Molekulare Nanomagnete: Quantenphysik zum Anfassen Molecular Nanomagnets

Molekulare Nanomagnete: Quantenphysik zum Anfassen ©Oliver Waldmann Molecular Nanomagnets

Content

I. Molecular Nanomagnets: Introduction

II. Magnetism: Definitions, Units, Overview II.1. What is Magnetism?

II.2. Magnetic Quantities and Units II.3. Types of Magnetism: Short Overview

a. Diamagnetism, b. Paramagnetism, c. Ferromagnetism, d. Antiferromagnetism, e. Ferrimagnetism,, f. Summary

III. The Spin

III.1. Angular Momenta in Condensed Matter Physics III.2. Fundamental Properties

III.3. Magnetic Moment

III.4. Magnetic Moments in a Magnetic Field III.5. Thermodynamic Relationships, Magnetization

IV. Magnetism of an Isolated, Localized Spin IV.1. Reminder

IV.2. High-Temperature Expansion IV.3. Low-Temperature Approximation IV.4. Full Calculation: Brillouin Function IV.5. A Classical Example: Atoms and Ions

IV.6. A Recent Exciting Example: The Case of the Mn19 Molecule IV.7. Excursion: Operator Equivalents and Effective Hamiltonian

V. Quantum Theory of Magnetism

V.1. The Microscopic Hamiltonian, Overview of its Derivation V.2. Magnetism f a Free Atom or Ion

a. Energy Considerations, b. Solution Approach, c. Formulation in Terms of Effective Hamiltonians V.3. Ligand-Field Theory

a. Introduction, b. Crystal-Field Theory, c. Ligand Field and Orbital Angular Momentum, d. Effective Spin Hamiltonian for Orbitally Non-Degenerate 3d Ions,

e. Energy Spectrum of Uniaxial Spin Clusters and Kramer's Theorem

VI. Single-Molecule Magnets I: Basic Properties and Giant-Spin Model VI.1. Basic Properties of the SMM Mn12ac

VI.2. Giant-Spin Model and Resonant Quantum Tunneling of Magnetization in Mn₁₂ac VI.3. Slow Relaxation of the Magnetization and Blocking Temperature

VI.4. Summary of Characteristics of SMMs

VII. Magnetic Coupling

VII.1. Phenomenological Approach to Magnetic Coupling VII.2. Origin of Isotropic Magnetic Interaction

a. Direct Exchange Interaction, b. Super-Exchange Interaction VII.3. General Recipe to Solve a Spin Hamiltonian

VIII. Heisenberg Spin Clusters VIII.1. Spin Dimers

a. Spin-1/2 Dimer in Zero Field, b. Spin-1/2 Dimer in Magnetic Fields, c. Magnetic Susceptibility, d. Magnetization, e. Dimer as Effective Model for AF Wheels

VIII.2. Spin Trimers

a. Classification of Types of Trimers, b. Basis States, c. Spin Trimer in Zero Field, d. General Heisenberg Cluster in Magnetic Fields, e. The V15 Molecule

VIII.3. Short Summary

IX. Single_Molecule MAgnetsi II: Quantum Tunneling and Two-Level System IX.1. reminder on Properties of SMMs

IX.2. Spin Hamiltonians for SMMs

a. Microscopic Spin Hamiltonian, b. From Microscopic to Giant-Spin Hamiltonian, c. Giant-Spin Hamiltonian, d. From Giant-Spin to Two-Level Hamiltonian

IX.3. The Two-Level System

a. Energies and Eigenfunctions, b. Time Dependence I: Quantum Oscillations, c. Time Dependence II: Landau-Zener-Stükelberg Transitions,

d. Quantum Oscillations in the Tunnel Splitting in Fe8

- 2 - Vorl. #1 (22. Okt. 2010) I. Molekulare Nanomagnete

Die molekularen Nanomagnete zeichnen sich gegenüber den "konventionellen" Magneten durch einige strukturelle Besonderheiten aus, die sich natürlich direkt in Ihren Eigenschaften, insbesondere den magnetischen Eigenschaften, widerspiegeln. Dies wird am Beispiel des Einzelmolekülmagneten Mn12 dargestellt (zur Notation was Einzelmolekülmagnet heisst kommen wir noch). Es werden dabei einige Begriffe ohne grosse Erklärung eingeführt, die Erklärung wird im Laufe der Vorlesung gegeben.

Die vollständige chemische Formel für das Mn12 lautet [Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H2O Die Notation ist so:

Der Teil zwischen den eckigen Klammern die Summenformel für den Cluster (das Molekül was uns interessiert) angibt, wobei manchmal noch Zusatzinformationen wie der Redoxzustand des Metallions oder Bindungsmoden mit eingeflochten werden.

Der Teil welcher direkt hinter der ]-Klammer kommt gibt die evtl vorhandenen Gegenionen an, welche für einen evtl Ladungsausgleich vorhanden sein müssen. Beim Mn12 ist keine solche Angabe zu finden, Mn12 ist also bereits elektrisch neutral und es sind keine Gegenionen vorhanden.

Die Teile welche mit dem Punkt · angefügt sind, sind die evtl. vorhandenen Kristallmoleküle (oder etwas lax Lösungsmittelmoleküle), welche bei der Kristallisation mit eingebaut werden und in der Regel mit darüber entscheiden wie der Cluster kristallisiert.

Der Einfluss der Gegenionen und der Kristallmoleküle auf die magnetischen Eigenschaften ist oft verschwindend gering. Zumindest werden wir uns hier in dieser Vorlesung auf Systeme beschränken bei denen das so ist.

Im folgenden Bild ist die Kristallstruktur des Clusters [Mn12O12(CH3COO)16(H2O)4], also dem eigentlichen Molekül, dargestellt. Kristallstrukturen werden üblicherweise mittels X-Ray Diffraktometrie bestimmt (Stichwort: Bragg). Seit dem Aufkommen der computerbasierten Methoden muss man aufpassen ob es sich um eine experimentelle oder theoretische Struktur handelt, hier werden wir immer nur experimentelle Strukturbilder gezeigt

Mn O C H Mn⁴⁺

Mn³⁺

OAc

Weinland, Fischer 1921 T. Lis 1980

[Mn₁₂O₁₂(CH3COO)₁₆(H₂O)₄]·2CH₃COOH·4H₂O

Mn O C H Mn Mn O O C C H H Mn⁴⁺

Mn³⁺

Mn⁴⁺

Mn³⁺

OAc

Weinland, Fischer 1921 T. Lis 1980

[Mn₁₂O₁₂(CH3COO)₁₆(H₂O)₄]·2CH₃COOH·4H₂O Mn12

erste Synthese: Weinland, Fischer 1921 Kristallstruktur: Lis 1980

Magnetismus: Sessoli et al, 1993

Wie zu erkennen ist Mn12 aus organischen Liganden (hier das Acetat CH3COOH, bzw MeCOOH, oder OAc) und Metallionen aufgebaut. Dies kann als typisches Merkmal aller molekularen Nanomagnete angesehen werden. Der Chemiker bezeichnet solche Objekte als Metallkomplexe, man kann also sagen

Metallkomplex = Metallionen + (organische) Liganden

Die Metallionen sind notwendigerweise positiv geladen, man schreibt dies z.B. als Mn(III) oder Mn³⁺ um eine dreifache Ladung anzudeuten. Im Mn12 Molekül sind sowohl Mn(III) wie auch Mn(IV) Ionen vorhanden.

Die Metallionen sind entscheidend für die magnetischen Eigenschaften, den wie wir noch lernen werden kann jedes Metallion als eine Art magnetische Kompassnadel betrachtet werden, physikalisch/quantenmechanisch spricht man von einem lokalisierten magnetischen Moment bzw. lokalisiertem Spin, welchen man dementsprechend auch durch einen Spin- Operator beschreibt.

Metallionen ↔ magnetisch ⇒ magnetisches Molekül

Der Spin des Metallions, sowie dessen Eigenschaften hängen von dem Typ des Metalions sowie dessen Ladungszustands ab. Für die Mn-Ionen in Mn12 gilt z.B.

Mn(III) (Mn³⁺): d⁴ S = 2

V Cr Mn Fe Co Ni Cu d³s² d⁵s d⁵s² d⁶s² d⁷s² d⁸s² d¹⁰s Mn(IV) (Mn⁴⁺): d³ S = 3/2

Also:

Typ des Metalions + Ladungszustand → Elektronenkonfiguration → Magnetismus Die Metallionen sind also ursächlich für die magnetischen Eigenschaften, das heisst aber nicht dass die Liganden keinen Einfluss hätten. Im Gegenteil, obwohl diese selber eigentlich unmagnetisch sind, haben sie drei sehr wichtige Auswirkungen.

Betrachten wir dazu zunächst die Liganden, hier das Acetat, etwas genauer.

Schreibweisen und chemische Strukturformel:

CH3COOH: Acetat = Essigsäure = Ethansäure (IUPAC) = Lebensmittelszusatzstoff E260 MeCOOH

MeCO2H Ac

Sehr häufig wird der Ligand aber nicht so im Cluster eingebaut wie gezeigt, da manche der H- Atome (Protonen) chemisch sehr aktiv sind, häufig liegen die Liganden in deprotonierter Form vor (ohne einige der H-Atome). Im Falle des Acetats ist es genau so, das H-Atom am Sauerstoff wird abgegeben:

O OH

O O

4 5 6 7 8 C N O F Ne Oktettregel!

CH3COO^-: deprotoniert in wässeriger Lösung = Essig

- 4 - Vorl. #1 (22. Okt. 2010) Die Atome des Liganden sind untereinander durch kovalente Bindungen aneinandergebunden, die Liganden widerum gehen mit den Metallionen chemische Bindungen ein, es handelt sich hierbei jedoch um Komplexbindungen (daher auch der Name Metallkomplexe).

Übersicht über typische chemische Bindungsarten

Bindungsstärke:

→ Bindungslänge

→ optische Eigenschaften

→ mechanischer Eigenschaften stark:

- ionisch - metallisch - kovalent mittelstartk:

- komplex schwächer ⇒ ca. 2-3 Å Bindungsabstand, sehr farbig schwach:

- Wasserstoffbrücken

- π-π (Wechselwirkung zwischen aromatischen Ringen) sehr schwach

- Van der Waals

Die "Stärke" der Bindung äussert sich in auch in den Eigenschaften, wie z.B., den Bindungslängen, der "Härte" von Materialien, Schmelztemperaturen, den optischen Eigenschaften, etc.

Die Komplexbindung ist typischerweise etwas schwächer als z.B. kovalente Bindungen, dementsprechend sind die Bindungslängen typisch 2-3 Å, und die Komplexe sind sehr farbig, gehen jedoch bei vergleichsweise niedrigen Temperaturen (100 °C) schon kaputt.

Aufgrund der chemischen Eigenschaften, sind die Liganden offensichtlich entscheidend für die Struktur der resultierenden Moleküle. Also:

Liganden → Struktur des Moleküls

Daneben haben die Liganden, obwohl selber unmagnetisch, entscheidenden Einfluss auf die magnetischen Eigenschaften. Einmal beeinflussen sie das Verhalten des an den Metallionen lokalisierten magnetischen Moments bzw Spins. Dies führt zu einer sogenannten magnetischen Anisotropie, welche mittels der Ligandenfeldtheorie beschrieben wird.

Darüber hinaus können die Liganden einen magnetische Wechselwirkung zwischen den verschiedenen Metallionen in dem Cluster vermitteln, d.h. die einzelnen Metallionen, bzw., deren Spins, in einem Cluster sind nicht unabhängig voneinander, sondern "sprechen"

sozusagen miteinander. Der Mechanismus welcher für diese Wechselwirkung verantwortlich ist, ist im Regelfall der sogenannte Superaustausch. Da diese magnetischen Wechselwirkungen sich auf Wechselwirkungen zwischen den Metallionen innerhalb eines Clusters/Moleküls beziehen, spricht man von intramolekularer magnetischer Wechselwirkung. Ohne diese magnetischen Wechselwirkungen wären die magnetischen Eigenschaften der Moleküle sehr langweilig.

Also:

Liganden → magnetische Anisotropie, intra-molekulare magnetische Wechselwirkungen

Nun kommen die Moleküle typischerweise im Experiment nicht einzeln vor, sondern in kristalliner Form, z.B. als Einkristall oder als mikrokristallines Pulver. Die Kristallstruktur für Mn12 ist im Folgenden dargestellt (die Kristallmoleküle wurden nicht eingezeichnet)

Der Kristall wird jetzt nicht wie z.B. bei Metallen oder Salzen durch starke chemische Bindungen zusammengehalten, sondern im Regelfall durch die schwächste chemische Bindung, nämlich der Van-der-Waals-Bindung; man spricht von Molekülkristallen. Dies liegt an dem Aufbau der Liganden, d.h., die Details der Kristallstruktur sind wieder durch die Liganden wesentlich mitbestimmt. Diese Tatsache, dass die Clusters im Kristallverbund nur durch diese schwachen Bindungen zusammengehalten werden, äussert sich wieder in den Eigenschaften, wir z.B. dem "Schmelzpunkt" der Kristalle, deren Härte, etc, und auch in den magnetischen Eigenschaften. Die sehr schwache chemische Bindung bedeutet unter anderem, dass auch evtl. magnetische Wechselwirkungen zwischen den Metallionen verschiedenere Cluster, man spricht von den intermolekularen magnetischen Wechselwirkungen, verschwindend gering sind. Also:

Liganden → keine inter-molekulare magnetische Wechselwirkung

Dies ist nun für die physikalischen/magnetischen Eigenschaften entscheidend, da es sich aus magnetischer Sicht um null-dimensionale (0D) Systeme handelt, um echte Nanocluster. Dies macht den grossen Unterschied zu den in den letzten 80 Jahren untersuchten

"konventionellen" Magneten, auch dem Quantenmagnetismus aus, welche sich immer auf ein- dimensionale (1D), zwei-dimensionale (2D), und drei-dimensionale (3D) Systeme bzw Anordnungen von miteinander wechselwirkender Metallionen bezog.

Dies bringt die molekularen Nanomagnete physikalisch in einen neuen Grenzbereich. Dies hat aber auch experimentelle Vorteile, da es nun möglich wird einfach durch Messung an einer makroskopischen Probe (Pulver/Kristall) die Eigenschaften einzelner Moleküle zu bestimmen. Zusammenfassend:

Molekulare Nanomagnete:

- Metallionen → lokaliserte magnetische Momente/Spins, ursächlich für Magnetismus - Liganden → Molekülstruktur

→ intra-molekulare mag. Wechselwirkungen, magnetische Anisotropie

→ verschwindende inter-molekulare magetische Wechselwirkungen

⇒ null-dimensionales Quantensystem

- 1 - Vorl. #2 (27. Oct. 2010) Molecular Nanomagnets ©Oliver Waldmann

II. Magnetism: Definitions, Units, Overview II. 1. What is Magnetism?

If one aims at finding a general definition for which effects to be considered a part of

"Magnetisms", one may consider

Magnetism embraces all effects which emerge as a response of a material on application of an external or internal magnetic field.

Examples

1) magnetization Mr

, magnetic moment mr

Upon application of a magnetic field, a material may become magnetic, i.e., develop a magnetic moment. Obviously, this is the primary response

2) magneto resistance:

The electric resistance of a material depends on the strength of an applied magnetic field.

Usually these effects are very small, but can be made to be large. Examples are the Hall effect, the quantum Hall effect, or the recently discovered giant magneto resistance (GMR), colossal magneto resistance (CMR), tunnel magneto resistance TMR (XMR), which nowadays find many applications, such as hard disc drives, automotive applications, etc.

3) magneto striction:

Also the shape of a material, e.g., the volume or the length, in general depends on an applied magnetic field. This effect may be noticeable in particular in ferromagnetic materials, e.g., the humming of a transformer is due to magneto-striction.

4) gyro magnetism:

Since magnetism is related to circulating electrical currents or angular momentum, respectively, changes in for instance an applied magnetic field may induce a rotation of a magnetic material.

In this lecture we are only concerned with the magnetization or magnetic moment which develops in an applied magnetic field.

II.2. Magnetic Quantities and Units

Three quantities will be of key importance to us, namely the magnetization, magnetic moment, and the magnetic susceptibility, which will be defined below. Unfortunately, the issue of the units is a quite complicated one in magnetism, and even experts sometimes may step into the traps. We will not elaborate on this but simply mention the units to be used here.

Magnetization

The magnetization is defined in the electro dynamics of materials as follows:

SI ) M H ( Br ₀ r r

+ μ

=

[T] [A/m] [A/m]

cgs M 4 H Br r r

π +

=

[G] [Oe] [G] oder [Oe]

B: magnetic induction H: magnetic field M: magnetization

The distinction between the B and the H field is introduced for convenience. The B field is, so to say, the "real" field in the sense that e.g. an electron would react according to the strength of the B field. The H field is, so to say, the hypothetical field which would be present in the absence of the material. It is the field which is generated by the sources of magnetic field which lie outside of the material, such as e.g. a solenoid.

Useful relation Comments

Although the above definition is clear and generally agreed on, usage is usually somewhat different:

1) If one uses the SI system one often does not use H directly for the magnetic field, but rather expressed it as μ₀H, i.e., uses the unit Tesla instead of A/m. Similarly, in the cgs system one often uses the unit Gauss for H too.

2) It is often common practice to use "magnetic field" for denoting the B field and not the H field. This may be justified then the magnetization is very small, such that the H and B field almost agree numerically, but in strongly magnetized materials this is jargon.

Except in this chapter, we will refer to the B field as the magnetic field, use the unit Tesla for it, and not use the H field. In the case of the molecular nanomagnets this may be justified because the magnetization is usually small: only few of the many atoms in a molecule are magnetic, the binding lengths are typically on the order of 3 Å and hence significantly longer than in inorganic crystals, and the packing of the molecules in the unit cell is low.

Magnetic Moment

In an experiment one would, however, not observe the magnetization, since the magnetization is defined to be an intrinsic property and where with independent on the amount, i.e., on the volume of the material. Clearly, in experiment one would measure a twice as large signal if one would use a twice as large material. That is, one measures the magnetic moment

V M mr = r SI

[Am²]

cgs [emu]

V: volume Useful relation Comment

The unit Am² of the magnetic moment indicates its electro-dynamical origin, namely what it is associated to circulating currents. For a conductor loop, e.g., it is known to be given by

IA m=

1 G = 10^-4T

1 emu = 10^-3Am²

earth ≈ 5 10^-5 T = 0.5 G NMR tomograph ≈ 1 T high-field lab ≈ 30 - 100 T neutron star ≈ 10⁸ T

m = I A I A

- 3 - Vorl. #2 (27. Oct. 2010) Magnetic Susceptibility

For the magnetic susceptibility several definitions are possible and useful. For our purposes the two most important definitions are the following:

Definition i) M(H)=χ_V(H)H Definition ii)

0 H

V H

M

∂ →

= ∂ χ

They coincide iff the magnetization response is linear at small magnetic fields (it also has to be in phase in case of time-varying fields, which will be discussed at a later pointer).

Comment

The above definitions are not appropriate for cases in which the magnetization exhibits a hysteric behavior, the magnetic fields are time-varying, or the magnetic fields are not constant over space.

However, unfortunately measuring the susceptibility as defined in the above is practically very difficult, because it would require very precise determination of the volume of the material, which obviously is difficult e.g. for powder or microcrystalline samples, or very small single crystals. Hence, it is much more practical to refer to e.g. the mass of the sample or the amount of substance (molarity), which results in the definitions

magnetic susceptibility per mass:

H mass / m

g = χ molar magnetic susceptibility:

H mol / m

mol = χ

SI cgs

χV dimension less dimension less (pay attention to the 4π)

χg [m³/kg]

χmol [m³/mol] or [NAμB/T] [emu/mol], [emu/cm³], [cm³/mol], [emu/molG], [emu//cm³G], ...

Unfortunately, the issue of the units for the magnetic susceptibility is a very tricky one, especially in the cgs systems, which however is quite commonly used. It is tricky because inconsistent units are used to denote the very same quantity, namely molar susceptibility. In order to not elaborate into the details, we simply agree on a system here

In experimental data, we will always use the molar magnetic susceptibility χmol; the unit can be either [emu/mol], [emu/cm³], or [cm³/mol], which however are identical.

For the connection to the theoretical equations, the following relation is useful

mol K emu 8 1 mol

K 12505emu

. k 0

3 N

B 2 B

Aμ = ≈

II.3. Short Overview: Types of Magnetism

II.3.a. Diamagnetism

value of susceptibility: χ < 0

physical origin: induced circulating currents Comments

1) A negative susceptibility means that the "real" magnetic field B is reduced in the material, i.e., is smaller inside the material than in the outside.

2) That induced circulating currents do weaken the magnetic field B in the interior of the material can be understood to be a consequence of the Lenz rule.

• Diamagnetism in atoms and molecules

⇒ Larmor diamagnetism χ ≈ -10^-6

χ(T) = constant

atoms: ^χ=−

∑

i 2 i e

2

A r

m 6

e N

molecules: χ ≈ - 0.4..0.5 × molar mass × 10^-6 emu/mol Comments

1) Any material exhibits diamagnetism, even the so called non-magnetic materials. E.g. are frogs, which due to their diamagnetism can be made to levitate iff the magnetic field is strong enough, see photo.

2) The formula for the atoms involves the area of the circulating current, as expected for the magnetic moment of a circulating current.

3) The sum in the formula for the atoms indicates that every electron contributes about a similar amount to the total susceptibility. This qualitative argument justifies the estimate for the molecules, since the number of electrons in a material is roughly related to the molar mass of the material.

4) In some molecules further paths for circulating currents exist, for instance in aromatic molecules such as benzene. Here a current may be induced to flow in the ring, which due to the large enclosed area results in a large diamagnetic response for fields perpendicular to the plane of the benzene molecule. This gives also rise to a significant anisotropy in the magnetic behavior.

• Diamagnetism of itinerante electrons or delocalisied magnetic moments

⇒ Landau-Diamagnetismus in metals χ ≈ -10^-6... -10^-5

χ(T) = constant

• Superconductors

⇒ Meissner-Ochsenfeld effekt χ ≈ -1

This is called an ideal diamagnet since a superconductor repels essentially all magnetic field from its interior, i.e., B ≈ 0 in the material.

- 5 - Vorl. #2 (27. Oct. 2010) II.3.b. Paramagnetism

value of susceptibility: χ > 0

physical origin: permanent magnetic moments Comments

1) A positive susceptibility means that the "real" magnetic field B is enhanced in the material, i.e., is larger inside the material than in the outside.

2) Permanent magnetic moments do not exist within classical physics; they do exist only in quantum mechanics. That is, any magnetism is intrinsically of quantum nature.

• Paramagnetism of unpaired electrons or localized magnetic moments

⇒ Curie susceptibility, Langevin equation, Brillouin function χ ≈ 10^-3 at room temperature

T ) C T

( =

χ 3k T

) 1 S ( S g ) N

T (

B 2 2 B

Aμ +

=

χ (Curie law)

C: Curie constant

The temperature dependence of the Curie law may be presented in three ways. Which one is useful depends on the particular question at hand. Often it can be useful to inspect experimental data in two or all representations.

χ vs T χT vs T 1/χ vs T χ

T

χT

T

1/χ

T

χ

T

χT

T

1/χ

T

In most cases, however, we will find the χT vs T representation of most interest. It exhibits two very important characteristics

1) χT is independent on temperature T

2) the magnitude of χT is given by the Curie constant, which in turn is determined by the g factor and the spin S of the localized magnetic moment

Hence, from analyzing the χT curve one can immediately draw some important information on a material, namely if it behaves like a simple paramagnet or not, and the magnitudes of the g factor and the spin S.

The Curie behavior is observed only in weak magnetic fields, which corresponds to a linear increase with increasing field at low fields. The magnetization curve, that is the magnetization or magnetic moment, respectively, as function of a magnetic field is a bit more involved. We will see soon that it is determined by the Brillouin function. However, qualitatively the behavior can be understood by realized that there are two energy scales,

which are involved in the game, namely the magnetic field (energy μ_BB) and the temperature (energy k_BT), which have opposite effects on the magnetic moments in a material:

magnetic field → tends to order the magnetic moments, i.e. to align them along the field temperature → tends to disorder, i.e., randomize the orientation of the moments

zero field weak field strong field

zero field weak field strong field

The qualitative aspects in the magnetization curve, which looks as follows, can then be understood by the competition of these two energies:

Sättigung

= C/T

χ

m0 = gS

T₂

The analysis of the magnetization curve provides some further important insight into the behavior of a material, for instance, from the saturation magnetization the quantity gS can directly e determined. With the available magnetic fields of several Tesla in typical magnetism labs, one has to perform the experiment at low temperatures of few Kelvins in order to observe saturation.

T1

m

B

Useful relations k 0.67171

B

B =

μ 1.4887

k_B

B = μ

Usage 1 K = 1.4887 T

• Van Vleck or temperatur independent paramagnetism (TIP) χ ≈ 10^-5

χ(T) = constant

This contribution exists in any metal complex, and hence in particular also in the molecular nanomagnets. It may be relevant for analyzing experimental data, but will not be considered in these lectures.

• Paramagnetism of itinerant electrons or delocalised magnetic moments

⇒ Pauli paramagnetism in metals χ ≈ 10^-6...10^-5

- 1 - Vorl. #3 (29. Oct. 2010) Molecular Nanomagnets ©Oliver Waldmann

II.3.c. Ferromagnetism

In a ferromagnet, magnetic interactions exist between the magnetic moments in the material, which favor a parallel alignment of the moments (= ferromagnetic interactions). A ferromagnet is characterized by a magnetic phase transition at a critical temperature, the Curie temperature Tc. Ferromagnetism is known for systems with localized magnetic moments as well as itinerant electrons.

T > Tc

At temperatures above the Curie temperature, a ferromagnet behaves similar to a paramagnet in as much as the magnetic moments are thermally disordered. The magnetic susceptibility follows (approximately) the Curie-Weiss law:

θ

= − χ T

C (Curie-Weiss law)

θ: Weiss constant, for which holds θ > 0, θ ∝ zJ Tc: Curie temperature

z: coordination number

J: magnetic coupling strength, J > 0

χ

Tc T

paramagnetic

ferro- magnetic

Due to the magnetic interactions, the magnetic susceptibility already approaches infinity, χ →

∞, at the critical temperature, which is achieved via the Weis constant in the Curie-Weis law.

The Weis constant is related to the strength of the magnetic interactions, J, and the number z of surrounding magnetic moments.

1/χ

T

T

paramagnetic ferro-

magnetic

T < Tc

Below the Curie temperature a ferromagnet spontaneously becomes fully magnetized, i.e., all magnetic moments become mutually parallel to each other. The dependence on a magnetic field is complicated for real ferromagnets, but for an ideal ferromagnet (which, however, is not existing) the magnetization would jump to maximum by application of even a minor magnetic field.

M

B

M₀

II.3.d. Antiferromagnetism

Similar to a ferromagnet, also in an antiferromagnet magnetic interactions exist between the magnetic moments in the material. Further, an antiferromagnet also exhibits a magnetic phase transition at a critical temperature, the Neél temperature TN, and antiferromagnetism is known for systems with localized magnetic moments as well as itinerant electrons. The key difference to a ferromagnet, however, is the very different nature of the magnetic interactions, which in an antiferromagnet favor an antiparallel alignment of the moments (=

antiferromagnetic interactions).

T > TN

As for ferromagnets, above the critical temperature an antiferromagnet at temperatures also behaves similar to a paramagnet in as much as the magnetic moments are thermally disordered, and the magnetic susceptibility follows the Curie-Weiss law:

θ

= − χ T

C (Curie-Weiss law)

θ: Weiss constant, for which holds θ < 0, θ ∝ zJ J: magnetic coupling strength, J < 0

1/χ

T

T

paramagnetic anti-

ferro

0

χ_||

χ_⊥

χ

T

T

paramagnetic antiferro-

magnetic

However, the antiferromagnetic interactions reveal themselves by a negative Weis constant.

Hence, at the Neél temperature the magnetic susceptibility remains, finite.

T < T_N

Below the critical temperature, an antiferromagnet also undergoes a phase transition to a long- range ordered ground state, in which, however, the magnetic moments are ordered such that neighboring moments are antiparallel to each other. At zero field, the magnetization is zero, because the moments of the antiparallel spin cancel. However, this antiparallel configuration is pretty "stiff", hence is not destroyed easily by a magnetic field, hence, the magnetic susceptibility is rather small. Furthermore, the magnetic susceptibility becomes anisotropic, i.e., depends on whether the magnetic field is applied parallel or perpendicular to the orientation of the magnetic moments.

M

B

B⊥

B||

- 3 - Vorl. #3 (29. Oct. 2010) II.3.e. Ferrimagnetism

A ferrimagnet is essentially an antiferromagnet, i.e., antiferromagnetic interactions exist between the magnetic moments. However, the neighboring magnetic moments are different, i.e., have different spin lengths. As a result, a net magnetic moment remains even for a perfect antiparallel arrangement of the magnetic moments, and magnetism-wise a ferrimagnet behaves very similar to a ferromagnet. This is hence a neat "trick" to obtain ferromagnetic behavior from antiferromagnetic interactions.

II.3.f. Summary

Type spin structure M(B) χ(T) Diamagnet

Paramagnet

induced moments

Ferromagnet (Ferrimagnet)

Antiferromagnet

M

B

χ

T

M

B

M0

paramagnetic χ

T

paramagnetic 1/χ

T

M

B

M₀ χ

Tc T

paramagnetic ferro-

magnetic 1/χ

T_c T

paramagnetic ferro- magnetic

M

B

χ_||

χ_⊥

χ

T T_N

paramagnetic antiferro-

magnetic 1/χ

T_N T

paramagnetic anti- ferro

0

Type spin structure M(B) χ(T) Diamagnet

Paramagnet

induced moments

Ferromagnet (Ferrimagnet)

Antiferromagnet

M

B

χ

T

M

B

M0

paramagnetic χ

T

paramagnetic 1/χ

T

M

B

M₀ χ

Tc T

paramagnetic ferro-

magnetic 1/χ

T_c T

paramagnetic ferro- magnetic

M

B

χ_||

χ_⊥

χ

T T_N

paramagnetic antiferro-

magnetic 1/χ

T_N T

paramagnetic anti- ferro

0

Comment

Besides the mentioned types of magnets, many more cases are known, such as weak ferromagnets. metamagnets, spiralmagnets, spin glases, cluster glases, and so on. They are characterized ever more complicated magnetic interactions and hence spin structures.

Moleculare Nanomagnets

How would the molecular nanomagnets fit in the above scheme?

They are characterized by localized moments, between which there may be ferro- and/or antiferromagnetic interactions. In this sense they are similar to ferro/ferri/antiferromagnets.

However, because of their small (finite) size (number of magnetic moments), they may not exhibit a magnetic phase transition and long-ranged ordered ground state, which distinguishes them markedly from ferro/ferri/antiferromagnets. However, they are also paramagnets because of the presence of the magnetic interactions.

Although ferro- and/or antiferromagnetic interactions are present in molecular nanomagnets (and are most important for their magnetic behavior), they are NOT ferro/ferri/antiferromagnets.

III. The Spin

III.1. Angular Momenta in Condensed Matter Physics

In condensed matter physics, one gets into touch with angular momentum at several places.

The maybe most important are

orbital angular momentum Lr rr pr

×

=

spin Sr

total angular momentum Jr Lr Sr +

=

nuclear spin rI

Clearly, they all are of very different physical origin. However, as regards their quantum mechanical description they all are similar, since the quantum behavior of an angular momentum is completely specified by the mere fact that the quantity under consideration is an angular momentum. Hence, in the following we need not to be concerned with the physical origins, but will concentrate just on the quantum aspects. We will agree on the following Conventions

1) Unless the physical nature of an angular momentum is relevant, we will consider the spin as example, with the understanding that all statements equally hold for

Sr

Lr , Jr,

Ir , etc.

2) We define , such that angular momentum quantum numbers, e.g. L, S, J, I become dimensionless

r h r S/ S≡

3) Iff appropriate, we will indicate a quantum mechanical operator by an additional hat, e.g. Sˆr Comment

Angular momentum is a vector operator, that is, the symbol Srˆ

is a shorthand notation for a triple of three operators Sˆ_x, Sˆ_y, , which are called the components of the vector.

Sˆz

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

=

z y x

Sˆ Sˆ Sˆ Srˆ

With respect to the components x, y, and z, a vector operator behaves exactly as an ordinary vector. For instance, the scalar product reads Sˆr⋅Br =Sˆ_xB_x +Sˆ_yB_Y +Sˆ_zB_z

(there B could also be a vector operator).

III.2. Fundamental Properties

The following properties of spins (= angular momenta) should be well known from the basic quantum mechanics lectures. They are reviewed here for convenience.

Algebraic Definition

[Sˆ_α,Sˆ_β]=iε_αβγSˆ_γ (α,β,γ = x,y,z)

- 5 - Vorl. #3 (29. Oct. 2010) Consequences

1) The components of the spin cannot simultaneously be diagonal ⇒ traditionally one chooses S_z to be diagonal

Sˆα

2) For the square of the spin holds Sˆ ,Sˆ ] 0 [r² _α =

for all components α ⇒ and can be diagonalized simultaneously

Sˆr2

Sˆz

3) ⇒ the wave function of a spin can be classified by two quantum numbers, one for Sˆr² and one for Sˆ_z

Spin Wave Function (or Spin Function in short)

The spin functions are defined as the eigen functions of and Srˆ²

and Sˆ_z M

, S M M , S

Sˆ_z =

M , S ) 1 S ( S M , S

Srˆ2 = + Rasing and Lowering Operators

y

x iSˆ Sˆ

Sˆ₊ = + Sˆ_x = ₂¹(Sˆ₊ +Sˆ₋)

y

x iSˆ Sˆ

Sˆ₋ = − Sˆ_y = ₂¹_i(Sˆ₊ −Sˆ₋)

1 M , S ) 1 M ( M ) 1 S ( S M , S

Sˆ_± = + − ± ±

Consequences

1) ⇒ the raising and lowering operators are NOT hermitian and hence cannot be observables

Sˆm

) Sˆ ( _± ^* =

2) [Sˆ_z,Sˆ_±]=±Sˆ_±, [Sˆ₊,Sˆ₋]=2Sˆ_z 3) all matrix elements of Sˆ_± are real

4) SMSM cannot be negative ⇒ S(S+1)-M(M±1) ≥ 0 ⇒ |M| ≤ 0 or M = -S, ..., S

⇒ S-(-S) = n or S = n/2 Matrix Representation

Quantum mechanical states and operator are deeply connected to vectors and matrices, and every thing can be expressed in matrix form (especially for finite dimensional Hilbert spaces).

S SS S− S,−S+ M

S Oˆ

SM ′ _S−S

1 S S− +

SS Comment

For any matrix representation it is important to specify a) the basis

b) the order of the basis states

III.3. Magnetic Moment

Since the magnetic moment is related to moving charges, which produce currents, it can physically be expected that a relation between magnetic moment and angular momentum exists. Indeed, for a charged and rotating sphere classical mechanics yields the relation

mc S 2

m q r

r = h Definition

c m 2

e

e B

= h

μ (for electrons, Bohr magneton)

However, in classical mechanics permanent currents and hence magnetic moments cannot exist, but they do in quantum mechanics, with a modification called the g factor. This yields the definition

magnetic moment operator Sˆ g mˆ

B

r =− μ r Comments

1) the g factor depends on the nature of the particle, e.g.

e: g = 2.0023 p: g = 5.59 n: g = -3.83

2) for electronic systems the g factor is in fact "negative" because of the negative charge. This is absorbed by the "-" sign in the above definition. it should be noted, however, that for nuclear particles, for which the proton is so to say the reference particle, the g factors are so to say "positive", i.e., the nuclear magnetic moment is defined as μr =gμ_KrI with a "+" sign.

3) because of the "-" sign, spin and magnetic moment are aligned ANTIPARALLEL to each other!

III.4. Magnetic Moments in a Magnetic Field

(potential) Energy in a Magnetic Field

Following classical electrodynamics, the potential energy of a magnetic moment in a magnetic field is

B m H=−r ⋅r.

Hence, the appropriate Hamiltonian operator to describe the system is B

Sˆ g Hˆ

B

r r⋅ μ

= Comments

1) This energy is just the potential energy in a field, it does not include the energy which is required to cerate the magnetic moment mr and/or magnetic field Br

, nor to maintain the magnetic field strength during a rotation of the magnetic moment. The details of this may be found in electrodynamics text books.

2) This Hamiltonian will be referred to as the Zeeman term.

- 7 - Vorl. #3 (29. Oct. 2010) Energies of H

The above Hamiltonian can trivially be solved or diagonalized, respectively.

The calculations starts from realizing that the system described by it is magnetically isotropic, at is, the energies do NOT depend on the actual direction of the magnetic field in space, they only depend on the angle between magnetic moment and magnetic field. Hence, we may choose the direction of the field at will, and we chose the z direction. Then, for this particular choice, the Hamiltonian simplifies to

B Sˆ g Hˆ

z

μB

=

where it should be noted that now only the z component of the spin operator enters.

Fortunately, we know already the eigen values and eigen functions of Sz, hence we directly obtain the energies

MB g E= μ_B

and the eigen functions as S,M . For the case of a spin with S = 1, the resulting energy spectrum is as this:

E

B M=-1

M=0 M=1

S = 1

Comments

1) The above argumentation for solving the Hamiltonian is VERY IMPORTANT; it is hence worthwhile to understand it well

2) In the ground state the magnetic quantum number M is minimal, i.e., M = -S, which expresses the fact that the magnetic moment mr wishes to align parallel to the magnetic field, but that because of the "-" sign in the definition of the magnetic moment this corresponds to an antiparallel spin orientation.

Molecular Nanomagnets ©Oliver Waldmann

Motion in a Magnetic Field (COMMENT: this part has been deferred until later)

The motion of a magnetic moment in a magnetic field may be discussed by different techniques, e.g., the time-dependent Schrödinger equation. Interestingly, the quantum and classical dynamics are not very different, and the dynamics is very often discussed in terms of the equation of motion, which for the quantum and classical cases look identical. For a spin an exact one-to-one correspondence can in fact be established. The equation of motion is derived from the mechanical torque, which is exerted by a magnetic field on the moment, the equation of motion of an angular moment, and putting everything together.

2 / 1 s=

torque on a magnetic moment rτ=mr ×Br equation of motion of angular moments r =τr

dt S d

B dt m

m

dr r r

γ

×

=

γ: gyro-magnetic ratio, γ=−gμ_B for electrons Comment

For the electrons, the gyro-magnetic ratio is negative.

Because of the cross product, the change with time of the magnetic moment is both perpendicular to the orientation of the magnetic moment and the magnetic field, i.e.,

, m dt / m

dr ⊥ r dmr /dt ⊥γBr

The solution of this equation is known from classical mechanics. For a time-constant magnetic field, the moment exhibits a precession around the magnetic field with a frequency

B gμ_B

=

ω (Larmor frequency)

Importantly, during this precession, the component of the magnetic field parallel to the magnetic field is not changed in time, i.e., is constant

B ⇒

0 dt /

dm_||B = m_||B =const The potential energy of the system is given by

B m B m

E=−r ⋅r =− _||B ⇒ E =const Consequences

1) Any change in energy of the precessing moment is related to a change in or the angle θ between magnetic moment and magnetic field, respectively, and vice versa.

B

m||

2) A change in involves a change in energy, while a change in does not, there is thus a fundamental difference between and . This gives rise to the Bloch equations.

B

m|| m_⊥B

B

m|| m_⊥B

dm/dt m

m_||B

θ

- 2 - Vorl. #4 (5. Nov. 2010) III.5. Thermodynamic Relationships, Magnetization

Quantum Statistics

The appropriate frame work to calculate the thermodynamic properties of a quantum system is given by quantum statistics. Here we will not discuss its justification and/or derivation, but instead just compile some useful equations. The canonic ensemble is used, with the associated thermodynamic potential, the free energy , as it is the temperature T, which is under our control in experiment

) N , V , T ( F

Comment

The thermodynamic potentials and state variables are micro canonical ensemble: E(S,V,N)

canonical ensemble: F(T,V,N) grand canonical ensemble: Ω(T,V,μ)

In the canonical ensemble, the thermodynamic expectation value, i.e., the quantity as it would be measured in an experiment, is given as

thermodynamic expectation value in the energy representation n

T k

1

B

=

∑

=

n

E_n

e n Oˆ Z n

Oˆ 1 ⁼

∑

n E_n

e Z n : n^th eigen function of the Hamilton operator

n: any set of (quantum) numbers which allows to uniquely count all states in the Hilbert space En: energy or expectation value of the Hamiltonian of the n^th eigen state

Z: partition functionβ: inverse temperature Comment

Z P e

En

n β

= − yields the thermodynamic occupation probability of the n^th eigen state.

thermodynamic expectation value in a general basis k

) Pˆ Oˆ ( Tr k e Oˆ Z k

Oˆ 1

k

Hˆ

=

=

∑

k

Hˆ −β

β

− =

=

∑

k : one of the basis functions of the chosen basis

k: any set of (quantum) numbers which allows to uniquely count all states in the basis Comments

1) The operator

Z Pˆ e

Hˆ β

= − is the density operator in thermodynamic equilibrium.

2) The thermodynamic occupation factor of an energy eigen state is then P_n = nPˆn .

Magnetization or Magnetic Moment

The magnetization in the thermodynamic equilibrium is obtained by simply inserting the operator for the magnetic moment in the above equations, and dividing the result by the volume V. However, as discuss in chapter I, it is more convenient to directly consider the magnetic moment directly, which we will always do in theoretical calculations.

magnetic moment operator Sˆ

g mˆ

B

r =− μ r

thermodynamic magnetic moment ^{α = α =− μ}

∑

n

B nSˆ n e En

Z mˆ g

m Definitions

1) mˆ _α is the α component of the magnetic moment operator

2) mˆ_α is the α component of the thermodynamic magnetic moment

3) m_α is the α component of the magnetic moment as it would be measured in experiment Comments

1) In the last equation, the meaning of the vector arrow has been made explicit.

2) The experimentally observed magnetic moment vector would hence formally correspond to the following

⎟⎟

⎟⎟

⎠

⎞

⎜⎜

⎜⎜

⎝

⎛

=

=

z y x

mˆ mˆ mˆ mˆ

mr r

Further Thermodynamic Relations

Useful equations can also be obtained from the thermodynamic potential, from which the measurable quantities are determined as first-order and second-order (in case of susceptibilities) derivatives. In our case, the free energy is a function of temperature and magnetic field, or more precisely, since the magnetic field is a vector and hence involves three quantities, it is a function of temperature and the three components of the magnetic field:

) B , B , B , T ( F

F= _{x y z}

The connection of the free energy to quantum statistics is made through the equation Z

ln T k F=− _B

such that the magnetic moment may alternatively be determined from

α

α ∂

− ∂

= B m F

⇒

∑

α α

α ∂

− ∂

∂ =

∂

=β

n

n E_n

B e E Z 1 B

Z Z m 1

The last equation will often be also very useful for practical calculations.

- 4 - Vorl. #4 (5. Nov. 2010) Comments

1) Sometimes it is useful to also have the following equations in mind, which complete the connection between thermodynamics and quantum statistics

, TS U

F= − U= Hˆ , S=−k_BTr(PˆlnPˆ)

2) In thermodynamics it generally holds that the response, which is related to a (generalized) field is given by the first derivative of the Hamiltonian in this field. In the present case it is revealing to note the consistency of the three equations

α α =∂ ∂ μ Sˆ Hˆ / B

g _B mˆ _α =−∂Hˆ /∂B_α Hˆ =−mrˆ ⋅Br 3) The relation

α α =−∂Hˆ /∂B mˆ

may be considered as the general definition of the appropriate magnetic moment operator for a system under consideration.

The above alternative equation for the magnetic moment is quite useful, also because it shows:

Since the partition function Z is completely determined iff all energy eigen values are know, also all thermodynamic quantities are already completely known if the energy eigen values and their variation with magnetic field, are known. A precise knowledge of the energy eigen functions is, in principle, not required. This can be a great technical advantage in evaluating e.g. the magnetic moment.

) B , B , B (

E_{n x y z}

Comment

Since it is sufficient to know the energy eigen values En, the solution of the time-independent Schrödinger equation Hˆ n E n

= n will be the subject of our efforts for most of the time. The concept of a so-called effective Hamiltonian will often be extremely useful.

Excursion I on the Effective Hamiltonian

As just mentioned, for much of our time it is sufficient to know the energy eigen values.

However, very often we are in fact not interested in knowing ALL eigen values, very often only a subset of them are of relevance for a question at hand. Thus, it would be convenient if it were possible to simplify the problem by throwing out all the unwanted energy eigen states.

this is in fact the idea of the effective Hamiltonian, namely, to construct, by whatever means, an "artificial" Hamiltonian Hˆ such that it reproduces as accurately as possible the energies of the wanted eigen states, but does ignore all the unwanted states from the outset.

Hˆ Hˆ

Comment

Which states are of relevant and which not is generally determined by the experiment which is considered. A typical case could be obtained from the temperature. At low temperatures only the lowest-lying states are thermally populated, and hence of relevance.

IV. Magnetism of an Isolated, Localised Spin

In this section the magnetization of a single, isolated, localized magnetic moment will be calculated and discussed. This is the simplest possible model, and fortunately can be treated analytically. It will hence serve as a yardstick in any discussion of magnetic properties of molecular nanomagnets. The discussion will be developed in three steps, in order to gain a better insight into the resulting behavior.

As always, the system one wishes to study is precisely specified by writing down the corresponding Hamiltonian. For the present case it is nothing else than the Zeeman term

B Sˆ g Hˆ

B

r r⋅ μ

= .

In other words, this Hamiltonian is the precise definition of what is meant by the phrase

"single, isolated, localized magnetic moment".

IV.1. Reminder

Hamilton operator Sˆ B

g Hˆ

B

r r⋅ μ

=

energy eigen values E_M =gμ_BMB energy eigen functions S,M

magnetic moment ^{=− μ}

∑

M

E z

B _M

e M , S Sˆ M , Z S

m g

Sättigung

= C/T

χ ^{T1 < T2} T2

T1

m

B

Comments

1) Qualitatively the behavior of the magnetization and magnetic susceptibility was already discussed previously. It was inferred by realizing that they are governed by the interplay of two mechanisms, namely the disordering by the temperature and the ordering by the magnetic field. Each mechanism may be associated to an energy scale, which here would be kBT for the temperature and μ_BB for the magnetic field.

2) It is may be useful to recall how the above Hamiltonian was solved, in particular how in the solution one took advantage of the fact that the Hamiltonian is isotropic in space

3) Because of isotropy, it is irrelevant which component of the magnetization vector we consider, we hence write m and drop the index α=x,y,z.

IV.2. High-Temperature Expansion

As a first step, the magnetization at high temperatures is calculated, since the so-called high- temperature expansion is in fact a very general concept, and useful for any, even very complicated molecular nanomagnets.

The high-temperature expansion is nothing else than an expansion of the thermodynamic quantities in powers of the inverse temperature β. Technically it may be obtained from the Taylor expansion of the exponential factor

e^−x =1−x+...

- 6 - Vorl. #4 (5. Nov. 2010) Limiting ourselves to the expansion up to first order in β, one obtains the general result

) Hˆ Oˆ ( Tr ) Oˆ ( Tr n Oˆ n E n

Oˆ n e

n Oˆ n

n n n

n

E_n ≈

∑

∑

∑

This may be evaluated for Oˆ =mˆ_z =−gμ_BSˆ_z and Oˆ =1,

3 , ) 1 S 2 )(

1 S ( BS g

BM g g

M E M

g e

M , S mˆ M , S

2 B 2

M

2 B B

M M M

B M

E z

M

+ β +

μ

=

⎟⎠

⎜ ⎞

⎝

⎛β μ μ

⎟=

⎠

⎜ ⎞

⎝

⎛ −β μ

−

≈

∑ ∑ ∑

∑

), 1 S 2 ( M B g ) 1 S 2 ( E

1 e

M B M

M M

M

E_M ⎟= +

⎠

⎜ ⎞

⎝

⎛ + − μ β

⎟=

⎠

⎜ ⎞

⎝

⎛ −β

≈

∑ ∑ ∑

∑

which yields the final result for the magnetization at high temperatures T B

k 3

) 1 S ( S m g

B 2 B 2 T

+

= μ

∞

→ .

Comments

1) At high temperatures, the magnetization will be linear in the magnetic field, i.e., can be described by a magnetic susceptibility.

2) The magnetic susceptibility at high temperatures is given by the Curie law.

3) These two comments do not only hold for isolated spins, but in fact hold exactly for ANY molecular nanomagnet, i.e., the Curie law is in general the first term of the high-temperature expansion.

4) Physically this makes sense, because at high enough temperatures, any additional mechanism, such as magnetic couplings, will be "overruled" by the temperature.

5) The high-temperature expansion may also be developed up to second order in β T ....

C T C

2 + + θ

≈

χ ,

which agrees with the expansion of the Curie-Weiss law up to second order T ....

C T ... C 1 T

T C T

C

Weiss 2

Curie θ+

+

⎟=

⎠

⎜ ⎞

⎝

⎛ θ + + θ ≈

= − χ ₋

Thus, the Curie-Weiss law can be, or should be, understood as an approximation to the high- temperature expansion up to second-order.

Range of validity of the high-temperature expansion

The above derivation was based on the Taylor expansion of the exponential. In the present case it is given as exp

(

)

(

)

(

)

T k

B y g

B

μB

= .

Hence, the range of validity is actually y→0, which may be realized in two ways:

⇔ 0

y→ T→∞ and/or B→0

IV.3. Low-Temperature Approximation

Next, the magnetization at very low temperatures is calculated. Again, this so-called low- temperature approximation is a very general concept and usefully applied to any type of molecular nanomagnets.

It exploits the behavior of the thermal population or Boltzmann factor, respectively, at low temperatures. At very low temperatures, only the lowest-lying states will be thermally populated, which in turn means that the Boltzmann factors for the higher-lying states become negligibly small, such that these states do not have to be considered anymore in the calculation of the thermodynamic sums.

In particular, the zero temperature, or temperatures smaller than an energy gap in the energy spectrum, only the ground state will be thermally populated, such that

( )

e M , S mˆ M ,

S _{B B}

M

E z

M ≈− μ − = μ

∑

1 e

M

E_M ≈

∑

At very low temperature the magnetic moment is hence given by ,

g m_T→0 = μ_BS

which we will call the saturation magnetization.

Comments

1) At very low temperatures, the spins are fully polarized, such that the magnetization is determined simply by maximal magnetic moment of the spin.

2) Physically this makes sense, because at low enough temperatures the disordering tendency of the temperature will not be effective anymore

Range of validity of the low-temperature approximation

The above derivation was based on the behavior of the exponential at low temperatures.

Hence, the range of validity is actually determined as y→∞, which may be realized in two ways:

⇔ and/or

∞

→

y T→0 B→∞

IV.4. Full Calculation: Brillouin Function

For our isolated spin, the magnetic moment may exactly (analytically) be calculated. The calculation starts most conveniently from the equation

B Z Z m 1

∂

∂

=β .

since the partition function, and hence the derivative, may exactly be obtained using the rules for geometrical sum: