physical inorganic chemistry (principles, methods, and models) || magnetochemical methods and models...

3 Magnetochemical Methods andModels in Inorganic Chemistry

PAUL KOGERLER

3.1 INTRODUCTION

Magnetochemistry focuses on the magnetic properties of compounds, in particular

those that are characterized by electronic configurations giving rise to magnetic

moments. Pioneered in the beginning of the twentieth century by scientists such as

Paul Langevin, the discipline expands the traditional approach of condensed matter

physics toward magnetism and aims for chemical design parameters that allow to

control magnetic phenomena at the atomic level, as well as to establish systematic

structure–property correlations.1 Over the past two decades, molecular magnetism

has emerged as a strong new direction in materials sciences. The goal is to investigate

the magnetic properties that are specific to quasi-zero-dimensional magnetic arrays

and their networks.2 One primary motivation for investigating molecule-based

materials is the general difficulty of controlling bulkmagnetic properties, for instance,

in alloys. In this context, a bottom-up approach that starts from functionalized

molecules promises a more rational and systematic control over the magnetic

properties resulting from various spin–spin interactions. In addition, it is of great

importance in this context to probe whether, or to what degree, magnetic phenomena

typically associated with the bulk state, such as cooperative effects, can already be

realized within single molecules.

In most instances, the magnetic structure of a compound can be understood to be

based on interacting localized spin centers, such as classical 3d/4d/5d transitionmetal

ions and 4f lanthanide or 5f actinide cations with unpaired electrons. Note that while

the assumption of localized moments is valid for many compounds comprising such

spin centers, even partial electron delocalization in mixed-valence coordination

compounds renders many localized spin models inapplicable.

As the example of molecular magnetism shows, modernmagnetochemistry is now

seeing more efforts toward the directed design and synthesis of novel magnetic

materials, usually based on molecular building blocks as opposed to traditional two-

or three-dimensional solid-state networks (bulk metals, metal oxides, etc.).

Physical Inorganic Chemistry: Principles, Methods, and Models Edited by Andreja BakacCopyright � 2010 by John Wiley & Sons, Inc.

69

In this regard, the primary objective in magnetochemistry is to relate a set of

experimental observables to a suitable quantitative model. For example, a discrete

(molecular) magnetic system of interacting spins will be characterized by a set of

multiplet states, each characterized by its eigenenergy and level of degeneracy. The

exact knowledge of this manifold of states, summarized as magnetic excitation

spectrum, is crucial for the determination of a parameterizedmodel system.Aswill be

seen, in several instances only a combination of several experimental techniques can

pinpoint all relevant aspects of the magnetic excitation spectrum.

Yet, apart from its core competence, that is, the characterization and the under-

standing of magnetic phenomena, magnetochemical techniques can also be fruitful

for a wide range of other aspects of inorganic chemistry, ranging from tracing short-

lived radical transition states to elucidating structural disorder.

This chapter is oriented toward the nonspecialist. It tries to illustrate some standard

experimental techniques and describe selected theoreticalmodels used to interpret the

observed magnetic data, along with their inherent limitations. Given the sheer com-

plexity and diversity ofmagnetic phenomena of inorganic compounds, it is beyond the

scope of this chapter to provide a complete overview. Instead, this chapter aims to

highlight selected basic methods and models and, where reasonable, hands-on rules.

3.2 MAGNETIC QUANTITIES AND BASIC RELATIONS

Magnetochemical characterization and analysis usually focuses on correlating the

experimentally observed data with the parameters that characterize the appropriate

theoretical model. The relevant parameters that primarily determine the observed

magnetic properties can be divided into the following:

1. The characteristics of the individual spin centers, such as the spin quantum

number, and parameters that arise from their interaction with their chemical

environments: g tensor elements, zero-field splitting (ZFS) effects, and spin–

orbit coupling.

2. The coupling between spin centers (in the case of molecule-based compounds,

it is helpful to distinguish between intra- and intermolecular coupling), that is,

the interaction energies.

3. Critical limits of phenomena or magnetic states that coincide with phase

transitions, for example, magnetic fields or temperatures in spatially ordered,

cooperative spin effects.

In thefollowing, the fundamental typesofmagnetismaresummarized,emphasizing

the relations between the most accessible observables and their atomic parameters.

3.2.1 Diamagnetism and Paramagnetism

On the level of single atoms and ions, only two fundamental types ofmagnetism exist,

diamagnetism and paramagnetism. All other magnetic mechanisms (in particular,

70 MAGNETOCHEMICAL METHODS AND MODELS IN INORGANIC CHEMISTRY

magnetic behavior of bulk ferromagnetism, etc.) are based on these two fundamental

magnetic phenomena.

Phenomenologically, the difference between diamagnetism and paramagnetism

becomes immediately apparent in the presence of an external homogeneous

magnetic field, quantified by its magnetic field strength H or its magnetic induction

(or magnetic flux density) B0¼m0H, where m0 is the magnetic permeability

of vacuum. SI and CGS units for these and other quantities are listed in

Appendix 3.A.

When a sample of a substance is introduced into this field, themagnetic flux density

within the substance B differs from B0 by the magnetizationM. IfM is negative, that

is, if the magnetic flux density decreases within the sample, the substance is called

diamagnetic. IfM is positive (increase of magnetic flux density), then the substance is

paramagnetic. The change in magnetic flux density manifests itself in a repulsion of a

diamagnetic sample out of the magnetic field and an attraction of a paramagnetic

sample into the magnetic field. This electromagnetic–mechanical response forms the

basis for the historic susceptibility measurement methods, the Faraday and the Gouy

balances.

Thereby, the field dependence of M defines the magnetic susceptibility w (which

can be referred to volume, mass, or molar units) in the following way:

w ¼ @M

@Hð3:1Þ

At low fields, w is virtually independent of the magnetic field and its definition

simplifies to M¼ wH.

Diamagnetism is observed in the absence of unpaired electrons and in the presence

of fully occupied molecular or atomic orbitals when no genuine magnetic moment

exists. Classically interpreted, the external field yields precession of the atoms around

the field direction axis. This implies circular currents of the individual electrons and

consequently induces magnetic moments that according to the Lenz rule are anti-

parallel to H. All compounds exhibit diamagnetism, even if other dominating

magnetic effects are present. The diamagnetic susceptibility wdia represents an

additive quantity from all constituents of a substance, and several tabulated constants

(the so-called Pascal’s constants) exist that allow estimation of wdia from atomic and

bond increments. wdia is temperature and field independent.3

Paramagnetism is observed in the presence of unpaired electrons when intrinsic

magnetic moments exist a priori, which stem from both the spin angular moment Sand orbital angular moment L. The angular momenta and the magnetic moment

m¼ ge(L þ 2S) are quantized with respect to the field axis z. Here, ge represents thegyromagnetic factor. In the absence of an externalmagnetic field, there is no preferred

orientation of z. Consequently, the possible orientations of the magnetic moments are

energetically degenerate. This degeneracy is, however, lifted in the presence of a

magnetic field, resulting in different energies for the quantized orientations of the total

angular momentum vector J¼L þ S. This so-called Zeeman effect can be described

by an interaction energy operator H:

MAGNETIC QUANTITIES AND BASIC RELATIONS 71

H ¼ �gJgeJH with gJ ¼ 1þ JðJþ 1Þþ SðSþ 1Þ�LðLþ 1Þ2JðJþ 1Þ ð3:2Þ

If the total angular momentum is derived only from an isotropic electron spin, that

is, J¼ S, the Zeeman operator can be rewritten as

H ¼ �gmBMSH ð3:3ÞHere, the quantization axis z is chosen along the field direction. Therefore, J¼ S

is replaced by the Sz eigenvalues�h MS and

�h is included in the Bohr magneton

mB¼�h ge. The magnetic quantum numbersMS take on the values þ S, þ S� 1, . . .,�(S� 1),�S. Correspondingly, an external fieldwill cause an even, linear splittingof theMS substates belonging to an Smultiplet. In a bulk sample at a given temperature, the

population of these states will follow a Boltzmann distribution. With increasing field,

the energetically favorable states, especially the energetically lowestMS¼ þ S state,

will be increasingly populated, thus reducing the overall energy of the system and

explaining the attractive force F� @H/@x toward stronger fields experienced by

paramagnetic samples.

For the ensemble of microscopic magnetic moments mn that correspond to the

eigenvalues En of their respective spin states, we thus obtain

mn ¼ �@En

@Hð3:4Þ

These individual magnetic moments combine to yield the macroscopic molar

magnetization via the Boltzmann distribution scheme:

M ¼ NA

Pn�ð@En=@HÞe�En=kBTP

ne�En=kBT

ð3:5Þ

with the Boltzmann constant kB¼ 1.38066� 10�23 J/K (SI) or 1.38066� 10�16 erg/K(CGS). The knowledge of the field dependence of En is therefore essential for deriving

M. A simple perturbation theory ansatz results in theVanVleck equation that is central

to the description of the molar magnetization of magnetic materials, as long as these

materials do not exhibit spontaneous magnetization.

Empirically, the temperature dependence of the susceptibility of paramagnetic

materials (or, more accurately, wpara¼ wtotal� wdia) in low fields follows the Curie law

wpara ¼ C=T ð3:6Þ

where C represents the Curie constant. This implies that the product wT and the

effective magnetic moment meff

meff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3kB

NAm0m2B

s ffiffiffiffiffiffiwT

pð3:7Þ


(withffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3kB=NAm0m

2B

p¼ 2.8279 (cm3K/mol)�1/2 (CGS) or 797.74 (m3K/mol)�1/2

(SI)) are both temperature independent.

Note that the omission of orbital angular momentum and the consequent reduction

to spin-only magnetism for transition metal complexes is a consequence of the ligand

field that lifts the orbital degeneracy of the L> 0 (P, D, F) terms, quenching the orbital

momentum. However, this lifting of orbital degeneracy is not complete for certain

configurations (e.g., octahedral complexeswith T1g or T2g ground states) that then still

retain a residual orbital moment.

If the orbital momentum is completely quenched (spin-onlymagnetism), the Curie

constant can be derived from the total spin quantum number S:

wT ¼ C ¼ NAm0

3kBg2SðSþ 1Þm2

B ¼NAm0

3kBm2eff ð3:8Þ

When allmicroscopicmagneticmoments are alignedwith the external field vector,

we speak of saturation. To describe the saturation effect, observable at higher fields,

the so-called Brillouin functions are used:

M

Msat

¼ 2Sþ 1

2Scoth

2Sþ 1

2S

gSmBH

kBT

� �� 1

2Scoth

gSmBH

kBT

� �ð3:9Þ

Here, the saturationmagnetizationMsat is defined asMsat¼NAgSmB. TheBrillouin

function is frequently used to establish the spin ground state of amagnetic material by

recording the magnetization as a function of magnetic field (over a wide range) at low

temperatures. Note that the Brillouin function strongly depends on the value of S, and

saturation occurs at lower fields for higher S (Figure 3.1). At finite temperatures, the

FIGURE 3.1 Brillouin functions describing the magnetization saturation of S¼ 1/2, S¼ 5/2,

and S¼ 7/2 centers in an applied field at T¼ 2.0K (according to Equation 3.9; g¼ 2.0).


magnetization of a bulk samplewill approach the saturation limit only asymptotically

since thermal modes will counteract a fully ordered saturated spin configuration. For

SH/(kBT)� 1, the function becomes linear with H again. For systems that are better

described by the quantum number J due to significant orbital momentum, g and S are

replaced by gJ and J.

Table 3.1 offers a quick assessment whether a certain 3dn configuration in an

octahedral ligand field can be treated as a spin-only magnetic center displaying

Curie-type magnetism or if a more complicated temperature dependence of the

susceptibility exists. For 4dn and 5dn systems as well as for actinide complexes,

more pronounced ligand field effects and stronger spin–orbit coupling render the

spin-only models moot. For lanthanide complexes, only the 4f7 configuration (8S7/2;

Eu(II), Gd(III)) results in spin-only behavior. As a rule of thumb, due to the

increasing spin–orbit coupling, the wT and meff values for homologous 4dn and

5dn complexes are lower than those of the corresponding 3dn configurations. For

example, for V3þ , Nb3þ , and Ta3þ , the spin–orbit coupling constants z increase

from 158 to 475 and 1657 cm�1.It is important to mention that a reduction of ligand field symmetry can strongly

alter the magnetic properties of ions: If the symmetry of a 3d1 system with an Oh-

symmetric ligand field is reduced (e.g., by an orthorhombic distortion), the entire

orbital moment will be quenched and spin-only magnetism is observed.

Several simple models exist5 that approximately describe the temperature depen-

dence of w for transition metal cations that do not represent spin-only centers. As one

example that is applicable to coordination complexes at low temperatures, the Kotani

theory6 incorporates the effects of spin–orbit coupling into the Van Vleck equation

and describes w(T) as a function of the spin–orbit coupling energy z.Some compounds exhibit considerable positive and temperature-independent

susceptibilities despite their singlet (S¼ 0) ground states, for example, several

octahedral Co(III) complexes (1A1 ground state). This temperature-independent

paramagnetism (TIP) arises as a result of mixing between the singlet ground state

and excited states with orbitalmoments that otherwise cannot be thermally populated.

Note that this second-order perturbation effect can also be present in paramagnetic

compounds (wtotal¼ wdia þ wTIP þ wpara).

TABLE 3.1 Estimates for the Low-Field Susceptibilities of Octahedrally

Coordinated 3dn Centers

Configuration Ground State Expected Susceptibility

3d5 (HS) 6A1 Curie paramagnetism; w¼C/T

3d1, 3d2, 3d6 (HS), 3d7 (HS) T Complex temperature dependence of w3d4 (HS), 3d9 E w¼C�/T þ wTIP3d3, 3d8 A2 w¼C�/T þ wTIP3d6 (HS) 1A1 wTIP

C� represents a modified Curie constant deviating from the equation due to spin–orbit and ligand field

effects. HS: high spin; LS: low spin.4


A similar temperature-independent paramagnetism, yet of different origin, is

observed for metallic compounds and here caused by the electrons of the conduction

band.

3.2.2 Magnetically Condensed Systems: Ferromagnetism

and Antiferromagnetism

In general, magnetic moments of paramagnetic centers are not strictly isolated and

will interact, even over longer distances, yielding bulk magnetic properties that can

strikingly differ from paramagnetic phenomena discussed above. On the microscopic

level, these spin–spin interactions can be caused by two types of mechanisms: via

direct dipole–dipole interactions, where the interaction energy decreases with r-3, and

(2) via indirect superexchange interactions involving interactions with orbitals of

nonmagnetic centers.

Regardless of the actual mechanisms involved, bulk magnetic properties can be

phenomenologically categorized into antiferromagnetic and ferromagnetic coupling.

Note that although magnetic interactions in solid-state structures occur in three

dimensions, both interaction type and strength can differ for each spatial direction.

This results in anisotropic phenomena, and usually the bulk material is characterized

by the dominating, energetically strongest interaction.

Ferromagnetic coupling between two spins causes their parallel alignment and a

high net magnetic moment. Correspondingly, antiferromagnetic coupling causes

antiparallel alignment and compensation of magnetic moments. Both types of

interactions are characterized by a coupling or exchange energy Jex. Empirically,

such interactions lead to susceptibilities whose temperature dependence can be

described by the Curie–Weiss law:

w ¼ C

T�u ð3:10Þ

Here, theWeiss temperature u is a function of the exchange energies Jex that can beobtained from a high-temperature series expansion:

u ¼ 2SðSþ 1Þ3kB

Xni¼1

aiJex;i ð3:11Þ

where n sets of neighboring spins interact so that a given spin center couples with aineighboring sites with a site-specific exchange energy Jex,i. Positive and negative

Weiss temperatures correspond to ferromagnetic and antiferromagnetic coupling,

respectively.

It is important to observe the applicability limits of the Curie–Weiss law and the

implied physicalmeaning of theWeiss temperature: It can only be applied tomagnetic

materials containing spin-only magnetic centers, and it only applies to magnetically

condensed systems (as opposed to magnetically dilute systems) at sufficiently high

temperatures (kBT� Jex).


For systems comprising magnetic centers that are both exchange coupled and

affected by ligand field effects (i.e., non-Curie paramagnetic centers), the suscept-

ibility (above an ordering temperature) can be approximated by molecular field

models.

Long-range coupling causes collective spin order within spatially extended

domains, resulting in cooperative phenomena such as bulk ferromagnetism or bulk

antiferromagnetism. These ordered phases are characterized by stability limits,

especially the critical temperatures (the Ne�el temperature TN for antiferromagnets

and the Curie temperature TC for ferromagnets) above which the compounds exhibit

paramagnetic Curie–Weiss behavior.

Typical examples for ferromagnets are metallic phases such as Fe, Co, Ni, Tb, Dy,

and certain oxides such as CrO2. For example, EuO displays a Curie temperature of

TC¼ 69K and a Weiss temperature of u¼ 74.2 K, with a saturation magnetization of

Msat¼ 6.94mB. Ferromagnets display spontaneous magnetization within Weiss dis-

tricts (which are separated by Blochwalls) even in the absence of an external field. An

external field change induces the diffusion of the Blochwalls and the fusion of aligned

Weiss domains, resulting in pronouncedmagnetization hysteresis. Themagnetization

hysteresis curves are parameterized by the remnantmagnetization at zero field and the

coercive field strengths that are necessary to quench the remnant magnetization. The

area enclosed by the hysteresis graphs is proportional to the magnetic energy

associated with the collective spin order. The so-called hard and soft ferromagnets

display large and small hysteresis areas, respectively.

Many ferromagnets are metals or metallic alloys with delocalized bands and

require specialized models that explain the spontaneous magnetization below TC or

the paramagnetic susceptibility for T> TC. The Stoner–Wohlfarth model,6 for ex-

ample, explains these observed magnetic parameters of d metals as by a formation of

excess spin density as a function of energy reduction due to electron spin correlation

and dependent on the density of states at the Fermi level. However, a unified model

that combines explanations for both electron spin correlations and electron transport

properties as predicted by band theory is still lacking today.

Upon reduction of the size of ferromagnetic systems, superparamagnetic systems

(“particles”) are obtained that exhibit single-domain ferromagnetic behavior below a

blocking temperature and superparamagnetic behavior above the blocking tempera-

ture. In the latter state, the collective magnetic moment of the particle freely rotates,

and an ensemble of superparamagnetic particles acts as a paramagnet, with the

difference that the constituent moments here stem from particles of ferromagnetic

materials and not from atomic moments as in the case of a normal paramagnet.

Several oxides and fluorides such asMnO, CoO, NiO, FeF2, andMnF2 (TN¼ 74K,

u¼�113K, Msat¼ 4.98mB), represent antiferromagnets in which below TN the

susceptibility quickly approaches zero with decreasing temperature, to result in a

diamagnet at 0K. Usually the antiparallel alignment of spins is energetically

favorable and antiferromagnetism is themost commonly observed cooperative effect.

If a compound comprises two or more different spin centers with different

magnetic moments (e.g., Fe(II), S¼ 2, and Fe(III), S¼ 5/2) that are antiferromagne-

tically coupled and if the crystal lattice features a pattern of alternating spins of


different magnitude, a net magnetic moment remains, giving rise to ferrimagnetism.

Ferrimagnetism is common for garnets such as Y3Fe5O12 or spinels such as Fe2CoO4,

Fe3O4, or Na2NiFeF7 (TC¼ 88K, u¼�113K). Figure 3.2 summarizes the observed

magnetic properties of ferromagnets, ferrimagnets, and antiferromagnets.

Similar to ferrimagnets, canted antiferromagnets of uniform spin moments that

cannot assume antiparallel orientations give rise to netmoments aswell. Examples for

such materials are FeF3 or FeBO3.

In addition, the aforementioned types of magnetic order can be combined to result

in metamagnets that display field-induced anomalies, if layers of a two-dimensional

ferromagnet couple antiferromagnetically along the third dimension in the presence

of low external fields and ferromagnetically in the presence of high magnetic fields.

3.2.3 Exchange Coupling of Magnetic Centers

The interaction between localized spin centers is usually modeled with a spin

Hamiltonian that takes into account single-ion effects, exchange coupling effects,

and interactions of the spin systemwith an external magnetic field. All of these effects

determine the resulting stationary spin wavefunctions of the system. Central to the

description of such systems is the exchange coupling between pairs of spin centers

that can be either antiferromagnetic or ferromagnetic. For example, the antiferro-

magnetic coupling between two spin-1/2 centers results in states that are character-

ized by a total spin quantumnumber S¼ 0 and S¼ 1,whereby the energy of the singlet

S¼ 0 state will be lower than that of the triplet S¼ 1 states. The difference ES¼ 0�ES¼ 1¼ Jex is due to the electronic exchange interaction. Here, a negative Jexindicates antiferromagnetic exchange and a positive Jex indicates ferromagnetic

exchange. Note that this description makes no assumption as to which exchange

mechanism is involved, and Jex represents a fully isotropic parameter. In general,

0

(a) (b) (c)M

/ µ B

M /

µ B00TC TC TCT (K) T (K) T (K)θ θ

χ m

(cm

3 /m

ol)

-1

χ m

(cm

3 /m

ol)

-1χ m

(cm

3 /m

ol)

-1

χ m

(cm

3 /m

ol)

-1

FIGURE 3.2 Schematic temperature dependence of magnetic properties of a ferromagnet

(a), a ferrimagnet (b), and an antiferromagnet (c). The behavior below the critical temperature

of ferri- and ferromagnetic materials is described by the magnetization M (dashed lines),

whereas the susceptibility (or w�1) can be used for T> TC (solid lines). For antiferromagnets, w(or w�1) can also be used to describe the system below TN. Curie–Weiss-type fits to the high-

temperature regimes (extrapolated as dashed-dotted lines) indicate the negative Weiss tem-

perature of ferri- and antiferromagnets.


superexchange interactions are stronger thanmagnetic dipole–dipole interactions, but

both have a different dependence on the distance of the involved spin centers: The

superexchange energy decreases faster with increasing distance than the dipole–

dipole interaction energy.

While the singlet ground state will be unaffected by an external magnetic field, the

S¼ 1 state will become Zeeman split into the MS¼�1, 0, 1 sublevels. Thus, the

energies of the four possible |S,MSi states are known: E(|0, 0i)¼ Jex; E(|1, þ 1i)¼�gmBH; E(|1, 0i)¼ 0; E(|1, �1i)¼ þ gmBH. When these four expressions are

introduced into the Van Vleck equation, the Bleaney–Bowers equation7 is obtained.

This equation describes the temperature dependence of the susceptibility (for the

zero-field limit), independent of the sign of Jex:

wT ¼ 2NAg2m2

B

kBð1þ 3eJex=kBTÞ ð3:12Þ

The resulting graphs for an arbitrary spin-1/2 dimer for Jex< 0, Jex¼ 0, and Jex> 0

are shown in Figure 3.3. A plot of wT (or meff) versus T is especially useful in this

context as one can immediately detect ferro- or antiferromagnetic interactions as a

deviation from the horizontal graphs of an uncoupled spin-only system. Yet care must

be taken in interpreting these plots, as orbital contributions can effectively give rise to

very similar curves.

In addition, magnetic level crossing effects can be observed in antiferromagne-

tically coupled spin systems. While the resulting multiplet states are energetically

higher at zero field than the singlet ground state, Zeeman splitting causes the

MS¼ þ S levels of these excited states to decrease linearly with the field and to

eventually undercut the field-independent singlet ground state at a certain crossing

field. For example, for an antiferromagnetically coupled spin-1/2 dimer with an

exchange energy of Jex¼�3 cm�1, this level crossing of the singlet ground statewitha |1, þ 1i state occurs at 3.2 T. In this case, the system is then magnetically saturated.

Correspondingly, at 0 K the system’s magnetization would remain zero up until Bsat,

where it jumps to the saturationmagnetization value. For systems with more excited

multiplet states, additional level crossings can take place as the slope of the

|S,MS¼ þ Si lines differs. At low temperatures (usually mK) one can then observe

step structures in M versus H plots, and from the corresponding crossing fields the

energetic spacing between the zero-field multiplet states can be determined. As

such, low-temperature/high-field magnetization measurements can be instrumental

in verifying the magnetic excitation spectrum, for example, as predicted by a

model calculation, as long as the crossing fields are still achievable (currently up to

ca. 60 T).

Likewise, the decreasing spacing between the ground state and the excited states

can have a considerable impact on the field dependence of susceptibility data

(Figure 3.4) that can be simulated for a given field using a number of programs.8

The quantum mechanical mechanisms that underlie exchange coupling are com-

plex, but can be modeled by a phenomenological Hamiltonian that involves the

coupling of local spin operators SA and SB, the so-called Heisenberg–Dirac–Van


FIGURE 3.3 Temperature dependence of wT (a), w (b), and 1/w (c) for a ferromagnetically,

antiferromagnetically, and uncoupled spin-1/2 dimer as calculated by the Bleaney–Bowers

equation (Equation 3.12).


FIGURE 3.4 (a) Energies of |S,MSi states as a function of the applied field B resulting from

an antiferromagnetically coupled spin-1/2 dimer (Jex¼�3 cm�1). (b) Corresponding w versusT curves at static fields of 0.01, 3.0, and 5.0 T. Model calculations based on

H¼�JexS1�S2� gmBBzSz.

Vleck Hamiltonian, which for two spin centers A and B takes on the form

H ¼ �JexSA � SB ¼ �JexðSAzSBzþ SAxSBxþ SAySByÞ ð3:13Þ

S denotes a vector operator comprising three components Sx, Sy, and Sz. Note that

generally a spinHamiltonian replaces all the orbital coordinates required to define the

system by spin coordinates. With the total spin operator S¼ SA þ SB and

S2 ¼ S2Aþ S2Bþ 2SASB, the Hamiltonian can be rewritten as

H ¼ � Jex

2ðS2� S2A� S2BÞ ð3:14Þ

S2 possesses the eigenvalues S(S þ 1). For a spin-1/2 dimer (S can be 0 or 1), the

same values are obtained from the derivation of the Bleaney–Bowers equation. For

spin systems in an external magnetic field, the Zeeman operator Hmag¼�gmBBzSzaccounts for Zeeman splitting. The isotropic Heisenberg Hamiltonian for multiple

spin centers can be expanded by adding the individual coupling pairs:

H ¼ �Jex;1SA � SB� Jex;2SC � SD� � � � ð3:15ÞTheHeisenberg approach remains valid as long as themagnetic centers act as spin-

only centers and represents an entirely empirical model. Orbital contributions can be

accommodated as perturbations. For example, ligand field effects can be effectively

approximated by the following anisotropic spin Hamilton operator:

Hani ¼ S �D � S ð3:16Þwhere D represents an anisotropy tensor (matrix), and upon appropriate coordinate

transformation Equation 3.16 is usually rewritten as

Hani ¼ DS2z þE S2x�S2y� �

ð3:17Þ


In systems of axial symmetry, Equation 3.17 simplifies to Haxial ¼ DS2z . Both

parameters D and E are phenomenological zero-field splitting energies.

As stated above, no information about the actual microscopic couplingmechanism

can be extracted from the exchange energy Jex. As a consequence, numerous studies

have explored the relevant structure–property correlations of dinuclear transition

metal complexes LaM(m-Lb)M0Lc, namely the relation between the value of Jex and

the geometry of the M(m-Lb)M0 link.Nevertheless, a simple qualitative model can explain antiferromagnetic and

ferromagnetic superexchange between two spin centers that involves the valence

orbitals of bridging ligands. Key to this model is the assumption of a partial spin

pairing between the unpaired electron of a “magnetic orbital,” usually a d-orbital of a

magnetic transitionmetal cation, and the electrons of the fully occupied ligand orbital

that overlaps with the metal d-orbital. If the ligand orbital, for example, a p-orbital,

then overlapswith symmetry-equivalent d-orbitals of bothmetal sites, this partial spin

pairing will result in an antiferromagnetic orientation of the two unpaired electrons

localized on the two metal cations. Here, both s and p binding modes can cause the

same result.

Ferromagnetic coupling, on the other hand, requires that no overlap exists between

the involved magnetic orbitals of the M andM0 sites. This then leads to a parallel spinorientation of the two unpaired electrons according to Hund’s rule. To this end, the

magnetic orbitals of the metal centers need to be orthogonal, or the coupling involves

two orthogonal orbitals on the same ligand donor position, for example, a px- and a py-

orbital (Figure 3.5).

If several exchange pathways exist in polynuclear coordination complexes,

ferromagnetic and antiferromagnetic pathways can compete for a given spin dimer,

usually resulting in a dominant antiferromagnetic exchange.

(a) (b)

FIGURE 3.5 Orbital overlap diagrams for M(m-L)M0 units: two paramagnetic metal centers

with single-occupied d-orbitals bridged by a (monoatomic) ligand featuring fully occupied p-

orbitals (one orbital lobe of each is highlighted in gray). (a) Overlap scenarios leading to

antiferromagnetic exchange; (b) overlap scenarios resulting in orthogonal singly occupied d-

orbitals, that is, ferromagnetic coupling.


To model several other electronic situations (e.g., in mixed-valence compounds),

the Heisenberg Hamiltonian can be augmented by specific exchange operators that

accommodate, for example, double exchange, antisymmetric exchange, anisotropic

exchange, or biquadratic exchange.

3.3 MEASUREMENT TECHNIQUES

At the core of magnetochemical analysis is the eventual determination of parts of (or

the entire) magnetic excitation spectrum, that is, of the manifold of magnetic

electronic states that are realized in a magnetic material, their energies, and degen-

eracies. The magnetic excitation spectrum is in the focus of both experimental

methods that shed light on its features and theoretical models that aim to reproduce it.

Several “thermodynamical” methods, for example, the measurements of suscept-

ibility or specific heat capacity, yield values that reflect the Boltzmann population of

various multiplet states at a given field and temperature (and possibly at other

specified external parameters such as pressure or irradiation) and essentially are

Boltzmann-weighted averages. Other techniques such as high-field electron para-

magnetic resonance (EPR) spectroscopy or inelastic neutron scattering, on the other

hand, result in energy parameters that correspond directly to gaps between multiplet

states and are thus useful to directly establish the magnetic excitation spectrum, at

least within the applicable selection rules and as long as probabilities of the involved

transitions are known.

3.3.1 General Considerations and Potential Pitfalls

Sample purity is one of the foremost criteria for reproducible magnetic measure-

ments. Paramagnetic impurities become especially dominant at low temperatures in

magnetization/susceptibility measurements and in EPR studies. Moreover, ferro-

magnetic impurities will affect the measured susceptibilities even if present only in

microscopic amounts. Under certain circumstances (for instance, if the actual sample

is characterized by a singlet ground state), these types of impurities that tend to

saturate already at low fields can be detected by field-dependent measurements. The

level of ferromagnetic impurities can be extracted, for example, from plots of w versusH�1, where the measured susceptibility will be the sum of the susceptibility of

the sample and the saturation magnetization of the impurity divided by the field,

w(H)¼ w(sample) þ Msat(impurity)/H. Diamagnetic impuritiesmainly cause an error

in the sample mass.

The selection of an appropriate applied field for susceptibility measurements

represents a crucial requirement that is frequently overlooked. For example, an

external field that is too strong can result in magnetic saturation or quenching of weak

antiferromagnetic and ferromagnetic spin–spin coupling, complicating the interpre-

tation of the obtained data (or consequently causing misinterpretation if field-

dependence relationships are not taken into account).


3.3.2 Magnetization Measurements

A magnetochemical characterization usually starts with determining the magnetiza-

tion (and its corresponding molar susceptibility) as a function of field and tempera-

ture. For this purpose, magnetometers utilizing a superconducting quantum inter-

ference device (SQUID) are widely used. These highly sensitive commercial instru-

ments allow programmed measurements in fields up to 7 Tand temperatures down to

ca. 1.8K (using rapidly evaporating liquid helium) or even down to ca. 400mK (using

a 3He cryostat insert). Only small amounts of samples are required (usually less than

10mg, depending on signal strength).

A SQUIDmagnetometer consists of a second-order gradiometer within the bore of

a solenoidal superconducting magnet. The superconducting gradiometer acts as a

pickup coil and a magnetic flux change caused by a sample that is moved through its

coils induces a current (Figure 3.6). The SQUID (usually realized as a thin film) acts as

an extremely sensitive current-to-voltage converter. The voltage measured at the

SQUID is then proportional to the magnetic moment of a sample within the applied

field.

However, great care must be taken in choosing an appropriate sample holder and a

suitable sample geometry.9 Cylindrical capsules machined from high-purity PTFE,

enclosed in long polyethylene straws, allow the measurement of compressed poly-

crystalline samples or even solutions. Polycrystalline samples need to be pulverized

so that preferential orientation can be avoided. The sample should be placed into a

cylindrically shaped sample holder. Air-sensitive compounds can be sealed in small

tubes from synthetic quartz glass. The field- and temperature-dependent contribution

of the sample holder must be known precisely.

Moreover, the sample should be perfectly centered within the sample tube, as even

small deviations toward the inner walls of the sample tube can cause significant

deviations in the measured signal.10

Z

FIGURE 3.6 Geometry of pickup coils in second-order gradiometers used in SQUID

magnetometers. The sample is moved along the z-axis through the coils.

MEASUREMENT TECHNIQUES 83

Importantly, a SQUID magnetometer allows both DC (equilibrium values at

constant field) and AC (modulated field) magnetization measurements. In the latter

configuration, the static fieldH0 ismodulated by a sinewave of a givenfield amplitude

HAC and frequencyw:H¼H0 þ HAC sin(wt). This allows the direct measurement of

dM/dH, that is, the differential susceptibility. At higher frequencies, the spin

dynamics may lag behind the modulation, causing a phase shift j from which both

the in-phase (w0, real) and the out-of-phase (w00, imaginary) components of the

susceptibility can be determined (for frequencies of up to 1.5 kHz in commercial

magnetometers):

w0 ¼ w cosðjÞ; w00 ¼ w sinðjÞ; w ¼ ðw02þ w002Þ1=2 ð3:18ÞAC susceptibility measurements are frequently used to identify thermodynamic

phase transitions and to characterize spin-glass behavior and superparamagnets such

as the single-molecule magnets discussed below.

At higher temperatures (up to ca. 1000 C), a vibrating sample magnetometer

represents an alternative to SQUID magnetometers. Here, a sample in a static and

homogeneous field is coupled to a membrane and vibrates as a harmonic oscillator,

usually at ca. 85Hz. The change in magnetic flux caused by the vibrating sample is

measured as an induced current in a pickup coil.

An important aspect of reliable susceptibility measurements is the calibration of

the magnetometers with respect to field and temperature; a number of high-purity

compounds exist to calibrate (1) the applied field via measurement of the suscept-

ibility (palladium metal, HgCo(NCS)4), (2) the temperature linearity via determina-

tion of the Curie constant of (NH4)2Mn(SO4)2�6H2O, or (3) the absolute values of the

sample temperature that are especially important for low-temperature measurements

and for which the critical temperatures of pure-element low-Tc superconductors can

be used, for example, lead (7.20K) or indium (3.41K).11

3.3.3 Specific Heat Capacity

The heat capacity Cm of a magnetic material comprises a magnetic component in

addition to lattice, electronic, and rotational contributions, among others. The

magnetic component is ideally derived as the difference between a measurement of

the magnetic sample and a diamagnetic derivative (e.g., by substituting Fe(III) with

Ga(III) ions). If this is not possible, the lattice contribution of nonconducting

crystalline compounds can be assessed as

C ¼ 12p4

5nkB

T

uD

� �3

ð3:19Þ

where uD is the Debye temperature and n the total number of atoms in the crystal.

Inmany cases, it is helpful tomeasure the specific heat as a function of temperature

with and without an applied magnetic field. Like the susceptibility, the magnetic

specific heat capacity Cm at a given temperature is averaged over the contributions of


all thermally populated levels of the magnetic excitation spectrum that basically

reflect their degeneracies and the overall magnetic entropy DSm:

DSm ¼ðCm

TdT ð3:20Þ

Note that DSm is proportional to ln(2S þ 1); therefore, a high-spin state will

contribute a larger entropic amount due to its higher degeneracy level.

Broad maxima in Cm versus T (the so-called Schottky anomalies) frequently

indicate partially populated discrete levels that are separated by an energy difference

DE in the range of kBT. For a simple two-level system, the maximum occurs at

kBTmax 0.42DE. Phase transitions, for example, transitions between a long-range

ordered ferromagnetic phase and a paramagnetic phase, produce a characteristic peak

in Cm versus T graphs.

Importantly, mononuclear impurities have virtually no effect on the magnetic

specific heat capacity, in stark contrast to the susceptibility.

3.3.4 Neutron Scattering

Neutron scattering can be due to two effects: the direct (nuclear) interaction of the

neutron with the atomic nuclei and the interaction of the neutron’s magnetic moment

with the local fields of unpaired electrons.12 In the case of elastic scattering, the energy

of the scattered neutron does not change: the kinetic energy of the emitted neutron is

equal to its energy prior to reaching the material. On the other hand, if the interaction

of the neutron causes an excitation (or induces a relaxation) in the scattering material,

the neutron’s energy is reduced or increased, which represents inelastic scattering.

If the magnetic material exhibits magnetic ordering, for example, antiferromag-

nets, the observed scattering pattern displays additional signals that by their intensity

and position indicate the magnitude and direction of a magnetic moment in the

crystalline lattice, which allows the construction of spin density maps.

Inelastic neutron scattering, on the other hand, usually employs a monochromatic

neutron beam and records the intensity of the scattered neutron beam as a function of

neutron kinetic energy. Such inelastic collision spectra are monitored as a function of

the applied field and the (usually low) temperature. The observed peaks then represent

the energy differences of thermally populated and excited unpopulated multiplet

states. Inelastic neutron scattering experiments can be conducted using triple-axis,

backscattering, or time-of-flight spectrometers.

3.3.5 Electron Paramagnetic Resonance Spectroscopy

Electron paramagnetic resonance spectroscopy, also known as electron spin reso-

nance (ESR) spectroscopy, detects the excitation of electron spins in an applied

external magnetic field.13 Conventional continuous-wave (CW) EPR is based on

resonance of a fixed-frequency standing microwave to excite some of the electrons in

Zeeman-split spin multiplets to undergo a transition from a lowerMS level to a higher


MS level (selection rule: DMS¼� 1). EPR is applicable to solid (powder, single

crystal, and frozen solutions) as well as liquid samples, and to electronic ground-state

systems as well as to photoexcited and carrier injection-induced systems (i.e., device

structures). The information contained in an EPR spectrum also reflects hyperfine

interactions and magnetic anisotropy (zero-field) effects.

Variable-temperature electron paramagnetic resonance spectroscopy represents a

resonance technique to scan transitions between the Zeeman-split MS states that are

induced by microwave radiation. Measurements require a minimum of sample

(usually less than a milligram), either in solid form or in solution. EPR is broadly

used since it allows direct determination of energy spacings within the thermally

populated part of themagnetic excitation spectrum. Themicrowave radiation induces

transitions within the MS states belonging to a given S multiplet split by an external

magnetic fieldH, and EPR also allows for the determination of the spin ground state.14

Note that in a crystal field environment the Land�e g factor is not necessarily an

isotropic scalar, but in general a tensor, g. For a static magnetic field parallel to the

z-axis of the g tensor, in the simplest case of S¼ 1/2, the resonance condition for a

transition between MS¼�1/2 and MS¼ 1/2 is met at

hv ¼ gzmBH ð3:21Þ

Traditionally, the available microwave frequency determines the resonance field:

For g¼ ge¼ 2.0023 (the isotropic g factor of a free electron) and a frequency of

9GHz, the resonance field equalsHe¼ 0.3211 T (X-band), while 35GHz correspond

to He¼ 1.2489 T (Q-band). Note that EPR spectra are usually evaluated using a spin

Hamiltonian, and orbital contributions are effectively accommodated in the g tensor.

The crucial importance of EPR is based on its potential to elucidate the anisotropy

effects of the chemical environments of the paramagnetic centers (usually their

coordination environments) that are associated with the spin–orbit coupling con-

tribution. EPR thus produces the g and the anisotropy D tensors, and the A tensors.

The latter, associated with the hyperfine interactions due to the magnetic moments of

the nuclei, are effectively quenched in polynuclear (i.e., magnetically nondiluted)

systems by the dipolar and exchange interactions between the magnetic centers. The

anisotropy of the remaining g and D tensors can be directly obtained from aligned

single crystals and, within limits, also from polycrystalline powders. For an aniso-

tropic g, the three principal components gx, gy, and gz are easily obtained from spectra

of the polycrystalline sample (Figure 3.7), but their relative orientation to the

molecular frame can only be determined from single-crystal measurements.

Zero-field splitting adds complexity to the spectra. For the case of an axial

symmetry distortion, the resonance fields for various MS ! MS þ 1 transitions can

be derived from a perturbation approach:14

HðMS!MSþ 1Þ ¼ ðge=gÞ½He�ðD=2gmBÞð2MSþ 1Þð3 cos2 a�1Þ ð3:22Þ

where a denotes the angle between the unique anisotropy axis and the external

magnetic field. For a given orientation, each Zeeman line is split into a total of 2S


components. Neighboring lines are separated by a field shift of (D/gmB)(3 cos2a� 1);

therefore, the ZFS parameter D can be directly extracted from the spectrum.

EPR configurations operating at higher microwave frequencies (i.e., Q- or W-band

setups)areespeciallyhelpfulsince this increase in frequencycorresponds toanincreased

resolution and sensitivity, thereby frequently simplifying the spectra. Furthermore, the

use of higher frequencies and fields generally also allows the observation of otherwise

“EPR silent” systems characterized by an integer total spin S for which the separation

between neighboringMS levels in the absence of an appliedmagnetic field is larger than

themicrowave energy (�0.5KkB at 9GHz,�2KkB at 35GHz), which in turn limits the

number of transitions that can be experimentally observed.15

The analysis of the low-temperature EPR spectra of aligned single crystals of the

molecular magnet {FeIII4 }¼ [Fe4(OCH3)6(dpm)6] (dpm: deprotonated dipivaloyl-

methane) exemplifies the potential of single-crystal EPR at higher fields/-

frequencies.16,17 The triangular {FeIII4 } molecule (see Figure 3.8, inset) consists of

four s¼ 5/2 spin centers (three outer centers binding to a central site) that are

antiferromagnetically coupled to yield a net S¼ 5 ground state. Since the octahedral

ligand fields of oxo positions around the iron sites are axially distorted, the degeneracy

of the |MS|¼ 0, 1, . . ., 5 states at zero field is lifted. As the principal axes of theD tensor

are largely determined by the local symmetry distortions of the iron sites in the

molecule, the spectra of aligned {Fe4} single crystals show 10 approximately

FIGURE 3.7 X-band EPR spectrum of a polycrystalline powder of an S¼ 1/2 molecule,

characterized by gx 6¼ gy 6¼ gz (a), gx¼ gy 6¼ gz (b), and gx¼ gy¼ gz (c).


equidistant transitions along the crystallographic c axis (a¼ 0) that yield a pattern ofnear-equidistant lines with half the splitting in the b direction (a¼ 90 ) (Figure 3.8).From the spectrum, the parameters gx¼ 1.995, gy¼ 1.997, gz¼ 2.009, and D¼ 0.2

cm�1 can be derived with the help of least-squares fitting software.17

Furthermore, the short scan times of EPR (usually 500ms or less) and the ability to

measure species in diamagnetic matrices, for example, aqueous solutions, enable the

time-resolved monitoring of chemical reactions involving radical reactants or inter-

mediates. In this way, kinetics of such reactions can be studied even if multiple

magnetic species are involved, as their characteristic signals typically differ suffi-

ciently to deconvolute the resulting EPR spectra. Commercial pulsed EPR spectro-

meters are also available, enabling the study of spin dynamics, that is, the relaxation of

the excited system via spin–spin and spin–lattice mechanisms.

Note that the very high sensitivity of EPR spectroscopy also implies that these

measurements are very susceptible to the presence of magnetic impurities. The

sensitivity and resolution of EPR are greatly enhanced in pulsed EPR spectroscopy

that entails applying a short (<20 ns) and intense (>300W)microwave pulse and then

measuring the microwave response signals generated by the samples’ magnetization

in the probe head. Upon Fourier transforming the signal, a frequency spectrum from

the sample is obtained.

3.3.6 Other Common Methods

One other important measurement technique that focuses on the electronic

surrounding of atomic nuclei is M€ossbauer spectroscopy, based on the

FIGURE 3.8 W-band EPR spectra of aligned single crystals of [Fe4(OCH3)6(dpm)6] at 5K

(only the S¼ 5 ground state is populated). Inset: coupling scheme of the four Fe(III) (s¼ 5/2)

centers. Additional smaller effects and structural isomers cause the spectrum to be more

complex than anticipated.17


recoilless nuclear resonance absorption of highly monochromatic g radiation (see

Chapter 2).18 Hyperfine interactions between the nuclei and the magnetic and

electric fields caused by the electrons interacting with the nuclei give rise to a

number of measured parameters: (1) isomer shifts, originating from electric

monopole interactions between protons of the nucleus and electrons, reflecting

oxidation and spin states as well as bonding properties; (2) quadrupole splitting of

the nuclear quadrupolemoment and an inhomogeneous electric field at the site of the

nucleus, again a function of oxidation and spin state as well as molecular symmetry;

(3) magnetic splitting caused by the interaction of the nuclear magnetic dipole

moment and the local magnetic field that is influenced by magnetic ordering in bulk

ferromagnets and antiferromagnets below their critical temperatures. Note that

M€ossbauer spectroscopy is limited to certain nuclei and mostly used for 57Fe-

containing samples.

Optical spectroscopy (UV/Vis/NIR) is of importance inmagnetochemistry aswell,

as it yields information about ligand field splitting effects.

3.4 INTERPRETATION OF EXPERIMENTAL DATA

As illustrated above, the microscopic explanation of observed magnetic properties

hinges on the construction of an appropriate model. In most instances, simplifications

have to be weighed and phenomenological models can be employed, such as the

Heisenberg spin Hamiltonian.

3.4.1 Model Systems for Extended Spin Systems

Inmagnetically condensed compounds, the interactions between spin centers can lead

to cooperative effects that significantly influence the magnetic properties. Even for a

wide variety of paramagnetic substances, magnetic ordering is observed at very low

temperatures, caused either by direct magnetic dipole–dipole interactions or by weak

superexchange via bridging entities or conducting electrons. A coupling of the

magnetic centers in particular dimensionalities is obtainedwhen these centers interact

predominantly with neighbors that are situated in chains (1D), layers (2D), or

networks (3D). While numerous theoretical models are known for one-dimensional

spin systems, no complete expressions exist (with very few exceptions) for higher

dimensional systems such as layers and networks. Thus, for these systems, approx-

imation methods are necessary.19

To this effect, for the three different dimensionalities of extended spin systems (1D

to 3D), the spin dimensionality needs to be distinguished and categorized. Depending

on mostly the single ion anisotropy, one can distinguish between the fully isotropic

Heisenberg model, the anisotropic two-dimensional XY model, and the strongly

anisotropic one-dimensional Ising model (Table 3.2).

The differences between these three models are evident from the exchange

operator Hex. This operator describes the interaction between localized magnetic

centers with spin operators Si and Sj and their components Sia and Sja (a¼ x, y, z) in

INTERPRETATION OF EXPERIMENTAL DATA 89

the following Hamiltonian:

Hex ¼ �2Xi < j

Jij ½aSizSjzþ bðSixSjxþ SiySjyÞ� ð3:23Þ

The coefficients a and b (see Table 3.2) take into account the restrictions in spin

dimensionality. For a¼ b¼ 1, the Heisenberg model with isotropic exchange inter-

action and isotropic susceptibility results. The combination of a¼ 1 and b¼ 0 yields

the strictly anisotropic Isingmodel, inwhich the orientation of the spins is restricted to

the z-axis. Consequently, the susceptibility is strongly orientation dependent and one

needs to differentiate between w|| in the direction of the z-axis (“easy axis”) and w?

perpendicular to z. The molar susceptibilities are then related as

wm ¼1

3wjjmþ

2

3w?m ð3:24Þ

The XY model is characterized by a¼ 0 and b¼ 1, and all spins are oriented

perpendicular to the z-axis. The anisotropy interaction can originate from single-

ion anisotropy, magnetic dipole–dipole interactions, and anisotropic crystal

morphologies.

3.4.2 Chains (1D)

The simplest spin chain consists of equivalent spin centersMwith local spins S¼ 1/2

and uniform exchange energies Jex:

� � � Mi � � � � � � Miþ 1 � � � � � � Miþ 2 � � � � � � Miþ 3 � � �

In real systems, the number of spin centers N is, of course, too large to determi-

nistically solve the corresponding eigenvalue problem of a corresponding spin

Hamiltonian. Thus, no exact solution exists. Several approximate expressions were

developed, such as the Bonner Fisher finite-chain model for equidistant antiferro-

magnetically coupled S¼ 1/2-based chains.20 Here, the susceptibility is calculated for

finite chain segments (ca. N¼ 10 spin centers) and extrapolated to an infinite chain

(N ! 1). The extrapolated expression for the susceptibility is as follows:

wm ¼ m0

NAg2m2

B

kBT

0:25þ 0:074974xþ 0:075235x2

1þ 0:9931xþ 0:172135x2þ 0:757825x3ð3:25Þ

TABLE 3.2 Models and Their Spin Dimensionality

Model Allowed Spin Orientation

Heisenberg a¼ b¼ 1 Isotropic

Ising a¼ 1, b¼ 0 Restricted to the z-axis

XY a¼ 0, b¼ 1 Restricted to the xy-plane


with x¼ |2Jex|/(kBT), which can be used to fit experimental w versus T data sets. As in

discrete (e.g., molecular) antiferromagnetic spin systems, w versus T shows a max-

imum; however, themaximumobtained fromEquation 3.25 is very broad, and c retains

a finitevalue atT¼ 0K.No corresponding equation exists for analogous ferromagnetic

chains; here, high-temperature series expansion methods (see below) offer an

alternative.

An equivalent procedure can be used for chains containing centers with S 1.

Since the matrix dimension of the eigenvalue problem increases significantly with

higher S, smaller chain segments are used to extrapolate to N ! 1, increasing the

uncertainty of this approach. Therefore, for chains with S 5/2 centers, it is better to

use an expression derived for classical spins (i.e., spin vectors that are not quantized

with respect to spatial directions):

wm ¼ m0

NAg2m2

B

kBT

1þ u

1�u with u ¼ coth2JexSðSþ 1Þ

kBT

� �� kBT

2JexSðSþ 1Þ ð3:26Þ

If the exchange interactions between nearest neighbors in the chain are not uniform

(equidistant chains) but are alternating between twovalues (e.g., caused by alternating

bridging ligands or nonequidistant spacings of the spin centers), “alternating chains”

are formed. Note that all uniform chains will form alternating chains at sufficiently

low temperatures (formation of dimers), as long as no three-dimensional order sets in

at higher temperatures. This so-called spin Peierls transition is equivalent to the

Peierls transition in quasi-one-dimensional metals.

The simplest alternating chain is defined by coupled S¼ 1/2 centers with two

exchange energies Jex and J0ex ¼ aJex, described by the Hamiltonian

Hex ¼ �2JexXN=2i¼1

S2i�1 � S2i þaS2i � S2iþ 1 ð3:27Þ

Here, a is called the alternation parameter specifying the ratio of the exchange

energies Jex and J0ex ¼ aJex. For 0<a< 1, both exchange energies have the same

sign, a¼ 1 results in the equidistant chain, and a¼ 0 describes isolated dimers. As

with uniform chains, the susceptibility of such Heisenberg systems can only be

approximated.

3.4.3 High-Temperature Series Expansion

Exact solutions do not exist for the exchange Hamiltonian of two-dimensional layers

and three-dimensional networks. These situations can, however, be described using

approximation methods such as molecular field approximation and high-temperature

series expansion.

The latter case implies that a meaningful calculation of the susceptibility applies

only to high temperatures in the case of three-dimensionally ordered compounds above

TC or TN. Already in 1958, Rushbrooke and Wood developed a high-temperature

INTERPRETATION OF EXPERIMENTAL DATA 91

expansion method to describe magnetic properties at high temperatures, for which

magnetic exchange only results in short-range order and no long-range order forms.21

Upon approaching an order transition (e.g., a critical temperature), the short-range

order affects ever larger areas, and the temperature dependence of the inverse

susceptibility shows a characteristic deviation from linear Curie–Weiss behavior.

High-temperature expansion tries to utilize this discrepancy to calculate exchange

interaction parameters.

Starting with a Heisenberg Hamiltonian in which Zeeman and exchange interac-

tions (limited to nearest neighbors) are accumulated, the susceptibility is derived as

follows:

wm ¼ m0

NAm2Bg

2

3kBT

� 1

ZSp

Xi

SizXj

Sjz exp �bHð Þ" #

� 1

Z2Sp

Xi

Siz exp �bHð Þ" # !2

8<:

9=;ð3:28Þ

Since the susceptibility cannot be evaluated in a deterministic way, the exponential

function is developed at b¼ 0 (which implies T ! 1) as a Taylor series:

expð�bHÞ ¼X1k¼0

1

k!ð�bHÞk ð3:29Þ

The sum runs over an infinite amount of terms, of which only a limited number can

be calculated due to the rapidly increasing computational demand. However, a

sufficiently high order of the series is a requirement in order to describe the

susceptibility in proximity to a critical temperature and determine the exchange

parameters. If the series is cut short to zeroth order, a Curie expression follows for the

susceptibility; if the series is developed to first order, the Curie–Weiss law follows.

Thus, bothCurie andCurie–Weiss expressions can be regarded as development stages

in the high-temperature expansion, which also implies that these expressions should

be valid only for high temperatures.

For small magnetic fields, wm is virtually independent of the external field B in the

paramagnetic region, and the zero-field limit B ! 0 can be used, resulting in

wm B! 0ð Þ ¼ m0

NAm2Bg

2

3kBT

SpP

iSizP

jSjzP1

k¼0 �2bð Þk=k! Pi < jJex;ijSi � Sj� �k� �

SpP1

k¼0 �2bð Þk=k! Pi < jJex;ijSi � Sj� �h i

ð3:30Þ

The values of Jex;ij, on the one side, represent the spin–spin coupling parameters

and, on the other side, define the coupling connectivity, that is, whichmagnetic centers


of a given lattice interact with each other. This separation leads to a lattice-

independent calculation of the spin productsP

Sj � Si and to a lattice-dependent

calculation as to how many times a graph (spin product) can be projected onto the

given layer (the so-called lattice count). Evaluation of the spin products proceeds via

the graphical procedure developed byRushbrooke andWood, inwhich the interaction

between two centers is represented by a line and the centers are represented as knots in

these lines.

Computer program packages are available for the simulation of the susceptibility

based on thismodel, based on the following information: the positions of themagnetic

centers of a given lattice, their spins, and the connectivities, based on a finite number

of exchange energies.

3.4.4 Monte Carlo Simulations

Oneof the limiting problems of numerically solving the eigenvalue problemof a given

spin Hamiltonian for a system with a finite number of spin centers is that the Hilbert

space dimension Q that translates into the dimension of matrices that need to be

diagonalized increases with

Q ¼YN

i¼1ð2Siþ 1Þ ð3:31Þ

where Si denotes the spin quantum number of the ith spin center and N is the total

number of spin centers. For a molecule containing 12 Cr(III) centers, for example, Q

would grow to 412¼ 16,777,216.

A number of techniques are known that can reduce the dimension of the involved

matrices that either utilize symmetry arguments (in the case of the irreducible tensor

operator method)22 or effectively limit the set of results to the lowest eigenvalues.

A different approach simulates the thermodynamic parameters of a finite spin

system by usingMonte Carlo statistics. Both classical spin and quantum spin systems

of very large dimension can be simulated, and Monte Carlo many-body simulations

are especially suited to fit a spin ensemble with defined interaction energies to match

experimental data. In the case of classical spins, the simulations involve solving the

equations of motion governing the orientations of the individual unit vectors, coupled

to a heat reservoir, that take the form of coupled deterministic nonlinear differential

equations.23 Quantum Monte Carlo involves the direct representation of many-body

effects in a wavefunction. Note that quantumMonte Carlo simulations are inherently

limited in that spin-frustrated systems can only be described at high temperatures.24

3.5 CASE STUDIES

The following examples of magnetochemical phenomena are selected to showcase

typical phenomena of inorganic compounds and their characterization. Molecule-

based systems are chosen that illustrate various phenomena on the basis of finite spin

architectures.

CASE STUDIES 93

3.5.1 High Spin–Low Spin Transitions

Transitionmetal ions in octahedral ligand fieldswith dn (n¼ 4� 7) configurations can

assume different populations of the t2g and eg orbitals, depending on the relation

between the spin-pairing energy P and the ligand field splitting DLF. If DLF�P, high-

spin (HS) configurations with a maximum number of unpaired d electrons will result;

if DLF�P, then low-spin (LS) configurations with a reduced number of unpaired

d electrons will represent an energetically more stable situation. If D and P are of

approximately the same magnitude, and if the difference in free energy of the lowest

vibronic levels of the HS and LS states is comparable to kBT, transitions between the

HS and LS states can be thermally induced, as evidenced in particular for several

Fe(II)-based compounds (usually comprising FeN6 environments) inwhich 1A1g (LS;

(t2g)6) and 5T2g (HS; (t2g)

4(eg)2) states are involved (Figure 3.9).25 The spin state

change is accompanied by a structural change due to the difference in ionic radii ofHS

and LS Fe(II), a consequence of an anisotropic redistribution of d electron density.26

This causes the Fe--L bonds to widen by up to ca. 0.2A. Note that HS–LS transitions

are characterized by both an enthalpy component caused by the change of M--L bond

strengths and an entropic component due to the different degeneracy levels of the spin

states involved. If structural changes induced by HS–LS transitions can propagate

throughout the crystal lattice as elastic tensions, these can in turn cause cooperative

effects resulting in very sharp LS–HS transitions that sometimes exhibit hysteresis—

and thus are a function not only of a critical temperature and external field but also of

external pressure.27

The local LS–HS transition frequently corresponds to drastic changes in the visible

absorption spectrum, and the spin states can also be switched by absorption of light.

FIGURE 3.9 Thermally induced spin crossover in [FeII(1-propyl-tetrazole)6](BF4)2, plotted

as the percentage of HS configuration. Note the slight transition hysteresis. Redrawn from

Ref. 28.


For example, if [FeII(1-propyl-tetrazole)6](BF4)2 that shows a LS–HS transition at ca.

130K is irradiated by a 514 nm source at 20K, corresponding to the energy of the1A1 ! 1T1 band, the complex undergoes a LS ! HS transition, and the resulting HS

configuration remains metastable at sufficiently low temperatures when phonon-

mediated relaxation is slowed down. Notably, this process can be reversed if the HS

complex absorbs 820 nm radiation corresponding to the 5T2 ! 5E2 band:28

½Feð1-propyl-tetrazoleÞ6�ðBF4Þ2 ðLSÞ !hv;514 nm

hv;820 nm½Feð1-propyl-tetrazoleÞ6�ðBF4Þ2 ðHSÞ

Both photoinducedLS ! HSandHS ! LS transitions involve transition through

a 3T1 state, from which the system can relax into the LS and HS ground states via

intersystem crossing processes. This reversible state switching has been summarized

as light-induced excited spin state trapping (LIESST) effect,29 and especially for Fe-

based compounds it can be conveniently traced by M€ossbauer spectroscopy.

3.5.2 Single-Molecule Magnets

An intensely investigated topic in molecular magnetism concerns so-called single-

molecule magnets that exhibit slow relaxation of the magnetization below a blocking

temperature and magnetization hysteresis of purely molecular origin (strictly speak-

ing, this naming is not entirely correct as a magnet is characterized by divergence of

the spin correlation length that cannot be the case in a cluster with a limited number of

spin centers).30 To explain the source of these phenomena, two historical

archetypes of such single-molecule magnets can be considered, the mixed-valence

{Mn12} complex [MnIII8 MnIV4 O12(CH3COO)16(H2O)4] and the {Fe8} complex

[FeIII8 O2(OH)12(H2O)(tacn)6]8þ (tacn¼ 1,4,7-triazacyclononane). Both feature an

S¼ 10 ground state due to intramolecular ferrimagnetic ordering and a uniaxial

magnetic anisotropy due to spin–orbit and ligand field interactions, parameterized as

zero-field splitting parameter D. For D< 0, a parabolic barrier of |MS|< S states

separates the energetically most favorable MS¼�S states so that the energy of the

MS¼ 0 state, that is, the cusp of the barrier, lies by an amount of DS2 above the

MS¼�S states (Figure 3.10), according to the ground-state Hamiltonian

H ¼ DS2z � gmBSH ð3:32Þ

An external field changes this symmetrical distribution ofMS states: according to

the Zeeman operator, the energy of the positive MS states decreases and that of the

negativeMS states increases. On the macroscopic scale, a repopulation of these states

is hindered by this anisotropy barrier as phonon-induced transitions are limited to

DMS¼�1, �2, causing slow relaxation of the magnetization characterized by

Arrhenius-type kinetics, as evidenced, for example, in AC susceptibility data

(Figure 3.11). However, several quantum tunneling mechanisms exist that enable

direct tunneling between positive and negative MS states if their energies match,

which happens at specific external fields. These tunneling effects can be seen as

CASE STUDIES 95

FIGURE 3.11 Schematic frequency-dependence diagram of out-of-phase (w00) AC suscept-

ibilities measured for the single-molecule magnet {Fe8} for a range of frequencies, indicating

the slowing down of spin relaxation at low temperatures.

FIGURE 3.10 (a) Ball-and-stick representation of the molecular structure of the {Mn12}

complex (MnIV: dark gray; MnIII: gray large spheres) with the central MnIV4 O4 cubane

highlighted. (b) Energies of the zero-field split S¼ 10 ground state resulting in an

anisotropy barrier, here shown in the absence of an external field. Apart from phonon-

induced transitions (curved arrows) quantum tunneling transitions (dashed arrow) are also

possible.


discontinuous steps at specific fields in the hysteresis curves at low temperatures

(Figure 3.12). However, the presence of not only an axial D term but also significant

nonaxial magnetic anisotropy allows additional direct transitions, effectively low-

ering the blocking temperature.

3.5.3 Geometric Spin Frustration and Spin-Phase Transitions

Geometric frustration describes a situation in which a set of physical or geometric

variables is incompatible with certain spatial or connectivity constraints. For ex-

ample, regular pentagons cannot tile a two-dimensional plane—an example of purely

geometric frustration.31 In a system of magnetic spin vectors, spin frustration can

result from competing antiferromagnetic interactions between the spin centers.

Physical phenomena caused by spin frustration—in network structures and within

discrete molecules—include plateaus of the magnetization as a function of the

external fields (M versus H), anomalously strong magnetic anisotropy, susceptibility

minima in w versus H, or pronounced magnetocaloric effects, such as giant adiabatic

cooling rates at magnetic fields for which large magnetization jumps occur.32

Frustration can even be associated with metamagnetic phase transitions demonstrat-

ing that single-molecule hysteresis can be exhibited by purely isotropic systems.33

If we assume classical spin vectors, that is, magnetic moments that are not

quantized in their orientation, such classical spin moments minimize their interaction

energy by maintaining an antiparallel orientation with their neighbors, which cannot

be strictly fulfilled, for example, for spin centers within a crystal latticewith triangular

symmetry or three spin centers forming a spin triangle. In the ground state of

the corresponding classical spin system, not all such interactions can be

satisfied simultaneously, resulting in noncollinear orientations of the spin vectors,

FIGURE3.12 Magnetization hysteresis of {Mn12} at 2.1K. Note the characteristic steps that

are caused by quantum tunneling processes. Redrawn from Ref. 30a.

CASE STUDIES 97

and therefore in geometrical spin frustration. For example, in the case of a spin

triangle of antiferromagnetically coupled classical spins, all interactingwith the same

exchange energy, the ground state is characterized by three spin vectors in a plane that

assume a 120 orientation toward each other. Note that the orientation of the plane isonly defined in the presence of an external field, when the plane is oriented

perpendicular to the field vector. Although this descriptive definition is limited to

a classical spin system, we call a quantum spin system frustrated if its classical

analogue is geometrically frustrated. In the following, we focus on molecular

polyoxometalate systems that comprise diamagnetic molybdate-based groups form-

ing a structural scaffolding into which spin-frustrated triangles M3 (M¼ FeIII, VIV)

are integrated.34

Frustration effects are especially pronounced in so-calledKeplerate structures of Ihsymmetry in which 12 pentagonal fMoIV6 g groups are interlinked via 30 hetero-

metallic linker positions that can be occupied by a range of magnetic transition metal

cations, for example, Fe(III). These 30 Fe(III) sites define the vertices of an

icosidodecahedron.23,35 This body of 20 equilateral triangles arranged around regular

pentagons can be considered as a spherical analogue of a classical Kagom�e lattice inwhich equilateral triangles are grouped around regular hexagons (Figure 3.13).

Importantly, intramolecular superexchange is limited to nearest neighbors, so that

60 Fe–Fe contacts result in which the spin sites are bridged by a molybdate bridge,

Fe–O–Mo–O–Fe. Next-nearest neighbor interactions are thought to be at least two

orders of magnitude smaller. The Fe(III) ions reside in near-perfect octahedral FeO6

coordination environments resulting in 6A1 states with no significant zero-field

splitting and therefore represent spin-only S¼ 5/2 centers.

Low-field susceptibility measurements on the magnetic Keplerate species

abbreviated as {Mo72Fe30} (isolated as the neutral cluster hydrate [MoVI72FeIII30O252

(CH3COO)12{MoVI2 O7(H2O)2}{H2MoVI2 O8(H2O)}(H2O)91]�150H2O) reveal anti-

FIGURE 3.13 (a) A section of a Kagom�e lattice in which equilateral triangles (gray) are

arranged around regular hexagons in a two-dimensional plane. (b) Arranging equilateral

triangles around regular pentagons causes a fold into the third dimension and ultimately yields

an icosidodecahedron in which 20 triangles share edges with 12 pentagons.


ferromagnetic coupling between the 30 s¼ 5/2 FeIII centers, and down to ap-

proximately 5 K the compound displays near-perfect Curie–Weiss behavior with

u¼�21.6 K.23 To establish a microscopic model that explains how 30 spins arrange

on the surface of this highly symmetric cluster structure requires an approximate

approach based on several simplifying assumptions as the underlying Hilbert space

dimension for this system, (2s þ 1)N¼ 630, renders futile any direct attempt to

numerically determine the energy eigenvalues of the spin states.

Note that, at least in a first approximation, all superexchange interactions are

confined to the 60 edges of the Fe30 icosidodecahedron and that all Heisenberg

interactions are characterized by a single exchange constant.

The spin operators Sn of the Fe(III) centers are replaced by scaled classical vectors,

[S(S þ 1)]1/2en, where en is a classical unit vector, free to point in any direction of

space.36 Heisenberg-type interaction energies between two such moments n and m

then amount to –Jexs(s þ 1)enem. In this context, s¼ 5/2 denotes the spin quantum

number of an individual Fe(III) center. By avoiding the quantization constraints of the

spin moments, one can use a quantitative method to characterize the {Mo72Fe30}

system: Classical Monte Carlo simulations of the thermodynamic properties of a

molecular ensemble as a function of temperature and external magnetic field (note

that due to spin frustration, quantum Monte Carlo simulations cannot be used here).

Applied to the low-field susceptibility data (corrected for paramagnetic impurity and

diamagnetic contributions) down to 120mK, this method yields an excellent fit with

Jex¼�1.09 cm�1 and an isotropic g factor of 1.974.23 When the temperature

approaches absolute zero, the simulations also yield a highly symmetrically frustrated

classical ground state (Figure 3.14) that is characterized by a finite susceptibility

w0(T¼ 0K) and a spin configuration in which the 30 spins are grouped into three

sublattices of 10 spins each, whereby any given spin interacts with two neighbor spins

of each of the other two sublattices. Within each sublattice, all 10 spin vectors assume

the same parallel orientation. In the zero-field limit, all spin vectors are coplanar and

nearest neighbors differ in angular direction by 120. The same geometrically

frustrated ground-state configuration also follows from techniques of analytical

graph theory that exploits that the icosidodecahedron is three-colorable (i.e., the

vertices have to be classified by a minimum of three types so that each position does

not neighbor a position of the same type) and can be decomposed into triangles. For

{Mo72Fe30}, this method also yields a rigorous zero-temperature limit of w0 that

coincides exactly with the extrapolated value for w0 from the aforementioned spin

simulations.

To account for quantum mechanical effects, an approximate quantum model that

reproduces the findings of the two classical spin-based approaches was constructed in

a next step.37One foundation of thismodelwas the finding that several (nonfrustrated)

molecular antiferromagnets of N spin centers s (which can be decomposed into two

sublattices) have as their lowest excitations the rotation of the Ne�el vector, that is, aseries of states characterized by a total spin quantum number S that runs from 0 to

N� s. In plots of these magnetic levels as a function of S, these lowest S states form

rotational (parabolic) bands with eigenvalues proportional to S(S þ 1). While this

feature is most evident for nonfrustrated systems, the idea of rotational bands can be

CASE STUDIES 99

used to approximate the lower section of the magnetic excitation of {Mo72Fe30}.

Then, to replicate the relative spin orientation of the classical three-sublattice

configuration and the rigorous classical ground-state energy, the classical vector

operators SA, SB, and SC for the three sublattices are exchanged for quantum

mechanical angular momentum operators, and the resulting effective Hamiltonian

results now are reduced to interaction between the sublattice A, B, and C net spins:

Heff ¼ �J AN

SA � SBþ SB � SCþ SC � SAð Þ ð3:33Þ

where the factor A is a function of the spin quantum numbers of the individual spin

centers and is specific to the particular spin polygon (A¼ 6 for {Mo72Fe30}) and

N¼ 30 is the total amount of spin centers.

The resulting eigenvalues E again are a function of the net spin moments of the

sublattices A, B, and C:

E ¼ J

10SðSþ 1Þ�

Xa¼A;B;C

SaðSaþ 1Þ !

ð3:34Þ

and, as seen in Figure 3.15, the lowest state for any given value of S forms a rotational

band that is followed by first, second, and so on excited rotational bands.

FIGURE3.14 Representation of the highly symmetrically frustrated classical ground state of

the Fe30 spin icosidodecahedron in {Mo72Fe30} in the absence of an external magnetic field.

The 10 classical spin vectors of each of the three sublattices (green, red, and blue) assume 120

relative orientation.Next to the Fe positions, only the bridging oxygen (small black spheres) and

molybdenum (pink) positions of the {Mo72Fe30} cluster framework are shown for clarity.

(See the color version of this figure in Color Plates section.)


An external field, via the Zeeman term in the Hamiltonian, lifts the degeneracies

of the individual MS¼�S, . . ., þ S substates belonging to an S state. Here, the

parabolic character of the first rotational band causes equidistant spin-level crossing:

the S¼ 1, MS¼�1 substate undercuts the singlet ground state at 0.24 T, only to be

undercut by a S¼ 2,MS¼�2 state at 0.48 T, and so forth until the system eventually

saturates in an S¼ 75, MS¼�75 state at Bsat¼ 12/7|Jex|s(s þ 1)/(gmB)¼ 17.7 T. In

the hypothetical case of T¼ 0K, these crossings cause the magnetization to rise in

discrete steps of 1Bohr with an increasing external field. With increasing tempera-

tures, these quantum mechanical features continually erode until a Brillouin curve

results. However, even at temperatures one order of magnitude lower than the

phenomenological threshold temperatures below which quantum effects are typi-

cally observed in antiferromagnetically coupled magnetic molecules (approximately

3.4K, given the single exchange value Jex for {Mo72Fe30}), this compound does not

exhibit such quantum steps: Whereas high-field magnetization measurements

(Figure 3.16) at 0.46K, nearly one order of magnitude smaller than this threshold,

confirm the saturation field Bsat, they also display a continuous, that is, nonquan-

tized, increase in M versus B without a step-like structure reflecting the spin-level

crossings. Importantly, such behavior complies with the purely classical description

in which the three sublattice spin vectors fold up toward the external field vector

while maintaining their 120 orientation in the plane perpendicular to the field vector(Figure 3.16).23

The underlying reason for this is that the lowest singlet–triplet energy difference is

very small since the exchange energy factors into this difference only as Jex/10 (see

Equation 3.4). The energy gaps between the first and the first excited rotational band

states and the degeneracies of the lowest involved levels resulting from the approx-

imate quantum model for {Mo72Fe30} were recently directly confirmed by inelastic

neutron scattering measurements (Figure 3.17).38

FIGURE 3.15 (a) Lowest section of the magnetic excitation spectrum as derived from an

approximate quantum model. Note that the resulting lowest S states, the first excited S states,

and even higher S states each form a parabola. (b) If an external field lifts the degeneracies of the

MS substates, this results in a “forest” ofMS levels, the lowest of which intersect at equal field

increments (n� 0.24 T for {Mo72Fe30}).37

CASE STUDIES 101

The field-dependent susceptibility at low temperatures displays a metamagnetic

feature:40 While a continuous increase inM versus H due to the continuous alignment

of the sublatticevectors should translate into a constant differential susceptibility dM/dB

untilsaturation(wheredM/dBshoulddroptozero),alocalminimumindM/dB isobserved

atapproximately5T(Figure3.18).Thisfeatureiscausednotbytransitionsbetweenstates

derived from the 120 three-sublattice “umbrella” configuration of spin vectors, but by

thermal population of states that belong to an entirely different three-sublattice config-

uration. The latter is again characterized by a frustrated arrangement, the so-called

“up–up–down” phase. In this phase, two sublattice vectors are aligned parallel (“up”) to

the external field vector, one is aligned antiparallel (“down”), regardless of the actual

magnitude of the external field. This becomes evident fromplots of the field dependence

of the energy of any given spin phase: While the “umbrella” spin phase represents the

energeticallymost stable one up until saturation (where it transforms into a fully aligned

“up–up–up”phase), theenergyof the“up–up–down”spinphasegets sufficientlyclose to

thatof the“umbrella” phaseatBsat/3,whichenables it toenergeticallycompete, that is, to

get populated at finite temperatures following a Boltzmann distribution (Figure 3.18).

Since the “up–up–down” phase is, as per definition, magnetically stiff (i.e., the total

magnetic moment remains independent of changes of the magnetic field), it is char-

acterized by dM/dB¼ 0; therefore, a partial population of its states subsequently

decreases the total differential susceptibility in the proximity of Bsat/3.

FIGURE 3.16 (a) Low-temperature (0.42K) magnetization data for the {Mo72Fe30} com-

pound (squares, only a small fraction of data points is shown), compared to the expected

quantum mechanical result (staircase graph), which is characterized by 75 discrete magnetiza-

tion steps until reaching saturation at Bsat¼ 17.7 T. No such spin-level crossing steps are

observed for {Mo72Fe30} at 0.42K. (b) Shown on a section of the Fe30 icosidodecahedron is a

simplified representation of the “umbrella” spin phase, which is defined by a continuous

alignment of the classical spin vectors of the three sublattices (light gray, gray, and dark gray

arrows) with an external magnetic field vector Bwhile retaining their 120 relative orientationin the plane perpendicular to the field vector.


FIGURE 3.18 (a) Experimental field dependence of the differential susceptibility of

{Mo72Fe30} at 0.42K (pulsed field measurements). Note the minimum slightly below 1/3 of

the saturation field Bsat, indicated by a vertical line. Bsat¼ 17.7 T is marked. (b) Competing

three-sublattice spin phases in {Mo72Fe30}. Whereas the “umbrella” phase is the most stable

phase below saturation, at Bsat/3 (vertical line) the “up–up–down” phase energetically nearly

matches the “umbrella” phase so that at finite temperatures it is populated significantly, thereby

decreasing the net differential susceptibility.

FIGURE 3.17 (a) Lowest energy levels calculated from the quantum rotational band model

for {Mo72Fe30}. Arrows indicate allowed transitions from the S¼ 0 ground state to states with

S¼ 1. The shading of the energy levels of the first excited rotational band represents potential

deviations from the ideal single-Jexmodel due to the fact that the high degeneracy of these levels

can be partially lifted due to a statistical variation in Jex.39 (b) Temperature dependence of the

magnetic contribution of the inelastic neutron scattering intensity of deuterated {Mo72Fe30},

compared to the theoretical scattering patterns for different temperatures (inset), calculated

using the quantum rotational band model and assuming identical probabilities for all involved

transitions.

CASE STUDIES 103

3.5.4 Structural Information from Magnetic Parameters

Aderivative of theKeplerate-type {Mo72Fe30} cluster discussed above illustrates that

simple low-field susceptibility data can be employed to elucidate disorder problems

that otherwise are difficult to solve by standard X-ray crystallography.

Thecompound{Mo72Mo6Fe24} contains linear polymers of cluster spheres inwhich

6 out of the 30 Fe positions in {Mo72Fe30} are replaced by effectively diamagnetic Mo

positions.41 The relative positioning of theseMo sites can take on a variety of scenarios:

from a purely statistical distribution, with the six nonmagnetic positions evenly dis-

tributed over the icosidodecahedral sites, to a clustering pattern, where all six positions

are situated adjacent to each other on one spot of the icosidodecahedron.

The difference in magnetic properties emerging from these different scenarios

stems from the number of Fe–Fe coupling interactions that are eliminated if one

Fe spin center is replaced by one nonmagnetic center. In a hypothetical sequence,

the change of the first spin center to a nonmagnetic center means that four Fe–Fe pairs

are eliminated (i.e., the number of nearest neighbors of each given position on the

icosidodecahedron). The next replacement can then again eliminate four further

Fe–Fe pairs if it takes place on a distant site, or it only eliminates three Fe–Fe pairs if

the two nonmagnetic sites are placed next to each other. If this pairing is continued,

successively less Fe–Fe pairs will be eliminated.

{Mo72Mo6Fe24} exhibits a temperature dependence of the susceptibility that

is similar to that of {Mo72Fe30}, indicative of weak antiferromagnetic coupling

within and between the cluster sphere units. Virtually identical geometries of the

Fe–O–Mo–O–Fe exchange pathways mean that the single nearest neighbor Heisen-

berg exchange constant Jex can be assumed to be equal to that in {Mo72Fe30}. Yet, the

Weiss temperature for {Mo72Mo6Fe24} of u¼�13.0K is significantly lower than the

value for {Mo72Fe30}, u¼�21.6K.The value of the ratio u({Mo72Mo6Fe24})/u({Mo72Fe30}) can be explained as

follows: u is proportional to the absolute number of nearest neighbor Fe��Fe contactsper sphere, which in {Mo72Mo6Fe24} is decreased in comparison to {Mo72Fe30} as

explained above. Assuming a uniform statistical distribution of these nonmagnetic

centers over the 30 icosidodecahedron vertices results in 38 intact Fe��Fe contacts

compared to the 62 contacts of a hypothetical all-iron configuration (60 edges of the

icosidodecahedron and two distinct bridges to the neighboring cluster spheres in the

chain).We therefore obtain u({Mo72Mo6Fe24})¼ 38/62� u({Mo72Fe30})¼�13.2K,in excellent agreement with the experimentally determined value. Therefore, the

additional Mo positions in {Mo72Mo6Fe24} must be distributed statistically, as a

clustered pattern would decrease the number of Fe–Fe contacts and thus the value of uby a much smaller degree.

Similar magnetochemical arguments have been used inmixed-metal derivatives of

dodecanuclear fNiII12g and fCoII12g coordination complexes in which one of the 12

metal centers is replaced, resulting in fNiII11 CoIIg and fCoII11 NiIIg stoichiometries in

which the heterometal position can assume one of two symmetry-equivalent sites. In

this case, simulations of the susceptibility data for the two possible scenarios of each

{M11M0} derivatives were able to identify the preferred substitution site.42


APPENDIX 3.A UNITS AND CONVERSION FACTORS

Two principal unit systems are in use for electromagnetic quantities, the SI and the

historic CGS-emu system.11Whereas the latter system is still being used in the current

literature, some quantities can be employed that are identical in value in both unit

systems: (1) the effective Bohr magneton number meff and (2) the magnetic dipole

moment m as well as the atomic magnetic dipole moment ma, divided by the Bohr

magneton mB (Tables 3.3 and 3.4).

Note that the dimensionless magnetic volume susceptibility w is related between

the two systems by a factor of 4p.

TABLE 3.3 Frequently Used Magnetic Quantities in the SI and the CGS System11

Quantity SI CGS Conversion Factor

m0 Permeability of

vacuum

4p� 10�7 V s/(A m) 1

B Magnetic

induction

B¼m0(H þ M) B¼H þ 4pMT¼V s/m2 G 10�4 T/G

H Magnetic field

strength

A/m Oe 103/4p (A/m)/Oe

M Magnetization A/m G 103 (A/m)/G

m Magnetic m¼MV m¼MV

Am2, J/T G cm3 10�3Am2/(G cm3)

m/mB

dipole moment

1 1 1

mB Bohr magneton mB¼ e�h/2me mB¼ e�h/2me

A m2 G cm3 10�3Am2/(G cm3)

Mm Molar Mm¼MM/r Mm¼MM/rmagnetization Am2/mol G cm3/mol 10�3Am2/(G cm3)

ma Atomic magnetic ma¼Mm/NA ma¼Mm/NA

dipole moment A m2 G cm3

ma/mB 1 1 1

w Magnetic volume M¼ wH M¼ wH

susceptibility 1 1 4p

wg Magnetic mass wg¼ w/r wg¼ w/r 4p/103

susceptibility m3/kg cm3/g (m3/kg)/(cm3/g)

wm Molar magnetic wm¼ wM/r wm¼ w M/rsusceptibility m3/mol cm3/mol 4p/106m3/cm3

meff Effective Bohr [3kB/m0NAmB2]1/2

[wmT]1/2

[3kB/NAmB2]1/2

[wmT]1/2magneton number

1 1 1

To obtain a quantity in SI units, the CGS quantity has to be multiplied with the conversion factor.

APPENDIX 3.A UNITS AND CONVERSION FACTORS 105

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Symbol Constant SI CGS

h Planck constant 6.626 08� 10�34 J s 6.62608� 10�27 erg s

kB Boltzmann constant 1.380 66� 10-23 J/K 1.38066� 10�16 erg/K

mB Bohr magneton 9.274 02� 10�24 Am2 9.27402� 10�21 G cm3

c0 Speed of light, vacuum 2.99792458� 108m/s 2.99792458� 1010 cm/s

me Electron mass 9.10939� 10�31 kg 9.10939� 10�28 gNA Avogadro constant 6.022 14� 1023mol�1

e Elementary charge 1.602 18� 10�19 C


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