The
theory of ESR spectroscopy begins with a quantum analysis of the energy
associated with the magnetic dipole moments of electrons in a molecule.
Magnetic dipoles have two components - one that is due to spin angular
momentum (arising from the electron spinning about its axis), and
one due to orbital momentum. In the vast majority of cases, the spin
angular momentum accounts for about 99% of the total magnetic dipole.
The
strength of the magnetic dipole is characterized by the magnetic dipole
moment, which is defined in terms of the interaction of the magnetic
dipole with a magnetic field, H. The energy, E, of the magnetic moment
is given by:
Electrons
also have an intrinsic spin angular momentum, P, which is an internal
property of the particle. The dipole moment is proportional to P,
and can be written:
A
proportionality constant, called the magnetogyric ratio, contains
an important factor, g, which will be discussed in more detail in
subsequent paragraphs. Inserting the magnetogyric ratio into the equation
above, the dipole moment becomes:
Since the quantum mechanical angular momentum is quantized, it is
helpful to analyze the properties of an electron by considering the
behavior of a particle restricted to motion about a ring. There is
a deBroglie wavelength associated with the momentum, P. In order for
the probability to be time-independent, the wave function must be
single-valued, which limits the circumference of the ring to be an
integral number times the deBroglie wavelength. That is:
Therefore:
where
P(theta) is the magnitude of the angular momentum of the
particle in the theta direction, and M is an integer (M = 0, 1, 2,
3, ...). This quantization of angular momentum allows electron orbitals
to be designated by the following notation: for M = 0, the electrons
are said to be in a sigma orbital, and for M = 1, they are in a pi
orbital. This derivation assumes that the potential energy is constant.
The
spin angular momentum of a particle is characterized by the spin quantum
number, MS, where the allowed values of MS range in unit increments
from -S up to +S giving 2S + 1 components. For a system with a single
electron, S=½, so the allowed values for MS are +/-½.
The
quantized angular momentum in the direction of the external magnetic
field (the Z direction) can be written in terms of the spin quantum
number:
The dipole moment becomes:
where
beta is the Bohr magneton. The energy becomes:
These two possible values for energy are called Zeeman energies. These
transitions are shown schematically in the figure below.
The
figure to the right is a pictoral representation of the sample cavity.
In the presence of an external magnetic field, dipoles will align
themselves either parallel or opposed to (antiparallel) the external
field as shown in the figure. Since parallel alignment is a lower
and more stable energy condition, the population of electrons in this
state is greater than in the antiparallel state. Energy transitions
are initiated by supplying an electromagnetic field with high frequency
(usually in the microwave range). If the energy of the microwave field
corresponds to delta E, then the field is said to be at resonant frequency
and transitions occur.
Dipoles
in parallel alignment to the magnetic field absorb energy from the
microwave field, sending them to higher energy states. Similarly,
antiparallel dipoles release the same amount of energy to the electromagnetic
field. Since there are slightly more parallel dipoles when resonance
occurs, there will be a net absorption of energy by the dipoles. The
energy from the electromagnetic field that is lost to the dipoles
is detected and amplified yielding the ESR signal for the sample being
analyzed.
The
g value is a universal constant and is a characteristic of electrons
(ge = 2.00232). Whenever an external magnetic field is applied to
a sample, an internal orbital magnetic moment can be introduced. This
internal magnetic moment is caused by mixing in of excited states
into the ground state, which is brought about by a coupling of the
electron spin and orbital angular momenta. This phenomenon is characterized
by the atomic spin-orbit coupling constant gamma.
The
internal magnetic field may add to or subtract from the external field.
Since Hr is defined to be the external magnetic field at resonance,
g must be allowed to vary to account for any local magnetic fields.
The effective g value is given by:
