6.4 Molecular Spectra: Nuclear Spin Effects
323
35
30
25
20
15
10
5
0
2287
2287.5
2288
2288.5
2289
2289.5
2290
2290.5
2291
2291.5
Intensity (arbitrary units)
Wavenumber/cm -1
Fig. 6.13 Rotational fine-structure lines corresponding to initial rotational quantum numbers j =
1 to j = 15 (from right to left) of the Q-branch of the vibration–rotation Raman spectrum of the
14 N 15 N heteronuclear isotopologue of nitrogen. From Bendtsen [32]. Reproduced by permission
of John Wiley & Sons, Ltd.
p j (T ) =
(2j + 1)e −j (j+1)) rot /T
z rot (T )
.
(6.4.1)
If we express the number of molecules in each rotational level as N j = p j N,
with N the total number of molecules in the sample, then the relative intensities
for rotational spectroscopic transitions will be given directly, all else being equal,
by the fractional populations N j (T )/N = p j (T ) of the initial rotational states.
This means, in particular, that the intensity envelope for a specific set of P -branch
or R-branch transitions (lines) in an infrared spectrum, similarly, of O-, Q-, or Sbranch transitions in a Raman spectrum, will have a maximum that coincides, as can
be seen in Fig. 6.13, with the most populous initial rotational state corresponding to
the temperature of the sample. 3
From Eq. (6.2.62b) we see that as a function of rotational quantum number j , the
fractional population p j (T ) will have a maximum value for j = j max determined
by the condition
3 This is, however, no longer the case for a pure rotational transition, as has been pointed out by
Le Roy [31].