An exact analytical solution to the master equation for the vibration–dissociation process of Morse oscillators

Abstract

The coupled vibration–dissociation process for Morse oscillators and structureless particles has been examined. A method to solve the appropriate master equation is developed. The case has been studied carefully where the dissociation process starts mainly from the bound level nearest to the dissociation limit. The master equation has an exact analytical solution in this case upon many-quantum transitions and for an arbitrary amount of energy transferred per collision. Simple expressions for the steady-state dissociation rate and for the incubation time are obtained.

Appendix: Orthogonalization procedure

The general theory of orthogonal polynomials offers the following recurrent equation for three neighboring polynomials [22]

$$\begin{aligned} P_{k+1} (n)= \left( {n-\frac{\left\langle {nP_k (n)^{2}} \right\rangle _T }{\left\langle {P_k (n)^{2}} \right\rangle _T }} \right) P_k (n)-\frac{\left\langle {nP_{k-1} (n)P_k (n)} \right\rangle _T }{\left\langle {P_{k-1} (n)^{2}} \right\rangle _T } P_{k-1} (n), \end{aligned}$$

(41)

where $P_{-1} (n)=0$ and $P_0 (n)=1$. The angular brackets indicate the averaging with a weight function. In this case, this is the thermal distribution from Eq. (2). The orthonormalized polynomials are given by the equation

$$\begin{aligned} \Phi _k (n , \theta )= \frac{P_k (n)}{\sqrt{\left\langle {P_k (n)^{2}} \right\rangle _T }} \end{aligned}$$

(42)

The orthogonalization procedure allows one to sequentially find all the necessary polynomials beginning with $\Phi _0 (n , \theta )=1$. The following two polynomials are of the form

$$\begin{aligned} \Phi _1 (n , \theta )= \frac{n-\left\langle n \right\rangle _T }{\sqrt{d_2 }} \end{aligned}$$

(43)

and

$$\begin{aligned} \Phi _2 (n , \theta )= \left( {\frac{d_2 }{d_4 d_2 -d_3^2 }} \right) ^{\frac{1}{2}}\left[ {n^{2}-\left\langle {n^{2}} \right\rangle _T -\frac{d_3 }{d_2 }\left( {n-\left\langle n \right\rangle _T } \right) } \right] \end{aligned}$$

(44)

The coefficients $d_k$ are defined as

$$\begin{aligned}&\displaystyle d_2 = \left\langle {\left( {n-\left\langle n \right\rangle _T } \right) ^{2}} \right\rangle _T ,\end{aligned}$$

(45)

$$\begin{aligned}&\displaystyle d_3 = \left\langle {\left( {n^{2}-\left\langle {n^{2}} \right\rangle _T } \right) \left( {n-\left\langle n \right\rangle _T } \right) } \right\rangle _T ,\end{aligned}$$

(46)

$$\begin{aligned}&\displaystyle d_4 = \left\langle {\left( {n^{2}-\left\langle {n^{2}} \right\rangle _T } \right) ^{2}} \right\rangle _T \end{aligned}$$

(47)

All the other polynomials were calculated numerically.

In the high-temperature limit, where $\theta =\hbar \omega _e /k_B T$ is less than unity, the recurrent equation makes it possible to sequentially calculate all the necessary polynomials up to $N\approx 100$, which corresponds to the anharmonicity constant of bromine ($x_e =0.005$). Of particular interest is the case of high temperatures where the vibrational nonequilibrium is produced by a shock wave.