4 Vibrational Relaxation of Molecular Ions
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H. SHIN University of Nevada, Reno, N e v .
The vibrational relaxation of simple molecular ions M in the M+-M collision (where M = O , N , and CO) is studied using the method of distorted waves with the interaction poten tial constructed from the inverse power and the polarization energy. For M-M collisions the calculated values of the col lision number required to de-excite a quantum of vibrational energy are consistently smaller than the observed data by a factor of 5 over a wide temperature range. For M -M collisions, the vibrational relaxation times of M (τ ) are estimated from 300° to 3000°Κ. In both N and CO, τ 's are smaller than τ's by 1-2 orders of magnitude whereas in O τ is smaller than τ less than 1 order of magnitude except at low temperatures. +
2
2
+
+
+
+
2
+
?
Τ η studying ion-molecule reactions it is important to know the details ·*• of energy transfer between translational and internal motions of the collision partners. In the charge transfer process (e.g., 0 + + 0 0 + 0 ) , which is often a glancing one and occurs at comparatively large impact parameters except at high energies, in ion-neutral collisions in aftergrows, or in the ionosphere process (e.g., N + 0 NO+ + N O ) , the molecular ions are probably found in vibrationally excited states (4,11,19,31,32). Since the states of both the reactant and product involve molecular ions, the collisional characteristics can be markedly different from reactions involving neutral molecules, owing to the enormous electric field near the ions. Since the fate of the ions and hence the overall process will depend upon the vibrational state of the molecular ions, it is important to evaluate the vibrational relaxation times or the number of collisions required for the molecular ions to return to the Boltzmann dis tribution for the given gas temperature. F o r example, in an electrical discharge in helium gas, the molecular ion H e may be formed in a vibra tionally excited state, and the rate of dissociative recombination of He + e H e + H e may depend on the vibrational state. Similarly, since 0 has an appreciably small internuclear distance (1.123 A.) compared with 0 (1.207 Α . ) , the ionization of 0 will probably lead to a vibrationally excited molecular ion. 2
2
2
2
2
2
2
+
+
2
+
+
2
+
2
2
In recent years measurements of cross-sections for ion-molecule collisions have become one of the active fields in physics and chemistry, 44
Ausloos; Ion-Molecule Reactions in the Gas Phase Advances in Chemistry; American Chemical Society: Washington, DC, 1967.
4.
SHIN
Vibrational Relaxation
45
and continued efforts have led to ingenious experimental techniques. In spite of the concentrated effort in this area, relatively few theoretical studies deal with estimating the vibrational transition probabilities, relaxation times, or collision numbers for excited molecular ions (29,31). T o gain a thorough insight into ion-molecule reactions, these collision quantities must be estimated from relevant interaction parameters. Our study deals with this aspect, extending our previous work (29) on the effect of the polarization energy on ion-molecule collisions to calculate the vibrational collision numbers of simple molecular ions and to estimate their vibrational relaxation times. Vibrational
Transition
Probability
T h e main difference between a molecule-molecule ( M - M ) collision and an ion-molecule ( M - M ) collision is the presence of a polarization force in the latter system owing to the attraction between the static charge on M and the dipole moment induced on M . F o r a large intermolecular separation, the polarization energy is known as +
+
where a is the angle-average polarization of M , e is the electronic charge, and r is the distance between centers of mass of the collision partners. T h e polarization energy can significantly increase the depth of the potential well between an ion and a molecule so that the relative kinetic energy is increased. Obviously, such a deep potential well will modify the slope of the repulsive part of the potential on which the probability of energy transfer depends. A t present, however, the short range value of the polarization energy is not known; therefore, it is difficult to analyze the role of polarization forces in ion-molecule inelastic collisions, and this prevents us from evaluating the vibrational transition probabilities and related quantities with accuracy. When Equation 1 is introduced into a typical intermolecular interaction energy potential such as a Morse type or an inverse power type, as a Lennard-Jones (LJ) potential, the resulting potential energy curve appears strongly repulsive with a deep attractive well at a close separation so that it may be used to describe M - M collisions. Therefore, we express the intermolecular interaction by introducing Equation 1 into an inverse power potential. T h e short range exponential potentials represent inelastic collisions well because strong repulsive terms give better insight into the collision than do inverse power potentials. However, as shown below the results of the inverse power potential can be reduced easily to those of the exponential potentials. T h e assumed form of the potential for this study is +
(2) where O (a) LJ
=» 0, and D is the depth of the potential well of the 12-6 func-
Ausloos; Ion-Molecule Reactions in the Gas Phase Advances in Chemistry; American Chemical Society: Washington, DC, 1967.
46
ION-MOLECULE REACTIONS IN T H E GAS PHASE
tion. Although the inverse power function or the Lennard-Jones (LJ) 12-6 function is known to be inadequate for "close-in" inelastic collisions, it is the potential used most often for inelastic collisions i n a thermal energy range because the perturbation integral for this potential is easy to evaluate (6,23,30). A s discussed below this method evaluates the per turbation integral of the method of distorted waves essentially b y con sidering only the collision partners close to each other where the repulsive interaction potential varies rapidly. We consider a general inverse power potential function of the form U(r) =
Σ
Ci
(3)
τη,η,ρ...
F o r the present 12-6-4 potential we then have C = 4Ζ)σ , C = — 4Ζ)σ , and C = — « e / 2 . T h e W K B evaluation of the perturbation integral of the method of distorted waves results in the (temperature) average transition probability i n the form (29): i 2
12
6
6
2
4
2kT
(4) where ρ = m /(m + ™>B), if we express M by A B μ = reduced mass of the collision partners e = magnitude of the change in the oscillator's energy owing to the transition, hv (v is the frequency of the oscillator) M = "effective" reduced mass of the oscillator and B
A
Ausloos; Ion-Molecule Reactions in the Gas Phase Advances in Chemistry; American Chemical Society: Washington, DC, 1967.
4.
Vibrational
SHIN
47
Relaxation 2m 2 + 3m
Xm
For the 12-6-4 potential, this reduces to: /ni/7 13/7 5/7\ u
/
e
1.346 ^
σ
\ ΐ 2 / 7
/
\l/2
+ 0.254 f - ° ^ - Λ
Γ
γ
+ 0.362 °-
^
+
+ -1-1
(5)
with ' Γ ( / ι ) Λ/2ττμ 19
2
. r(Vi2)
(4D) / Ve^Tl / 1
1
1 2
Λ
1 9
~J
T h e time constant r, appearing i n the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1 / r = P — Pd, where P is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. T h e reciprocal of this quantity is the number of collisions Ζ required to deexcite a quantum of vibrational energy e = hv. T h i s number can be explicitly calculated from Equation 4 since Z = 1 / P , and it can be experi mentally derived from the measured relaxation times. Since there are no experimental values of Ζ for M - M collisions available at present, we first calculate Ζ for the M - M collisions for N , C O , and 0 whose experimental values are well established (7,22). After establishing the usefulness of this approach for the M - M collisions, we then calculate the collision numbers for the M - M systems, Z . e
e
+
2
2
+
Molecule-Molecule
+
Collisions
For numerical illustrations we use the potential parameters (9,14) and vibrational frequencies (12) given in Table I. In Figures 1-3 the calculated values of Ζ for M - M and for M - M are shown as (log Ζ vs. T ) for M = N , 0 , and C O , respectively. We will first discuss the M - M collisions. F r o m 500° to 760 °K. the impact tube method of Huber and K a n trowitz (15) used nitrogen containing 0.05% water vapor. Their work on the relaxation of N in water yields Ρ ( N - H 0 ) / P ( N - N ) = 1100 at 560 °K., and their data are smaller than the calculated values b y a factor of 4. T h e values of Lukasik and Young (17) are obtained from the resonance method between 770° and 1190 °K. using the sample con taining r° / σ > 0.83 or ae /2r kT is larger than ae /2a kT by a factor of 1.46 to 2.07. Table I V shows the values of the polarization term in the exponent for the potentials given by Equations 2, 13, and 15 for 0 . When the depth of the potential well D is small compared with kT, the potential assumed by Equation 15 may adequately describe the M + - M collision process. Since the " h a r d core" limiting potentials ignore the effect of attractive energies on the slope of the repulsive part of U(r), they generally give smaller Ρ values compared with the analytical expression given by Equation 5. T a k a yanagfs result obtained by assuming the polarization energy ae /2R with R c^l a.u. seriously underestimates the importance of the energy. 0
12
0
6
2
2
2
04
A
2
+
2
A
c
c
Table IV.
T
O
R
300 400 600 800 1000 1500 3000
Calculated Values of the Polarization Energy T e r m i n the Exponent of Ρ 0.254
/
2
\
αβ
χ
1/3
1
ae 2r°* kT 2
V Z ) / ^ / kT
2σ
4
Α
5.357 4.268 3.098 2.469 2.070 1.503 0.869
kT
3.206 2.405 1.603 1.202 0.962 0.641 0.320
5.996 4.498 2.998 2.248 1.799 1.198 0.598
As shown i n Figure 6 the most probable distance for the vibrational transition is r * = 2.223 A . for 0 - 0 , and the potential energy at this distance is U(r*) = 1.06 X 10 ~ ergs. T h i s distance is significantly small compared with r°. T o justify using the inverse power potentials for the repulsive part of U(r) we first note that the potential at the most probable distance for the vibrational transitions is much larger than the average kinetic energy of the relative motion. F o r example, i n 0 - 0 U(r*)/kT = 25.6 even at 3000 °K.; this implies that the incident waves which reach this region control the over-all energy transfer process. T h e incident particles with sufficiently high collision velocities will reach this region, but it is also possible that the incident waves can reach this region b y the potential barrier penetration—i.e., the quantum nature of the translational motion is essential i n the problem. A t high collision velocities the ordinary perturbation methods (such as this one) fails because the probability becomes too large, and exchange of more than one quantum in a single collision becomes increasingly probable. Y e t our approach shows that the collision process is essentially controlled by the partners reached at such close proximity. According to the argument of R a p p and Sharp (25), if energetic collisions must control the over-all process, we should have obtained transition probabilities not much dif ferent from unity. However, we obtained the probabilities
