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

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physical size is much larger than the mean wall sep-

aration (Young and Chapman 1993). At higher mag-

niﬁcation attention shifts to the boundaries between

adjacent grains, as shown in Fig. 8. The Foucault im-

age here shows how the domain structures distort as

walls approach the boundaries. Continuity or other-

wise across a boundary then provides important in-

formation as to whether there is exchange coupling

across boundaries or if interactions between grains are

primarily magnetostatic in nature.

Figure 7

(a)–(d) Foucault images and accompanying LAD patterns from a CoPt ﬁlm in different remanence states following

application of the ﬁelds shown in the insets.

490

Magnetic Materials: Transmission Electron Microscopy

5. Conclusions

The techniques of Lorentz microscopy provide a

powerful and direct means of obtaining experimental

micromagnetic information from magnetic ﬁlms and,

to a slightly lesser extent, from thinned sections of

bulk magnetic materials. Many of the techniques

possess complementary advantages. The experiment-

er can choose between an imaging mode that is easy

to implement and which is suitable for rapid speci-

men assessment and real-time studies, or one which

yields a detailed quantitative description of the dis-

tribution of induction at very high spatial resolution.

It is noteworthy that the phenomenon of magnetiza-

tion reversal varies markedly in different material

systems. Hence, the ability to observe directly how

reversal proceeds, and to study the nature of domain

walls involved, provides invaluable insight into the

overall magnetic behavior.

6. Acknowledgments

I would like to thank Drs. G Yi, M F Gillies, J Rose

and J P King for Figures 4–8.

See also: Kerr Microscopy; Magnetic Force Micro-

scopy; SQUIDs: Magnetic Microscopy

Bibliography

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Buschow K H J 1998 Permanent-magnet Materials and Their

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Chapman J N 1984 The investigation of magnetic domain

structures in thin foils by electron microscopy. J. Phys. D

Appl. Phys. 17, 623–47

Chapman J N, Heyderman L J, McVitie S, Nicholson W A P

1995 Direct observation of magnetic domain structures by

ﬁeld emission Lorentz microscopy. In: Bando Y, Kamo M,

Haneda H, Aizawa T (eds.) Proc. 2nd NIRIM Int. Symp.

Advanced Materials. NIRIM, Japan, pp. 123–30

Chapman J N, McFadyen I R, McVitie S 1990 Modiﬁed dif-

ferential phase contrast Lorentz microscopy for improved

imaging of magnetic structures. IEEE Trans. Magn. 26,

1506–11

Chapman J N, Scheinfein M R 1999 Transmission electron

microscopies of magnetic microstructures. J. Magn. Magn.

Mater. 200, 729–40

Dekkers N H, de Lang H 1974 Optik. 41, 452

Dieny B 1994 Giant magnetoresistance in spin-valve multi-

layers. J. Magn. Magn. Mater. 136, 335–59

Dunin-Borkowski R E, McCartney M R, Kardynal B, Smith D

J 1998 J. Appl. Phys. 84, 374

Gillies M F, Chapman J N, Kools J C S 1995 Magnetization

reversal mechanisms in NiFe/Cu/NiFe/FeMn spin-valve

structures. J. Appl. Phys. 78, 5554–62

Gutﬂeisch O, Harris I R 1996 Fundamental and practical

aspects of the hydrogenation, disproportionation, desorption

and recombination process. J. Phys. D 29, 2255–65

Hale M E, Fuller H W, Rubinstein H 1959 J. Appl. Phys. 30,

789

Hefferman S J, Chapman J N, McVitie S 1990 In situ magnet-

izing experiments on small regular particles fabricated by

electron beam lithography. J. Magn. Magn. Mater. 83, 223–4

Jakubovics J P 1994 Magnetism and Magnetic Materials, 2nd

edn. Institute of Materials, London

Johnston A B, Chapman J N 1995 The development of coher-

ent Foucault imaging to investigate magnetic microstructure.

J. Microsc. 179, 119–28

Kools J C S 1996 Exchange-biased spin-valves for magnetic

storage. IEEE Trans. Magn. 32, 3165–84

Mankos M, Scheinfein M R, Cowley J M 1994 J. Appl. Phys.

75, 7418

McFadyen I R, Chapman J N 1992 Electron microscopy of

magnetic materials. EMSA Bull. 22, 64–75

McVitie S, Chapman J N, Zhou L, Heyderman L J, Nicholson

W A P 1995 In situ magnetizing experiments using coherent

magnetic imaging in TEM. J. Magn. Magn. Mater. 148, 232–6

Paganin D, Nugent K A 1998 Noninterferometric phase imag-

ing with partially coherent light. Phys. Rev. Lett. 80, 2586–9

Reimer L 1984 Springer Series in Optical Sciences, Vol. 36.

Springer, Berlin

Richter H J 1999 Longitudinal recording at 10 to 20 Gbit/inch

and beyond. IEEE Trans. Magn. 35, 2790–5

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Young S, Chapman J N 1993 An investigation of domains and

walls in two NdFeB alloys by transmission electron micro-

scopy. IEEE Trans. Magn. 29, 2779–82

J. N. Chapman

University of Glasgow, UK

Magnetic Measurements: Pulsed Field

Pulsed high ﬁeld systems are important for studying

physical properties in high magnetic ﬁelds. A large

variety of experiments is possible here. Therefore,

generation and permanent increase of the available

Figure 8

Foucault image of the boundary regions between grains

in a thinned section of a NdFeB-type alloy.

491

Magnetic Measur ements: Pulsed Field

ﬁelds is an important topic. The production of high

magnetic ﬁelds dates back to 1924 where ﬁelds up to

200 kOe were produced by discharging a large

number of accumulators (Kapitza 1924). Later, as a

most common technique, the discharge of large con-

denser batteries was used in order to achieve ﬁelds up

to 1 MOe (100 T), but for a short time period, typ-

ically several microseconds (Foner and Fisher 1967).

Static ﬁelds are nowadays generated using super-

conducting solenoids. With this technology, ﬁelds up

to 21 T are available (Brown et al. 1996, Kiyoshi et al.

1996). Higher static ﬁelds can be produced using re-

sistive magnets with a huge power supply of about

10 MW in a polyhelix (Schneider-Muntau 1989) and

with 13.5 MW in a combined system using a bitter

magnet with a polybitter-type helix, which allows

static ﬁelds up to 27 T (Bird et al. 1996). Hybrid sys-

tems combining a superconducting magnet with a

resistive magnet generate static ﬁelds up to about

40 T. Such machines were constructed at the High

Magnetic Field Laboratory (HMFL) in Grenoble, at

the National High Magnetic Field Laboratory

(NHMFL) in Tallahassee, and also in Japan (Sendai

and Tsukuba); for a review see Herlach and Peren-

boom (1995). Beyond 40 T, transient ﬁelds are used.

One can differentiate between reproducibly—non

destructive—methods and single-shot experiments.

Using a condenser battery in connection with a

reinforced and well-optimized wire wound magnet,

pulsed ﬁelds between 53.6 T (Asano et al. 1995) and

72.6 T are reproducibly achieved (Herlach 1994). The

typical pulse duration is of the order of 5–10 ms, the

amplitude of the produced ﬁeld is limited by the me-

chanical properties of the conductors, such as Cu–

Ag, Cu–SS (SS: stainless steel), Cu–Nb, etc. (Dupouy

et al. 1995, Frings and van Bockstal 1995, Foner et al.

1988). Fields up to 200 T are produced in single-turn

coils (von Ortenberg et al. 1996) and up to 550 T by

ﬂux compression (Miura et al. 1996). These are sin-

gle-shot experiments with a typical pulse duration

of 10 ms. A pulse duration below 1 ms is, for metallic

samples, generally too short to obtain reliable results.

A survey of the magnet laboratories worldwide using

nondestructive pulsed ﬁelds is given by Herlach and

Perenboom (1995).

The disadvantages of short pulsed ﬁelds are eddy

currents, which limit the experimental studies of me-

tallic samples. In order to overcome this problem, in

several laboratories, systems with quasi-static ﬁeld

pulses (100 ms to 1 s) were built. Examples of such

high ﬁeld facilities are located at the University of

Amsterdam (Roeland et al. 1989, Gersdorf et al.

1965), the new installation at the Technical Univer-

sity of Vienna (Gro

ssinger et al. 1999), and the CNRS

in Toulouse (Askenazy et al. 1989). In these systems

quasi-static ﬁelds between 40 T and 60 T are achieved.

The systems of Amsterdam and Vienna use a regu-

lated current supply, which is directly connected to

the line delivering 5.4 MW and 10 MW respectively,

at times up to 1 s. In Toulouse a huge condenser

battery of 1 MJ is in operation. A larger battery of

12 MJ is just under installation. A new high quasi-

static ﬁeld center is under construction in Dresden. A

condenser battery of 1 MJ is used to produce ﬁelds up

to 50 T in a wire-wound magnet. Additionally, a

larger installation with a condenser battery of 30 MJ

is expected to allow ﬁelds up to 100 T.

1. Applications of High Magnetic Fields to

Magnetism

Experiments that need high magnetic ﬁelds cover ar-

eas from semiconductor physics (e.g., quantum dots)

and superconductivity (organic superconductors,

high-T

super-conductors) to magnetism (phase dia-

gram of rare earth 3d compounds, study of actinide

compounds, organic ferromagnets, etc.). A compre-

hensive survey of applications of high magnetic ﬁelds

has been compiled (MRS Bulletin 1993). In the area

of magnetism and magnetic materials, there are many

publications with open questions, owing to the ne-

cessity for experiments with magnetic ﬁelds higher

than those available. Some examples are: ﬁeld-in-

duced magnetism, e.g., in YCo

and similar com-

pounds (Misawa 1995, Yamada 1995); ﬁeld-induced

changes of the spin structure, e.g., in the R

and

based materials (Sinnema et al. 1986); and

also topics related to the application of hard mag-

netic materials such as the study of the magnetocrys-

talline anisotropy covering a broad temperature

range, e.g., the Sm

based permanent magnets

(Tellez-Blanco et al. 1996), R

B (Gro

ssinger et al.

1985), R(Fe,Ti)

(Kou et al. 1993, Gro

ssinger et al.

1994) and other materials with high anisotropy.

2. Set-up for a High Field System

2.1 Block Diagram

Figure 1 shows the block diagram of a high ﬁeld sys-

tem based on a condenser discharge. It consists of a

charging unit, a condenser battery, a discharge switch

(e.g., thyristors þ diode), a pulse magnet, a meas-

uring system and some measuring electronics. The

stored energy (units J or Ws) is given by C U

/2. In

other words, the power (W) can be varied by chang-

ing the pulse time. The pulse duration is determined

by the design of the magnet, which also determines

the volume where the ﬁeld is available and the max-

imum achievable ﬁeld. The pulsed ﬁeld technique can

be used for measurements of magneto-resistance,

magnetization, magnetostriction and also for more

sophisticated techniques such as de Haas–van Alphen

or Schubnikov–de Haas effect but also for the deter-

mination of elastic constants in high magnetic ﬁelds

using an ultrasonic method.

492

Magnetic Measurements: Pulsed Field

2.2 The Magnet

High ﬁeld magnets need high current densities. They

are limited mainly by two factors: the heating of the

magnet and stresses in the magnet windings.

The adiabatic heating can be calculated from:

ðtÞdt ¼

D  cðTÞ

rðTÞ

dT ¼ G ð1Þ

where j(t) is the current density, D is the density of the

conductor, c(T ) is the specific heat and r(T ) is the

specific resistivity of the conductor. The temperature

rise of a pulse magnet can be calculated assuming

that thermal conductivity within the short time of the

high-ﬁeld experiment can be neglected (adiabatic

condition). The right-hand side of Eqn. (1) shows

that the heating of the magnet is determined only by

material parameters of the conductor such as D, c(T )

and r(T ). This expression is called the current car-

rying capability function (‘‘action integral’’) G (van

Bockstal 1994). The magnet is generally cooled with

liquid nitrogen (77 K) in order to reduce the Joule

losses.

However, the more serious limiting factor is the

stress in the magnet, which is caused by the Lorentz

forces. The stress equilibrium condition of the mag-

net is given by the derivative of the stress in the radial

direction:

 s

 l  F

 l  j  B

ðrÞð2Þ

where s

, s

, and s

are the stress components in

cylindrical coordinates, s

is the yield strength of the

wire material, l is the ﬁlling factor, r is the distance

from the axis, and F

¼j B

is the radial body force.

Solving this differential equation including boundary

conditions (reinforcement, etc.) leads to stress in the

z ¼0 plane, which allows a decision about the me-

chanical stability of the magnets to be made. Addi-

tionally, for stability of the magnets the plastic

deformation of the wire material as well as the re-

inforcement must be considered (Gersdorf and Frings

1995, Liang and Herlach 1995).

The maximum achievable ﬁeld using a certain con-

ductor material is determined, therefore, by consider-

ing both the stresses and the ‘‘action integral’’ G

(Bockstal 1994). So far, the maximum ﬁeld that can be

achieved using standard copper conductors is 40 T.

However, it is possible to obtain higher ﬁelds when

a conductor with a higher yield strength, and/or a

stress-adjusted reinforcement of each layer (Askenazy

1995) and/or a variation of the current densities in

each layer (Askenazy 1996, Gersdorf and Frings 1995)

are realized.

3. Magnetization Measurements

For magnetization measurements three main meth-

ods exist which can be used in high pulsed ﬁelds:

(i) Measurement of the force acting on a magnetic

dipole

(ii) Magnetization measurement using the law of

induction

(iii) Magneto-optical methods

The ﬁrst two methods are focused on here.

3.1 Force Method

A magnetic dipole recognizes a force in an inhomo-

geneous ﬁeld. For static magnetization measurements

this method is commonly known as the Faraday bal-

ance. In fast-pulsed high magnetic ﬁelds, the meas-

urement of magnetic forces is technically difﬁcult,

owing to the occurrence of mechanic resonances.

However, for long pulsed ﬁelds a so-called cantilever

method has been developed (Naugthon et al. 1998). A

small sample is mounted on a semiconducting can-

tilever, which bends owing to the magnetic force. The

bending of this cantilever is proportional to the force

on the magnetic dipole. The bending is determined

using integrated sensor electronics. In the case of a

high magnetic ﬁeld system, the signal is relatively

large, owing to the ampliﬁcation factor of high ﬁelds.

3.2 Induction Method

Generally, magnetization is measured using a com-

pensated pick-up coil system, in order to measure the

magnetization M instead of the inductance B. For

this purpose different arrangements of pick-up coils

are used. The general idea is always to get a signal

which is proportional to dM/dt. Such systems consist

at least of two coils.

Figure 1

Block diagram of a high pulsed ﬁeld system (CD, charge

device; CU, control unit; PU, pick-up system; ME,

measuring electronics including ampliﬁer, integrator,

differentiator and input protector; TRC, transient

recorder; PC, personal computer; R, resistance from

wire; C, condenser battery; L, ﬁeld generating coil

‘‘Magnet’’; D, diode, TH, thyristor).

493

Magnetic Measur ements: Pulsed Field

The induction voltage in such coils can be given by:

ðtÞ¼m



ðtÞ¼m



ðtÞ¼u

ðtÞþu

ðtÞð3Þ

where dH/dt is the applied ﬁeld rate, u

, N

, K

, r

(i ¼1, 2) are induction voltage, number of windings,

coupling factor or radius of the outer (i ¼1) and inner

(i ¼2) pick up coil, respectively. The addition of u

(t)

and u

(t) shall be performed in such a way that u

(t)is

proportional to dM/dt. One can get this condition in

the following cases:

(i) An axial two-coils system N–N: two identical

coils are wound on the same diameter, then we may

have N

¼N

; r

¼r

and K

¼K

(see Fig. 2). This

system is sensitive to the local position and therefore

sensitive to vibrations.

(ii) An axial three-coils system

=N=

: compensat-

ing coils, u

(t), are formed by two N/2 coils. In this case

we have: N

¼N; N

¼N/2; r

¼r

;andK

¼K

(see

Fig. 2(b)). This system needs a long homogeneous ﬁeld.

(iii) A coaxial system where N

; r

; K

(see Fig. 2(c)). The coaxial systems, which are based on

Maxwell coils (a centric spherical coil system) are best

suited (Gersdorf et al. 1967). The requirement of a

larger volume is one of the disadvantages of this sys-

tem. In the compensated system, from Eqn. (3), we can

get:

¼ N

ð4Þ

In order to achieve an optimum pick-up geometry

(length) for a given experiment, also cancelling of the

quadrupolar contribution, the homogeneity ﬁeld

range according to the size of the sample and the

inductivity of the two coils has to be considered.

Finally, a good degree of compensation of the

ﬁeld, which also holds for higher frequency compo-

nents, should be achieved. However, the compensa-

tion is very sensitive to deviations of the radii or

lengths of the induction coils. Therefore, all dimen-

sions of the pick-up coil have to be of the highest

possible accuracy (E0.001 mm). The sensitivity of

measuring magnetization is limited by the remaining

unbalance of the pick-up system. It is worth noting

that for measuring paramagnetic substances a com-

pensation of 1 10

is desired! The dynamic behavior

of such systems is described by the transfer function.

Especially in this case, possible resonance peaks have

to be avoided.

It should be mentioned that the degree of com-

pensation changes with temperature, owing to the

thermal expansion of the supporting materials.

Therefore, in this case, the use of supporting mate-

rials with a very low thermal expansion (as, e.g.,

quartz glass, ceramics) or additional position coils

which allow a readjustment of the system at each

temperature, is suggested. Below 20 K, the magneto-

resistance effect of the wire, which may change

the calibration constant of the pick-up coils, must

be considered.

The magnetization of the magnetic materials is

obtained by integrating dM/dt. Generally, in quasi-

static systems, owing to the low frequency integra-

tion, this method presents a bad signal-to-noise ratio,

resulting in low sensitivity. To improve the signal-to-

noise ratio, two different methods for measuring the

magnetization are proposed: modulation and vibra-

tion techniques. In the ﬁrst method, the ﬁeld pulse

H(t) will be modulated by a small a.c. ﬁeld of the type

¼H

sin ot (with H

5H and T5t; where T is the

period of a.c. ﬁeld and t is the pulse duration). The

disadvantage of this method is that the technique of

modulating a high ﬁeld pulse is very complicated. In

the second method, the sample can be mounted on

a vibrating piezoelectric actuator. The frequency is

chosen according to the characteristics of the quasi-

static pulse ﬁeld system. For example, 100 kHz is

sufﬁcient for a 100 ms to 1 s quasi-static system.

However, generally the amplitude of a piezoelectric

system decreases with increasing frequency.

In both cases a periodic induction voltage is pro-

duced in pick-up coils, allowing use of a lock-in

technique which combines a high Q band pass with a

phase-locking method, improving the signal-to-noise

ratio drastically. However, for fast a.c. ﬁeld meas-

urements of a metallic sample in a short pulse mag-

netic ﬁeld, the skin effect (penetration of the magnetic

ﬁeld into the material) has to be considered.

Figure 2

Scheme of the different types of pick-up systems for

magnetization measurements. See text for details.

494

Magnetic Measurements: Pulsed Field

The skin depth, d, is given by:

d ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

o  m

 m

 s

ð5Þ

where m

is the relative permeability, m

is the vacuum

permeability and s is the conductivity of the material.

To compare the skin depth values between good and

bad conductors (Cu: 1/s ¼5 10

8

Om; Nd–Fe–B:

1/s ¼200 10

8

Om) Table 1 shows d values ob-

tained for a pulse duration of 10 ms and for a mod-

ulating frequency of 100 Hz and 100 kHz.

Two factors are extremely important for an accu-

rate magnetization measurement in a pulsed high-ﬁeld

magnetometer, namely: (a) ﬁeld and magnetization

calibration of the magnetometer and (b) eddy current

effect in conducting magnetic materials.

(a) Calibration of pulsed ﬁeld magnetometer

In a magnetometer the ﬁeld and the magnetization

have to be calibrated. For pulsed magnetic high-ﬁeld

calibration, the ‘‘calibration sample’’ method is often

used. For low temperature magnetization measure-

ments, samples with well known critical ﬁelds at

which a magnetic transition occurs are chosen for the

ﬁeld calibration. One example is the anti-ferromag-

netic MnF

where at a ﬁeld of 9.2 T a spin-ﬂop tran-

sition occurs at 4.2 K. For high temperatures, the

ﬁeld is usually calibrated to the internal anisotropy

ﬁeld, H

, of barium ferrite (H

¼1.65 T, at room

temperature). This material is chosen because H

changes only slightly around room temperature

(Kojima 1982). The magnetization M is usually cal-

ibrated using the M(H) of pure nickel or iron.

The reliability of pulsed ﬁeld measurements was

tested by comparison with static measurements per-

formed at the CNRS in Grenoble. Figure 3 shows

hysteresis loops measured on an isotropic barium

ferrite sample obtained in the static system of Gre-

noble and that measured in a well-calibrated pulsed

ﬁeld system (TU Vienna). The use of barium ferrite

has the advantage that there are no eddy currents,

because this material is insulating. The difference is

within the line thickness.

(b) Eddy current effect

The effect of eddy currents in the sample depends on

the sample geometrical dimensions and the pulse

time. For a cylinder of nickel of about 4 mm in dia-

meter and a pulse time of about 15 ms the eddy

current effect in the magnetization is about 2%. The-

oretically, for a well deﬁned sample geometry, it is

possible to correct the eddy current effect to the

magnetization, but use of a nonconducting soft mag-

netic calibration is preferred (Gro

ssinger et al. 1999).

For this type of calibration, material such as Fe

proposed.

In pulsed ﬁeld measurements, the effect of eddy

currents that occur in a metallic sample is shown by

measuring the mean magnetization that is generated

by these currents. Figure 4 shows, as an example,

the magnetization curve measured on a cylindrical

copper sample at room temperature applying a ﬁeld

pulse with m

max

¼75.7 T at two different pulse

Figure 3

Hysteresis loop measured on the isotropic BaFeO HF

8/22 spherical sample.

Table 1

Skin depth of conducting materials.

Frequency d

(mm) d

Nd-Fe-B

(mm)

100 Hz 11 69

100 kHz 0.35 2.2

Figure 4

Hysteresis loops measured on a cylindrical copper

sample with m

max

¼75.7 T and two different pulse

durations of 9.1 ms and 15.7 ms.

495

Magnetic Measur ements: Pulsed Field

durations, namely 9.1 ms and 15.7 ms. The increase

of the eddy current moment with decreasing pulse

duration is clearly visible. The phase shift between

magnetization and ﬁeld is approximately 901. The

experimentally determined eddy currents agree well

with a Finite Element calculation. It was shown that

plotting the eddy current magnetization versus dH/dt

delivers a linear relation which can also be explained

theoretically (Gro

ssinger et al. 1999).

4. The SPD method

A very important magnetic property of hard mag-

netic materials is the magneto-crystalline anisotropy

(see Magnetic Anisotropy). Generally, to investigate

this anisotropy a direction-dependent study of single

crystals is needed. In many cases such single crystals

of technically relevant materials are not available.

Often, hard magnetic materials cannot be aligned,

owing to special preparation methods. In this case the

so-called SPD method (Asti and Rinaldi 1974) is a

unique technique for determining the anisotropy.

The SPD method is based on the idea that in higher

derivatives of the magnetization with respect to the

ﬁeld, d

M/dH

, a singularity appears just at that ﬁeld

where H

ext

¼H

. In the case of a uniaxial material

the second derivative of the magnetization is already

sufﬁcient. The amplitude of the singularity is pro-

portional to the ratio of anisotropy constants K

and also proportional to the distribution function of

the crystallites. Additionally, the amplitude is deter-

mined by o

max

H

). This means that a maxi-

mum external ﬁeld sufﬁciently higher than the

anisotropy ﬁeld is necessary. Moreover, the pulse

time t (oE2p=t) should be as short as possible taking

into account that the scaling is proportional to o

and that the noise is reduced with increasing fre-

quency. An example of the use of the SPD method is

shown in Fig. 5. It shows singularities obtained by

measurement (a) and by calculation (b) for the po-

lycrystalline Sm

at room temperature. Although

the theoretical peak is sharper than that obtained by

experiment, an excellent agreement in the peak

position is observed.

The SPD method delivers the physically real

anisotropy ﬁeld, H

, from which, knowing the sat-

uration magnetization, the technically relevant

anisotropy energy can be obtained.

See also: Magnetism: High-ﬁeld; Magnetic Measure-

ments: Quasistatic and ac

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Technische Universita

t Wien, Vienna, Austria

Magnetic Measurements:

Quasistatic and ac

It is of great importance to develop and to improve

measurement systems for groups of magnetically

ordered materials, which have special demands with

respect to typical sample shapes and also to the ﬁeld

range. A magnetically ordered material is character-

ized by its critical temperature, the Curie temperature

, which is proportional to the exchange energy. In

rare earth–3d based materials at low temperatures

a spin reorientation may occur due to the interplay

between the different sub-lattice magnetization or

because of the crystal electric ﬁeld (see, e.g., Kou

et al. 1993, 1995a, 1998, Gro

ssinger et al. 1996). One

of the usual methods determining these critical tem-

peratures is the measurement of a.c. susceptibility in

low ﬁelds as a function of temperature.

The most important technique to characterize

magnetic materials is the measurement of their hys-

teresis loop (see Magnetic Viscosity). Figure 1 shows

a typical hysteresis loop of a ferromagnetic material

with all the significant parts and points of the loop. A

loop like this is obtained by applying a cycle of mag-

netic ﬁeld H to a magnetic material and by recording

the change of the magnetization M or of the magnetic

induction B along the ﬁeld direction. Hysteresis loops

may take many different shapes and are important

for obtaining parameters that give a ﬁrst character-

ization of loop properties such as the coercive ﬁeld

, and the remanence M

. The technically relevant

materials differ only in the special properties of the

loop (see Table 1). For hard magnetic materials

the coercive ﬁeld lies between 80 and 1600 kAm

1

whereas for soft magnetic materials the coercive ﬁeld

is below 100 Am

1

An additional parameter of basic importance in

the characterization of soft magnetic materials is the

power loss, which is proportional to the area of the

hysteresis loop. The most important parameter char-

acterizing the strength of a hard magnetic material is

the maximum energy product, which gives a measure

of the energy density available in the magnet for ex-

ternal work. The coercive ﬁelds of magnetic recording

materials lie somewhere between those of hard and

soft materials. In both cases the saturation magnet-

ization, M

, should be as high as possible. Generally,

Figure 1

Hysteresis curve of a ferromagnetic material.

497

Magnetic Measurements: Quasistatic and ac

for obtaining the ‘‘true’’ saturation magnetization

the law of approach to saturation extrapolating for

H-N or ﬁtting the M(H) curve (Akulov 1931, Gans

1932, Gro

ssinger 1981) may be applied.

In addition, the loop itself as well as parts of the

hysteresis (initial curve, minor loops, recoil curves)

is used for modeling, and this yields an understand-

ing of the magnetization process (Mayergoyz 1991,

Bertotti 1996, Hauser 1994).

A hysteresis loop can be created in terms of B(H)

or of M(H), where B ¼m

(H þM). For technical

applications B(H) is more important whereas for

scientific investigations M( H) is generally used. In

contrast to hard materials where H and M have

comparable orders of magnitude, in a soft material

(for low H) plotting B(H)orM(H) makes no differ-

ence due to a very good approximation, BDm

M.In

hard magnetic materials

, whereas in soft

magnetic materials both coercivities are equal (

coercivity where I ¼m

M ¼0,

is coercivity where

B ¼0). Therefore a hysteresigraph for soft magnetic

materials is technically different from that of a system

suited for hard magnetic materials or for recording

materials.

Consequently, within this contribution two groups

of measurements will be proposed which are im-

portant for characterizing hard and soft magnetic

materials:

A.c. susceptibility. The temperature dependence of

the susceptibility yields the Curie temperature, spin

reorientation temperature, freezing temperature for

spin glasses, magnetic relaxation, etc. and also do-

main parameters such as the Rayleigh constant and

the pinning ﬁeld;

Magnetization and hysteresis loop. This delivers

the coercive force, remanence, saturation magnetiza-

tion, and energy losses (or stored energy) and also

gives information about the magnetization process.

A special type of instrument used here is a torque

magnetometer which allows determination of the

anisotropy constants.

1. Susceptibility Measurements

1.1 Theoretical Considerations

Starting from a demagnetized state of the hysteresis

loop, the inclination of the initial curve is called the

initial susceptibility w

(if one considers M ¼w

H)or

the initial permeability m

(if one considers B ¼m

H).

For a soft magnetic material the initial susceptibility

is very high and m

In randomly oriented particle systems, polycrystal-

line bulk materials where due to the presence of dif-

ferently oriented crystal grains a random variation of

anisotropies exists, or in magnetically soft materials

where a ﬂuctuating internal stress couples magneto-

elastically to the magnetization, the response to small

ﬁelds is mainly controlled by the local anisotropy

ﬁelds. In this simple case the initial susceptibility can

be written as:

ð1Þ

where K stands for all kinds of anisotropy ener-

gies including the magnetoelastic energy and c is a

dimensionless factor dependent on the specific distri-

bution of the anisotropy axes of anisotropy con-

stants. Very often an additional contribution comes

from the actual microstructure. However, this is very

difﬁcult to estimate.

With increasing ﬁelds the ﬁrst irreversible losses

have to be considered. If local inhomogeneities (local

stress centers, etc.) occur a well-deﬁned pinning ﬁeld

can be observed as will be discussed below. The

different ﬁeld dependencies of the permeability are

shown in Fig. 2. The hysteresis loop of low ﬁelds,

where the maximum ﬁelds involved are much smaller

than the coercive ﬁeld, can be described by the

Rayleigh law (Fig. 2(a); see, e.g., Jordan 1922 or

Chikazumi 1964). Within the Rayleigh law, under

small but ﬁnite ﬁelds, the permeability consists of

two contributions: the reversible magnetization is

Table 1

Comparison of important magnetic properties between soft and hard magnetic materials.

Soft magnetic material Hard magnetic material

Saturation

magnetization

As high as possible (0.8–2 T) As high as possible (0.2–1.5 T)

Coercive force As low as possible (o100 Am

1

) As high as possible (80–400 kAm

1

)

Permeability As high as possible 10 000–200 000 Not important

Losses As low as possible, frequency-dependent Area of loopDstored energy as high as possible

Shape of loop Important because it determines

application

Important and should be rectangular

Remanence Not important As high as possible

Conductivity Determines a.c. losses Important for magnetizing procedure

498

Magnetic Measurements: Quasistatic and ac

described by the ﬁrst term (initial permeability m

)

and the irreversible part that depends on the square

of the ﬁeld:

B ¼ m

H þ nH

ð2Þ

where n is the Rayleigh constant. This law is impor-

tant in principle as a description of the initial steps of

irreversibility.

If a magnetic material is magnetized by an a.c. ﬁeld

H ¼H

iot

, the magnetic ﬂux density B is delayed by

the phase angle d because of the presence of losses

and is then expressed as B ¼B

i(otd)

. The perme-

ability is given by:

iðotdÞ

iot

id

¼ m

 im

ð3Þ

where the real part is m

¼(B

) cos d and the

imaginary part m

¼(B

) sin d. The imaginary part

is related to the losses. Another important loss con-

tribution for ferromagnetic metals and alloys is the

eddy current loss. Since a power loss of this type in-

creases proportionally to the square of the frequency,

it plays an important role in the high-frequency

range.

Generally in soft magnetic materials due to the

existence of well-deﬁned local pinning centers a crit-

ical ﬁeld, the pinning ﬁeld H

, is often observed (see

Fig. 2(b)). Up to H

the permeability is constant and

for an applied ﬁeld higher than H

the permeability

increases. If a linear behavior is observed the permea-

bility can be described by a modiﬁed Rayleigh law:

m ¼

¼ m

þ nðH  H

Þð4Þ

If one considers now the magnetization curves in

more ordinary materials where the presence of struc-

tural defects or disorder plays a dominant role under

most circumstances, the measurement of the a.c. sus-

ceptibility is one of the techniques to characterize

these materials. In hard magnetic materials the

shape of the virgin magnetization curve as well as

the shape of the loop with increasing external ﬁeld

is used to determine roughly the different types of

magnetization processes (nucleation or pinning

controlled) (Becker 1976).

1.2 Experimental Determination of the a.c.

Susceptibility

As shown above the measurement of the initial sus-

ceptibility (in a ﬁeld which should be small compared

with the coercivity) as a function of temperature is

important. Because the susceptibility is roughly pro-

portional to 1/K (Eqn. (1)) it shows a dramatic

change at any spin reorientation temperature, crys-

tallization temperature, or the Curie temperature,

because the anisotropy goes much faster to zero than

. For this purpose a small a.c. ﬁeld is applied to a

sample which is surrounded by a pick-up coil. In

principle the a.c. susceptibility can be measured in

most hysteresigraph systems (using an air coil), where

one has just to drive the magnet with a low a.c. ﬁeld.

The pick-up coil system is then connected to a lock-in

ampliﬁer which can measure the real and imaginary

parts of the susceptibility simultaneously. A block

diagram suitable for soft magnetic ribbons is shown

in Fig. 3. In this device an external stress can also be

applied to the sample which additionally causes an

effect on the domain conﬁguration (Holzer 1998). In

the case of a hard magnetic material the ribbon is

replaced by a (spherical) sample inside a compensated

pick-up system.

Figure 3

Setup for a.c. susceptibility measurements (after Holzer

1998).

Figure 2

Field dependence of the permeability: (a) assuming the

Rayleigh law; (b) showing an initial range below H

499

Magnetic Measurements: Quasistatic and ac