Page 325 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN009N-447 July 19, 2001 23:3

842 Microwave Molecular Spectroscopy

the Zeeman effect. Except for molecules with nonzero along the spin axis). This nonspherical nuclear charge can
electronic angular momentum and consequently a perma- interact with the surrounding unsymmetrical molecular
nent magnetic dipole moment, the effect is small, but ob- charge distribution arising from all the other charges in
servable nonetheless with kilogauss magnetic fields. The the molecule. This latter distribution is measured by the
2 2
2
Zeeman effect Hamiltonian for the interaction of a mag- electric field gradient ∂ V/∂ z , with V the electrostatic
netic dipole with an applied field H is given by potential. In such cases the nuclear spin I is coupled to
the overall rotation J, and a nuclear quadrupole hyperfine
H =−µ · H. (118)
structure results. The total angular moment F = J + I has
√
For molecules in singlet ground electronic states, the the magnitude h F(F + 1), and a new quantum number
magnetic moment µ is generated by the molecular ro- F is now required to characterize the energy levels, or
tation (rotation of charges). With the assumption that the hyperfine states,
molecular magnetic moment is proportional to the angular
momentum, it can be shown that the Zeeman splitting of F = J + I, J + I − 1, J + I − 2,..., |J − I|. (120)
the rotational levels is given by
Each rotational level is hence split into a number of differ-
E H =−g J,τ β I HM, (119) ent levels labeled by the values of F. When J > I, there
are in general (2I + 1) values of F. The projection of F
where g J,τ is the rotational g-factor and depends on the
along an axis fixed in space, hM F , is specified by the
inner quantum numbers K and τ for symmetric and asym-
quantum number
metric tops, respectively. Here β I is the nuclear magneton,
H is the applied magnetic field, and M is the orienta- M F = F, F − 1, F − 2,..., −F. (121)
tion quantum number. The selection rules are M = 0or
±1 depending on whether the magnetic field is applied This quantum number becomes important when an ex-
parallel or perpendicular to the electric radiation vector. ternal field is applied to a molecule with a quadrupolar
Analysis of the Zeeman splittings provides values of the nucleus. Since the rotational states are split by the nuclear
molecular rotational g-factors. These g-factors, for exam- interaction, a given rotational transition splits into a num-
ple, when measured for two isotopic species, can be used ber of components and a group of closely spaced lines
to determine the vector direction of the electric dipole is observed. The selection rules are as before, with the
moment. additional rules
Besides the molecular magnetic moments generated by
rotation, smaller magnetic moments are induced by the F → F, F → F ± 1, I → I. (122)
external field. These additional effects are expressed in
This splitting, in general, increases for some of the more
terms of a magnetic susceptibility tensor χ, and analysis
14
37
common nuclei in the order N(I = 1) < Cl(I = 3/2) <
of these effects yields the elements of the magnetic sus-
81
35 Cl(I =3/2)< Br(I =3/2)< Br(I = 3/2) < 127 I(I =
79
ceptibility tensor. Considerable progress has been made in 12 16
5/2). Many common nuclei have I = 0 (e.g., C, O,
the study of magnetic properties of molecules and in de-
32 S) or I = 1/2 (e.g., H, 13 C, 15 N, 19 F) and hence do
1
riving information on the electronic structure of molecules
not give rise to a quadrupole coupling interaction. The
from these studies. However, their description is outside
quadrupole splittings decrease with increasing J, often
the scope of this presentation.
becoming unresolvable at sufficiently high J. Nuclear
quadrupole hyperfine structure in the rotational spectrum
indicates the presence of a quadrupolar nucleus such as
X. NUCLEAR QUADRUPOLE
HYPERFINE STRUCTURE Cl; it can be useful in the assignment of a spectrum and it
provides information on chemical bonding.
The effects of nuclear coupling can give rise to hyperfine
structure in the rotational spectrum. The most important A. Linear and Symmetric-Top Molecules
type of nuclear interaction occurs when a nucleus with a
The nuclear quadrupole energy for a linear molecule with
nonzero nuclear quadrupole moment (I > 1/2) is present
a single coupling nucleus in the absence of external fields
in the molecule. The nuclear quadrupole moment Q mea-
is given by
sures the deviation of the nuclear charge distribution from
a spherical distribution. Generally, Q becomes larger as E Q =−χY(J, I, F), (123)
the nucleus becomes heavier. A positive Q indicates a
prolatelike distribution (elongated along the spin axis) and where χ = eQq is the nuclear quadrupole coupling con-
2
2
a negative Q indicates an oblatelike distribution (flattened stant in frequency units with q = ∂ V/∂z , the electric