Introduction

1.1 What is ESR Spectroscopy?

Electron spin resonance spectroscopy (ESR), also known as electron paramagnetic resonance (EPR) or electron magnetic resonance (EMR), was invented by the Russian physicist Zavoisky in 1945. It was extended by a group of physicists at Oxford University in the next decade. Reviews of the Oxford group's successes are available and books by Abragam and Bleaney and by Abragam cover the major points discovered by the Oxford group. In the present book, we focus on the spectra of organic and organotransition metal radicals and coordination complexes. Although ESR spectroscopy is supposed to be a mature field with a fully developed theory, there have been some surprises as organometallic problems have explored new domains in ESR parameter space. We will start in this chapter with a synopsis of the fundamentals of ESR spectroscopy. For further details on the theory and practice of ESR spectroscopy, the reader is referred to one of the excellent texts and monographs on ESR spectroscopy. Sources of data and a guide to the literature of ESR up to about 1990 can be found in ref. 16a. The history of ESR has also been described by many of those involved in the founding and development of the field.

The electron spin resonance spectrum of a free radical or coordination complex with one unpaired electron is the simplest of all forms of spectroscopy. The degeneracy of the electron spin states characterized by the quantum number, mS = [+ or -]1/2, is lifted by the application of a magnetic field, and transitions between the spin levels are induced by radiation of the appropriate frequency (Figure 1.1). If unpaired electrons in radicals were indistinguishable from free electrons, the only information content of an ESR spectrum would be the integrated intensity, proportional to the radical concentration. Fortunately, an unpaired electron interacts with its environment, and the details of ESR spectra depend on the nature of those interactions. The arrow in Figure 1.1 shows the transitions induced by 0.315 cm-1 radiation.

Two kinds of environmental interactions are commonly important in the ESR spectrum of a free radical: (i) To the extent that the unpaired electron has residual, or unquenched, orbital angular momentum, the total magnetic moment is different from the spin-only moment (either larger or smaller, depending on how the angular momentum vectors couple). It is customary to lump the orbital and spin angular momenta together in an effective spin and to treat the effect as a shift in the energy of the spin transition. (ii) The electron spin energy levels are split by interaction with nuclear magnetic moments – the nuclear hyperfine interaction. Each nucleus of spin I splits the electron spin levels into (2I + 1) sublevels. Since transitions are observed between sublevels with the same values of mI, nuclear spin splitting of energy levels is mirrored by splitting of the resonance line (Figure 1.2).

1.2 The ESR Experiment

When an electron is placed in a magnetic field, the degeneracy of the electron spin energy levels is lifted as shown in Figure 1.1 and as described by the spin Hamiltonian:

Hs = gμBBSz (1.1)

In eqn (1.1), g is called the g-value (or g-factor), (ge = 2.00232 for a free electron), μB is the Bohr magneton (9.274 × 10-28 J G-1), B is the magnetic field strength in Gauss, and Sz is the z-component of the spin angular momentum operator (the magnetic field defines the z-direction). The electron spin energy levels are easily found by application of Hs to the electron spin eigenfunctions corresponding to mS = ±1/2:

Hs|±1/2) = ±1/2gμBB|±1/2) = E±|±1/2)

Thus

E± = ±(1/2)gμBB (1.2)

The difference in energy between the two levels:

ΔE = E± - E- = gμBB = hw (1.3)

corresponds to the energy, hv, of a photon required to cause a transition; or in wavenumbers by eqn (1.4), where geμB/hc = 0.9348 x 10-4 cm-1 G-1:

v = λ-1 = v/c = (gμB/hc)B (1.4)

Since the g-values of organic and organometallic free radicals are usually in the range 1.9–2.1, the free electron value is a good starting point for describing the experiment.

Magnetic fields of up to ca. 15000 G are easily obtained with an iron-core electromagnet; thus we could use radiation with n~ up to 1.4 cm-1 (v < 42 GHz or λ > 0.71 cm). Radiation with this kind of wavelength is in the microwave region. Microwaves are normally handled using waveguides designed to transmit radiation over a relatively narrow frequency range. Waveguides look like rectangular cross-section pipes with dimensions on the order of the wavelength to be transmitted. As a practical matter for ESR, waveguides can not be too big or too small -1 cm is a bit small and 10 cm a bit large; the most common choice, called X-band microwaves, has λ in the range 3.0–3.3 cm (v ≈ 9–10 GHz); in the middle of X-band, the free electron resonance is found at 3390 G.

Although X-band is by far the most common, ESR spectrometers are available commercially or have been custom built in several frequency ranges (Table 1.1).

1.2.1 Sensitivity

As for any quantum mechanical system interacting with electromagnetic radiation, a photon can induce either absorption or emission. The experiment detects net absorption, i.e., the difference between the number of photons absorbed and the number emitted. Since absorption is proportional to the number of spins in the lower level and emission is proportional to the number of spins in the upper level, net absorption, i.e., absorption intensity, is proportional to the difference:

Net absorption [varies] N_ - N+

The ratio of populations at equilibrium is given by the Boltzmann distribution:

N+/N_ = exp(-ΔE/kT) = exp(-gμB/kT) (1.5)

For ordinary temperatures and ordinary magnetic fields, the exponent is very small and the exponential can be accurately approximated by the expansion, e-x ≈ 1 - x. Thus

N+/N_ ≈ 1 - gμBB/kT

Since N_ ≈ N+ N/2, the population difference can be written:

N_ - N+ = N_ [1 - (1- gμBB/kT)] = NgμB/2kT) (1.6)

This expression tells us that ESR sensitivity (net absorption) increases with the total number of spins, N, with decreasing temperature and with increasing magnetic field strength. Since the field at which absorption occurs is proportional to microwave frequency, in principle sensitivity should be greater for higher frequency K- or Q-band spectrometers than for X-band. However, the K- or Q-band waveguides are smaller, so samples...