CW-EPR Theory

This section of the Bruker EPR webpage is an introduction to the basic theory of continuous wave EPR spectroscopy. It gives you sufficient understanding of the properties which Bruker's EPR spectrometers read and translate into statistical data.

Introduction to Spectroscopy

During the early part of this century, when scientists began to apply the principles of quantum mechanics to describe atoms or molecules, they found that a molecule or atom has discrete (or separate) states, each with a corresponding energy. Spectroscopy is the measurement and interpretation of the energy differences between the atomic or molecular states. With knowledge of these energy differences, you gain insight into the identity, structure, and dynamics of the sample under study. We can measure these energy differences, ΔE, because of an important relationship between ΔE and the absorption of electromagnetic radiation. According to Planck's law, electromagnetic radiation will be absorbed if:

ΔE = hν

equation 1.1

where h is Planck's constant and ν is the frequency of the radiation.

: Figure 1.1: Transition associated with the absorption of electromagnetic energy

The absorption of energy causes a transition from the lower energy state to the higher energy state. (See Figure 1.1) In conventional spectroscopy, ν is varied or swept and the frequencies at which absorption occurs correspond to the energy differences of the states. (We shall see later that EPR differs slightly.) This record is called a spectrum. (See Figure 1.2) Typically, the frequencies vary from the megahertz range for NMR (Nuclear Magnetic Resonance) (AM, FM, and TV transmissions use electromagnetic radiation at these frequencies), through visible light, to ultraviolet light. Radiation in the gigahertz range (the same as in your microwave oven) is used for EPR experiments.

: Figure 1.2: A spectrum

Zeeman Effect

The energy differences we study in EPR spectroscopy are predominately due to the interaction of unpaired electrons in the sample with a magnetic field produced by a magnet in the laboratory. This effect is called the Zeeman effect. Because the electron has a magnetic moment, it acts like a compass or a bar magnet when you place it in a magnetic field, B₀ . It will have a state of lowest energy when the moment of the electron, µ, is aligned with the magnetic field and a state of highest energy when µ is aligned against the magnetic field. (See Figure 1.3.) The two states are labelled by the projection of the electron spin, M_s, on the direction of the magnetic field. Because the electron is a spin ½ particle, the parallel state is designated as M_s = - ½ and the antiparallel state is M s = + ½.

: Figure 1.3: Minimum and maximum energy orientations of µ with respect to the magnetic field B₀

From quantum mechanics, we obtain the most basic equations of EPR:

E = gµ_BB₀MS = ±½gµ_BB₀

equation 1.2

and

ΔE = hν = gµ_BB₀

equation 1.3

g is the g-factor, which is a proportionality constant approximately equal to 2 for most samples, but which varies depending on the electronic configuration of the radical or ion. µ_B is the Bohr magneton, which is the natural unit of electronic magnetic moment.

Two facts are apparent from equations Equation 1.2, Equation 1.3, and their graph in Figure 1.4:

The two spin states have the same energy in the absence of a magnetic field.
The energies of the spin states diverge linearly as the magnetic field increases.

These two facts have important consequences for spectroscopy:

Without a magnetic field, there is no energy difference to measure.
The measured energy difference depends linearly on the magnetic field.

Because we can change the energy differences between the two spin states by varying the magnetic field strength, we have an alternative means to obtain spectra. We could apply a constant magnetic field and scan the frequency of the electromagnetic radiation as in conventional spectroscopy. Alternatively, we could keep the electromagnetic radiation frequency constant and scan the magnetic field. (See Figure 1.4) A peak in the absorption will occur when the magnetic field tunes the two spin states so that their energy difference matches the energy of the radiation. This field is called the field for resonance. Owing to the limitations of microwave electronics, the latter method offers superior performance. This technique is used in all Bruker EPR spectrometers.

: Figure 1.4: Variation of the spin state energies as a function of the applied magnetic field

The field for resonance is not a unique fingerprint for identification of a compound because spectra can be acquired at several different frequencies. The g-factor,

g = hν / (µ_BB₀)

equation 1.4

being independent of the microwave frequency, is much better for that purpose. Notice that high values of g occur at low magnetic fields and vice versa. A list of fields for resonance for a g = 2 signal at microwave frequencies commonly available in EPR spectrometers is presented in table 1.1:

Table 1.1: Field for resonance, B_res, for a g = 2 signal at selected microwave frequencies

Microwave Band	Frequency (GHz)	B_res(G)
L S X Q W	1.1 3.0 9.75 34.0 94.0	392 1070 3480 12000 34000

Hyperfine Interactions

Measurement of g-factors can give us some useful information; however, it does not tell us much about the molecular structure of our sample. Fortunately, the unpaired electron, which gives us the EPR spectrum, is very sensitive to its local surroundings. The nuclei of the atoms in a molecule or complex often have a magnetic moment, which produces a local magnetic field at the electron. The interaction between the electron and the nuclei is called the hyperfine interaction. It gives us a wealth of information about our sample such as the identity and number of atoms which make up a molecule or complex as well as their distances from the unpaired electron.

: Figure 1.5: Local magnetic field at the electron, B₁, due to a nearby nucleus

Figure 1.5 depicts the origin of the hyperfine interaction. The magnetic moment of the nucleus acts like a bar magnet (albeit a weaker magnet than the electron) and produces a magnetic field at the electron, B₁. This magnetic field opposes or adds to the magnetic field from the laboratory magnet, depending on the alignment of the moment of the nucleus. When B₁ adds to the magnetic field, we need less magnetic field from our laboratory magnet and therefore the field for resonance is lowered by B₁. The opposite is true when B₁ opposes the laboratory field.

For a spin ½ nucleus such as a hydrogen nucleus, we observe that our single EPR absorption signal splits into two signals which are each B₁ away from the original signal. (See figure 1.6.)

: Figure 1.6: Splitting in an EPR signal due to the local magnetic field of a nearby nucleus

If there is a second nucleus, each of the signals is further split into a pair, resulting in four signals. For N spin 1/2 nuclei, we will generally observe 2^N EPR signals. As the number of nuclei gets larger, the number of signals increases exponentially. Sometimes there are so many signals that they overlap and we only observe the one broad signal.

Signal Intensity

: Figure 1.7: integrated intensity of absorption signals. both signals have same intensities

So far, we have concerned ourselves with where the EPR signal is, but the size of the EPR signal is also important if we want to measure the concentration of the EPR active species in our sample. In the language of spectroscopy, the size of a signal is defined as the integrated intensity, i.e, the area beneath the absorption curve. (See figure 1.7) The integrated intensity of an EPR signal is proportional to the concentration.

Signal intensities do not depend solely on concentrations. They also depend on the microwave power. If you do not use too much microwave power, the signal intensity grows as the square root of the power. At higher power levels, the signal diminishes as well as broadens with increasing microwave power levels. This effect is called saturation. If you want to measure accurate linewidths, lineshapes, and closely spaced hyperfine splittings, you should avoid saturation by using low microwave power. A quick means of checking for the absence of saturation is to decrease the microwave power and verify that the signal intensity also decreases by the square root of the microwave power.