5.1 Radicals containing n equivalent nuclei with I = 1/2

The energy diagram and the EPR spectrum of a radical that contains a nucleus with $I=1/2$ and an unpaired electron are showed in Fig. 2.

Both resonance fields ($H_1$ and $H_2$) are deduced from Eq. (5) replacing $m_I$ by $1/2$ and $-1/2$:

$$H_1 (m_I=1/2) = H_0 - a/2 \hspace{0.5cm} and \hspace{0.5cm} H_2(m_I=-1/2) = H_0 + a/2.$$

Figure 2: Energy levels of a radical with a nucleus with I = 1/2 and its corresponding EPR spectrum.

The spectrum consists of two lines of equal intensity (1:1) called doublet. The separation between these two lines of the doublet is the hyperfine splitting a.

Usually in EPR spectroscopy the first derivative of the absorption curve is represented. This is showed in the lower part of Fig. 2.

If a radical contains $n$ equivalent nuclei (identical chemical surrounding) of spin $I$ they will give the same hyperfine splitting (equal value of a); its EPR spectrum will be formed by $2nI + 1$ equidistant lines to the resonance fields:

$$H_k (m_{k,1}, \ m_{k,2},\ \ldots ,\ m_{k,n})\ =\ H_0 - a \sum_{i=1}^{n} m_{k,i}\ = \ H_0 - a M_k$$ (6)

where $m_{k,i}$ are the quantum numbers corresponding to the Z component of the angular momentum of nuclear spin of each nucleus $i$ in the state $k$. It should be noted that we have eliminated the subscript $I$ in $m_I$ to simplify the nomenclature. $M_k$ it is the quantum number corresponding to the total Z component of the angular momentum of nuclear spin. It is the sum of the individual Z components $(m_{k,i})$ of each equivalent nucleus for a certain $k$ state. There are $(2I + 1)^n$ nuclear spin states and, therefore, there are $(2I + 1)^n$ different possibilities for $M_k$. Nevertheless, the EPR spectrum has only $2nI + 1$ resonance fields or $H_k$ lines.

For example, a radical with two equivalent protons (n = 2) has $(2\cdot 1/2 + 1)^2 =4$ possibilities:

$$\begin{array}{llllllllll} H_1 (m_{1,1}=\frac{1}{2},m_{1,2}=\... ...,1}=-\frac{1}{2},m_{4,2}= -\frac{1}{2}) &=& H_0 + a \end{array}$$

However, as we indicated above, there are only $2nI + 1$ different resonance fields, i.e. different lines of EPR; because all lines excepting the first and the last one are degenerated (same position) and their relative intensities increase towards the centre of the spectrum. In the example of two protons there are only three different lines $(2\cdot 2 \cdot 1/2 + 1)$. The resonances $H_2$ and $H_3$ are degenerated and the intensity of this line is double of that of $H_1$ and $H_4$. The spectrum is a triplet of relative intensities 1:2:1 and the separation between two consecutive lines is the hyperfine splitting $a$ (see fig. 3).

The relative intensities for nuclei with spin $I=1/2$ are given by the coefficients of the binomial $(a + b)^n$ that can be obtained easily from the Triangle of Pascal (Table 2); where each number is equal to the sum of the two numbers placed over it. In Table 2, $N$ is the total number of lines and $S$ is the sum of each row of the Triangle of Pascal, $2^n$ (or total number of transitions).

Table 2: Relative intensities for the lines generated by n nuclei with I = 1/2.

n	Relative intensities																	N	Multiplet	S
0									1									1	Singlet	1
1								1		1								2	Doublet	2
2							1		2		1							3	Triplet	4
3						1		3		3		1						4	Quartet	8
4					1		4		6		4		1					5	Quintet	16
5				1		5		10		10		5		1				6	Sextet	32
6			1		6		15		20		15		6		1			7	Septet	64
7		1		7		21		35		35		21		7		1		8	Octet	128
8	1		8		28		56		70		56		28		8		1	9	Nonet	256

Triangle of Pascal $I=1/2$; $N$ (number of lines) $=n+1$; $S$ (number of transitions) $=2^n$.

Figure 3: Energy levels and the EPR spectrum of a radical with two equivalent nuclei with I = 1/2.

The following subsections include, in increasing order of difficulty, a series of EPR spectra of radicals that have n equivalent nuclei (n = 2, 3, $\ldots$, 8) with spin $1/2$.

In all the cases you must perform the following operations:

• Load the simulator program pressing the shown link.
• Measure the hyperfine splitting value in the experimental spectrum (clicking with the mouse in the centre of the Lorenztian lines, the distance in mT between two consecutive clicks will be shown on the screen).
• Use the measured value to simulate the spectrum.
• When you think that the simulation is correct, overlap both spectra (simulated and experimental) and if you detect differences (they do not overlap correctly) refine the hyperfine splitting and/or change the peak to peak linewidth (to see Appendix A).
• Measure the heights (clicking with the mouse) of each line and divide the result by the value of the first line (for the radicals [a006] and [a007] is better to divide by the value of the second line since first one is very small).
• Observe the number of equivalent nuclei and the signals that they generate. Compare the results obtained with the data from Table 2.
• Print out the simulated and experimental spectra with the parameters used for the simulation.

When you finish the simulation, close the window of the simulator to return to the tutorial.