# Spontaneous emission

**Spontaneous emission** is the process by which an atom or molecule in an excited state drops to the ground state, resulting in the creation of a photon.

If the atom is in the excited state with energy **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2}**
, it may spontaneously decay into the ground state, with energy **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_1}**
, releasing the difference in energies between the two states as a photon. The photon will have frequency **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu}**
and energy **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \nu}**
, given by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_2 - E_1 = h \nu}**,

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:

Before emission After emission --------O--------- ------------------ E2 | Atom in | excited state | ~~~> | Photon hν | V ------------------ ---------O-------- E1 Atom in ground state

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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial N}{\partial t} = -A_{21} N}**,

where *A*_{21} 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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(t) = N(0) e^{ - \frac{t}{\tau_{21} } } }**,

where *N*(0) is the initial number of atoms in the excited state, and τ_{21} is the *lifetime* of the transition, τ_{21} = (*A*_{21})^{-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.