Lecture21-Doppler pdf
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lecture21-doppler
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broadening. If the gas is at a temperature T , the probability M
(v)dv of finding an atom with velocity between v and v + dv is given by the Maxwell-Boltzmann distribution M (v) = m 2πk
B T r e − m v2 2k BT (12)
where m is the mass of the Hydrogen molecules k B is the Boltzmann constant. At rest, a Hydrogen atom would have an emission at λ 0 = 656 nm or equivalently at a frequency of ν 0 = 4.5 × 10 17 Hz. If the molecules are moving towards the observer, the frequency will go up due to the Doppler effect, and if they are moving away, it will go down. The probability of finding the emission at ν is then P (ν)dν where P (ν) =
c ν 0 M
c ν ν 0 − 1
= m c 2 2πk B Tν 0 2 s exp − mc 2 (ν − ν
0 ) 2 2k B Tν 0 2 (13) This is a Gaussian distribution with width ∆ν ν
= ∆λ λ = 8k B T mc 2 r = 8 γ r c s c (14) where c s = γ k B T m q is the speed of sound. For example, at room temperature, where c s = 343 m s and γ = 7 5 for Hydrogen gas, ∆λ λ = 2 × 10 −6 and the emission line at λ = 656nm will have a width of ∆λ = 0.002nm. Doppler spectroscopy 7 Document Outline
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