Page 321 - 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

838 Microwave Molecular Spectroscopy

The correction factor utilizes a bond elongation param- evaluation of the dipole components along the principal
˚
∼
eter δr D (=0.003 A). Simple correction procedures have axes.
ρ D
been given for various molecule types. The (I ) and In the general case, the molecular dipole moment fixed
corr
the other scaled moments of the complete set are then in the molecule can have three nonvanishing components
used in a least-squares determination of the structural pa- µ x ,µ y ,µ z alongthemolecule-fixedprincipalaxissystem.
rameters. The results found are quite good. The scaled The Stark effect Hamiltonian is then expressed by
method represents a significant step in the improvement

of spectroscopic structure determinations of polyatomic =− µ g Zg , g = x, y, z, (104)
molecules.
where Zg are the direction cosines of the x, y, z axes rel-
ative to the space-fixed Z axis, the direction along which
IX. STARK EFFECT AND ZEEMAN EFFECT
the electric field is applied. For linear and symmetric-
top molecules only a dipole component along the z axis
Applied electric or magnetic fields modify the rotational
is present: µ z = µ and µ x = µ y = 0. Usually, the above
spectrum. The most important of these is the effect of elec-
interaction can be adequately treated by first- and second-
tricfields,commonlycalledtheStarkeffect.Intheabsence
order perturbation theory. When levels that interact via
of an electric field, the rotational states are (2J + 1)-fold
are degenerate or near degenerate, the frequency dis-
degenerate since the energies are independent of M,
placement ν of the Stark components is linear in the
electric field and depends on M, that is, ν = B Jτ M .
M = 0, ±1, ±2,..., ±J, (102)
Here B Jτ are functions of J and other possible quantum
which specifies the possible orientations of the angular numbers. In this case, a rotational line splits into 2J + 1
components. On the other hand, when levels interact via
momentum vector relative to a fixed direction in space.
which are widely separated, the frequency displace-
When an electric field is present, the field interacts with 2
2
ment varies as ν = (A Jτ + B Jτ M ) , and only J + 1
the molecular dipole moment and the rotational energy
components are obtained for J =±1 and J components
levels split into a number of sublevels. The degeneracy in
for J = 0. Here J refers to the smaller J value involved
the space orientation quantum number M is thus partially
in the transition.
or completely lifted by the interaction. A schematic illus-
The relative intensities of the Stark components depend
tration is provided in Fig. 3. A rotational line, therefore,
on the transition type
splits into a number of components. The general selection
rules for rotational transitions are 
2
2
I M = P[(J + 1) − M ], J → J + 1 

2
M → M, M → M ± 1, (103) I M = QM , J → J M → M.
2
2
I M = R[J − M ], J → J − 1 

in addition to those already given for the different types
(105)
of rotors (Section IV). Usually, the electric field is ap-
plied parallel to the electric vector of the radiation, and For asymmetric tops all three expressions apply, while
the selection rule M = 0 applies. for linear and symmetric-top molecules only the first in-
The Stark effect is usually used to modulate rotational tensity expression is needed. The P, Q, R coefficinets
lines to improve their detection. This is the basis of the are independent of M but depend on the intensity of the
Stark-modulation spectrometer (Fig. 2) discussed earlier. unsplit line. Note that the M = 0 component is forbid-
With this type of spectrometer both the Stark lines and den for a J = 0 transition, and for a second-order effect
2
the zero-field transitions are displayed. The Stark effect ( ν ∼ M ), a factor of 1/2 must be included in the inten-
pattern can be a valuable aid in the assignment of rota- sity expression for the M = 0 component since the +M
tional spectra, particularly for asymmetric tops. Specifi- and −M degeneracy is lost.
cally, by counting the number of Stark components, one The general features of the second-order effect for the
can obtain an indication of the smaller J value involved linear molecule OCS are illustrated in Fig. 3. Clearly, as
in the transition. Another particularly important applica- the electric field is increased, the field-dependent Stark
tion of this effect is in the evaluation of very accurate components move further away from the zero-field line.
electric dipole moments. These are determined by care- The three components |M |= 0, 1, 2 are as expected for a
ful measurement of the displacement of the Stark com- J = 2 → 3 transition. The component nearest the unsplit
ponents from the zero-field absorption line as a func- line requires a high field before it shows up clearly. By
2
tion of the applied field. The analysis of these Stark plotting ν against , the effect is easily classified as a
splittings by means of the appropriate expression allows second-order effect. The appearance of a first-order Stark