Theory
Energy Level Structure, g-factor and
hyperfine interaction
In
EPR, because of the interaction of the unpaired electron spin moment (given by
two projections, ms = � 1/2, for a free
electron) with the magnetic field, the so-called Zeeman effect, different
projections of the spin gain different energies, as shown below and on the
figure to the right:
Here Bo is the field strength of the
external magnetic field. The SI units for magnetic field is tesla,T, but,
historically in EPR, gauss (1 G = 0.0001 T) is still used. Other terms in Eq.(1):
ms - is a spin projection on the field (ms
= � 1/2 for a free electron), mB
is the Bohr magneton:
| |
mB
= |eh/ 4pme|=
9.2740 x 10-24 J/T |
(2) |
with e and me beeing
electron charge and mass, respectively, and h-Planck's constant.
Parameter g for free electron, ge, has the
value close to two: ge = 2.0023193. If the electron
has nonzero orbiatl angular moment, L, the g-value (sometimes
called factor Land�) becomes:
|
g = 1 + |
S(S + 1) - L(L + 1) + J(J + 1) |
(3) |
|
2J(J + 1) |
The overall magnetic momentun, meff
, can be expessed via overall angular momentum, J, and the
g-value:
| |
meff
= gmB[J(J
+ 1)]1/2 |
(4) |
For most of organic radicals and radical ions, unpaired electrons have L
close to zero and the total electron angular momentum quantum number J
is pretty much the spin quantum number, S. As result, their g-values
are close to 2. Situation becomes much more complicated with transition metals.
Not only they have large L's and S's, but these values depend
on the surrounding electric fields of ligands, making everythings messier but
also more interesting. In this case, Eq.(1) should be written as
EmJ = gJmBBomJ,
stating that the Zeeman splitting appears in accordance with the total angular
momentum projection, mJ. If L = 0 then
J = S, and Eq.(1) will define the energies of all the possible projection
of ms from -S to S - 1, S (2S
+ 1 of such).
If the molecule contains nuclei with magnetic moments, such as protons, their
interaction with external field and the electronic magnetic moment will change
stationary energies of Eq.(1). The nuclear angular momentum quantum number I
determines the nuclear magnetic moment the same way as for the electron:
| |
m = gNmN[I(I
+ 1)]1/2 |
(5) |
|
