Spontaneous emission

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In physics, spontaneous emission is the process by which an atom, molecule or nucleus in an excited state drops to a lower-energy state, resulting in the creation of a photon.

1 Overview
2 Theory
3 Spontaneous emission in semiconductors
4 See also
5 External links

[edit] Overview

If the atom is in the excited state with energy $E 2$ , it may spontaneously decay into the ground state, with energy $E 1$ , releasing the difference in energies between the two states as a photon. The photon will have frequency $ν$ and energy $h ν$ , given by:

E 2 - E 1 = h ν

where h is Planck's 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 is shown below:

In a group of such atoms, if the number of atoms in the excited state is given by N, the rate at which spontaneous emission occurs is given by:

$\frac{\partial N}{\partial t} = -A_{21} N$ ,

where A₂₁ is a proportionality constant for this particular transition in this particular atom. (The constant is referred to as an Einstein A co-efficient.) The rate of emission is thus proportional to the number of atoms in the excited state, N.

The above equation can be solved to give:

$N(t) = N(0) e^{ - \frac{t}{\tau_{21} } }$ ,

where N(0) is the initial number of atoms in the excited state, and τ₂₁ is the lifetime of the transition, τ₂₁ = (A₂₁)^-1.

It can be seen that spontaneous emission occurs in a way rather similar to the decay of radioactive particles, in particular that the lifetime is analogous to a half-life.

There are two different ways in which decay or relaxation can occur: radiative and nonradiative. In nonradiative relaxation, the energy is absorbed as phonons, more commonly known as heat. Nonradiative relaxation is nearly impossible to measure and cannot be inferred except in very small particles because the difference in the temperature before and after a relaxation is so small that it is in the noise of any measurement for practical systems.

Nonradiative relaxations occur when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance, semiconductors), electrons move quickly from a high energy level to a meta-stable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition (This final transition is the transition over the bandgap in semiconductors.). Large nonradiative transitions do not occur frequently because the crystal structure generally can not support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Meta-stable states form a very important feature that is exploited in the construction of lasers. Specifically, since electrons decay slowly from them, they can be piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal.

[edit] Theory

Quantum mechanics explicitly prohibits spontaneous transitions. That is, using the machinery of ordinary first-quantized quantum mechanics, if one computes the probability of spontaneous transitions from one stationary state to another, one finds that it is zero. In order to explain spontaneous transitions, quantum mechanics must be extended to a second-quantized theory, wherein the electromagnetic field is quantized at every point in space. Such a theory is known as a quantum field theory; the quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics.

In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the vacuum state, which can mix with the excited stationary states of the atom. As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it.

Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may transition from ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the directions in which the photon can move off in. Equivalently, one might say that the phase space offered by the electromagnetic field is infinitely larger than that offered by the atom. Since one must consider probabilities that occupy all of phase space equally, the combined system of atom plus electromagnetic field must transition from electronic excitation to photonic excitation; the atom must decay by spontaneous emission.

This process is very similar to stimulated emission.