Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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for example, in Co(Fe)-based amorphous alloys. This

is again a consequence of the averaging effect of ex-

change interaction for structural correlation lengths

much smaller than the domain wall width. The sit-

uation thus contrasts with that for large-grained

crystalline systems, where an average zero saturation

magnetostriction does generally not imply stress-

insensitivity of the hysteresis loop.

3. Applications

Another requirement for soft magnetic materials is

that the shape of the hysteresis loop should be able to

be varied according to the demands of various ap-

plications. Like in other soft magnetic materials, this

can be realized in nanocrystalline alloys by magnetic

ﬁeld annealing, which induces an easy axis parallel to

the applied ﬁeld by atomic pair ordering. The result-

ing almost perfectly shaped loops indicate that the

induced anisotropy clearly dominates over the other

anisotropy contributions. Still, the anisotropy con-

stant, K

, can be tailored small enough in order to

achieve the highest permeabilities (Fig. 4).

The anisotropy is primarily induced in the b.c.c.

grains and depends upon the crystalline fraction, the

silicon content, and the annealing conditions. In par-

ticular, the existence of a DO

superlattice structure at

high silicon contents is the key factor such that K

relatively small despite of the high T

. The kinetics of

anisotropy formation is significantly slower than in

the amorphous state. As a consequence, the nano-

crystalline alloys exhibit an excellent thermal stability

of their magnetic properties surpassing by far

those of amorphous alloys and even permalloys,

which allow elevated application temperatures up to

about 150 1C (see Amorphous and Nanocrystalline

Materials).

The electrical resistivity of nanocrystalline materi-

als is around 50 mOcm for Fe–(Cu)–Zr–B alloys, but

as high as 115 mOcm in the case of Fe–Cu–Nb–Si–B

alloys owing to the high silicon content in the b.c.c.

grains. Combined with the low ribbon thickness in-

herent to the production process, this yields a favor-

able frequency dependence of permeability and low

losses even at elevated frequencies, as in amorphous

cobalt-based alloys.

The relatively inexpensive base composition (iron)

and an established production technique coupled

with the extraordinary combination of material prop-

erties discussed above means these materials can be

used for a wide range of applications, such as mag-

netic cores for ground-fault interrupters, common

mode chokes, and high-frequency transformers.

See also: Amorphous and Nanocrystalline Materials;

Magnets, Soft and Hard: Magnetic Domains

Bibliography

Alben R, Becker J J, Chi M C 1978 Random anisotropy in

amorphous ferromagnets. J. Appl. Phys. 49, 1653–8

Buschow K H 1997 Magnetism and processing of permanent

magnet materials. In: Buschow K H (ed.) Handbook of

Magnetic Materials. Elsevier, Amsterdam, Vol. 10, pp.

463–593

Hasegawa N, Saito M 1991 Structural and soft magnetic prop-

erties of carbide-dispersed microcrystalline ﬁlms with high

thermal stability. In: Proc. Int. Symp. 3d Transition–Semi

Metal Thin Films, Magnetism and Processing. Japan Society

for the Promotion of Science, Sendai, Japan, pp. 165–74

Herzer G 1990 Grain size dependence of coercivity and perme-

ability in nanocrystalline ferromagnets. IEEE Trans. Magn.

26, 1397–402

Herzer G 1997 Nanocrystalline soft magnetic alloys. In: Bus-

chow K H (ed.) Handbook of Magnetic Materials. Elsevier,

Amsterdam, Vol. 10, pp. 415–62

Kneller E 1969 Fine particle theory. In: Berkowitz A E, Kneller

E (eds.) Magnetism and Metallurgy. Academic Press, New

York, pp. 365–471

Lachowicz H K, Slawska-Waniewska A 1994 Coexistence of

various magnetic phases in nanocrystalline Fe-based metallic

glasses. J. Magn. Magn. Mat. 133, 238–42

Luborsky F E 1961 High coercive materials. J. Appl. Phys. 32,

171S–83S

Pfeifer F, Radeloff C 1980 Soft magnetic Ni–Fe and Co–Fe

alloys—some physical and metallurgical aspects. J. Magn.

Magn. Mat. 19, 190–207

Scha

fer R, Hubert A, Herzer G 1991 Domain observation on

nanocrystalline material. J. Appl. Phys. 69, 5325–7

Suzuki K, Makino A, Inoue A, Masumoto T 1991 Soft mag-

netic properties of nanocrystalline bcc Fe–Zr–B and

Figure 4

DC hysteresis loops of nanocrystalline

73.5

13.5

annealed for 1h at 540 1C without

(R) and with a magnetic ﬁeld applied parallel (Z) and

transverse (F2; K

¼20Jm

3

, m ¼30 10

) to the

magnetic path. Sample F1 (K

¼6Jm

3

, m ¼100 10

)

was ﬁrst crystallized at 540 1C and subsequently

transverse ﬁeld annealed at 350 1C.

970

Nanocrystalline Materials: Magnetism

Fe-M–B–Cu (M ¼transition metal) alloys with high satura-

tion magnetization. J. Appl. Phys. 70, 6232–7

Yoshizawa Y, Oguma S, Yamauchi K 1988 New Fe-based soft

magnetic alloys composed of ultraﬁne grain structure. J.

Appl. Phys. 64, 6044–6

G. Herzer

Vacuumschmelze GmbH, Hanau, Germany

Nanosized Particle Systems,

Magnetometry on

New techniques for performing and interpreting

magnetization measurements can be used on nano-

sized particle systems. In this article measurements

using magnetometers such as a Faraday balance,

SQUID, or vibrating sample magnetometer are

considered. Their aim is to obtain information about

the magnetic properties of the entire sample but also

if possible about the system constituents—the parti-

cles. This article will focus on two measurement

strategies.

First, the magnetization M (B ¼ const., T )is

measured vs. temperature T at a constant external

magnetic ﬁeld B. Before the measurement the sample is

placed in a (high) magnetic ﬁeld to ‘‘prepare’’ a well-

deﬁned low-temperature magnetic state (LTMS). As

an example of LTMS which can be ‘‘prepared’’ in a

deﬁnite and reproducible way we introduce the new

idea of PHFC (positive high-ﬁeld cooling) which for

particles exhibiting magnetic anisoptropy energy opens

attractive possibilities for measuring the magnetic

moment of the particles and their anisotropy energy.

After reaching this LTMS a (weak) magnetic ﬁeld is

applied and the magnetization during the warming up

process (ﬁeld warming ¼ FW) is recorded.

The susceptibility vs. temperature derived from

adjacent FW magnetization curves M(B, T ) and

MðB þ DB; TÞ exhibits a maximum at a temperature

determined only by the particles’ magnetic moment

and the mean ﬁeld applied, i.e., B þ DB=2.

The theory of superparamagnetism is applicable

after PHFC in the limit of high enough magnetic

ﬁelds and leads to a simple formula allowing the

determination of the magnetic moment of the

particles without any calibration of the magnet-

ometer. After the PHFC process the magnetic

anisotropy energy inﬂuences the starting magnetiza-

tion for each applied (low) magnetic ﬁeld differently

before the FW measurement.

Second, the magnetization M(T¼ const., B)is

measured vs. ﬁeld at a constant temperature. Before

this, the sample must be completely demagnetized.

The initial magnetization curve is measured till

reaching saturation. Subsequently, the hysteresis loop

is measured. A useful criterion for the particle–

particle interaction can easily be obtained relating the

initial magnetization with the branches of the

hysteresis loop.

1. Nanosized Single-domain Particles

Every modern textbook on magnetism (e.g., Craik

1995) deals with single-domain particles. The max-

imum size for such single-domain particles is

determined by the so-called magnetic exchange length

as summarized by Hernando (1999). From data

obtained on popular ferromagnets it follows that

this typical length scale is of the order of nanometers.

Such nanosized particles considered in this article are

of great technical importance because of their wide

use in magnetic data storage technology.

1.1 Ideal Particle Systems

The properties of ideal particles considered in this

article are as follows. All particles are single domain.

The magnetization of one particle should (in a ﬁrst

approximation) be temperature independent. This

means that the exchange coupling between the atoms

inside the particle is very high. The magnetization of

a particle will change its direction with respect to the

particle body in a homogeneous way (coherent

reversal). It will ﬂuctuate like a single magnetic

moment with a value of the order of (10

–10

. The

particles do not exhibit any anisotropy energy (no

easy axes of magnetization) and there is also no

magnetic particle–particle interaction. Additionally it

is assumed that the particles are all of the same size,

i.e., no size distribution and therefore no distribution

of particle magnetic moments. Such particle systems

are superparamagnetic. For such particle systems

there has for a long time existed a theory for their

magnetic properties in thermodynamic equilibrium.

Their magnetization will be described by the Lange-

vin function considered later. The calculations pre-

sented in this article are performed with such ideal

particle properties.

It has been shown that one of the features of real

particle systems is that in the limiting case of high

enough external magnetic ﬁelds this Langevin theory

can be used to describe and evaluate their properties

from measured magnetization and/or susceptibility

curves (Hesse et al. 2000).

1.2 Stoner–Wohlfarth Particle Systems

A strong improvement in the physical understanding

of the magnetic properties of particle systems can be

achieved by introducing ideal particles with aniso-

tropy energy, the Stoner–Wohlfarth model (Stoner

and Wohlfarth 1948, Hesse et al. 2000).

971

Nanosized Particle Systems, Magnetometry on

The system may consist of particles with a homo-

geneous magnetization and a prolate ellipsoid shape.

Without an external magnetic ﬁeld the magnetization

will point into the long direction (polar axis) of an

ellipsoid of revolution. Both directions along this

polar axis are equivalent. This fact is called uniaxial

or bidirectional anisotropy (see Magnetic Anisotropy).

From the shape of the particle there follows a

difference in the magnetic stray dipole ﬁeld energy

when the magnetization is pointing parallel to the

polar axis or perpendicular to it (see Magnetic Layers:

Anisotropy). This anisotropy energy is described by

the product KV where the constant K is the aniso-

tropy energy density and V is the volume of the

particle (Stoner and Wohlfarth 1948, Craik 1995,

Hesse et al. 2002).

The angle-dependent part of the energy E concern-

ing the particle’s magnetization is given by a

competition of the anisotropy energy (see Magnetic

Anisotropy) and the Zeeman energy with maximum

values KV and BM

V, respectively. With a magnetic

moment of the particle equal to m ¼ M

V (M

is the

saturation value of the magnetization) in the external

magnetic ﬁeld B and a ﬁxed easy direction this energy

becomes:

Eða; gÞ¼KVcos

ðg  aÞM

VB cosðaÞB

switch

; g ¼ 0 ð1Þ

where a is the angle between the magnetization vector

M of the particle and the applied magnetic ﬁeld B,

and g represents the angle between the easy direction

and B (see also Fig. 2, below). If 0oBoB

switch

the

energy E(a,g) in Eqn. (1) exhibits two minima where

the deeper one corresponds to the parallel orientation

of the magnetization vector to the external ﬁeld. If

B4B

switch

only one minimum exists and in thermal

equilibrium all magnetization vectors point in the

ﬁeld direction. This situation will be used in the

PHFC process described later.

1.3 Real Particle Systems

When performing magnetization measurements, ex-

perimentalists generally have to deal with real particle

systems. Even if these particles satisfy the condition

of being single domain, many complicating features

in comparison to the ideal particles will appear from

the following facts. There will be a distribution of

particle sizes. As a consequence the system will

exhibit a distribution of magnetic moments and a

distribution of particle anisotropy energies. In the

case of anisotropy there will be a distribution of easy

axis orientations. Additionally, particle–particle in-

teractions can be expected. The magnetization of a

particle will not change its orientation by a uniform

rotation process but will more or less perform a

curling process (Aharony 1966).

All these complications will disturb the picture of

an ideal superparamagnet. In real particle systems the

magnetization of one particle will depend on tem-

perature (internal particle excitations) which must be

considered when the temperatures for magnetization

measurements approach the Curie temperature of the

particle material.

2. Basic Superparamagnetism

The basic theory of (super)paramagnetism deals with

the magnetization MðB; T Þ of a very large number of

identical particles (number density per unit volume

N) with a magnetic moment m measured in the

direction of an external magnetic ﬁeld B in the state

of thermodynamic equilibrium at a temperature T.

For particles with large magnetic moments, the

quantization of angular momentum can be neglected

and therefore the Langevin function LðxÞ can be used

for describing the system. M

is the saturation value

of the magnetization for T-0orB-N and

x ¼



Bfield

All relevant formulas are summarized as follows:

MðT; BÞ¼M

LðxÞ;

¼ mNLðxÞ¼cothðxÞ

;

x ¼



Bfield

ð2Þ

In the following examples experiments on real

particle systems (described in detail in Hesse et al.

2000, 2002) will be presented. The results are

compared with calculations based on the Langevin

paramagnetism. The parameters assumed for calcula-

tions are chosen to be close to the properties of the

real particle system. For simplicity the following

abbreviations are introduced: in cases where the

anisotropy energy KV is in competition with the

magnetic energy mB ¼ M

VB we use their tempera-

ture equivalents deﬁned as KV ¼ k

aniso

and

mB ¼ k

B2field

3. Susceptibility Versus Temperature

Besides the magnetization, the susceptibility is a very

important and informative quantity. If the system in

question is in thermodynamic equilibrium at the

temperature T in the magnetic ﬁeld B the suscept-

ibility w is a measure of the change of magnetization

when we increase the magnetic ﬁeld by an amount

DB. In the exact mathematical limit DB must tend to

zero. In laboratory practice a ﬁnite small DB can be

accepted. Then, the susceptibility w

lab

is obtained.

The dimensionless susceptibility is in the case of the

972

Nanosized Particle Systems, Magnetometry on

Langevin function describing the paramagnetism

accompanied by a susceptibility function HðxÞ. The

theory is summarized in the following equations:

wðT; B ¼ B

applied

Þ¼m

lim

DB-0

MðT; B þ DBÞMð BÞ



wðT; B ¼ B

applied

Þ¼m



B¼B

applied

¼ m

HðxÞHðxÞ¼x 1  coth x

ðÞ

(



)

lab

T; B þ



¼ m

MðT; B þ DBÞMð BÞ



 w

initial

ðTÞ¼wðT ; B

applied

¼ 0Þ

¼ m



B¼0

ð3Þ

Considering the course of the susceptibility vs.

temperature T (at constant magnetic ﬁeld B)itis

obvious that this function exhibits a maximum. This

maximum corresponds to a temperature T

max

. This

fact is not well known because for ‘‘classic’’ para-

magnets with atomic moments of the order of one

Bohr magneton and an external magnetic ﬁeld of 1 T

we expect T

max

¼ 0:35 K. Kunkel et al. (1988)

demonstrated the existence of a maximum in experi-

ments on paramagnets exhibiting magnetic moments

of the order of 1 m

by performing measurements in

high magnetic ﬁelds.

The situation changes dramatically if we consider

modern nanosized materials with magnetic particles

exhibiting moments of the order of (10

–10

From the properties of the susceptibility function

HðxÞ it follows that

1:9k

max

¼ mB

max

½K

¼ n

Bohr

1:9k

½T

¼ 0:354n

½T

ð4Þ

where the square brackets [ ] serve to indicate that the

units used to measure temperature and ﬁeld are

kelvin and tesla, respectively. The (dimensionless)

number of Bohr magnetons is indicated by n. For

example, if n ¼ 10

and B ¼ 1 mT we ﬁnd T

max

35:4 K. Of course, T

max

is proportional to the mag-

netic ﬁeld applied. A useful feature of the formula is

that it allows the determination of the particles’

magnetic moment from T

max

without any calibration

of the magnetometer.

This simple and valuable law holds for ideal

paramagnets, for particles without any anisotropy

energy, without interaction, and without any size

distribution. Can such a theory be of value for real

particle systems, for real experiments? It has been

shown (Hesse et al. 2000) that in the limit of high

applied external magnetic ﬁelds the theory of ideal

superparamagnets becomes applicable. To achieve

this it is necessary to prepare the low-temperature

magnetic state (LTMS) of the particle system in a

quasiparamagnetic state by the process of positive

high-ﬁeld cooling (PHFC).

4. The Positive High-ﬁeld Cooling Process and the

Quasi-paramagnetic Low-temperature State

As mentioned above, the particles considered must be

ﬁxed in space. We cool down the sample to low

temperatures in an applied ﬁeld B4B

switch

starting

from high temperatures. The result is that all

magnetic moments are pointing parallel to the

external magnetic ﬁeld (in the positive direction).

For the magnetization measurement we lower the

ﬁeld value to B

low

switch

and start the FW (ﬁeld

warming) process. As a consequence of the PHFC the

projections of all magnetic moments show a compo-

nent in the (positive) ﬁeld direction.

This way the blocking behavior is eliminated.

Heating up this LTMS persists up to rather high

temperatures, which exceed the usual blocking

temperature due to the stabilization of the magnetic

moment direction by the magnetic ﬁeld. Such LTMS

is named the quasiparamagnetic state.

After the PHFC process performing FW magneti-

zation measurements in different B

low

magnetic ﬁelds

the maxima in the susceptibility (Fig. 1) and the

particles’ anisotropy energy (Fig. 3) become visible. In

Fig. 1(a) two calculated magnetization curves MðBÞ

and MðB þ DBÞ vs. temperature are presented. The

resulting susceptibility is derived as w

lab

from the

difference ðMðB þ DBÞ2MðBÞÞ and clearly shows

the expected maximum. In the calculation a magnetic

moment of m ¼ 10

was assumed, together with B ¼

15 mT and B þ DB ¼ 18 mT corresponding to values

of T

B2field

of 100 K and 120 K, respectively. In

Fig. 1(b) results of measurements performed on nickel

nanoparticles ﬁxed in an aluminum oxide matrix are

presented. The susceptibility was derived from adja-

cent curves of magnetization. One can clearly observe

the maximum of the susceptibility and its shift towards

higher temperatures with increasing magnetic ﬁeld.

The details are described by Hesse et al.2000).

5. Magnetization and Anisotropy

Next it will be shown that the PHFC process offers a

rather simple way to measure the particles’ magnetic

anisotropy energy. For simplicity in the ﬁrst step it is

assumed that the sample is textured. This means that

all particles are oriented parallel to each other with

respect to their easy axis of magnetization. All the easy

axes have the same angle with respect to the magnetic

ﬁeld (Fig. 2). The Stoner–Wohlfarth model is chosen

973

Nanosized Particle Systems, Magnetometry on

for a more detailed description. If the angle between

the applied external magnetic ﬁeld and the easy axis is

g, the projection of the magnetization vector at very

low temperature along the applied magnetic ﬁeld

(in the quasiparamagnetic state) becomes M

cos a

(see Fig. 2). In order to calculate the temperature

dependence of the magnetization observed in the

following FW process intuitive formulae (Eqns. (5))

are proposed. Here a function L

ðxÞ is used:

MðB; TÞ¼Nm cosðaÞL

res



res

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðB þ B

aniso

cosgÞ

þðB

aniso

singÞ

cosa

B þ B

aniso

cosg

ðB þ B

aniso

cosgÞ

þðB

aniso

singÞ

aniso

¼k

aniso

¼ KV ð5Þ

The anisotropy inﬂuence (after PHFC) is taken into

account by an anisotropy ﬁeld B

aniso

. The inﬂuence of

the anisotropy energy is represented by two factors,

the anisotropy ﬁeld B

aniso

and the angle g between the

applied external magnetic ﬁeld and the easy axis. The

thermal ﬂuctuation of the magnetization vectors will

be described by a Langevin-like function L

ðxÞ

labelled with an asterisk. This asterisk means that

this function is not identical to LðxÞ deﬁned in Eqn.

(2). It is necessary to take into consideration thermal

ﬂuctuations of the magnetization vector within a

limited range of angles due to the conﬁnement of the

magnetization vector in the potential minimum

caused by the anisotropy energy. In addition, ‘‘back

jumps’’ of the magnetization vector into the second

energy minimum (where the magnetization vector has

a negative component with respect to the applied

magnetic ﬁeld) must be considered.

In Fig. 3(a) calculated magnetization curves are

presented using LðxÞ instead of the analytically

unknown L

ðxÞ. The experimental results presented

Figure 2

The vector model to explain the inﬂuence of the

anisotropy after the PHFC process in the

quasiparamagnetic state. (a) A Stoner–Wohlfarth

particle with its easy direction, the external magnetic

ﬁeld B, and the equilibrium position of the particle’s

magnetization vector. (b) Here the anisotropy energy is

replaced by an anisotropy ﬁeld. The resulting magnetic

ﬁeld acting in the direction of the particle’s

magnetization vector can be calculated using Eqns. (5).

Figure 1

(a) Calculated magnetization MðBÞ and MðB þ DBÞ vs.

temperature and the resulting susceptibility exhibiting

a maximum. In the calculation a magnetic moment of

m ¼ 10

and B ¼ 15 mT, B þ DB ¼ 18 mT were

assumed corresponding to T

B2field

of 100 K and 120 K,

respectively. (b) The susceptibility w

Lab

derived from FW

magnetization measurements after PHFC (presented in

Fig. 3) performed on nickel nanoparticles. The ﬁeld

differences used for w

lab

calculations are indicated. The

maximum of the susceptibility and its shift with

increasing magnetic ﬁeld towards higher temperatures

can be observed. The sample details are described by

Hesse et al. (2000).

974

Nanosized Particle Systems, Magnetometry on

in Fig. 3(b) prove the validity of this approach. A

Stoner–Wohlfarth particle system with easy axes

oriented at 451 with respect to the magnetic ﬁeld

applied was assumed. The ﬁeld values and the particle

anisotropy are given in equivalent temperatures as

deﬁned in the text. In the lower part of Fig. 3(b) the

magnetization measured on nickel nanoparticles after

PHFC in the ﬁelds indicated (and their equivalent

temperatures) are presented. One can clearly observe

offsets in the magnetization values due to the

presence of anisotropy. Details are described by

Hesse et al. 2000).

6. The Hysteresi s Loop

Using the property that a Stoner–Wohlfarth particle

has no explicit initial magnetization curve, Thamm

and Hesse (1996) proposed an alternative method

to the well known Henkel plot (Henkel 1964) in order

to study the interactions between single–domain

particles. This proposal is justiﬁed exactly for

particles with uniaxial anisotropy. For such systems

it is only necessary to measure ﬁrst the initial

magnetization curve of the completely demagnetized

sample and then the hysteresis loop (see Magnetic

Hysteresis). In an ensemble of a large number of

spatially ﬁxed noninteracting particles with uniaxial

anisotropy and randomly distributed easy axes, there

are always pairs of particles with magnetization

vectors in opposite directions. The resulting normal-

ized hysteresis loop of such a pair represents the mean

value caused by the summation of the magnetization

of each particle. As a consequence an initial curve

will appear which is exactly half of the sum of the

upper m

and the lower branch m

þ m

ðÞDm ¼ m



þ m

ðÞð6Þ

Dm is zero for noninteracting Stoner–Wohlfarth

particles.

A ﬁnite value of Dm may be caused by interactions

between the particles or by the violation of the criteria

for Stoner–Wohlfarth particles. An example presenting

a measured initial magnetization curve together with

the hysteresis loop is given in Fig. 4. The details are

Figure 3

(a) Calculated magnetization M versus temperature for

a Stoner–Wohlfarth particle system with easy axes

oriented at 451 with respect to the magnetic ﬁeld applied.

The ﬁeld values and the particle anisotropy are given in

equivalent temperatures as deﬁned in the text. (b) The

FW magnetization measured after PHFC on nickel

nanoparticles in the ﬁelds indicated. T

B2field

also is given

calculated for particles with m ¼ 4300m

. The offsets in

the lowest temperature value of the magnetization due

to the anisotropy can be observed. The sample details

are described by Hesse et al . (2000).

Figure 4

(a) Measured (normalized) initial magnetization curve

together with the hysteresis loop. (b) The Thamm–Hesse

plot as derived from the measurement in the upper part

indicating particle–particle interaction.

975

Nanosized Particle Systems, Magnetometry on

described by Thamm and Hesse 1998. There is still no

general theoretical justiﬁcation for Dm in the case of

particles with multiaxial anisotropy energy. Never-

theless a nonzero deviation Dm and its dependence on

the applied magnetic ﬁeld DmðBÞ will bear information

about the particle–particle interaction. The depen-

dence of DmðBÞ can be derived from one measured

initial magnetization curve and the hysteresis loop,

which is much faster and easier to do than the mea-

surement of the rather popular Henkel plot (Henkel

1964), which hides the ﬁeld dependence of Dm.

7. Concluding Remarks

Nanosized single-domain magnetic particles open new

possibilities in performing and interpreting magneti-

zation measurements. If the particles are ﬁxed in space

and exhibit magnetic anisotropy energy it is possible

to bring them into a well-deﬁned new low-tempera-

ture magnetic state. This can be achieved by the

PHFC process. After that the particle’s magnetization

possesses a component pointing parallel to the applied

magnetic ﬁeld. This LTMS is called quasiparamag-

netic. When measuring the magnetization versus

temperature in a constant (weak) external magnetic

ﬁeld starting at low temperatures from a quasipar-

amagnetic state, the susceptibility exhibits a max-

imum. From the temperature of this maximum a

direct determination of the (mean) particle magnetic

moment is possible without any calibration. In the

magnetization values measured for the LTMS in

different (weak) magnetic ﬁelds the anisotropy

becomes visible and can be determined. This new

technique of performing magnetic measurements will

enable experimentalists to distinguish between rever-

sible superparamagnetic and irreversible or thermally

stable behavior as reported by Chantrell et al. (2000).

All measurement procedures and considerations

will hold for modern artiﬁcially produced arrays of

particles or for systems exhibiting self-organized

structures as well. Here a large variety of measure-

ment possibilities will appear depending on the

direction of the sample with respect to the magnetic

ﬁeld used for the PHFC process.

In the case when the hysteresis loop is measured at

constant temperature it is recommended to start the

measurement from a completely demagnetized state of

the sample. Then the initial magnetization curve must

be determined ﬁrst and subsequently both branches of

the hysteresis loop. A simple and useful criterion for

the particle–particle interaction can be obtained by

plotting the deviation Dm vs. applied magnetic ﬁeld.

See also: Ferroﬂuids: Introduction; Magnetic Re-

cording Materials: Tape Particles, Magnetic Proper-

ties; Magnetic Recording Technologies: Overview;

Nanocrystalline Materials: Magnetism; Thin Films:

Domain Formation

Bibliography

Aharony A 1966 Magnetization curling. Phys. Stat. Solidi 16,

3–42

Chantrell R W, Walmsley N, Gore J, Maylin M 2000

Calculation of the susceptibility of interacting superpara-

magnetic particles. Phys. Rev. B 63, 024410 (1-14)

Craik D 1995 Magnetism—Principles and Applications. Wiley,

Chichester, UK

Henkel O 1964 Remanenzverhalten und Wechselwirkungen in

hartmagnetischen Teilchenkollektiven. Phys. Stat. Solidi 7,

919–29

Hernando A 1999 Magnetic properties and spin disorder in

nanocrystalline materials. J. Phys.: Condens. Matter 11,

9455–82

Hesse J, Bremers H, Hupe O, Veith M, Fritscher E W, Valtchev

K 2000 Different susceptibilities of nanosized single-domain

particles derived from magnetization measurements.

J. Magn. Magn. Mater. 212, 153–67

Hesse J, Pu

tter S, Michele O, Bremers H, Hupe O, Hartung H,

Eichler M 2002 Probing magnetism of nanosized single

domain particle systems. Phys. Stat. Solidi (a) 189, 481–94

Kunkel H P, Roshko R M, Williams G 1988 Field-dependent

susceptibility of a paramagnet. Phys. Rev. B 37, 5880–3

Stoner E C, Wohlfarth E P 1948 A mechanism of magnetic

hysteresis in heterogeneous alloys. Philos. Trans. R. Soc.

London 240A, 599–642

Thamm S, Hesse J 1996 A simple plot indicating interactions

between single-domain particles. J. Magn. Magn. Mater. 154,

254–62

Thamm S, Hesse J 1998 The remanence of a Stoner–Wohlfarth

particle ensemble as a function of the demagnetization

process. J. Magn. Magn. Mater. 184, 245–55

J. Hesse

Technische Universita

t Braunschweig, Germany

NDT Techniques: Magnetic

Recent years have seen rapid developments in the use

of magnetic measurement techniques for materials

evaluation. These range from the use of macroscopic

hysteresis measurements through microscopic Bark-

hausen effect measurements down to nanometer scale

magnetic force microscopy. Magnetic methods can be

used to address two main classes of problems in ma-

terials evaluation: detection of defects, and determi-

nation of intrinsic properties. It is now one of the

fastest-developing ﬁelds in materials evaluation due

in part to the development of computer models that

have made it possible to interpret the results of mag-

netic measurements in terms of changes in stress or in

the structure of materials.

1. Macroscopic Scale: Magnetic Hysteresis

On the macroscopic scale, all ferromagnetic materials

exhibit hysteresis in the variation of ﬂux density B

976

NDT Techniques: Magnetic

with magnetic ﬁeld H (Jiles 1998). The hysteretic

properties such as permeability, coercivity, reman-

ence, and hysteresis loss are known to be sensitive to

stress, strain, grain size, heat treatment, and the pres-

ence of a second phase. In addition, the measurement

of hysteresis yields a number of independent para-

meters, each of which changes to some degree with

stress, strain, and microstructure. Since several inde-

pendent parameters are needed in general to separate

effects of stress from microstructure, hysteresis meas-

urements are ideally suited for materials evaluation

because the necessary parameters can be obtained

from a single measurement.

The effects of elastic stress on hysteresis have been

reported by a number of investigators. Stress changes

the shape of the hysteresis loop as shown in Fig. 1.

However, interpretation of results is difﬁcult with-

out a working model of hysteresis, since none of the

direct hysteresis parameters such as coercivity, reman-

ence, initial permeability, or hysteresis loss is uniquely

related to a single physical property. It is clear that any

model used for interpretation of nondestructive eval-

uation results such as effects of stress, elastic and

plastic strain, creep and fatigue should have as few

parameters as is reasonably possible. The stress or

strain dependence of these parameters can then be

determined. Some success has been achieved with a

magneto-mechanical model based on an effective ﬁeld

due to stress (Sablik et al. 1993) and this can be used

to predict and interpret the magnetic hysteresis prop-

erties under stress.

2. Microscopic Scale: Barkhausen Effect

On the microscopic scale the magnetic Barkhausen

effect (MBE) consists of discontinuous changes in

ﬂux density, known as Barkhausen jumps, as shown

in Fig. 2. These are caused by sudden irreversible

motion of magnetic domain walls when they break

away from pinning sites as the magnetic ﬁeld H

changes. There are ﬁve mechanisms by which MBE is

caused: (i) discontinuous, irreversible domain wall

motion; (ii) discontinuous rotation within a domain;

(iii) appearance and disappearance of Ne

el peaks; (iv)

inversion of magnetization in single-domain particles;

(v) displacement of Bloch or Ne

el lines in two 1801

walls with oppositely directed magnetizations.

Figure 1

Changes in the hysteresis curve caused by stress (after Makar and Tanner 1998).

977

NDT Techniques: Magnetic

The Barkhausen pulse height distribution is de-

pendent on stress and the number density and nature

of pinning sites within the material (Mandal et al.

1999). The pinning sites are inhomogeneities in the

material such as grain boundaries, dislocations, or

precipitates of a second phase with different magnetic

properties from the matrix material. Most Barkhau-

sen activity occurs close to the coercive ﬁeld H

Double peaks in the count rate can occur and the

location and size of the peaks can shift as a result of

changes in the defect distribution so that two spec-

imens of the same material with different defect dis-

tributions have different Barkhausen spectra.

Barkhausen signals are sensitive to stress as shown

in Fig. 3. MBE measurements are used for the non-

destructive evaluation of stress. All ferromagnetic

NDE methods are sensitive to both mechanical stress

and the microstructure of the material and therefore,

in order to determine stress, it is necessary to use

several independent measurement parameters. It has

been found that Barkhausen effect, incremental per-

meability, X-ray diffraction, and hardness measure-

ments were successful in estimating residual stress.

Changes in the density of dislocations affect the MBE

signals and MBE can be used to distinguish between

microstructures, which cannot be distinguished on

the basis of optical microscopy.

The effect of fatigue by cyclic loading on the root

mean square value of MBE signals has shown that

residual strains in unloaded specimens could be iden-

tiﬁed, although changes occurring under fatigue cy-

cling were complicated so that the Barkhausen signals

could not be simply related to the applied load.

However, the median Barkhausen pulse amplitude

can be used to assess whether the applied stress am-

plitude is above or below the fatigue limit.

3. Nanoscopic Scale: Magnetic Force Microscopy

In order to evaluate the variation of materials prop-

erties from one location to the next, magnetic imaging

techniques can be used. For example, the variation of

magnetic ﬁeld strength across the surface of a material

can be used to produce a two-dimensional image. This

could be used to identify unusual regions of a mate-

rial, which may have defects or be otherwise dam-

aged, or simply to investigate the spatial variation of

magnetization.

The magnetic force microscope samples the mag-

netic ﬁeld gradients close to the surface of the spec-

imen over a scanning range of typically 10–100 mm.

The method (see Magnetic Force Microscopy) is ba-

sically a scanning force probe technique detecting

magnetic forces between the magnetized tip of the

microscope and a test specimen, as shown in Fig. 4.

The magnetic force microscope can be used to detect

both static and dynamic magnetic ﬁelds on the sur-

face of a specimen. In the original device, a spatial

resolution of 100 nm was demonstrated. Today, spa-

tial resolutions of 50 nm are routine, and claims of

resolutions down to 10 nm are made in some cases

(Dahlberg and Proksch 1999). This method is there-

fore useful for characterizing magnetic materials on

length scales from typically 10 nm up to 100 mm.

The image formed as a result of magnetic force mi-

croscopy is obtained by measuring the force or the

Figure 2

Barkhausen signal shown as voltage against time. The

time dependence of magnetic ﬁeld H, voltage in ﬂux coil

V, and envelope of irreversible component of voltage in

ﬂux coil V

irr

are also shown on the same time scale.

Figure 3

Changes in Barkhausen emissions as a result of applied

tensile stress.

978

NDT Techniques: Magnetic

force gradient. The images formed can be used to study

magnetic structures such as domain walls, closure

domains, Bloch lines and, in the case of magnetic re-

cording media, intentionally magnetized regions of a

material. It is in the area of magnetic recording media

that the magnetic force microscope is ﬁnding most

applications. The ability to resolve features at length

scales down to 50 nm, with little or no surface prep-

aration required, and under ambient atmosphere

conditions makes this technique ideal for practical

high-resolution magnetic evaluation of material

surfaces.

4. Conclusions

The increasing demand for characterization of mag-

netic materials indicates that there will be a continual

growth of interest in the available magnetic evaluation

methods in the next few years. A variety of magnetic

methods for characterization and evaluation of mate-

rials is available. Some techniques are designed to de-

termine intrinsic properties of materials and these

include the Barkhausen effect for surface characteri-

zation and hysteresis measurements for determination

of bulk properties. These methods all offer wide scope

for future development because of their sensitivity to

material treatment, particularly the presence of differ-

ent material phases, chemical composition and mate-

rial deformation in its various forms: elastic and plastic

strain, fatigue, dislocation density, and creep stress.

Another major group of magnetic techniques has been

developed to detect and characterize inhomogeneities,

including defects, in materials. These techniques in-

clude such methods as magnetic particle inspection,

magnetic ﬂux leakage determination, and magnetic

force microscopy. Magnetic force microscopy, an ultra

high-resolution imaging method, can be used to pro-

duce an image of the magnetic ﬁeld at the surface of a

test specimen with resolutions down to 10 nm.

See also: SQUIDs: Nondestructive Testing

Bibliography

Dahlberg E D, Proksch R 1999 Magnetic microscopies: the new

additions. J. Magn. Magn. Mater. 200, 720

Jiles D C 1998 Introduction to Magnetism and Magnetic Ma-

terials. Chapman and Hall, London

Magni A, Beatrice C, Durin G, Bertotti G 1999 Stochastic the-

ory of hysteresis. Br. J. Appl. Phys. 86, 3253

Makar J M, Tanner B K 1998 The effect of stressses approach-

ing and exceeding the yield point on the magnetic properties

of high strength pearlitic steels. NDT&E Int. 31, 117

Mandal K, Dufour D, Atherton D L 1999 Use of magnetic

Barkhausen effect noise and magnetic ﬂux leakage signals for

analysis of defects in pipeline steel. IEEE Trans. Magn. 35,

2007

Sablik M J, Rubin S W, Riley L A, Jiles D C, Kaminski D A,

Biner S B 1993 A model for hysteretic magnetic properties

under the application of noncoaxial stress and ﬁeld. Br. J.

Appl. Phys. 74, 480

D. C. Jiles

Iowa State University, Ames, Iowa, USA

Non-Fermi Liquid Behavior: Quantum

Phase Transitions

The Landau theory of Fermi liquids has been very

successful in describing even complex metals such as

heavy-fermion systems. However, under certain con-

ditions this model is not capable of explaining the

observed thermodynamic and transport properties of

metals. Several origins of such non-Fermi liquid be-

havior are possible, e.g., a magnetic instability leading

Figure 4

(a) Magnetic probe tip scanning over the surface of a material. (b) Schematic of the cantilever arrangement supporting

the probe tip.

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Non-Fermi Liquid Behavior: Quantum Phase Transitions