Modern Spectroscopy

9.3 USES OF LASERS IN SPECTROSCOPY 383
where e(~n) is the molar absorption coefficient of the sample at a wavenumber ~n. The fact that
the absorbance is proportional to both the concentration c of the sample and the path length
of absorption constitutes the Beer–Lambert law. This was introduced in equation (2.16) and
illustrated in Figure 2.4.
Using napierian logarithms (log e ¼ ln), Equation (9.27) can be rewritten:
lnðI 0=IÞ ¼kc‘ ¼ a‘ ð9:28Þ
where k is the molar (napierian) absorption coefficient and a is the linear (napierian)
absorption coefficient. Converting Equation (9.28) to an exponential form gives:
I=I 0 ¼ expð a‘Þ ð9:29Þ
In the this form the Beer-Lambert law shows that the intensity of radiation transmitted by an
absorbing sample declines exponentially as the length over which the absorption takes place
increases. If the radiation, travelling with the speed of light c, takes time t ‘ to traverse the
absorbing path ‘ Equation (9.29) becomes:
I=I 0 ¼ expð act ‘Þ ð9:30Þ
The fact that the transmitted intensity decreases exponentially with time forms the basis of
cavity ring-down spectroscopy (CRDS).
The experimental method used for CRDS is illustrated in Figure 9.36. Radiation from a
pulsed laser enters the cavity formed between two mirrors M 1 and M 2. These mirrors are
plano-concave, with flat outer and concave inner surfaces, coated on the inner surfaces with
a material which is very highly reflecting at the wavelength being employed. A reflectivity as
high as 99.93% has been recorded at 515 nm but this decreases with wavelength.
Consequently, when a photon enters the cavity, it may be reflected backwards and forwards
many times between the mirrors before it emerges to the detector. In the near infrared it is
not unusual for it to be reflected 5 000 times so that, with a cavity length of 1 m, the effective
absorption path length is 10 km. Such large path lengths, obtainable in a relatively compact
space, make CRDS an extremely sensitive and convenient technique for detection of very
weak absorption. This may be weak because the transitions being studied have a very low
probability or because the molecule being studied is present in only very low concentration.
Figure 9.36 shows how the detected signal decays with time. Even with no sample in the
cavity, the signal decays slowly and smoothly due to the high reflectivity of M 1 and M 2.
When there is an absorbing sample in the cavity there is, after a few traversals of the cavity,
Figure 9.36 Schematic diagram showing how a cavity ring-down absorption spectrum is obtained