Page 326 - 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

Microwave Molecular Spectroscopy 843

field gradient relative to the principal axis system, and weaker F → F components (3/2 → 3/2, 5/2 → 5/2, and
where z is along the molecular axis and Q is consid- 7/2 → 7/2), and two even weaker F → F − 1 compo-
ered known and is characteristic of the coupling nucleus. nents (5/2 → 3/2 and 7/2 → 5/2). These considerations
Also, enable the identification of the hyperfine components
of a given rotational transition. Early applications of
(3/4)C(C + 1) − I(I + 1)J(J + 1)
Y(J, I, F) = microwave spectroscopy to the study of hyperfine
2(2J − 1)(2J + 3)I(2I − 1)
structure used the appearance of the fine structure, that is,
(124) the relative spacings and intensities of the components, to
determine unknown nuclear spins. For example, the spin
C = F(F + 1) − J(J + 1) − I(I + 1) (125)
of 33 S was found to be 3/2 by this method.
is a function of the various quantum numbers and the spin For a symmetric top, with the coupling atom on the
I. It has been tabulated for various I and J. It may be symmetry axis,
observed that Y(J, I, F) is undefined for I = 0 and 1/2,
" 2 #
which is consistent with the requirement I t for a nu- 3K
E Q = χ − 1 Y(J, I, F), (127)
clear quadrupole interaction. The E Q must be added to J(J + 1)
the rigid-rotor energy E r to give the total energy. Apply-
where χ is the coupling constant with reference to the
ing the selection rules J = 0, F = 0, ±1, one obtains
molecular axis of symmetry. From the above expression
the rotational frequencies including effects of quadrupole
it follows that each J, K level splits into a number of sub-
coupling:
levels of different F. The selection rules are J =± 1,
ν = ν r − χ[Y(J + 1, I, F ) − Y(J, I, F)], (126) K = 0, I = 0, F = 0, ±1. When K = 0, the hyper-

fine pattern is like that for a linear molecule. For other K
where ν r is the unperturbed rotational frequency and values, a similar pattern is obtained. However, when dif-
F = F, F ± 1. The coupling constant can be evaluated ferent K lines are separated by less than the quadrupole

from the splitting between any two hyperfine components. splitting, the individual patterns for each K overlap, and
To evaluate the rotational constant, the rigid-rotor fre- a quite complex overall structure can be obtained.
quency ν r is required, and this may be evaluated by cor- For coupling atoms off the symmetry axis we have a
recting the hyperfine components ν with the known χ via more complicated problem. An example would be HCCl 3 .
the above frequency expression. The hyperfine structure for molecules with two or more
To understand the appearance of the hyperfine pattern, a coupling nuclei is more complex but has been treated the-
knowledge of the relative intensities of the components is oretically and observed experimentally.
required. The explicit expressions require too much space
to give here. However, we may point out that for any class
B. Asymmetric-Top Molecules
of rotor when J > I, there are 2I + 1 components for
F → F + 1, 2I components for F → F, and 2I − 1 com- For an asymmetric prolate rotor with a single coupling
ponents for F → F − 1. Furthermore, the most intense nucleus we have
components are those where F = J. An approximate ( 2 ! 2 ! )
E Q = 3 P − J(J + 1) χ aa − σ P − W(b p ) ηχ aa
a
a
intensity rule is that for J → J + 1 transitions, the inten-
sities of the F → F + 1 components are proportional to Y(J, I, F)
× , (128)
F, while the F → F components are considerably weaker J(J + 1)
and the F → F − 1 components even weaker. For the
where σ =−1/b p , and W(b p ) is Wang’s reduced energy
J → J transition, the intensity of the F → F component 2 2
and P the average P in the asymmetric rotor basis.
a
Z
is proportional to F, while the F → F ± 1 components
The asymmetry parameter
are considerably weaker. In fact, the intensities of the
components for F = J decrease rapidly with increasing χ bb − χ cc
η = (129)
J. Also, the function Y(J, I, F) may be positive or nega- χ aa
tive, but for the maximum and minimum values of F, the
measures the departure of the field gradient from cylin-
function is positive. Moreover, the strongest component
drical symmetry about a. The coupling constants χ aa =
is usually not significantly displaced from the rigid-rotor
eQq aa , χ bb = eQq bb , and χ cc = eQq cc are relative to the
position. Consider, for example, the 2 → 3 transition 2 2
principal axis system; q aa = ∂ V/∂a and so on are the
with I = 3/2. For J = 2, F = 1/2, 3/2, 5/2, 7/2 and
corresponding field gradients. Since Laplace’s relation
for J = 3, F = 3/2, 5/2, 7/2, 9/2. We expect four F →
holds,
F + 1 components (1/2 → 3/2, 3/2 → 5/2, 5/2 → 7/2,
and 7/2 → 9/2 in order of increasing intensity), three χ aa + χ bb + χ cc = 0, (130)