in the collision of hard balls or material points. Again, hðDE A Þ
2 i is maximal when
I and I
0 are close, and this may be when the reduced masses BC and BC-A are
close. The (5.2.1) describes the T ! R transfer; but we must remember that if we
are able to describe this process, we can also use the principle of detailed equilibrium to describe the opposite, i.e., the R ! T process.
We will not go further into the jungle of semiclassical calculations of the
quantities hðDE A Þ
2 i during rotational relaxation of various classes of molecules,
because these processes are very fast, and very often we need to know only the
order of magnitude of the time of rotational relaxation, i.e., time required to
establish local thermodynamic equilibrium for these degrees of freedom. Below,
only the order of magnitude of the number of collisions required for complete
rotational relaxation, information on the order of magnitude of the energy transferred in the collision, and on the temperature dependence of the relaxation efficiency, are presented.
H 2 , D 2 . These molecules stand apart because their rotational quanta are large
(B
H 2
e = 61 cm
−1 ), and their dipole moment is equal to zero. The relatively large
magnitude of the quanta ‘makes relaxation quantized’, and has little resemblance to
a relaxation of non-quantized translational energy. Correspondingly, the probability
of relaxation is relatively small: to establish local thermodynamic equilibrium,
approximately Z rot = 200–300 collisions are required at T = 300–1000 K [1], p. 71.
In the process of rotational relaxation of any homonuclear diatomic molecules, if
it is described in the framework of quantum mechanics, rather than in the semiclassical approximation, the selection rule is strictly followed, which we will often
recall later: s $ s, a $ a, i.e., symmetric rotational levels commute in collision
with symmetric one, antisymmetric with antisymmetric one; the process s $ a is
forbidden. This rule is a consequence of the weakness of the hyperfine interaction,
i.e., interactions of nuclear spins with other types of intramolecular motion (see
Sect. 4.6.1). This rule is quite simply proved in the monograph of G. Herzberg [3],
p. 131: the probability of such a transition is determined by the integral hW s j b
V jW a i,
and it is clear that the operator b
V is symmetric with respect to the exchange of
nuclei; ‘it does not distinguish between them.’ Since W a changes its sign when
replacing the nuclei, this operation should also lead to a change of the integral sign.
Since the collision partner does not matter which of the same atoms is closer to it,
i.e., the result of the collision does not depend on this, it must be assumed that this
integral is generally 0, i.e., the process s $ a is forbidden. For the H 2 X
1 R
þ
g
state, in which even levels are symmetric and odd are antisymmetric, the selection
rule for rotational relaxation (excitation) DJ = 0, ± 2 follows from the selection
rule discussed above.
Di-, triatomic hydrides. The values of the rotational quanta of HCl, OH, and
H 2 O molecules are also relatively large (B
HCl
e
= 11 cm
–1 ). However, these molecules possess nonzero dipole moments, and in collisions with a similar molecule a
dipole–dipole interaction takes place. This circumstance leads to a high probability
of the R-T process: the number of collisions required for complete relaxation of
156
5 Energy Transfer in Collisions
2 i is maximal when
I and I
0 are close, and this may be when the reduced masses BC and BC-A are
close. The (5.2.1) describes the T ! R transfer; but we must remember that if we
are able to describe this process, we can also use the principle of detailed equilibrium to describe the opposite, i.e., the R ! T process.
We will not go further into the jungle of semiclassical calculations of the
quantities hðDE A Þ
2 i during rotational relaxation of various classes of molecules,
because these processes are very fast, and very often we need to know only the
order of magnitude of the time of rotational relaxation, i.e., time required to
establish local thermodynamic equilibrium for these degrees of freedom. Below,
only the order of magnitude of the number of collisions required for complete
rotational relaxation, information on the order of magnitude of the energy transferred in the collision, and on the temperature dependence of the relaxation efficiency, are presented.
H 2 , D 2 . These molecules stand apart because their rotational quanta are large
(B
H 2
e = 61 cm
−1 ), and their dipole moment is equal to zero. The relatively large
magnitude of the quanta ‘makes relaxation quantized’, and has little resemblance to
a relaxation of non-quantized translational energy. Correspondingly, the probability
of relaxation is relatively small: to establish local thermodynamic equilibrium,
approximately Z rot = 200–300 collisions are required at T = 300–1000 K [1], p. 71.
In the process of rotational relaxation of any homonuclear diatomic molecules, if
it is described in the framework of quantum mechanics, rather than in the semiclassical approximation, the selection rule is strictly followed, which we will often
recall later: s $ s, a $ a, i.e., symmetric rotational levels commute in collision
with symmetric one, antisymmetric with antisymmetric one; the process s $ a is
forbidden. This rule is a consequence of the weakness of the hyperfine interaction,
i.e., interactions of nuclear spins with other types of intramolecular motion (see
Sect. 4.6.1). This rule is quite simply proved in the monograph of G. Herzberg [3],
p. 131: the probability of such a transition is determined by the integral hW s j b
V jW a i,
and it is clear that the operator b
V is symmetric with respect to the exchange of
nuclei; ‘it does not distinguish between them.’ Since W a changes its sign when
replacing the nuclei, this operation should also lead to a change of the integral sign.
Since the collision partner does not matter which of the same atoms is closer to it,
i.e., the result of the collision does not depend on this, it must be assumed that this
integral is generally 0, i.e., the process s $ a is forbidden. For the H 2 X
1 R
þ
g
state, in which even levels are symmetric and odd are antisymmetric, the selection
rule for rotational relaxation (excitation) DJ = 0, ± 2 follows from the selection
rule discussed above.
Di-, triatomic hydrides. The values of the rotational quanta of HCl, OH, and
H 2 O molecules are also relatively large (B
HCl
e
= 11 cm
–1 ). However, these molecules possess nonzero dipole moments, and in collisions with a similar molecule a
dipole–dipole interaction takes place. This circumstance leads to a high probability
of the R-T process: the number of collisions required for complete relaxation of
156
5 Energy Transfer in Collisions