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

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(1970). However, the present analysis shows that this

interpretation is misleading and that D

(T) is nothing

but an indication of enhanced decrease of magnetic

moments near surfaces, superimposed from surface

and size effects. Measured at ﬁnite T, an experimental

value of D

(T) is roughly indicative of T

) ¼T.In

order to obtain absolute values of p

surf

for some

combination of surfaces (interfaces) from magneto-

metry, one needs to extrapolate to T ¼0, where p

size

disappears. This extrapolation is possible because

size

(T) is proportional to T (Gradmann 1974, Wag-

ner et al. 1997).

An alternative approach to determine p

surf

is given

by evaluation of spin polarized electron diffraction

(SPLEED) by dynamical diffraction theories. This

difﬁcult approach has been carried out for the two

clean surfaces Ni(111) and Fe(110) (Gradmann and

Alvarado 1985).

Experimental data for surface excess magneton

numbers are shown in Table 1, for selected surfaces.

Some theoretical results are included; they show that

ﬁrst principles theories are now able to calculate the

surface excess moments correctly. The enhanced sur-

face moment in Fe(110) is explained from band nar-

rowing as a result of reduced number of nearest

neighbors. Accordingly, a further reduction of near-

est neighbors in steps results in a further increase of

magnetic moment by some tenths of a Bohr magne-

ton (Albrecht et al. 1992). The coverage by noble

metal results in moderate changes of moment below

1 m

. Remarkably strong values of p

surf

are induced

by Pd as a substrate or coverage material. Appar-

ently, this results from a magnetic polarization of Pd

by the neighboring ferromagnetic ﬁlm, in tight sim-

ilarity with the formation of giant moments of po-

larized Pd around Ni, Fe, and Co impurities in bulk

Pd. Interestingly, the polarizability of Pd is enhanced

if Pd is not used as a coverage material, but as a

substrate which in turn is prepared as an ultrathin

epitaxial ﬁlm on Re(0001) and therefore strained by

the Re/PD misﬁt of 0.3% in the plane (Gradmann

1993, pp. 72–4).

3. Magnetic Anisotropies

For applications, by far the most important proper-

ties of UFF are their outstanding magnetic anisotro-

pies. In general, magnetic anisotropies are the

magnetic evidence of broken symmetries in the lat-

tice of any magnetic sample. Intrinsic magnetic an-

isotropies of UFF result from the strong intrinsic

anisotropies of their lattice: The anisotropic shape

results in the volume type anisotropy of magneto-

static energy, known as magnetic shape anisotropy,

which always supports in-plane magnetization. The

broken symmetry in both surfaces is the origin of the

surface-type magnetic surface anisotropy (MSA) that

may support in-plane or perpendicular magnetiza-

tion. In the latter case, MSA may overcome shape

anisotropy, in the very thinnest ﬁlms, and thus result

in perpendicular magnetization (see Magnetic Films:

Anisotropy).

To be quantitative, we restrict ourselves (i) to the

dominating anisotropies that depend on the tilt angle

of the magnetization with respect to the surface nor-

mal, y only (out-of-plane anisotropies) and (ii) to the

leading term which is proportional to cos

y. For a

discussion of in-plane anisotropies and of higher or-

der anisotropies see Fritzsche et al. (1994). The tem-

perature dependence of MSA has been discussed by

Farle et al. (1997).

In the second order out-of-plane approximation,

the anisotropic part of the free energy F of an UFF of

volume V and area A is given by

F ¼

þ K



 V þðK

ð1Þ

þ K

ð2Þ

ÞA



cos

y ð5Þ

The anisotropy constant in brackets is determined

for a given ﬁlm by magnetometric methods (torsion

magnetometry, torsion oscillation magnetometry,

ferromagnetic resonance, magnetic loop tracing).

The volume term is composed of the dipolar shape

anisotropy J

=2m

(always positive, easy plane) and

an additional crystalline term K

(of any sign). The

Table 1

Surface excess magneton numbers p

surf

for selected surfaces (interfaces).

Surface (interface) p

surf

experiment; (theory) Reference

Ni(111)/UHV þ0.0670.12; (0.11) Gradmann 1993

Ni(111)/Cu 0.4270.04; (0.42) Gradmann 1993

Ni(111)/Pd (Pd coverage) þ0.870.2 Gradmann 1993

Ni(111)/Pd (Pd substrate,

strained)

þ2.070.2 Gradmann 1993

Fe(110)/UHV þ0.8670.35; (0.66) Wagner et al. 1997 (Gradmann 1994)

Fe(110)/Pd þ1.670.4 Gradmann et al. 1997

Fe(100)/Ag þ0.870.2 Fullerton et al . 1995

Fe(100)/Pd þ1.770.4 Fullerton et al . 1995

940

Monolayer Films: Magnetism

surface term is composed by the surface anisotropy

constants K

ðiÞ

of both surfaces (i ¼1; 2) (we use the

notation s ¼K

cos

y as introduced by Ne

el (1954) in

his pioneering work on MSA; unfortunately, the op-

posite sign is used in part of the literature after 1980).

To separate the volume terms from surface terms, one

uses a ﬁlm series with variable thickness t, all other

parameters being ﬁxed (or assumed to be ﬁxed). The

separation then is done by plotting F versus t or

versus 1/t. In both cases, a linear plot conﬁrms the

Ansatz of Eqn. (5); volume and surface anisotropies

can be taken from slopes and axial sections, respec-

tively, in a straightforward manner.

The phenomenological anisotropy constants K

in-

clude all terms that are proportional to the ﬁlm sur-

face, or scale with 1/t. The following contributions

may be distinguished (Gradmann 1993). (i) Intrinsic

MSA results from the bare existence of the surface

(interface), by the modiﬁed electronic state of surface/

interface atoms. It is this contribution that Ne

el had in

mind when proposing the existence of MSA (Ne

1954). Intrinsic MSA is the subject of ﬁrst principles

band structure calculations. Usually, it is the leading

MSA contribution. (ii) Dipolar MSA: surface correc-

tions to dipolar energies result both from the atomic

structure of an ideal surface and from the roughness of

real surfaces. (iii) Crystalline roughness MSA: the

electronic state of step and kink atoms is different from

surface atoms, resulting in step and kink anisotropies

which roughly are collected in a crystalline roughness

contribution. Step anisotropies could be determined

for steps on Fe(110) (Gradmann et al. 1997). Step an-

isotropies may enforce or reduce the intrinsic MSA.

(iv) Apparent MSA from 1/t scaling shape anisotropy:

By the size effect discussed in section 1, J

decreases

with decreasing t. It turns out that this results in a

contribution to shape anisotropy which scales with 1/t.

(v) Apparent MSA from 1/t scaling relaxing strain.

For small misﬁt f, the epitaxial UFF is pseudomorphic

with the substrate up to a critical thickness t

which

increases with decreasing f.Fort4t

,themisﬁtisac-

commodated gradually by misﬁt dislocations, resulting

in a relaxing strain which roughly scales with t

/t,and

in a strain induced anisotropy which scales in the same

way. Apparently, this is the most important one of the

extrinsic MSA contributions. It disappears in the pseu-

domorphic range t4t

. Accordingly, intrinsic MSA

can be isolated if the analysis is restricted to the pseu-

domorphic range.

Table 2 shows magnetic surface anisotropy con-

stants K

for selected interfaces. For comparison with

bulk anisotropies, the constants are alternatively given

as anisotropy energies per atom, e

. Shape anisotro-

pies for Ni, Fe, and Co are included in this notation,

for comparison. Note that for a couple of interfaces e

is negative, supporting perpendicular magnetiza-

tion.Perpendicular magnetization is expected and

observed for e

ð1Þ

þ e

ð2Þ

þ D  e

shape

o0, if volume type

Table 2

Magnetic surface anisotropy constants, K

, and surface anisotropy energies per atom, e

, for selected surfaces

(interfaces). Data from sandwich samples on single crystal substrates. Surface energy is deﬁned as s ¼K

cos

y, where

y is the angle between magnetization and surface normal (Ne

el notation). Data for Ni(111)/Cu, Co(0001)/Cu and

Co(0001)/Au from the pseudomorphic range of thickness (intrinsic MSA); this is indicated by bold faces. Shape

anisotropies for Ni, Fe, and Co for comparison.

Surface (interface) K

(mJ/m

2

) e

(meV/atom) Reference

Ni(111)/UHV þ0.33 þ109 Gradmann 1993

Ni(111)/Re þ0.29 þ95 Gradmann 1993

Ni(111)/Cu þ0.08 þ27 Jungblut et al. 1994

Ni, shape þ11.5

Fe(110)/UHV 0.97 390 Gradmann et al. 1997

Fe(110)/W þ1.92 þ690 Gradmann et al. 1997

Fe(110)/Cr 0.12 40 Gradmann et al. 1997

Fe(110)/Cu 0.52 190 Gradmann et al. 1997

Fe(110)/Ag 0.82 300 Gradmann et al. 1997

Fe(110)/Au 0.72 260 Gradmann et al. 1997

Fe(110)/Au; T ¼10 K 0.69 250 Lugert and Bayreuther 1989

Fe, shape þ140

Co(0001)/UHV þ0.28 þ95 Kohlhepp et al. 1995

Co(0001)/Cu 0.18 61 Kohlhepp et al. 1993

Co(0001)/Ag 0.42 82 Kohlhepp et al. 1995

Co(0001)/Au 0.37 126 Kohlhepp et al. 1993

Co(0001)/Pd 0.92 312 Purcell et al. 1992

Co(0001)/Pt 1.15 392 McGee et al. 1993

Co, shape þ90

941

Monolayer Films: Magnetism

anisotropies can be neglected (e

ðiÞ

and e

shape

are an-

isotropy energies per atom for interface (i) and for

shape anisotropy, respectively). For Co(0001), nega-

tive volume type magnetocrystalline and strain an-

isotropies support perpendicular magnetization.

Co(0001) therefore is the most important magnetic

part of perpendicularly magnetized multilayers.

Whereas a separation of the different MSA con-

tributions discussed above is not possible in general,

the intrinsic MSA is usually considered as leading

contribution. For the systems shown in Table 2 in

bold faces, the pseudomorphic range has been used.

Accordingly, they represent the intrinsic MSA to a

good approximation. Higher order terms of MSA

(Kohlhepp and Gradmann 1995) are not considered

in the data. Because they enter the measurement dif-

ferently for different methods, the omission of higher

order anisotropies creates an uncertainty of the order

of 10% in the data. The MSA constants in Table 2

therefore must not be taken as high precision natural

constants, but rather as useful parameters to describe

a complex variety of anisotropy phenomena in a

reasonable model.

4. Conclusions

We have shown for representative examples that

magnetic order in ultrathin ferromagnetic ﬁlms

(UFF) deviates from the bulk case by an enhanced

thermal decrease of magnetic order, by modiﬁed mo-

ments in the surfaces and by strong surface-specific

anisotropies, which may induce the technologically

important phenomenon of perpendicular magnetiza-

tion. Other interesting aspects of UFF magnetism,

like the opportunity to epitaxially prepare magnetic

matter in metastable states which do not exist in bulk

(fcc Fe, bcc Co, for a review see Prinz (1991)), and to

study their magnetic properties, could not be dis-

cussed. We focused on metallic ferromagnets on me-

tallic substrates. There is presently strong interest in

extending the ﬁeld to nonmetallic, in particular semi-

conducting substrates.

New preparation methods like pulsed laser deposi-

tion (PLD), in combination with structural testing by

scanning tunnel microscopy (STM) open new aspects

of the ﬁeld (Ohresser et al. 1999). UFF are important

as building blocks for existing and forthcoming mag-

netic multilayers and nanostructures and the bewil-

dering rich magnetic ordering in them.

See also: Magnetic Layers: Anisotropy; Thin Film

Magnetism: Band Calculations; Thin Film Magnet-

ism: PEEM Studies

Bibliography

Albrecht M, Gradmann U, Furubayashi T, Harrison W A 1992

Magnetic moments in rough Fe-surfaces. Europhys. Lett. 20,

65–70

Elmers H J 1995 Ferromagnetic monolayers. Int. J. Mod.

Physics 9, 3115–80

Farle M, Platow W, Anisimov A N, Schulz B, Baberschke K

1997 The temperature dependence of magnetic anisotropy in

ultra-thin ﬁlms. J. Magn. Magn. Mat. 165, 74–7

Freeman A J, Wu R 1991 Electronic band structure theory of

surface, interface, and thin-ﬁlm magnetism. J. Magn. Magn.

Mat. 100, 497–514

Fritzsche H, Elmers H J, Gradmann U 1994 Magnetic an-

isotropies of Fe(110) interfaces. J. Magn. Magn. Mat. 135,

343–54

Fullerton E, Stoefﬂer D, Ounadjela K, Heinrich B, Celinski Z,

Bland J A C 1995 Structure and magnetism of epitaxially

strained Pd(001) ﬁlms on Fe(001): experiment and theory.

Phys. Rev. B 51, 6364–78

Gradmann U 1974 Ferromagnetism near surfaces and in thin

ﬁlms. Appl. Phys. 3, 161–78

Gradmann U 1993 Magnetism in ultrathin transition metal

ﬁlms. In: Buschow K H J (ed.) Handbook of Magnetic

Materials. Elsevier Science Publishers, Amsterdam, Vol. 7,

pp. 1–96

Gradmann U 1994 Magnetic properties of single crystal sur-

faces. In: Maddung O, Martienssen W (eds.) Landolt

rnstein New Series, III, 24 (5), 506–19

Gradmann U, Alvarado S F 1985 Elastic spin-polarized low

energy electron scattering from magnetic surfaces. In: Feder

R (ed.) Polarized Electrons in Surface Physics. World Scien-

tific, Singapore, pp. 321–52

Gradmann U, Du

rkop T, Elmers H J 1997 Magnetic moments

and anisotropies in smooth and rough surfaces and interfac-

es. J. Magn. Magn. Mat. 165, 56–61

Gradmann U, Mu

ller J 1968 Very thin (2–200 A

) ferromagnetic

NiFe ﬁlms. J. Appl. Phys. 39, 1379–81

Heinrich B, Bland J A C 1994 Ultrathin Magnetic Structures.

Springer, Berlin

Jungblut R, Johnson M T, de Stegge J, Reinders A, den

Broeder F J A 1994 Orientational and structural dependence

of magnetic anisotropy of Cu/Ni/Cu sandwiches: Misﬁt in-

terface anisotropy. J. Appl. Phys. 75, 6424–6

King D A, Woodruff D P (eds.) 1997 Growth and Properties of

Ultrathin Epitaxial Layers. Elsevier, Amsterdam

Kohlhepp J, Elmers H J, Gradmann U 1993 Magnetic interface

anisotropies of Co/Cu(111) and Co/Au(111) interfaces from

ultrathin Co-ﬁlms on Cu(111). J. Magn. Magn. Mater. 121,

487–9

Kohlhepp J, Gradmann U 1995 Magnetic surface anisotropies

of Co(0001)-based interfaces from in situ magnetometry of

Co-ﬁlms on Pd(111), covered with ultrathin ﬁlms of Pd and

Ag. J. Magn. Magn. Mat. 139, 347–54

Liebermann L, Clinton J, Edwards D M, Mathon J 1970 Dead

layers in ferromagnetic transition metals. Phys. Rev. Lett. 25,

232–5

Lugert G, Bayreuther G 1989 evidence for perpendicular mag-

netic anisotropy in Fe(110) epitaxial ﬁlms in the monolayer

range on gold. Thin Solid Films. 175, 311–6

McGee N W, Johnson M T, de Vries J J, aan de Stegge J 1993

Localized Kerr study of the magnetic properties of an ultra-

thin epitaxial Co wedge grown on Pt(111). J. Appl. Phys. 73,

3418–25

el L 1954 Anisotropie magne

tique superﬁcielle et surstruc-

tures d’orientation. J. Physique Rad. 15, 225–39

Ohresser P, Shen J, Barthel J, Zheng M, Mohan C V, Klaua M,

Kirschner J 1999 Growth, structure and magnetism of fcc

Fe ultrathin ﬁlms on Cu(111) by PLD. Phys. Rev. B 59,

3696–706

942

Monolayer Films: Magnetism

Prinz G A 1991 Metastability in epitaxial magnetic metal ﬁlms.

J. Magn. Magn. Mat. 100, 469–80

Purcell S T, Johnson M T, McGee N W E, Zeper W B, Hoving

W 1992 Spatially resolved magneto-optical investigation of

the perpendicular anisotropy in a wedge-shaped ultrathin

epitaxial Co layer on Pd(111). J. Magn. Magn. Mater. 113,

257–63

Wagner K, Weber N, Elmers H J, Gradmann U 1997 Magnet-

ization of free Fe(110) surfaces from thin ﬁlm magnetometry.

J. Magn. Magn. Mat. 167, 21–6

U. Gradmann

MPI Mikrostrukturphysik, Halle, Germany

ssbauer Spectr ometry

Since the discovery of recoil-free g-ray resonance

emission and absorption in solids by R. L. Mo

ssbauer

(1958), this phenomenon, termed the Mo

ssbauer ef-

fect, has been developed as a useful spectroscopic

method to study various solids. Although its princi-

ples look somewhat complicated, today’s Mo

ssbauer

spectroscopy is no longer a strange and difﬁcult

method. Usually, as sketched in Fig. 1, g-rays from a

radioactive source placed on a moving stage pass

through a foil specimen and an absorption spectrum

is recorded. The data are analyzed in terms of the

hyperﬁne interactions between the g-absorbing nuclei

in the specimen and the surrounding electrons and

atoms, and give useful information on the phases, the

crystalline and bonding states, and the atomic, mag-

netic, electronic, and defect structures.

Before introducing the principles and techniques of

ssbauer spectroscopy, here is a simple explanation

of the Mo

ssbauer effect. As is shown on the left-hand

side of Fig. 2, when an excited nuclear state drops to

the ground level by emitting a g-ray, a recoil motion

of the nucleus will reduce the g-ray’s energy, E

,by

the amount of the recoil energy, Mv

/2, where M is

the nuclear mass and v the recoil velocity. The g-ray

emitted in this way would not be absorbed by a

nearby nucleus of the same species in the ground level

because of the energy deficit. By the laws of momen-

tum and energy conservation are given

MV ¼ MðV þ vÞþ

and ð1Þ

þ E

MðV þ vÞ

þ E ð2Þ

where V is the initial velocity given to the nucleus

prior to the emission, E is the energy of the emitted

g-ray which is different from E

, c is the light velocity,

and E/c is the g-ray’s momentum. From the two

equations, the decrease of the g-ray energy is given as

dE ¼ E

 E ¼

2Mc



ð3Þ

The ﬁrst term on the right-hand side is the recoil

energy and the last term is the Doppler effect pro-

duced by V. The latter could compensate the former

to maintain the g-ray’s energy as E

, but the necessary

Doppler velocity will be of the order of 100 ms

1

ordinary cases; this is difﬁcult to reproduce and con-

trol at the usual laboratory scale. In addition, for

absorption, another Doppler effect-aided supplement

of energy is needed.

In the actual situation, however, a nucleus in a

solid is bound by the surrounding electrons, which

are also bound to its neighboring atoms by chemical

bonds. This will make the mass, M, in Eqn. (3)

extremely large so that the recoil energy will be

effectively zero. This is the essential feature of the

recoil-free emission and absorption of g-rays discov-

ered by Mo

ssbauer. More exactly, the problem must

be treated by lattice dynamics in quantum theory,

taking into account the coupling between the recoil

Figure 1

Alignment of apparatus for Mo

ssbauer spectroscopy.

Figure 2

Relation between g-ray emission and absorption by

nuclei and the recoil reaction.

943

Mo¨ssbauer Spectrometry

motion and lattice phonons to ﬁnd the probability, f,

of recoil-free reactions.

1. Principles and Elements of Mo¨ssbauer

Spectroscopy

The actual situation at nuclear levels is more com-

plicated than the above and the g-ray absorption will

not always take place, even in the recoil-free condi-

tion, because the levels’ small shifts and splits occur

to produce detuning against the g-ray energy. The

modiﬁed level structure, called the hyperﬁne struc-

ture, arises from the interactions with environmental

electronic conﬁgurations. This difﬁculty in tuning or

resonance was solved by Mo

ssbauer himself by

changing and controlling the incident g-ray’s energy

with a tiny Doppler motion to offset the shifts and

splits. As already stated, using the Doppler motion to

compensate the recoil energy is unnecessary, but the

Doppler velocity used by Mo

ssbauer is an indispen-

sable tool for the spectroscopy.

The necessary relative Doppler motion between the

g-ray source and the specimen is given by a moving

stage on a transducer shown in Fig. 1. A velocity of

710 mms

1

corresponds to about 710

7

eV in prac-

tical cases which is as small as 10

4

times the above

mentioned recoil energy but large enough to cover the

ordinary range of hyperﬁne energies. The given Dop-

pler velocity is registered in the abscissa of a display

screen in Fig. 1 as the energy scale, and the amount of

g-ray transmission at each velocity is recorded in the

ordinate. In the real experimental setup a function

generator, a high voltage supply, a multichannel

pulse height analyzer, and ampliﬁers are necessary.

Reﬂection-type spectroscopy is also possible by uti-

lizing conversion electrons or conversion x rays. The

data analysis is done by computer.

In addition to the most popular and useful

isotope with

Co source (

Co decays by electron

capture to an excited level of

Fe),

119

Sn,

121

Sb,

125

Te,

129

151

Eu,

161

Dy,

197

Au,

237

Np, and some ten

more isotopes are available, using suitable radioiso-

tope sources. In addition, monochromatized syn-

chrotron orbital radiation has become available as a

g-ray source.

In order to analyze the obtained spectra, at least

three elements of the hyperﬁne structure—the isomer

shift, the quadrupole splitting, and the nuclear Zee-

man splitting—must be considered. A downward ar-

row at the upper left corner of Fig. 3 indicates the

transition from the ﬁrst excited level to the ground

level of

Fe formed from

Co, resulting in the emis-

sion of a 14.4 keV g-ray to be absorbed by another

stable

Fe nucleus in the specimen.

However, Coulomb interactions between the nu-

clear and electron charges always exist, the amounts

of which are different between the excited and ground

states and between the source and the specimen. The

difference between the emission and absorption en-

ergy is the isomer shift, d, which is given as

d ¼

CDR

ð7C

ð0Þ7

 7C

ð0Þ7

Þð4Þ

where C is a certain constant, DR is the nuclear radius

difference between the excited and ground states, and

(0) and C

(0) are the wave functions of s electrons

of the absorber (specimen) and the source, respec-

tively. The isomer shift d can be measured as the de-

viation of the center of the absorption peak in the

Doppler velocity scale, as the lower left schematic

shows. It yields information, through the s electron

density 7C(0)7

, on the phases, atom coordinations,

chemical bonds, ionic states, alloying states, etc.

A nucleus with a spin state greater than

has an

electric quadrupole moment. When such a nucleus is

placed in an electric ﬁeld gradient, the electric quad-

rupole interaction makes the nuclear level split. In the

case of

Fe, the excited level with the spin I ¼

splits

into two and the ground level with I ¼

does not split,

as the level schematics in the upper part and the lower

middle spectrum in Fig. 3 show. The amount of

splitting is given by

D ¼ 2e ¼ 7

eQV

1 þ



1=2

ð5Þ

where e is the electron charge, Q the nuclear quad-

rupole moment, V

the principal axis component of

the ﬁeld gradient, and Z an asymmetry parameter of

the ﬁeld. By measuring the lattice asymmetry and

distortion D, the effects of coordinating atoms, a

charged impurity nearby, clustering of atoms, etc.,

can be studied.

Figure 3

Schematic presentation of hyperﬁne structures, possible

level transitions, Mo

ssbauer parameters, and Mo

ssbauer

spectra expected from them.

944

Mo¨ssbauer Spectrometry

Another important hyperﬁne interaction is the nu-

clear Zeeman effect, by which a nuclear level with a

spin I is split into (2I þ1) levels when placed in a

magnetic ﬁeld. In the case of

Fe, the excited level

splits into four and the ground level into two. There

are six possible transitions between them, giving rise

to a six-line spectrum, as the right-hand parts of Fig.

3 show. The peak height ratios are normally

3:2:1:1:2:3. The eigenvalues of the splits are given by

¼

ð6Þ

where m is the nuclear moment, H

is the internal

magnetic ﬁeld, and m

is the magnetic quantum

number. Equation (6) shows that H

is obtained from

the total breadth of the sextet spectrum. It is 330 kOe

in pure iron at room temperature. As the temperature

increases, the total breadth decreases due to the ﬂip-

ping of atom moments, and disappears at the Curie

point T

. When coupled with the quadrupole effect,

the sextet becomes asymmetrical as shown by the

right-hand lowest spectrum. The Zeeman splitting

gives information on various kinds of magnetic

states.

The lineshape of each spectral component is of the

Lorentzian type, and the linewidth is different for

different nuclides. It is inﬂuenced by alloying and in-

strumental errors too. The recoil-free fraction f and,

therefore, the peak height decreases with increasing

temperature. The temperature dependence of f and d

give us the Debye temperature of the material. When

thermally assisted atomic jumps take place, the line-

width exhibits another increase, DG, from which the

diffusion constant D can be measured.

The total absorption area, A, is another important

parameter directly connected with the amount of the

ssbauer atoms and the recoil free fraction, f.

When polarized g-rays are used to see the magnetic

spin orientations in the specimen, one more param-

eter, y

, the angle between the beam and the spin

orientation, is needed.

2. Typical Examples of Mo¨ssbauer Studies in

Material Science

Figure 4 shows a simple example of

Fe Mo

ssbauer

spectroscopy to study the tempering process of steel.

After being water-quenched from the f.c.c. austenite

phase, 1.6 wt.% carbon steel at room temperature

shows a spectrum consisting of a large paramagnetic

single absorption peak from the retained austenite

g phase (d ¼0.01 mms

1

) and six line ferromag-

netic peaks from the martensite a

phase ( d ¼

þ0.01 mms

1

, D ¼0.01 mms

1

, H

¼334 kOe). The

subpeaks and shoulders as well as line broadening

arise from the inﬂuence of the interstitial carbon

atoms. Tempering for 30 min at 210 1C makes the

spectrum sharper, and tempering at 280 1C causes a

noticeable decrease of the central peak due to the au-

stenite decomposition. Some new subpeaks emerge

too. After 410 1C annealing the spectrum shows no

austenite but clear ferromagnetic cementite (y;Fe

precipitates, corresponding to the separated and shad-

owed component (d ¼þ0.18 mms

1

, D ¼0.01 mms

1

¼208 kOe) in the ﬁgure. In such a way the structure

and behavior of materials can be studied.

Another popular material is magnetite, a-Fe

which appears not only in materials science but also

in earth, biological, and medical sciences. Magnetite

has the inverse spinel structure, in which the tetrahe-

dral (A) sites are occupied by Fe

3 þ

ions and the oc-

tahedral (B) sites by the mixture of Fe

3 þ

and Fe

2 þ

Figure 4

ssbauer spectra obtained from the water-quenched

and tempered plain carbon steel containing 1.6 wt.%

carbon.

945

Mo¨ssbauer Spectrometry

ions, effectively giving the averaged ionic value,

2.5 þ

As an unusual example, a Mo

ssbauer study of

magnetite under high pressures, utilizing a special

diamond anvil cell, is shown in Fig. 5 (Pasternak

et al. 1994). Under low pressures, the

Fe spectrum

exhibits a two-component ferromagnetic pattern,

corresponding to the two different ionic states, as

the uppermost spectrum shows. The outer smaller

component arises from Fe

3 þ

(d ¼þ0.26 mms

1

D ¼0 mms

1

, H

¼490 kOe), and the inner larger

one from Fe

2.5 þ

(d ¼þ0.67 mms

1

, D ¼0 mms

1

¼455 kOe). Under 26 GPa and higher pressures, a

paramagnetic new phase with two-site iron appears

(D ¼0.3 mms

1

). An in situ dispersion-type x-ray

analysis on the same specimen shows a monoclinic

two-site structure. By lowering the pressure from

66 GPa, the reverse transition takes place with hys-

teresis, as the lower part of the ﬁgure shows.

Noticeable progress of Mo

ssbauer spectroscopy

has been achieved not only in the automation of

measurement and analysis but also in methodology

(Fujita 1999) including high-pressure and high-

temperature measurement up to 1800 K (Fujita and

Yoshida 1997), and simultaneous measurement of

g-rays, conversion electrons, and x-rays (Gonser

1998), and use of synchrotron orbital radiation

(Gerdau et al. 1985). Many applications have been

carried out with metals, ceramics, semiconductors,

glasses, and other pure and compound materials,

with further extension to biology, medicine, archeol-

ogy, mineralogy, mining, etc.

Bibliography

Fujita F E 1999 Recent developments in Mo

ssbauer spectros-

copy. Contemp. Phys. 40, 323–37

Fujita F E, Yoshida Y 1997 Mo

ssbauer spectroscopy of high-

temperature materials. Prog. Mater. Sci. 42, 159–75

Gerdau E, Ru

ffer R, Winkler H, Tolksdorf W, Klages C P,

Hannon J P 1985 Nuclear Bragg diffraction of synchrotron

radiation in yttrium iron garnet. Phys. Rev. Lett. 54, 835–8

Gonser U 1998 Mo

ssbauer spectroscopy in materials science.

In: Fujita F E (ed.) Physics of New Materials, 2nd edn.

Springer, Heidelberg, Germany, Chap. 9, pp. 269–300

ssbauer R L 1958 Kernresonanzﬂuoreszenz von Gamma-

strahlung in

191

Ir. Z. Phys. 151, 124–43

Pasternak M P, Nasu S, Wada K, Endo S 1994 High-pressure

phase of magnetite. Phys. Rev. B 50, 6446–9

F. E. Fujita

Osaka University, Machikane, Toyonaka, Japan

MRI Contrast Agents

Magnetic resonance imaging (MRI) is a powerful

clinical modality in diagnostic medicine. The contrast

agents are paramagnetic materials, used for improv-

ing the efﬁciency of MRI investigations. The MRI

technique is based on the behavior of protons in

Figure 5

ssbauer spectra of magnetite (a-Fe

) under high

pressures (after Pasternak et al. 1994).

946

MRI Contrast Agents

the body (mainly water protons) when exposed to

magnetic ﬁelds (typically 0.5–1.5 T). The magnetic

moments of protons align parallel to or antiparallel

to the magnetic ﬁeld direction. Since the number of

spins aligned parallel is a little higher than the

number aligned antiparallel, a macroscopic magnet-

ization vector (MMV) is formed parallel to the ex-

ternal ﬁeld (B

). When a short radiofrequency pulse

) is applied at right angles to B

, some nuclei enter

a higher energy state and the MMV ﬂips into the

transverse plane. In a receiver coil situated in the

transverse plane, the precessing MMV induces a

voltage which produces the MRI signal.

When the radiofrequency pulse is switched off, the

excited protons lose energy and the MMV realigns

with B

. This process is called relaxation. The loss of

energy may take place via interaction between the

protons and the surrounding molecules, character-

ized by the time constant T

, the longitudinal relax-

ation time. The decay of the MMV in the transverse

plane occurs through energy exchange between the

neighboring protons and because of inhomogeneities

in B

, characterized by the time constant T

, the

transverse relaxation time. The reciprocals of the re-

laxation times, 1/T

and 1/T

, are referred to as the

relaxation rates.

The intensity of MRI signals is proportional to the

proton density and T

1

and inversely related to T

1

To obtain the MR image, the signals emitted by dif-

ferent volume elements of the body are spatially lo-

cated in three dimensions with the use of magnetic

ﬁeld gradients. The differences in the signal intensities

of the neighboring volume elements in a slice of the

body result in the image contrast. The parameters

inﬂuencing the signal intensity are often different in

normal and diseased tissues, which leads to differ-

ences in the image contrast.

MRI was introduced as a noninvasive diagnostic

tool, but certain clinical problems necessitated the use

of exogenous materials, called contrast enhancement

agents (CAs), which increase the differences in the

signal intensities. The CAs are paramagnetic com-

pounds which produce ﬂuctuating magnetic ﬁelds in

the proximity of protons in the body, resulting in

shorter T

and T

values (faster relaxation). The CAs

known so far can be divided into three groups (see

also Nuclear Magnetic Resonance Spectrometry).

1. Paramagnetic Metal Complexes

For shortening the relaxation times of protons the

most effective metal ions are Gd

3 þ

,Fe

3 þ

, and

2 þ

, each of which has a half-ﬁlled electron shell,

a large magnetic moment and a relatively long elec-

tron spin relaxation time (B10

9

s). The aquo-metal

ions are toxic; in order to decrease the toxicity, the

ions are used in the form of complexes, which are well

tolerated. In clinical practice, virtually only complexes

of Gd

3 þ

are used (the commercially available Mn

2 þ

dipyridoxyl-diphosphate is of merely marginal

importance).

The complexes used as CAs must meet the follow-

ing requirements: lack of toxicity, high water solu-

bility, low osmolality, high relaxation enhancement

effect, high thermodynamic and kinetic stability,

quick excretability, and specific in vivo distribution.

The ligands used for the complexation of Gd

3 þ

are

the open-chain DTPA, the macrocyclic DOTA, and

their derivatives (Fig. 1).

The stability constants of the Gd

3 þ

complexes

formed with the ligands L

–L

are very high (log

GdL

E22–26). The only exception is GdL

(log

GdL

¼16.85), but L

is highly selective for Gd

3 þ

as compared with endogenous ions such as Zn

2 þ

The ligands used are all octadentate in their Gd

3 þ

complexes. The ninth coordination site of the Gd

3 þ

occupied by an H

O molecule. The protons of this

molecule rapidly relax in the ﬁrst coordination sphere

and the fast exchange of this H

O with the bulk water

results in the transfer of the paramagnetic effect of

3 þ

to the surrounding water protons. To express

the relaxation effect of paramagnetic compounds,

the term relaxivity is used. The relaxivity (1/T

1,2

1

) is the increase in the proton relaxation rate

when the concentration of the CA increases by 1mM.

It is about 3.5–4.5 mM

1

(at 20 MHz) for the clin-

ically used CAs. The relaxivity of Gd

3 þ

complexes is

comprised of inner sphere relaxation (R

) and outer

sphere relaxation (R

). R

is proportional to the

number of H

O molecules in the inner sphere, the

magnetic moment of the central ion, and a correlation

time, which depends on the tumbling rate of the com-

plex, on the electron spin relaxation time of the Gd

3 þ

ion, and on the lifetime of water protons in the inner

sphere, while the relaxation decreases in proportion

to the sixth power of the Gd

3 þ

-proton distance. R

depends on the diffusion rates of the complex and

the water molecules. The contributions of the R

and

to the measured relaxivity are approximately

equal.

The Gd

3 þ

complexes are prepared by dissolving

III

oxide, hydroxide, or carbonate together with

Figure 1

Formulas of ligands used for complexation of Gd

3 þ

947

MRI Contrast Agents

the protonated ligands. The negative charges of

the complexes formed with L

, and L

are

neutralized by the protonated N-methyl-meglumine

cation. The commercial products contain about 5%

2 þ

complex, in order to diminish the release of free

3 þ

in the body. The CAs are administered to the

patients intravenously in 0.5MGdL solution, which

can be painful, particularly in the case of the charged

complexes, because of the high osmolality. To di-

minish this problem, nonionic compounds (GdL

GdL

, and GdL

) have been developed. The Gd

3 þ

complexes decrease the T

values of protons more

significantly than the T

values, and they are used

in T

-weighted imaging to increase the signal inten-

sity. The usual dose of GdL complexes is 0.1–

0.3 mmolkg

1

body weight. The low molecular

weight CAs distribute rapidly in the extracellular

space of the body. However, due to the fast diffusion

of the water molecules through the cell membranes,

the CAs shorten the T

and T

values of protons also

in the intracellular space.

The Gd

3 þ

complexes formed with the ligands L

, and L

are hydrophilic and are excreted

through the kidneys into the urine (t

1/2

¼1.6h). The

lipophilic GdL

and GdL

are excreted partly

through the liver and bile and can be used to image

the liver. The tolerance to the CAs depends on the

selectivity of the ligands for Gd

3 þ

over the endo-

genous ions (Ca

2 þ

,Zn

2 þ

,Cu

2 þ

, etc.), which is ex-

pressed as the ratio of the conditional stability

constants at physiological pH. A more important

factor is the kinetic stability. Both the spontaneous

and H

-assisted dissociation of the complexes GdL

and the direct exchange reactions with the endo-

genous ions and ligands must occur extremely slowly.

The Gd

3 þ

complexes formed with the macrocyclic

ligands L

, and L

are extremely inert under

physiological conditions. The complexes of DTPA

and its derivatives are not so sluggish but, because of

the rapid excretion, the amount of the Gd

3 þ

released

in the body is very low and the CAs are safe.

2. Superparamagnetic Particles

Superparamagnetic materials used as CAs are com-

posed of single-domain crystals (mainly Fe

) which

have much larger susceptibilities than those of para-

magnetic compounds. The particles used are coated

with a thin dextran or siloxane layer. They primarily

shorten the T

values by creating large magnetic ﬁeld

inhomogeneities.

The relaxation effect and biodistribution depend

greatly on the size of the particles. Particles larger

than 50 nm have a short half-life in the blood and

are localized mainly in the liver and spleen, from

where the exretion is slow (t

1/2

E3–4 days). Ultra-

small superparamagnetic iron oxide particles

(USPIO, o50 nm) reside in the blood pool for a

longer time (t

1/2

E200 min) and can be used as intra-

vascular CAs for both T

- and T

-weighted images.

USPIO agents have a more widespread tissue biodis-

tribution, as they are taken up by the liver, spleen,

marrow, and lymph nodes. Thus, the potential use

and development of the USPIO agents is of great

importance.

3. Stable Free Radicals

Nitroxide free radicals (NFRs) have one unpaired

electron, localized in a nitrogen–oxygen bond. The

six-membered ring piperidine nitroxides produce

larger relaxivities than the ﬁve-membered ring

pyrrolidine nitroxides. In animal experiments, how-

ever, pyrrolidine nitroxides are more resistant to re-

duction by chemical and enzymatic processes. The

relaxation enhancement of the NFRs is much lower

than that of paramagnetic complexes. Thus, the clin-

ical use of NFRs seems less probable.

4. Future Developments

The main directions in the research and development

of CAs are the increase of relaxivity and the creation

of organ-specific materials. The attachment of Gd

3 þ

complexes to macromolecules (proteins, dendrimers,

etc.) results in higher relaxivities due to the slower

rotation of the molecules. Such CAs are transferred

more slowly from the intravascular space and can be

used in MR angiography. The Gd

3 þ

complexes of

‘‘porphyrin-like’’ ligands produce very high relaxivi-

ties. Some MRI agents can be used to quantify spe-

cific enzymatic activity. The development of CAs for

-weighted images opens up new vistas. Potentially,

the complexes of Dy

3 þ

are the most interesting; these

result in large proton chemical shifts and significantly

shorten the T

values, and they are planned to be

used as ‘‘susceptibility agents.’’

Bibliography

Aime S, Botta M, Fasano M, Terreno M 1998 Lanthanide

chelates for NMR biomedical applications. Chem. Soc. Rev.

27, 19–29

Lauffer R B 1987 Paramagnetic metal complexes as water pro-

ton relaxation agents for NMR imaging: theory and design.

Chem. Rev. 87, 901–27

Tweedle M F 1989 Relaxation agents in NMR imaging. In:

nzli J-C G, Choppin G R (eds.) Lanthanide Probes in

Life Chemical and Earth Sciences. Elsevier, Amsterdam,

pp. 127–79

Westbrook C, Kaut C 1993 MRI in Practice. Blackwell Science,

Oxford, UK

E. Bru

cher

Lajos Kossuth University, Debrecen, Hungary

948

MRI Contrast Agents

Multilayer Optical Film Modeling

Optical multilayer thin ﬁlm technologies have many

applications ranging from complex high-performance

mirrors, ﬁlters, and polarizers, to simple single-layer

anti-reﬂection coatings. The design of optical coat-

ings and calculation of their properties is a well-es-

tablished science and several books (Macleod 1969,

Heavens 1991) and substantial articles (Lissberger

1970) have been written on the topic.

More recently, the properties of multilayers have

been exploited in the area of information storage with

the introduction of CD–RW/DVD (rewritable Com-

pact Disk/Digital Versatile Disk) optical technologies

and rewritable magneto-optical systems such as the

audio mini-disc. The optimization of these systems

requires an understanding of the interaction of light

with thin ﬁlm multilayer media and an appreciation

of the factors that determine the ultimate perform-

ance of any optical data readout system (see Magne-

to-optical Effects, Enhancement of). For such

applications calculation of the optical and magneto-

optical properties of the multilayers is crucial. The

important properties are the intensity reﬂectance and

transmittance in the optical case, and the Kerr and

Faraday effects in the magneto-optical case (see Fara-

day and Kerr Rotation: Phenomenological Theory). To

determine these it is necessary ﬁrst to calculate re-

ﬂection, transmission, Kerr, and Faraday amplitude

coefﬁcients (r, t, k, and f, respectively), which are

different for each of the orthogonal p and s compo-

nent polarization states.

In general, the calculation of optical and magneto-

optical properties of a magnetic multilayer stack is

complicated, particularly where the magnetization

vector may be in an arbitrary direction. However, it is

possible to simplify this problem considerably by

treating such a system as a linear superposition of

three principal magneto-optic conﬁgurations. These

are the polar, longitudinal, and transverse conﬁgura-

tions, which have been deﬁned elsewhere (see Faraday

and Kerr Rotation: Phenomenological Theory). A

further simpliﬁcation is possible for the polar conﬁg-

uration at normal incidence, which is a case partic-

ularly relevant to magneto-optical recording. In this

instance, the optical and magneto-optical properties

may be readily found using a simple reiterative tech-

nique normally applied to optical calculations.

In the space of this short article it is not possible

or appropriate to give a detailed derivation of the

mathematical formulations necessary for a complete

description of all linear, ﬁrst-order optical and mag-

neto-optical effects. However, it is intended to give

sufﬁcient information, together with appropriate

references, to enable the reader to understand the

basic methods used for thin ﬁlm and multilayer

calculations.

The discussion commences with the simplest case

of the reiterative reﬂectance formula and the normal

incidence polar Kerr effect. Following this, a more

general analysis is given that leads to a 4 4 char-

acteristic matrix method that, whilst applicable to

straightforward optical media, also applies to mag-

netic media and which is convenient for dealing with

many-layered systems.

1. The Permittivity Tensor

The interaction between an electromagnetic wave and

magnetic medium depends on the frequency-depend-

ent magnetic and electric response of the material. In

general, this response is quantiﬁed through material

equations expressed in terms of the relative permea-

bility, m

, and the relative permittivity, e

. At the high

frequencies of optical radiation, the magnetic inter-

action is negligible and m

is equal to its free space

value of unity. Consequently, the medium response is

described entirely by the permittivity. For an opti-

cally isotropic system, the relative permittivity is a

scalar quantity. For a magnetic material, the unique

direction of the magnetization vector imposes an an-

isotropy on the system and the permittivity becomes

a skew-symmetric tensor. In general, if the direction

of magnetization is deﬁned by the unit vector,

m ¼ m

i þ m

j þ m

k, then the ﬁrst-order permittiv-

ity tensor for any direction is

½e

¼n

1 im

Qim

Q 1 im

im

Qim

Q 1

ð1Þ

where n is the isotropic complex refractive index and

Q is the complex magneto-optical Voigt parameter,

which is responsible for magneto-optical effects. In

most cases 7Q7B1 10

3

, and for this reason higher

order terms in Q, that may appear in Eqn. (1) have

been omitted.

The effect of the above permittivity tensor is to

introduce an optical anisotropy that results in the

propagation of particular magneto-optical modes

within the medium. In general, these modes are rath-

er complicated to visualize. However, for the special

case of a light wave traveling parallel to the magnet-

ization vector there are two simple modes, both cir-

cularly polarized but with opposite senses, i.e., one is

right handed and the other is left handed. These

modes propagate within the medium experiencing

refractive indices n

and n



, respectively. The two

refractive indices differ from each other by an

amount of the order of Q and the medium is said

to exhibit circular birefringence. Consequently, the

circularly polarized modes propagate through the

medium at different velocities and are attenuated at

different rates. The result is a change in the overall

949

Multilayer Optical Film Modeling