EPR of Exchange Coupled Systems

Alessandro Bencini, Dante Gatteschi, Dante Gatteschi

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486310329
• Topic: Naturwissenschaften
• Subtopic: Chemie

From chemistry to solid state physics to biology, the applications of Electron Paramagnetic Resonance (EPR) are relevant to many areas. This unified treatment is based on the spin Hamiltonian approach and makes extensive use of irreducible tensor techniques to analyze systems in which two or more spins are magnetically coupled. This edition contains a new Introduction by coauthor Dante Gatteschi, a pioneer and scholar of molecular magnetism.
The first two chapters review the foundations of exchange interactions, followed by examinations of the spectra of pairs and clusters, relaxation in oligonuclear species, approaches to infinite lattices, and how EPR can provide firsthand information on spin dynamics. Subsequent chapters explore experimental data, magnetically coupled systems, low-dimensional materials, and the use of EPR to characterize excitons and exciton motion. More than 200 figures and tables appear throughout the book, which concludes with a pair of helpful appendices.

1 Exchange and Superexchange

1.1 The Exchange Interaction

The essential fundament of the exchange (or the superexchange) interaction is the formation of a weak bond. It is well known that spin pairing characterizes bond formation: two isolated hydrogen atoms have a spin S = 1/2 each, but when they couple to form a molecule, H₂, the result is a spin singlet state, because the two electrons must pair their spins to obey the Pauli principle. If the bond is strong enough, the possibility of having the two electrons with parallel spins is very low, and the triplet state has a much higher energy than the singlet (

, the singlet-triplet separation, is much larger than kT at room temperature). However, if the bonding interaction is weak, the singlet-triplet energy separation becomes smaller, and eventually of the same order of magnitude as kT. It must be recalled here that although the exchange interaction is a bond interaction, therefore, acting only on the orbital coordinates of the electrons, the spin coordinates are extremely useful for the characterization of the wave functions of the pair. In fact, the Pauli principle imposes that the complete wave function of a system is antisymmetric with respect to the exchange of electrons: in the above example of the hydrogen molecule the symmetric orbital function must be coupled to the antisymmetric spin singlet function, and the antisymmetric orbital function is coupled to the symmetric spin triplet. Therefore, spins act as indicators of the nature of the orbital states.

When the two centers in the pair have individual spins S_i different from

, as can occur when the number of unpaired electrons is larger than one, the states of the pair are classified by the total spin quantum number S defined by the angular momentum addition rules:

The exchange regime occurs when the interaction between two species, characterized by individual spins S₁ and S₂ before turning on the coupling, yields a number of levels characterized by different total spins, neglecting relativistic effects, which are thermally populated within the normal range of temperatures.

With regards to the intuitively simple example of two identical species with S_i = 1/2 (one unpaired electron on each noninteracting species), three cases can occur in the limit of weak interaction. When the interaction is vanishingly small, the two spins are completely uncorrected and the two centers can be described by their individual spin quantum numbers. A simple way of determining whether this situation holds is through measurements of the magnetic susceptibility which must be the sum of the individual susceptibilities. In principle, EPR as well can be used to this purpose, and one should observe the spectra of the individual spins. However, EPR is a much more sensitive technique than static magnetic susceptibility measurements, and even residual interactions, including magnetic dipolar interactions, as small as a fraction of wave number, can be enough to yield spectra very different from the spectra of the individual spins. In other words, EPR moves the limit for vanishingly small interaction to much lower energy than in the case of magnetic susceptibility. Indeed, for the latter the limit is always of the order of kT, and unless extremely low temperatures are reached, it cannot become much smaller than 1 cm⁻¹. EPR can easily detect interactions of 10⁻²–10⁻³ cm⁻¹ even at room temperature. Even two spins as far apart as 1000 pm can be found to be interacting by the EPR technique.

The second limiting case occurs when the two spins are coupled in such a way that the singlet is the ground state and the triplet is thermally populated. In this case the coupling is said to be antiferromagnetic.

The third limiting case occurs when the triplet is the ground state and the singlet is thermally populated. In this case the coupling is said to be ferromagnetic.

When the two individual spins have S_i

1/2, the situation is similar: the antiferromagnetic case is obtained when S = |S₁ — S₂| is the ground state, and the ferromagnetic case is obtained when S = S₁ + S₂ has the lowest energy. A simple picture of the three limiting cases for S₁ = S₂ = 1/2 is shown in Fig. 1.1.

The exchange regime can be rarely obtained when two paramagnetic atoms are directly bound, but generally this situation is found in more complex molecules. A rather common case is that of two paramagnetic metal ions which are bridged by some intervening, formally diamagnetic, atoms. A relevant example is shown in Fig. 1.2. The two copper(II) ions, which have a ground d⁹ configuration, and one unpaired electron each, are bridged by one oxalato ion. It has been found experimentally [1.1] that the Cu(ox)Cu moiety has a ground singlet and an excited triplet at ≈ 385 cm⁻¹. Since the copper-copper distance, > 500 pm, is too long to justify any direct overlap between the two metal ions, it must be concluded that the diamagnetic oxalato ion is effectively transmitting the exchange interaction. This situation, in which the paramagnetic centers are coupled through intervening diamagnetic atoms, or groups of atoms, is referred to as superexchange.

In order to put all the above qualitative conclusions on a more quantitative basis it is necessary to resort to some model for the description of the chemical bond intervening between the two individual species. The first successful attempt in this direction was made by Anderson [1.2], who used a Valence Bo...

Table of contents

• Cover
• Title Page
• Copyright Page
• Forward
• Preface
• Contents
• 1 Exchange and Superexchange
• 2 Spin Hamiltonians
• 3 Spectra of Pairs
• 4 Spectra of Clusters
• 5 Relaxation in Oligonuclear Species
• 6 Spectra in Extended Lattices
• 7 Selected Examples of Spectra of Pairs
• 8 Coupled Transition-Metal Ions-Organic Radicals
• 9 Biological Systems
• 10 Low Dimensional Materials
• 11 Excitons
• Appendix A. Second Quantization
• Appendix B. Properties of Angular Momentum Operators and Elements of Irreducible Tensor Algebra.
• Subject Index