4 Organic Radicals

Most molecules have all their electrons with paired spins $(\uparrow \downarrow)$, therefore, the resulting spin and the magnetic moment are zero and they have no application in EPR. Nevertheless, there are an important number of systems with unpaired electrons that can be studied by Electron Paramagnetic Resonance (EPR) also called Electron Spin Resonance (ESR). In this tutorial we will learn how to analyse first order EPR spectra of organic free radicals in dissolution. A brief description of the effects of second order is presented in the appendix B.

Free radicals have a magnetic moment and yield symmetrical spectra formed by a series of lines. The characteristic disposition of these lines, denominated ''hyperfine structure'', arises from the interaction between the magnetic moment of the unpaired electron with the neighbouring atomic nuclei with nonzero spin ($^{1}$H, $^{14}$N, $^{2}$H, $^{13}$C, $\ldots$).

Most of the nuclei have magnetic moment associated with the spin. The spin of the nucleus is characterised by the quantum number $I$, which can take values of 0, 1/2, 1, 3/2, $\ldots$ depending on the nucleus (to see Table 1).

A nucleus with spin $I$ has $2I + 1$ substates designated by the magnetic quantum number $m_I$, $(m_I = I$, $I-1$, $I-2$, $\ldots$, $-I)$. These substates correspond to different orientations of the nuclear moment in an external magnetic field.

In short, the electron and some nuclei behave like magnetic dipoles in the presence of an external magnetic field.

If a radical contains a nucleus with spin $I \ne 0$, there will be an interaction between the magnetic moment of the unpaired electron with the local magnetic field generated by the nuclear magnetic moment. Since there are $(2I + 1)$ possible values of $m_I$, there will be $(2I + 1)$ possible values of local field. For each value of $m_I$ there will be $2I + 1$ energy levels with a small separation, Fig. 2. If the external field is much more intense that the local fields and the radicals are dissolved in solvents of low viscosity in order to rotate quickly, the EPR transitions are obtained by following the ''resonance condition'':

$$h \nu = g \cdot \beta \cdot H + a' m_I$$ (4)

where: $a' m_I$ is the contribution of the local field and g is the factor g of the radical.

Table 1: Properties of some isotopes

Isotope	Natural	I	Isotope	Natural	I
	abundance (%)			abundance (%)
$^{1}$H	99.98	1/2	$^{19}$F	100.0	1/2
$^{2}$H	0.015	1	$^{35}$Cl	75.8	3/2
$^{12}$C	98.9	0	$^{23}$Na	100.0	3/2
$^{13}$C	1.1	1/2	$^{39}$K	93.1	3/2
$^{16}$O	99.8	0	$^{31}$P	100.0	1/2
$^{14}$N	99.6	1	$^{32}$S	95.0	0

If the microwave radiation is fixed in $\nu_0$, there will be in the spectrum $2I + 1$ equidistant lines of equal intensity (or height). Their position in militesla (mT) is given by:

$$H (m_I) = H_0 - a m_I$$ (5)

where $H_0 = h \nu_0 /g \beta$ is the resonance field without magnetic nuclei and $a = a'/g \beta$ is the hyperfine splitting (hfs). The constant $a$ (in mT) can be obtained with high precision from the spectrum and is very useful in identifying the radical.