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# Spectroscopic Transition Strengths

## Introduction

An atom or molecule can be stimulated by light to change from one energy state to another. An atom or molecule in an excited energy state can also decay spontaneously to a lower state. The probability of an atom or molecule changing states depends on the nature of the initial and final state wavefunctions, how strongly light can interact with them, and on the intensity of any incident light. This document discusses some of the practical terms used to describe the probability of a transition occuring, which is commonly called the transition strength. To a first approximation, transitions strengths are governed by selection rules which determine whether a transition is allowed or disallowed. Practical measurements of transitions strengths are usually described in terms of the Einstein A and B coefficients or the oscillator strength (f).

## Selection Rules

1. The parity of the initial and final wavefunctions must be different.
2. The spin can not change, S = 0.
3. The change in orbital angular momentum can be L = 0, ±1, but L=0 to L=0 transitions are not allowed.
4. The change in total angular momentum can be J = 0, ±1, but J=0 to J=0 transitions are not allowed.

## Transition Probability

The transition probability is R2 with units of J cm3, where R is the transition moment given by:

where j and i are the wavefunctions of the upper and lower states, respectively, and µ is the dipole moment operator. Basically what this equation indicates is that the strength of a transition is relative to how strongly the dipole moment of a resonance between energy states can couple to the electric field of a light wave.

## Einstein coefficients

For a two-level system (ground-state level i and upper level j), the rate of an upward stimulated transition (absorption, -dNi/dt or dNj/dt) is:

where Ni is the number density of atoms in the ground state, Uv is the light intensity, and the proportionality factor Bij is the Einstein B coefficient for absorption:

For stimulated emission the Einstein coefficient becomes:

where gi and gj are the degeneracies of the ground and excited states, respectively.

Atoms in the excited state can decay without the presence of an external light field due to stimulation due to "zero-point fluctuations." Zero-point fluctuations are the dynamic variations in the shape of an electronic orbital at any instant in time. These instantaneous orbitals can be described by a linear combination of the wavefunctions of the system, which provides the mechanism for transitions between different states of the system. The spontaneous decay rate (-dNj/dt or dNi/dt) is:

where Aji is the Einstein coefficient for spontaneous emission:

Since atoms in the upper level can decay by both spontaneous and stimulated emission, the total downward rate (-dNj/dt or dNi/dt) is given by:

## Oscillator strength

The oscillator strength of a transition is a dimensionless number that is useful for comparing different transitions. It is defined as the ratio of the strength an atomic or molecular transition to the theoretical transition strength of a single electron using a harmonic-oscillator model. For absorption:

and for emission:

`fji = fij gi/gj`
Oscillator strengths can range from 0 to 1, or a small integer. A strong transition will have an f close to 1. Oscillator strengths greater than 1 result from the degeneracy of real electronic systems.

Tabulations in the literature often use gf, where gf = gi fij = gj fji

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