# 都卜勒增寬

## 公式推導

${\displaystyle f=f_{0}\left(1+{\frac {v}{c}}\right)}$

${\displaystyle \ f}$ 是觀測者所見輻射的頻率、${\displaystyle \ f_{0}}$ 是輻射體靜止時發射輻射的頻率、${\displaystyle \ v}$ 是輻射體相對觀測者的運動速度、${\displaystyle c}$光速

${\displaystyle P_{f}(f)df=P_{v}(v_{f}){\frac {dv}{df}}df}$

 ${\displaystyle P_{f}(f)df={\frac {c}{f_{0}}}P_{v}\left(c\left({\frac {f}{f_{0}}}-1\right)\right)df}$

 ${\displaystyle P_{\lambda }(\lambda )d\lambda ={\frac {c}{\lambda _{0}}}P_{v}\left(c\left(1-{\frac {\lambda }{\lambda _{0}}}\right)\right)d\lambda }$

${\displaystyle P_{v}(v)dv={\sqrt {\frac {m}{2\pi kT}}}\,\exp \left(-{\frac {mv^{2}}{2kT}}\right)dv}$

${\displaystyle P_{f}(f)df=\left({\frac {c}{f_{0}}}\right){\sqrt {\frac {m}{2\pi kT}}}\,\exp \left(-{\frac {m\left[c\left({\frac {f}{f_{0}}}-1\right)\right]^{2}}{2kT}}\right)df}$

${\displaystyle P_{f}(f)df={\sqrt {\frac {mc^{2}}{2\pi kT{f_{0}}^{2}}}}\,\exp \left(-{\frac {mc^{2}\left(f-f_{0}\right)^{2}}{2kT{f_{0}}^{2}}}\right)df}$

${\displaystyle \sigma _{f}={\sqrt {\frac {kT}{mc^{2}}}}f_{0}}$

 ${\displaystyle \Delta f_{\text{FWHM}}={\sqrt {\frac {8kT\ln 2}{mc^{2}}}}f_{0}}$

## 參考資料

1. ^ Siegman, AE. Lasers. 1986 [2014-06-25]. （原始内容存档于2012-02-23）.
2. ^ Daryl W. Preston. Doppler-free saturated absorption: Laser spectroscopy (PDF). American Journal of Physics. November 1996, 64 (11): 1432–1436 [2014-06-27]. Bibcode:1996AmJPh..64.1432P. doi:10.1119/1.18457. （原始内容存档 (PDF)于2010-07-11）.
3. ^ Griem, Hans R. Principles of Plasmas Spectroscopy. Cambridge: University Press. 1997. ISBN 0-521-45504-9.
4. ^ Doppler broadening induced spectral shift effects on reactor safety. [2014-06-27]. （原始内容存档于2021-07-29）.