Elements of Modern Physics
164
=
2
I
J, J 1,2,... for ∆n = 0
(5.68)
This gives a band of spectral lines (Fig. 5.13), known as the rotational band,
with equally spaced frequencies
v =
2
4
h
I
π
J
(5.69)
The spacing is of the order of 10
12
s
–1
, which falls in the very far infrared
region. The spacing allows us to evaluate I and hence the equilibrium distance
[r 0 = (I/M)
1/2
]. For HCl, the spacing is ∆v ≈ 6.2 × 10
11
s
–1
which gives the
values of I ≈ 2.7 × 10
–47
kg. m
2
and therefore r 0 ≈ 1.29 Å.
For transitions with ∆n = 1,
hv = (k/M)
1/2
±
2
h
I
J, J = 1, 2,... for ∆n = 1 (5.70)
where the plus sign is for J′ = J – 1 and the minus sign is for J′ = J + 1. This
again gives us a band of spectral lines which have the same spacing as the lines
in the rotational band, but with the centre at v 0 =
1
2π
(k/M)
1/2
(which has a
value of above 8.67 × 10
13
s
–1
for HCl) and with the central frequency missing.
This is known as the vibrational-rotational band (Fig. 5.13).
Symmetric Molecules
Symmetrical molecules do not have an electric dipole moment and the associated
dipole transitions, and hence do not exhibit the pure rotational or vibrationalrotational bands just described. The changes in their states are due to higherorder effects, so that the radiation emitted is much weaker. These higher-order
transitions obey the selection rules:
∆J = 0, ± 1, ± 2
(5.71)
Since the nuclei of a symmetrical molecule are identical, the total nuclear
wave function must satisfy the requirements of exchange symmetry, i.e. the
total wave function must be symmetric for an integral nuclear spin I, and
antisymmetric for an half-integral nuclear spin I, under the interchange of the
nuclei. The exchange symmetry of the spatial part of the wave function is
determined by the rotational states (i.e. the
m
l
Y (θ, φ) functions) which for the
exchange of the nuclei (i.e. θ → π – θ, and φ → π + φ) are even for even l and
odd for odd l. Here l plays the role of J. Of the spin states of nuclei with spin I,
there are (I + 1) (2I + 1) states which are even and I (2I + 1) states which are
odd, under the exchange of spin. States with even spin state are called orthomodifications while those with odd spin state are called para-modifications.
164
=
2
I
J, J 1,2,... for ∆n = 0
(5.68)
This gives a band of spectral lines (Fig. 5.13), known as the rotational band,
with equally spaced frequencies
v =
2
4
h
I
π
J
(5.69)
The spacing is of the order of 10
12
s
–1
, which falls in the very far infrared
region. The spacing allows us to evaluate I and hence the equilibrium distance
[r 0 = (I/M)
1/2
]. For HCl, the spacing is ∆v ≈ 6.2 × 10
11
s
–1
which gives the
values of I ≈ 2.7 × 10
–47
kg. m
2
and therefore r 0 ≈ 1.29 Å.
For transitions with ∆n = 1,
hv = (k/M)
1/2
±
2
h
I
J, J = 1, 2,... for ∆n = 1 (5.70)
where the plus sign is for J′ = J – 1 and the minus sign is for J′ = J + 1. This
again gives us a band of spectral lines which have the same spacing as the lines
in the rotational band, but with the centre at v 0 =
1
2π
(k/M)
1/2
(which has a
value of above 8.67 × 10
13
s
–1
for HCl) and with the central frequency missing.
This is known as the vibrational-rotational band (Fig. 5.13).
Symmetric Molecules
Symmetrical molecules do not have an electric dipole moment and the associated
dipole transitions, and hence do not exhibit the pure rotational or vibrationalrotational bands just described. The changes in their states are due to higherorder effects, so that the radiation emitted is much weaker. These higher-order
transitions obey the selection rules:
∆J = 0, ± 1, ± 2
(5.71)
Since the nuclei of a symmetrical molecule are identical, the total nuclear
wave function must satisfy the requirements of exchange symmetry, i.e. the
total wave function must be symmetric for an integral nuclear spin I, and
antisymmetric for an half-integral nuclear spin I, under the interchange of the
nuclei. The exchange symmetry of the spatial part of the wave function is
determined by the rotational states (i.e. the
m
l
Y (θ, φ) functions) which for the
exchange of the nuclei (i.e. θ → π – θ, and φ → π + φ) are even for even l and
odd for odd l. Here l plays the role of J. Of the spin states of nuclei with spin I,
there are (I + 1) (2I + 1) states which are even and I (2I + 1) states which are
odd, under the exchange of spin. States with even spin state are called orthomodifications while those with odd spin state are called para-modifications.