128 INSTRUMENTAL METHODS
magnetic moment. The interactions may be of two types: (1) the hyperfi ne
interaction — the unpaired electron and the nucleus having a nuclear spin
belong to the same atom; and (2) the superhyperfi ne interaction — the unpaired
electron and the nucleus having a nuclear spin belong to different atoms in
the molecule.
The resonance expression relating the magnetic fi eld and energy (fre-
quency) as seen in equation 3.40 , ΔE = hν = gβBR , has its analog for hyperfi ne
interactions written as equation 3.43 , where A is the hyperfi ne coupling con-
stant in Hz anda is the hyperfi ne splitting constant — that is, the distance
between the split lines in the EPR spectrum:
hA=βg a (3.43)
For g equal to 2, A will equal 2.8 a (MHz). Usually A is expressed in reciprocal
centimeters using the relation that 30 GHz = 1 cm − 1. For g = 2, a hyperfi ne
splitting of 10 mT can be expressed as 280 MHz or 0.0093 cm − 1. (Use β =
9.2741 × 10 − 21 erg G − 1 , h = 6.626 × 10 − 27 erg s, 1 G = 10 − 4 T to calculate that
10 mT = 0.0093 cm − 1 .) Provided that the z direction is parallel to the magnetic
fi eld, one can write
EB=−μzL (3.44)
Assuming that BHF (where HF = hyperfi ne) is much smaller than BL , one can
write equation 3.45 from equations 3.34, 3.38, and 3.44. The constant h is often
omitted in this expression.
EB=−μβzL()−amI =gmBAhmmsL+ ()sI (3.45)
where ms is electron spin and mI is nuclear spin.
The hyperfi ne interaction is shown in Figure 21 of reference 29. The Ms = ± 1/2
states of anS = 1/2 paramagnet interact with an I = 1/2 nuclear moment to
create the hyperfi ne interaction. Interactions from ms = − 1/2 to MI = − 1/2 and
ms = − 1/2 MI = +1/2, for instance, create the magnetic fi eld specifi ed as the
hyperfi ne interaction A. Figure 21 of reference 29 describes the behavior for
anI = 1/2 nuclear moment. The number of hyperfi ne lines will be equal to 2 I + 1
for nuclear moments greater than 1/2. Each hyperfi ne line will be of equal
intensity when the electron is interacting with its own nucleus. For instance,
the Cu 2+ , I = 3/2 nucleus will produce four hyperfi ne lines as described in
Section 3.5.4.
The superhyperfi ne interaction is observed for metal complexes in cases
where the metal ligands have a nuclear moment. For instance, the nitrosyl (NO,
with an unpaired electron) complexes of iron(II) heme proteins have two
inequivalent axial nitrogen ligands. The^14 N( I = 1) nucleus of one ligand couples
strongly to NO ’ s unpaired electron, yielding a widely split triplet with each
component of equal intensity and separated by 2.1 mT. The second nitrogen
of the histidine proximal ligand (see Figures 7.4 and 7.7 in Section 7.2) shows
weaker coupling with the splitting about 0.7 mT. Diffi culties arise in trying to