 Magnetic units

The units and symbols used in this book are those of the Système International

d’Unités, designated SI. This system, which is based on the meter, kilogram,

second, ampere, kelvin and candela, is essentially the same as the Giorgi, or

MKSA, system in the ﬁeld of electromagnetism.

1.4. MAGNETIC PROPERTIES OF MATERIALS 11

Essentially all students today are taught in SI units and most scientiﬁc jour-

nals require their contributors to use SI units. Nevertheless, many papers on

magnetism - and magnetic separation is no exception - are still published in cgs

units. Occasionally, the cgs system is polluted with the emu ”unit”, which is

not really a unit to which dimensional analysis can be applied [S3]. Therefore,

even today the ﬁeld of magnetism is not blessed with a lucid system of quantities

and units and numerous panel discussions and articles on the subject illustrate

the reigning chaos [C1, S4, G2]. Those authors who follow the current trend

and convert cgs units to SI units ﬁnd that the task of conversion is gigantic and

that errors almost always creep in.

Table 1.3 lists the SI units in use for most important quantities, together

with their conversion factors to electromagnetic cgs units.

Conversion of magnetic induction in the cgs unit of Gauss into the SI unit of

Tesla should present no di!culty, with the values in Gauss divided by 10

.When

the magnetic ﬁeld strength is used, Oersted multiplied by 10

@4 (approximately

79.58) will give the value in the SI unit of A/m.

Magnetic susceptibility is a more complicated quantity. It is the volume mag-

netic susceptibility  that enters most equations used in magnetic separation.

Although dimensionless in both systems of units, volume magnetic susceptibil-

ity diers by a factor of 4 in the systems of units, as can be seen in Table

1.3. Experimentally, it is easier to determine the mass magnetic susceptibility

and therefore most values of magnetic susceptibility that can be found in the

literature are those of mass susceptibility, often in cgs units. Various names are

used for this unit, e.g. emu/cm, emu/g×Oe and cgs, but all of them usually

represent the same unit, namely cm

/g. Thus in SI the unit is m

/kg and con-

version factor from cgs to SI is 4×10

3

. The following equations summarize

relationships between volume and mass magnetic susceptibilities, in both SI and

cgs systems of units (density is expressed in g/cm

(VL)=4(fjv) (1.27)

(fjv)="(fjv) and (VL)=10

"(VL) (1.28)

(VL)=4"(fjv) (1.29)

"(VL)=

4

"(fjv) (1.30)

"(fjv)=

4

"(VL) (1.31)

1.4 Magnetic properties of materials

It has been stated at the outset that all materials display certain magnetic prop-

erties, regardless of their composition and state. According to their magnetic

12 CHAPTER 1. PRINCIPLES OF MAGNETIC TREATMENT

Table 1.3: Units of important magnetic quantities. The quantity is cgs units is

obtained by multiplying the quantity in SI by the conversion factor.

For instance, 1 G = 10e-4 T.

Quantity Symbol SI unit cgs unit Conversion

factor

Magnetic ﬁeld

strength

H A/m Oersted (Oe) 10

@4

Magnetic induc-

tion (ﬂux den-

sity)

B Tesla (T) Gauss (G) 10

4

Magnetization M A/m Oe,

erg/Gcm

Mass magneti-

zation

 Am

/kg emu/g 1

Magnetic ﬂux  Wb (Weber) Mx

(Maxwell)

8

Polarization J Tesla (T) G 10

4

Magnetic mo-

ment



erg/G 10

3

Volume mag-

netic suscepti-

bility

 dimensionless dimensionless,

emu/cm

4

Mass (spe-

ciﬁc) magnetic

susceptibility

" m

/kg cm

/g 4 · 10

3

Demagnetiza-

tion factor

N dimensionless dimensionless 1/4

Energy product BH J/m

GOe 1/40

Permeability of

vacuum



H/m 1 (dimen-

sionless)

4 · 10

7

Bohr magneton 

9.274×10

24

Nuclear magne-

ton



5.051×10

27

Anisotropy con-

stant

K J/m

erg/cm

1

1.4. MAGNETIC PROPERTIES OF MATERIALS 13

Paramagnetic

Ferromagnetic

Antiferromagnetic

Ferrimagnetic

Figure 1.5: Schematic diagram of the alignment of magnetic moments.

properties, materials can be divided into ﬁve basic groups: diamagnetic, para-

magnetic, ferromagnetic, antiferromagnetic and ferrimagnetic. The last three

groups have generally higher magnetic susceptibilities than the other groups and

are frequently termed ”ferromagnetic” sensu lato (s.l.). This broad deﬁnition

must be distinguished from one of its subdivisions, ferromagnetic sensu stricto

(s.s.) [T1] as discussed below. The alignment of magnetic moments in each

type of material is shown in Fig. 1.5. The classiﬁcation of magnetism in terms

of magnetic permeability is given in Table 1.4.

1.4.1 Diamagnetism

Diamagnetism has its origin in the modiﬁcation of the electron orbit magnetic

moment by an external magnetic ﬁeld. The currents thus induced give rise to

an extra magnetic moment. However, according to Lenz’s law, the resulting

magnetic moment is in the opposite direction to the ﬁeld that has induced the

current. Therefore, approaching a source of the magnetic ﬁeld, a diamagnetic

material is repelled from it. This eect is present in all materials, independently

of temperature. Diamagnets have a very small and negative volume magnetic

susceptibility, of the order of - 10

5

(SI).

While many inorganic and almost all organic molecules are diamagnetic,

in many materials their diamagnetism is hidden by a much stronger paramag-

netic/ferromagnetic (s.l.) eect. This is, for instance, the case of metals that

are normally associated with small negative susceptibility, often overwhelmed

by positive paramagnetic or ferromagnetic contribution. The values of selected

diamagnetic metals and non-metallic compounds are given in Table 1.5. Table

14 CHAPTER 1. PRINCIPLES OF MAGNETIC TREATMENT

Table 1.4: Magnetic susceptibility in various types of magnetism.

Parameter Diamagnetism Paramagnetism

Antiferromag-

netism

Ferromagnetism

Ferrimagnetism

Susceptibility ? 0  0 AA 0

Relative per-

meability

? 1 1 AA 1

Table 1.5: Magnetic susceptibilities of diamagnetic materials.

Material [SI] ×10

"[m

@kg] × 10

Copper -1.0 -1.1

Zinc -1.4 -1.96

Gold -3.6 -1.86

Lead -1.56 -1.38

Diamond -2.0 -5.6

Water -0.9 -9.0

Ethylalcohol -0.73 -9.3

1.6 summarizes the magnetic susceptibilities of various diamagnetic minerals.

1.4.2 Paramagnetism

In paramagnetic materials the magnetic atoms or ions have permanent intrinsic

magnetic moments and occur in low concentrations. Susceptibility arises from

the competition between the aligning eect of the applied magnetic ﬁeld and

the randomizing eect of thermal vibrations. In zero applied magnetic ﬁeld,

the moments point in random directions. When a ﬁeld is applied, a small

magnetization develops, but the magnetic susceptibility is very small and is

inversely proportional to the absolute temperature.

In the presence of the external magnetic ﬁeld, and under standard practical

conditions (i.e. when 

K@nW ?? 1, where 

is the magnetic moment, n is

the Boltzmann constant and T is the absolute temperature) the magnetization

of a paramagnetic material is



f

q

(1.32)

where f is a constant and q is the number of magnetic dipoles. The susceptibility

of paramagnetic materials is then given by

 =

f

q

(1.33)

1.4. MAGNETIC PROPERTIES OF MATERIALS 15

Table 1.6: Mass magnetic susceptibilities of assorted diamagnetic minerals.

Mineral Chemical formula Mass magnetic

susceptibility

/kg]×10

Quartz SiO

-6.0

Calcite CaCO

-4.8

Sphalerite ZnS -3.2

Galena PbS -4.4

Baddeleyite ZrO

-1.4

Barite BaSO

-3.8

Magnesite MgCO

-6.4

Apatite Ca

[PO

]

(F,Cl) -3.3

Fluorite CaF

-7.9

Halite NaCl -9.0

Kaolinite Al

(OH)

- 20.0

where F is the Curie constant.

Equation (1.33) is called the Curie law, which states that magnetic sus-

ceptibility of the paramagnetic substance is inversely proportional to the tem-

perature. The paramagnetic susceptibility is always small (  10

5

to 10

3

(SI)), although paramagnetism can be several orders of magnitude larger than

diamagnetism.

As follows from eq. (1.32), the dependence of the magnetization of a para-

magnetic material on the ﬁeld strength is expressed by a straight line. In most

cases, in standard conditions such as W = 300 K and E = 1 T, the above formulae

apply. In extreme conditions, such as at cryogenic temperatures and very high

magnetic ﬁeld, a saturation eect appears. Taking into account eq. (1.23), it

transpires from eq. (1.32) that for paramagnetic materials the magnetic suscep-

tibility is independent of the magnetic ﬁeld. Table 1.7 summarizes the values of

magnetic susceptibility for selected paramagnetic metals and compounds, while

Table 1.8 lists susceptibilities of assorted paramagnetic minerals.

Compilations of susceptibility of minerals are numerous [A1, S1, K2, D2]

and large variations can be observed. Conﬂicting data can be explained by the

inherent variability in mineral composition and the presence of inclusions and

contaminants. Dierent measuring techniques tend to give diering values of

susceptibility [M1] and the usage of dierent systems of units also contributes

to frequent inconsistency of the values of magnetic susceptibility found in the

literature.

1.4.3 Ferromagnetism

In ferromagnetic materials (s.s.) interaction between neighbouring atoms is

strong so that the magnetic moments of all atoms are aligned parallel to each

other against the randomizing force of thermal motion. The strong internal

16 CHAPTER 1. PRINCIPLES OF MAGNETIC TREATMENT

Table 1.7: Magnetic susceptiblities of paramagnetic materials.

Material Mass magnetic

susceptibility

/kg]×10

Volume magnetic

susceptibility

[SI]×10

Aluminium 7.67 2.1

Chromium 42.0 30.2

Titanium 42.2 19.0

Vanadium 73.0 44.5

Platinum 12.3 26.4

Zirconium 16.7 10.8

CuSO

·5H

O 76.7 17.5

NiSO

·7H

O 201 39.8

MnO 860 445.5

Table 1.8: Magnetic susceptibilities of paramagnetic minerals.

Mineral Chemical composition Mass magnetic suscep-

tibility [m

/kg]×10

Hematite Fe

250 - 3800

Siderite FeCO

350 - 1500

Goethite FeOOH 250 - 400

Ilmenite FeTiO

200 - 1500

Rutile TiO

10 - 50

Wolframite (Mn,Fe)WO

350 - 1200

Pyrite FeS

3 - 200

Limonite Fe

·nH

O 100 - 400

Manganite MnO

·Mn(OH)

100 - 500

Pyroxene (Mg,Fe)

40 - 1000

Dolomite CaMg(CO

)

5-20

ﬁelds, which align the spins, are called molecular or Weiss ﬁelds. These ﬁelds

are the result of a quantum mechanical process called exchange interaction.

Weiss assumed that the molecular ﬁeld is very large, its magnitude indepen-

dent of external magnetic ﬁeld, and its direction not ﬁxed, but always parallel

to the magnetization. The Weiss hypothesis predicts that a material can be

spontaneously magnetized even in the absence of an applied magnetic ﬁeld at

temperatures below the Curie temperature, at which the thermal agitation over-

comes the molecular ﬁeld. This tendency aids the external ﬁeld in producing

saturation, i.e. complete alignment.

There is an apparent contradiction between theory, which explains that mag-

netic materials are fully magnetized even in the absence of an external ﬁeld, and

practice, which generally shows that such materials exhibit no magnetization or

magnetization much smaller than saturation. In fact, theory is correct at a mi-

1.4. MAGNETIC PROPERTIES OF MATERIALS 17

Figure 1.6: Temperature behaviour of various classes of magnetic materials.

croscopic level, but at a macroscopic scale the ferromagnetic body is subdivided

into Weiss domains, each of which is spontaneously magnetized to saturation,

but not in the same direction, by virtue of a strong exchange magnetic ﬁeld.

Inside a domain the magnetization is at a maximum (saturation), but as all

domains compensate each other, on macroscale the mean magnetization is nil.

The value of the saturation magnetization varies with temperature, decreas-

ing from a maximum value at W = 0 K, becoming zero at the Curie temperature

.AboveW

, the behaviour is similar to that of a paramagnetic material, with

the magnetization being proportional to the ﬁeld and the susceptibility decreas-

ing with increasing temperature. However, the susceptibility does not follow

the Curie law, eq. (1.33), but varies as

 =

W  W

(1.34)

Equation (1.34) is called the Curie-Weiss law. The temperature behaviour

of various types of magnetism is shown in Fig. 1.6.

In contrast to diamagnetic and paramagnetic materials, magnetization of

ferromagnetic materials (v=o=) depends not only on the ﬁeld strength, but also

on the shape and the magnetic history of the sample. For instance, a ferromag-

netic material can remain magnetized after removal of the external ﬁeld. The

ferromagnetics (s.l.) are thus characterized by hysteresis as is illustrated in Fig.

1.7.

18 CHAPTER 1. PRINCIPLES OF MAGNETIC TREATMENT

Figure 1.7: Hysteresis loop of a ferromagnetic (s.l.) material.

It can be seen in Fig. 1.7 that two kinds of the magnetic curve (or hysteresis

loop) can be determined. The induction curve is E = E(K)> while the mag-

netization (or intrinsic) curve illustrates the P = P(K) dependence. Since

E = 

(P + K), and since P is much greater than K in most parts of the

initial magnetization curve (curves OP and OQ, respectively, in Fig. 1.7), both

curves are comparable. The dierence occurs mainly in the saturation region

where P(K) is asymptotic to P

, while E(K) approaches 

+ K).

It can be seen in Fig. 1.7 that after the initial (or virgin) magnetization

reaches its saturation value E

or P

, (points P and Q, respectively), irre-

versibility, resulting from imperfections in the material, prevents the magne-

tization (or the induction) from returning to zero when the external ﬁeld is

removed. Magnetization and induction at K =0are called remanent magneti-

zation, or remanent induction, respectively.

By reversing the direction of the external ﬁeld K, the residual induction

and magnetization decrease to zero for particular ﬁelds K

and K

called

coercive force and intrinsic coercive force, respectively. As can be seen in Fig.

1.7, K

is not the same for P(K) and E(K) curves. For low-K

materials (soft

magnetic materials) the dierence is generally negligible, while for hard material

or permanent magnets (high K

) the dierence becomes signiﬁcant.

Magnetic permeability, deﬁned as  = E@K, can vary widely at various

points on the magnetizing and demagnetizing portions of the hysteresis loop.

The initial permeability 

and the maximum permeability 

max

are important

1.4. MAGNETIC PROPERTIES OF MATERIALS 19

Figure 1.8: The initial magnetization curve and permeabilities of a ferromag-

netic material.

parameters that can be determined as the slopes of the magnetization curve as

shown in Fig. 1.8. The initial permeability illustrates the behaviour of the

permeability when the external magnetic ﬁeld approaches zero. With increas-

ing magnetic ﬁeld the permeability of the ferromagnetics (s.l.) increases and

reaches its maximum. With further increase of the magnetic ﬁeld permeability

decreases and the relative permeability asymptotically approaches unity. This

trend is shown in Fig. 1.9 which illustrates the dependence of the relative per-

meability on the external magnetic ﬁeld for decarburized steel used for magnetic

yokes [B1]. In a similar way we can deﬁne magnetic susceptibility of a ferromag-

netic material, taking account relationship (1.24). Magnetic susceptibility has

a similar trend as permeability, and at a high magnetic ﬁeld it asymptotically

approaches zero.

Ferromagnetism occurs in nine elements: three transition metals, iron, cobalt

and nickel, and six rare earth metals, gadolinium, terbium, dysprosium, holmium,

erbium and thulium. Most alloys consisting of the three transition metals are

ferromagnetic, and so are many of their alloys with non-magnetic elements.

Many alloys of manganese with non-magnetic elements are also ferromagnetic,

although manganese is not ferromagnetic in the pure state. On the other hand,

there is no mineral that would possess ferromagnetic properties. The values of

magnetization and of the Curie temperature for three ferromagnetic elements

(v=v) and other ferromagnetic materials are given in Table 1.9.