ESR Spectroscopy

with m_N now being the nuclear Bohr magneton:

much smaller value because of the ~2000 times more heavier proton m_p. The nuclear g factor, g_N, is obtained from a knowledge of the structure of the nucleus. Interaction with external field splits the nuclear sublevels due to Zeeman interaction the same way as for electron spin:

where m_I is one of the 2I + 1 projections of the nuclear spin. Electron-nuclear interaction will depend on the projections of both, electron and nuclear spins:

where coefficient A, a so-called hyperfine coupling constant/interaction (hfi), depends not only on the g-values for the electron and the nucleus but also on the distance between them and their orientation with respect to the external field (dipole-dipole interaction). In solutions, the anisotropic part of this interaction averages out because of the fast molecular rotation. The remaining isotropic part is given by the Fermi contact interaction in the form:

where r(0) = |y(0)|² is the unpaired electron density at the nucleus. For pure 1s electron on hydrogen atom, A equals h x 1420 MHz. As a matter of convinience, hyperfine constants are usually given in Gauss, a = A/g_em_B. For example, hydrogen it corresponds to a = 508 G. Summing up all energies, we arrive at modified Eq. (1):

You should note two important things:

• the signs in front of g_e and g_N are different because of the opposite charges of the electron and the proton, causing a spin of the electron to be higher in energy than b, but b_N spin of the proton to be higher in energy than a_N
• the magnetic field on the electron differs from the B_o by a integer times a_i and the sign of this cange is opposite for a and b electron spins

Hydrogen atom example is given on the right. In EPR, allowed transitions correspond to the change of only m_s:

i.e. electronic spin "flips" but the nuclear one remains unchanged. Thus, in the figure on the right for hydrogen atom, only two transitions can be observed for a single resonant frequency hn. (This is a typical mode of EPR signal detection: microwave frequency is kept constant and magnetic field is varied). These transitions are separated by the hyperfine constant a (508 G for hydrogen atom). If there are more than one hydrogens, each of them will

contribute to the change in magnetic field on the electron.

All combinations of the nuclear spin projections should be included but (EPR allowed) transitions only between sublevels with the same configurstions of nuclear spins will be realized. For the case of four equivalent protons, a simple ladder propagation scheme (Figure on the left) illustrates how to construct the EPR spectrum. Each new step corresponds to one more proton added. In the first step, two lines appear corresponding to the two different orientations of m_I = -1/2 and m_I = + 1/2. In the second step, each line splits into two, corresponding to the two different orientations of m_I = -1/2 and m_I = + 1/2, of the second proton.Since the two protons are equivalent, the splitting is the same on each step, resulting in three lines of 1:2:1 intesity ratio. For the four protons, there will be 5 lines with the 1:4:6:4:1 intensity ratio. Lines with the lengths of these ratio separated by appropriate constants are called to represent a stick diagram for the radical.

If the spin of the nucleus is greater than 1/2, as in the case of ¹⁴N ( I = 1), it splits each of the electronic levels, a and b, into the 2I + 1 sublevels resulting in 2I + 1 obeserved lines of equal intensity. In most cases, one observes a combination of different nuclei with different constants which sometimes is not as easy to interpret.

Example of a radical with two different groups of hfi constants and different nuclei is shown on the right for a radical with one nitrogen and two equivalent hydrogens (e.g. NH₂).

The magnitude of the hyperfine constants in a radical or radical ion reflects the extent of s character in its orbital (only s-orbital has nonzero density at the nucleus). Sometimes it is quite tricky how s -orbital gets populated. For example, we think of aromatic ion radicals as being purely p-systems, as you will see for yourselves, there are substantial hfi constants on there hydrogens. McConnell⁶ showed that a aprt of the unpaired p-electron density is transferred through the C-H sigma bonding electrons to the H nucleus via exchange interactions. He provided a simple formular which links the a_H on a hydrogen with the unpaired electron spin density on the nearest carbon atom r_p:

The signal intensity in EPR depends on how much absorption of the microwave power takes place. This value will be the larger the greater is the difference between populations of the a and b electron spin states. Following Boltzmann statistics, the latter can be estimated :

a small value for regular X-band EPR spectrometer (9.5 GHz), which can be substanially increased by lowering the temperatute.