# Spontaneous emission

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Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon. Spontaneous emission of light or luminescence is a fundamental process that plays an essential role in many phenomena in nature and forms the basis of many applications, such as fluorescent tubes, older television screens (cathode ray tubes), plasma display panels, lasers (for startup - normal continuous operation works by stimulated emission instead) and light emitting diodes.

## Contents

### Introduction

If a light source ('the atom') is in the excited state with energy E2, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy E1, releasing the difference in energy between the two states as a photon. The photon will have angular frequency ω and energy $\hbar \omega$ (= hν, where h is the Planck constant and ν is the frequency):

where $\hbar$ is the reduced Planck constant. The phase of the photon in spontaneous emission is random as is the direction the photon propagates in. This is not true for stimulated emission. An energy level diagram illustrating the process of spontaneous emission is shown below:

If the number of light sources in the excited state is given by N, the rate at which N decays is:

where A21 is the rate of spontaneous emission. In the rate-equation A21 is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the Einstein A coefficient, and has units s − 1[1]. The above equation can be solved to give:

where N(0) is the initial number of light sources in the excited state, t is the time and Γrad is the radiative decay rate of the transition. The number of excited states N thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% of its original value ($\frac{1}{e}$-time). The radiative decay rate Γrad is inversely proportional to the lifetime τ12: