326
6 Molecular Systems
molecule there is a significant difference between the intensities of the transitions
originating from the odd and even rotational levels, and associated with the nuclear
interchange symmetries of the electronic, rotational, and nuclear spin wavefunctions
as the odd/even fractional populations of the rotational levels are in the ratio
p j o
p j e
=
I a + 1
I a
(2j o + 1)e −j o (j o +1)) rot /T
(2j e + 1)e −j e (j e +1)) rot /T .
(6.4.6)
If we examine what happens for adjacent lines characterized by j o and j o − 1 for
pure rotational Raman scattering, we find that this ratio becomes
p j o
p j o −1
=
I a + 1
I a
2j o + 1
2j o − 1
e
−2j o rot /T ,
j o ≥ 1 .
(6.4.7)
Thus, we see that adjacent lines in the Raman scattering spectrum for a homonuclear
diatomic molecule have an intensity ratio given by the product of the ratio of the
nuclear spin degeneracies associated with the two nuclear spin modifications and a
factor that depends upon the rotational quantum number of the odd-j level and the
ratio of the characteristic rotational temperature and the temperature of the sample.
The second (temperature dependent) factor plays a larger role for small values of
the rotational quantum number, but otherwise provides only a slight skewing effect
on the intensity alternation across a sequence of rotational fine-structure lines.
As the 14 N nucleus has nuclear spin I a = 0 and the 15 N nucleus has nuclear
spin I a =
1
2 , the 14 N 2 and 15 N 2 nitrogen isotopologues are homonuclear diatomic
molecules whose nuclei are bosons, respectively, fermions. The appropriate forms
for the combined rotational-nuclear spin partition function z rot−nuc (T ) are thus
given by Eq. (6.2.108) and Eq. (6.2.109), with the consequence that the intensities
for adjacent j
o and j
e = j
o + 1 lines should be 1:2 and 3:1, respectively. While the
j = 0 transition for 15 N 2 only appears as a small shoulder on the j = 1 transition,
this transition is clearly distinguishable for 14 N 2 due to the preferential nuclear spin
weighting of its even rotational levels.
Example 6.7 Raman Q-branch fine-structure lines for T 2 .
The rotational fine structure of the Q-branch of the fundamental Raman vibrational scattering spectrum for molecular tritium, 3 H 2 , is shown in Fig. 6.15.
The 3 H isotope of hydrogen has nuclear spin I a =
1
2 , so that the intensity
alternation for rotational fine-structure lines in Raman scattering spectra of T 2
should be 3:1 for transitions associated with odd versus even j -values were it
determined solely by the nuclear spin degeneracy factors. However, because the
characteristic rotational temperature for 3 H 2 is of the order of 30 K, the temperaturedependent factor in Eq. (6.4.7) plays a greater role than it does for many other
Raman scattering spectra. We may therefore determine from Eq. (6.4.7) that the
intensity ratios I j o /I j o −1 for temperature 300 K will be approximately 8.1, 3.1, and
2.2 for j o = 1, 3, 5, respectively, which is roughly what is observed in Fig. 6.15.
Note also that the wavenumber axis increases from right to left in this figure.
6 Molecular Systems
molecule there is a significant difference between the intensities of the transitions
originating from the odd and even rotational levels, and associated with the nuclear
interchange symmetries of the electronic, rotational, and nuclear spin wavefunctions
as the odd/even fractional populations of the rotational levels are in the ratio
p j o
p j e
=
I a + 1
I a
(2j o + 1)e −j o (j o +1)) rot /T
(2j e + 1)e −j e (j e +1)) rot /T .
(6.4.6)
If we examine what happens for adjacent lines characterized by j o and j o − 1 for
pure rotational Raman scattering, we find that this ratio becomes
p j o
p j o −1
=
I a + 1
I a
2j o + 1
2j o − 1
e
−2j o rot /T ,
j o ≥ 1 .
(6.4.7)
Thus, we see that adjacent lines in the Raman scattering spectrum for a homonuclear
diatomic molecule have an intensity ratio given by the product of the ratio of the
nuclear spin degeneracies associated with the two nuclear spin modifications and a
factor that depends upon the rotational quantum number of the odd-j level and the
ratio of the characteristic rotational temperature and the temperature of the sample.
The second (temperature dependent) factor plays a larger role for small values of
the rotational quantum number, but otherwise provides only a slight skewing effect
on the intensity alternation across a sequence of rotational fine-structure lines.
As the 14 N nucleus has nuclear spin I a = 0 and the 15 N nucleus has nuclear
spin I a =
1
2 , the 14 N 2 and 15 N 2 nitrogen isotopologues are homonuclear diatomic
molecules whose nuclei are bosons, respectively, fermions. The appropriate forms
for the combined rotational-nuclear spin partition function z rot−nuc (T ) are thus
given by Eq. (6.2.108) and Eq. (6.2.109), with the consequence that the intensities
for adjacent j
o and j
e = j
o + 1 lines should be 1:2 and 3:1, respectively. While the
j = 0 transition for 15 N 2 only appears as a small shoulder on the j = 1 transition,
this transition is clearly distinguishable for 14 N 2 due to the preferential nuclear spin
weighting of its even rotational levels.
Example 6.7 Raman Q-branch fine-structure lines for T 2 .
The rotational fine structure of the Q-branch of the fundamental Raman vibrational scattering spectrum for molecular tritium, 3 H 2 , is shown in Fig. 6.15.
The 3 H isotope of hydrogen has nuclear spin I a =
1
2 , so that the intensity
alternation for rotational fine-structure lines in Raman scattering spectra of T 2
should be 3:1 for transitions associated with odd versus even j -values were it
determined solely by the nuclear spin degeneracy factors. However, because the
characteristic rotational temperature for 3 H 2 is of the order of 30 K, the temperaturedependent factor in Eq. (6.4.7) plays a greater role than it does for many other
Raman scattering spectra. We may therefore determine from Eq. (6.4.7) that the
intensity ratios I j o /I j o −1 for temperature 300 K will be approximately 8.1, 3.1, and
2.2 for j o = 1, 3, 5, respectively, which is roughly what is observed in Fig. 6.15.
Note also that the wavenumber axis increases from right to left in this figure.