Electron paramagnetic resonance

Electron paramagnetic resonance (discovered in 1945 [25]) is largely based on similar principles to NMR, except that it is electron spin transitions rather than nuclear transitions that are observed. There are also considerable experimental differences in actually carrying out the two spectroscopic techniques. Only a very brief summary of the relevant theoretical framework will be given here, with more detailed accounts of EPR theory available in References [26], [27], [28] and [29].

2.2.1 Electron spin and the Zeeman effect

The electron has a spin angular momentum S = 1/2, the magnitude of which is quantised in units of ¯h and given by pS(S+ 1)¯h. Similarly to nuclear spin, the z-component of the electron spinSz =mSh¯ can be in one of two states,mS =±1/2.

Associated with each electron spin, is a magnetic moment

µe=−gµBS (2.24)

where g is a dimensionless quantity known as the g-factor and µB =|e|h/4πme is

a constant called the Bohr magneton, here defined to be a positive quantity (me is

the electron mass).

In a magnetic field of strengthB0 (conventionally defined in thez-direction),

an electron spin will have an energy of

E =mSgµBB0=±

1

2gµBB0 (2.25)

and the degeneracy of the spin states will be lifted, resulting in the splitting of the energy level into two (the Zeeman effect) with an energy difference that gives the resonance condition for EPR:

∆E =gµBB0 =hν (2.26)

whereν is the frequency of electromagnetic field (usually in the microwave region) used to induce transitions.

2.2.2 The g-factor

The magnetic field actually experienced by each spin will differ from the externally appliedB0 field; there will also be a contribution from fields induced by (and hence dependent on the strength of)B0. This can be accounted for by replacing the field

in Equation 2.26 with an effective field:

Beff =B0+Blocal (2.27)

where Blocal is the induced field at the electron. However, it is more practical to

continue using the externally applied field B0 and use an effective g-factor g that varies from the g-factor of a free electronge:

Beff= (1−σS)B0 (2.28)

where σS is a term analogous to the chemical shielding constant σ in NMR (see

Section 2.1.6).

2.2.3 Hyperfine interaction

Measuring the g-factor provides useful information about electronic structure, but insight into the molecular structure of the sample comes from the hyperfine inter- action between the electron spin and nuclear spins in the molecule. The hyperfine interaction can be split into two contributions: a Fermi contact interaction, which accounts for the hyperfine field in the region inside the nucleus and is independent of direction; and dipole-dipole interaction in the region outside the nucleus whose strength has a 1/r3 dependence (whereris the electron-nuclear distance) and is also orientation dependent. In liquid samples, as considered in this thesis, the dipole- dipole contribution is averaged to zero by rapid molecular tumbling. So only the Fermi contact contribution needs to be considered.

The strength of the hyperfine interaction is characterised by the hyperfine coupling constant

a= 8π

3 gµBgNµN|ψ(0)|

2 _(2.29)

where the EPR notationgNµN is equivalent to the NMR notation γ¯h and |ψ(0)| is

the wavefunction describing the motion of the electron calculated at the nucleus. In the strong field approximation (|a| gµBB0), the energy of the electron spin is

given by:

E =gµBB0mS−gNµNB0mI+amSmI (2.30)

where the first and second terms on the right-hand side are the electron and nuclear Zeeman terms, respectively [27].

This interaction gives rise to the (hyperfine) splitting of EPR lines into 2I+1 components when the electron interacts with a nucleus of spinI. For example, if an

Figure 2.3: Energy levels resulting from the Zeeman effect and hyperfine splitting for an S = 1/2 electron and I = 1 nucleus. The 3 allowed EPR transitions are indicated by the blue arrows, corresponding to an energy gaphν.

electron withS = 1/2 is hyperfine coupled to a nucleus, such as14N, withI = 1 then there are 6 possible energy states, corresponding to mS =±1/2 and mI =−1,0,1.

The following selection rules then apply:

∆mS=±1, ∆mI= 0 (2.31)

for EPR transitions, and

∆mS= 0, ∆mI=±1 (2.32)

for NMR transitions. The EPR selection rule (Equation 2.31) is for allowed tran- sitions which apply in the strong field approximation, as already mentioned. If, however, the hyperfine interaction becomes very large and|a| gµBB0 is no longer

the case, so-called forbidden transitions (∆mS = ±1,∆mI = ±1) are observable

(though small compared to the allowed transitions). The hyperfine splitting and al- lowed EPR transitions for aS= 1/2,I = 1 coupled system are shown in Figure 2.3.

2.2.4 Electron-spin exchange

The rapid tumbling motion of radicals in low viscosity solutions results in the aver- aging out of anisotropies and results in narrow lineshapes. For the case considered in Figure 2.3, for example, three well-resolved hyperfine lines can be observed in dilute solution. However, as the radical concentration is increased, a quantum me- chanical effect known as electron-spin exchange broadens the hyperfine lines as the unpaired electrons on two different molecules swap spin orientations. Eventually this broadening results in the lines collapsing into a single broad resonance. Upon further increase of the radical concentration the line will begin to narrow (exchange- narrowing) since the spin exchange is occurring on such a rapid timescale that the average hyperfine field approaches zero [26].

2.2.5 Detection

In EPR spectrometers a resonant cavity (analogous to a coil in NMR) is used to amplify the sample signals. The efficiency of this cavity is expressed as its quality factor

Q= 2πEsto Edis