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 are also necessarily smaller and for the same concentration contain fewer spins. This usually more than cancels the advantage of a more favorable Boltzmann factor for samples of unlimited size or fixed concentration.

Under ideal conditions, a commercial X-band spectrometer can detect about 1012 spins (ca. 10-12 moles) at room temperature. This number of spins in a 1 cm3 sample corresponds to a concentration of about 10-9 M. By ideal conditions, we mean a single line, on the order of 0.1 G wide, with sensitivity going down roughly as the reciprocal square of the line width. When the resonance is split into two or more hyperfine lines, sensitivity decreases still further. Nonetheless, ESR is a remarkably sensitive technique, especially compared with NMR.

1.2.2 Saturation

Because the two spin levels are affected primarily by magnetic forces, which are weaker than the electric forces responsible for most other types of spectroscopy, once the populations are disturbed by radiation it takes longer for equilibrium population differences to be established. Therefore an intense radiation field, which tends to equalize the populations, leads to a decrease in net absorption which is not instantly restored once the radiation is removed. This effect is called ''saturation''. The return of the spin system to thermal equilibrium, via energy transfer to the surroundings, is a rate process called spin–lattice relaxation, with a characteristic time (T1), the spin–lattice relaxation time (relaxation rate constant = 1/T1). Systems with a long T1 (i.e., pin systems weakly coupled to the surroundings) will be easily saturated; those with shorter T1 will be more difficult to saturate. Since spin–orbit coupling provides an important energy transfer mechanism, we usually find that odd-electron species with light atoms (e.g., organic radicals) have long T1s, those with heavier atoms (e.g., organotransition metal radicals) have shorter T1s. The effect of saturation is considered in more detail in Chapter 5, where the phenomenological Bloch equations are introduced.

1.2.3 Nuclear Hyperfine Interaction

When one or more magnetic nuclei interact with the unpaired electron, we have another perturbation of the electron energy, i.e., another term in the spin Hamiltonian:

Hs = gμBBSz + AI x S (1.7)

where A is the hyperfine coupling parameter in energy units (joule). Strictly speaking we should include the nuclear Zeeman interaction, γBIz. However, in most cases the energy contributions are negligible on the ESR energy scale, and, since observed transitions are between levels with the same values of mI, the nuclear Zeeman energies cancel in computing ESR transition energies.

The eigenfunctions of the spin Hamiltonian [eqn (1.7)] are expressed in terms of an electron- and nuclear-spin basis set |mSmI), corresponding to the electron and nuclear spin quantum numbers mS and mI, respectively. The energy eigenvalues of eqn (1.7) are:

E(1/2, 1/2) = 1/2gμBB + 1/4A (1.8A)

E(-1/2, 1/2) = -1/2gμBB + 1/4A (1.8B)

E+(1/2, 1/2) = 1/2gμBB [1 + (A/gμBB)2]1/2 - 1/4 A (1.8C)

E_ = = -1/2gμBB[1 + (A/gμBB)2]1/2 - 1/4 A (1.8D)

The eigenfunctions corresponding the E+ and E_ are mixtures of |1/2; -1/2) and |-1/2, 1/2).

If the hyperfine coupling is sufficiently small, A < < gμBB, the second term in brackets in eqns (1.8C) and (1.8D) are negligible, which corresponds to first-order in perturbation theory, and the energies become:

E = [+ or -]1/2gμBB [+ or -] 1/2A (1.9)

These are the energy levels shown in Figure 1.2. The exact energies in eqn. (1.8), which were first derived by Breit and Rabi, are plotted as functions of B in Figure 1.3 for g = 2.00, A/hc = 0.1 cm-1. Notice that, at zero field, there are two levels corresponding to a spin singlet (E = -3A/4hc) and a triplet (E = +A/4hc). At high field, the four levels divide into two higher levels (mS = +1/2) and two lower levels (mS = -1/2) and approach Figure 1.2, the first-order result, eqn. (1.9) (the first-order solution is called the high-field approximation). To conserve angular momentum, transitions among these levels can involve changes in angular momentum of only one unit. At high fields this corresponds to flipping only one spin at a time; in other words, the selection rules are ΔmS = [+ or -]1, ΔmI = 0 (ESR transitions) or ΔmS = 0, ΔmI = [+ or -] (NMR transitions). The latter involves much lower energy photons, and, in practice, only the ΔmI = [+ or -] transitions are observed in an ESR spectrometer. At lower fields, or when A becomes comparable in magnitude to gμBB, the transitions may involve simultaneous flipping of electron and nuclear spins. This gives rise to second- order shifts in ESR spectra (see Chapters 2 and 3).

1.3 Operation of an ESR Spectrometer

Although many spectrometer designs have been produced over the years, the vast majority of laboratory instruments are based on the simplified block diagram shown in Figure 1.4. Plane-polarized microwaves are generated by the klystron tube and the power level adjusted with the Attenuator. The Circulator behaves like a traffic circle: microwaves entering from the klystron are routed toward the Cavity where the sample is mounted. Microwaves reflected back from the Cavity (which is reduced when power is being absorbed) are routed to the Diode Detector, and any power reflected from the diode is absorbed completely by the Load. The diode is mounted along the E-vector of the plane-polarized microwaves and thus produces a current proportional to the microwave power reflected from the cavity. Thus, in principle, the absorption of microwaves by the sample could be detected by noting a decrease in current in the Microammeter. In practice, of course, such a measurement would detect noise at all frequencies as well as signal and have a far too low signal-to-noise ratio to be useful.

The solution to the signal-to-noise problem is to introduce small amplitude field modulation. An oscillating magnetic field is superimposed on the dc field by means of small coils, usually built into the cavity walls. When the field is in the vicinity of a resonance line, it is swept back and forth through part of the line, leading to an a.c. component in the diode current. This a.c. component is amplified using a frequency selective amplifier tuned to the modulation frequency, thus eliminating a great deal of noise. The modulation amplitude is normally less than the line width. Thus the detected a.c. signal is proportional to the change in sample absorption as the field is swept. As shown in Figure 1.5, this amounts to detection of the first derivative of the absorption curve.

It takes a little practice to get used to looking at first-derivative spectra, but there is a distinct advantage: first-derivative spectra have much better apparent resolution than do absorption spectra. Indeed, second-derivative spectra are even better resolved (though the signal-to-noise ratio decreases on further differentiation). Figure 1.6 shows the effect of higher derivatives on the resolution of a 1:2:1 triplet arising from the interaction of an electron with two equivalent I = 1/2 nuclei.

The microwave-generating klystron tube requires explanation. A schematic drawing of the klystron is shown in Figure 1.7. There are three electrodes: a heated cathode from which electrons are emitted, an anode to collect the electrons, and a highly negative reflector electrode that sends those electrons which pass through a hole in the anode back to the anode. The motion of the charged electrons from the hole in the anode to the reflector and back to the anode generates an oscillating electric field and thus electromagnetic radiation.

The transit time from the hole to the reflector and back again corresponds to the period of oscillation (v). Thus the microwave frequency can be tuned (over a small range) by adjusting the physical distance between the anode and the reflector or by adjusting the reflector voltage. In practice, both methods are used: the metal tube is distorted mechanically to adjust the distance (a coarse frequency adjustment) and the reflector voltage is adjusted as a fine control.

The sample is mounted in the microwave cavity (Figure 1.8). The cavity is a rectangular metal box, exactly one wavelength long. An X-band cavity has dimensions of about 1 × 2 × 3 cm. The electric and magnetic fields of the standing wave are shown in the figure. Note that the sample is mounted in the electric field nodal plane, but at a maximum in the magnetic field. The static field, B, is perpendicular to the sample port.

The cavity length is not adjustable, but it must be exactly one wavelength. Thus the spectrometer must be tuned such that the klystron frequency is equal to the cavity resonant frequency. The tune-up procedure usually includes observing the klystron power mode. That is, the frequency is swept across a range that includes the cavity resonance by sweeping the klystron reflector voltage, and the diode detector current is plotted on an oscilloscope or other device. When the klystron frequency is close to the cavity resonance, microwave energy is absorbed by the cavity and the power reflected from the cavity to the diode is minimized, resulting in a dip in the power mode (Figure 1.9). The "cavity dip" is centered on the power mode using the coarse mechanical frequency adjustment, while the reflector voltage is used to fine tune the frequency.