This work uses findings to generate a set of simple dimensionless equations, one of which yields the maximum amplitude of oscillation of the system at dissociation. The equations incorporate experimental data which are widely available and a set of such data for simple molecules is presented at the conclusion of the work. The simplicity of the equations developed is exemplified by that for the maximum amplitude of oscillation, Xe, when the system encounters an incoming photon possessing the magnitude of the dissociation energy, De, of the molecule, i.e. (Xe/R) = 2 sqrt(De.Be)/We.

Here, R is the bond length, We, the energy of the photon which raises the system from the ground state to the first excited state, and Be is the bond rotational constant. It is suggested that the simple expressions developed here have their application in the violent events associated with dissociation in shock waves and/or high energy particles emanating from stellar collapse and hence may be of utility in these areas of astrophysical modelling.

Excerpt

List of Contents

Introduction

Analysis

Specimen calculations of the maximum dimensionless amplitude of a selection of simple diatomic molecules

Additional data for a selection of simple diatomic molecules

Discussion

References

Abstract

An expression was developed in 1 for the energy eigenstates of a diatomic molecule modelled as a simple quantum harmonic oscillator, in which the two-body system was reduced to the oscillation of a single body of reduced mass, In addition, a wave function was generated which was specified in such a fashion that the oscillating body could not be found outwith the envelope of the oscillation. In the course of analysis it was shown that the total potential energy of the system was equal to the energy of the incoming photon. Here, Einstein’s model of a photon stream in the photo-electric effect was invoked to argue that the oscillation of the body could be ascribed to a series of impacts by photons.

This work uses findings from 1 to generate a set of simple dimensionless equations which describe the maximum amplitude of oscillation of the system at dissociation.

The equations incorporate experimental data which are widely available and a set of the relevant data for a sample of simple molecules are displayed at the conclusion of the work.

The simplicity of the equations is exemplified by that for the maximum amplitude of oscillation, when the system encounters an incoming photon having a magnitude equal to that of the dissociation energy, of the molecule, i.e.

Here, R is the bond length, the energy of the photon which raises the system from the ground state to the first excited state and is the bond rotational constant.

It is proposed that the expressions developed here have their application in the dissociation of a molecule occasioned by the shock waves and/or the high energy particles emanating from stellar events and hence may be of utility in astrophysical modelling.

Introduction

As is well known in quantum mechanics, the vibrational energy states of a diatomic molecule correspond to the eigenstates of the internuclear potential energy.

The lowest energy states are well approximated by the potential energy states, of the so- called ‘classical quantum harmonic oscillator’, where n is the vibrational quantum number. This is demonstrated in 1, and it is widely recognised to be one of the very few analytical solutions of the Schrdinger equation.

An alternative perspective of the modelling of a simple quantum harmonic oscillator is presented in 1 and it is this approach which is adopted here in the determination of the amplitude of oscillation between energy eigenstates and at dissociation, of a diatomic molecule modelled as a quantum harmonic oscillator.

The Morse potential model of a diatomic molecule has many energy eigenstates, the difference between which becomes progressively smaller with increasing vibrational quantum number. This is in contradistinction to the model developed in 1 where the difference between the energy eigenstates becomes larger with increasing vibrational quantum number. Indeed, it may be posited that this model is more appropriate to the bombardment of a molecule in its ground state by a stream of high energy (relatively speaking), photons. This, on an astrophysical basis is perhaps a more realistic scenario than that of many spectroscopic investigations where conditions of equilibrium are imposed before each measurement to facilitate, amongst other things, the determination of the number of vibrational energy eigenstates to dissociation.

Analysis

The energy eigenstates for the model developed in 1 are given by the expression:

This formula is not valid for n = 0; it was shown in 1 that if n was equal to zero for all x in the range, , where is the amplitude of oscillation of the reduced mass body, then the wave function was also equal to zero in this range and this implied the absurdity that, for n = 0, the mass centre of the body could not be found within the above envelope of oscillation.

It was argued that the ground state of the molecule could be attained by the emission of a photon of magnitude, where is the natural frequency of the model and is given by the well-known expression, . Here, k and are the bond force constant and the reduced mass, respectively.

As noted by Irikura 2, the zero point energy cannot be measured directly since no molecule can be observed below its ground state. We posited in 1, following Einstein’s explanation of the photo-electric effect, that the observed strong response of a molecule in its ground state to the impact of a stream of discrete photons at frequency, could be explained by the setting of the system into resonance at this frequency, with the concomitant effect of raising the system to its first energy eigenstate. Indeed, this is implied in (1), for we may reach the ground state from the first state by the emission of a photon of energy,

In addition, it was shown in 1 that the maximum potential energy of the bond (spring) was equal to the energy of the incoming photon, i.e.

We argue that, between energy eigenstates the spring responds to the incoming photon in exactly the same manner as the response between the ground and the first, energy eigenstates.

It will be shown later that, although we can write an equation similar to that shown immediately above but which contains the vibrational quantum number, the spring still responds classically.

In the following, for the sake of ease of cross-reference between the results of Irikura, [ 2 ], Fan et al, 3, Darwent, 5 and Cardona et al, 4, we adopt the notations presented therein.

In the new notation we now write equation (1) as: