# Difference between revisions of "Einstein Coefficients"

Line 68: | Line 68: | ||

separations. $\phi(\nu)$ is called the line profile function. It | separations. $\phi(\nu)$ is called the line profile function. It | ||

describes | describes | ||

− | + | the relative absorption probability around $\nu_0$ (the | |

absorption frequency), and is subject to the requirement that: | absorption frequency), and is subject to the requirement that: | ||

− | + | $\int_0^\infty{\phi(\nu)d\nu}=1$. We can approximate the width of $\phi(\nu)$ as an effective width $\Delta\nu$. | |

− | + | $\Delta\n $ is affected by many factors: | |

− | + | \begin{itemize} | |

− | $\ato$ (the natural, uncertainty-based broadening of at atom in isolation), | + | \item $\ato$ (the natural, uncertainty-based broadening of at atom in isolation), |

− | $\nu_0{ | + | \item $\nu_0\frac{v_{\rm therm}{c}$ (the thermal, Doppler-based broadening), and |

− | $n_{coll}\sigma_{coll}v_{rel}$ | + | \item $n_{coll}\sigma_{coll}v_{rel}$ |

− | (collisional broadening, a.k.a. pressure broadening). | + | (collisional broadening, a.k.a. pressure broadening). |

− | + | \end{itemize} | |

+ | Line profile functions are of special interest for studying line emission/absorption, and we have more discussion in a separate section on line profile functions. | ||

+ | |||

+ | Suffice to say, the transition probability per unit time associated with spontaneous absorption is: | ||

$$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$ | $$R_{ex}^{-1}=\bot\int_0^\infty{J_\nu\phi(\nu)d\nu}\approx\bot\Jbar$$ | ||

## Revision as of 12:30, 18 September 2014

### Short Topical Videos

- LASER: Stimulated emission: part 1, part 2 (Nainani, Stanford)
- Einstein Coefficients Part 1, Part 2, Part 3 (kridnix, Bucknell)

### Reference Material

- Einstein Coefficients (Wikipedia)
- Einstein Coefficients (Christensen, BYU)
- Einstein coefficients, cross sections, f values, dipole moments, and all that (Hilborn, Amherst)
- Relations between the Einstein coefficients (Wood, U. St. Andrews)

# Einstein Coefficients

Einstein coefficients describe the absorption and emission of photons via electronic transitions in atoms. Say we have 2 energy levels with an energy difference of . Einstein coefficients govern the interaction of radiation with these discrete energy levels. There are three coefficients:

*Left: Photon absorption is described by . Center: Spontaneous photon emission is described by . Right: Stimulated photon emission is described by .*

### 1 Spontaneous Emission,

governs decay from energy state 2 to 1. It is the transition probability per unit time for an atom, and has units of . More specifically, the probability an atom undergoes spontaneous de-excitation and releases a photon is Poisson-distributed, with mean rate . So is the mean lifetime of the excited state. As an example, (n=3 to n=2 in hydrogen) has an Einstein A coefficeint of .

### 2 Spontaneous Absorption,

governs absorptions causing transitions . The transition probability per unit time is , where is the probability constant, and is:

It depends on (the intensity), but it does not depend on direction, so we integrate over all angles. The is a normalization constant which makes the mean intensity, instead of the total intensity. However, we have to remember that there are uncertainties in the energy-level separations. is called the line profile function. It describes the relative absorption probability around (the absorption frequency), and is subject to the requirement that: . We can approximate the width of as an effective width . **Failed to parse (unknown function "\n"): {\displaystyle \Delta \n }**
is affected by many factors:

- (the natural, uncertainty-based broadening of at atom in isolation),