The intensities of spectral lines
The ratio of the transmitted intensity, I, to the incident intensity, I0, at a given
frequency is called the transmittance, T, of the sample at that frequency :
T = I / I0
It is found empirically that the transmitted intensity varies with the length, L, of the
sample and the molar concentration, [J], of the absorbing species J in accord
with
the Beer-Lambert law ;
I = I0 10-Ɛ [J] L
The quantity Ɛ is called the molar absorption coefficient.
The molar absorption coefficient depends on the frequency of the incident
radiation
and is greatest where the absorption is most intense. Its dimensions are
1 / ( concentration x length), and it is normally convenient to express it in cubic
decimetres per mole per centimetre (dm 3 mol- 1 cm- I ). Alternative units are
square centimetres per mole (cm2 mol-I). This change of units demonstrates that
Ɛ may be regarded as a molar cross-section for absorption and, the greater the
cross -sectional area of the molecule for absorption, the greater its ability to block
the
passage of the incident radiation .
To simplify the previous equation we introduce the absorbance, A, of the sample
at a
given wavenumber as
A = log I / I0
or
A = - log T
Then the Beer-Lambert law becomes
A = Ɛ [J] L
The product Ɛ [J] L was known formerly as the optical density of the sample.
suggests that, to achieve sufficient absorption, path lengths through gaseous
samples must be very long, of the order of metres, because concentrations are
low.
Long path lengths are achieved by multiple passage of the beam between
parallel
mIrrors at each end of the sample cavity. Conversely, path lengths through liquid
samples can be significantly shorter, of the order of millimetres or centimetres.
The Beer-Lambert law implies that the intensity of electromagnetic radiation
transmitted through a sample at a given wavenumber decreases exponentially
with
the sample thickness and the molar concentration. If the transmittance is 0.1 for
a
path length of 1 cm (corresponding to a 90 per cent reduction in intensity), then it
would be (0.1)2 = 0.01 for a path of double the length (corresponding to a 99 per
cent reduction in intensity overall).
Selection rules and transition moments
For the molecule to be able to interact with the electromagnetic field and absorb
or
create a photon of frequency v, it must possess, at least transiently, a dipole
oscillating at that frequency. This transient dipole is expressed quantum
mechanically in terms of the transition dipole moment, µfi between states
ᴪf :
µfi =
∫ ᴪf
µ ᴪi
dt
ᴪi and
where µ is the electric dipole moment operator. The size of the transition dipole
can
be regarded as a measure of the charge redistribution that accompanies a
transition:
a transition will be active (and generate or absorb photons) only if the
accompanying
charge redistribution is dipolar (Fig. 1).
Fig.1 (a) When a Is electron becomes a 2s electron, there is a spherical
migration of charge; there is no dipole moment associated with this migration of
charge; this transition is electric-dipole forbidden. (b) In contrast, when a Is
electron becomes a 2p electron, there is a dipole associated with the charge
migration; this transition is allowed.
We know from time-dependent perturbation theory that the transition rate is
proportional to IµfiI2. It follows that the coefficient of stimulated absorption (and
emission), and therefore the intensity of the transition, is also proportional to
IµfiI2. A detailed analysis gives
The constant B is the Einstein coefficient of stimulated absorption.
Only if the transition moment is nonzero does the transition contribute to the
spectrum. It follows that, to identify the selection rules, we must establish the
conditions
for which µfi ≠ 0.
A gross selection rule specifies the general features a molecule must have if it is
to
have a spectrum of a given kind. For instance, we shall see that a molecule gives
a
rotational spectrum only if it has a permanent electric dipole moment. This rule,
and
others like it for other types of transition.
A detailed study of the transition moment leads to the specific selection rules that
express the allowed transitions in terms of the changes in quantum numbers.
Linewidths
A number of effects contribute to the widths of spectroscopic lines. Some
contributions to linewidths can be modified by changing the conditions, and to achieve
high
resolutions we need to know how to minimize these contributions. Other contributions cannot be changed, and represent an inherent limitation on resolution.
(a) Doppler broadening
The study of gaseous samples is very important, as it can inform our
understanding
of atmospheric chemistry. In some cases, meaningful spectroscopic data can be
obtained only from gaseous samples. For example, they are essential for
rotational
spectroscopy, for only in gases can molecules rotate freely.
One important broadening process in gaseous samples is the Doppler effect, in
which radiation is shifted in frequency when the source is moving towards or
away
from the observer. When a source emitting electromagnetic radiation of
frequency
v moves with a speed s relative to an observer, the observer detects radiation of
frequency
where c is the speed of light. For nonrelativistic speeds (s«
expressions simplify to:
c), these
Molecules reach high speeds in all directions in a gas, and a stationary observer
detects the corresponding Doppler-shifted range of frequencies. Some molecules
approach the observer, some move away; some move quickly, others slowly.
The
detected spectral 'line' is the absorption or emission profile arising from all the
resulting Doppler shifts.
The Doppler line shape is a Gaussian (Fig. 2), when the temperature is T and the
mass of the molecule is m, then the observed width of the line at half-height (in
terms
of frequency or wavelength) is
For a molecule like N2 at room temperature (T = 300 K), δv/v = 2.3 x 10- 6 . For a
typical rotational transition wavenumber of 1 cm- I (corresponding to a frequency
of
30 GHz), the linewidth is about 70 kHz. Doppler broadening increases with
temperature because the molecules acquire a wider range of speeds. Therefore, to
obtain
spectra of maximum sharpness, it is best to work with cold samples.
Fig. 2 The Gaussian shape of a Doppler- broadened spectral line reflects the
Maxwell distribution of speeds in the sample at the temperature of the
experiment. Notice that the line broadens as the temperature is increased.