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Doppler shift

  1. Nov 27, 2013 #1
    1. The problem statement, all variables and given/known data

    A gas of atoms, each of mass m, is maintained in a box at temperature T. The atoms emit
    light which passes (in the x-direction) through a window in the box and can be observed
    as a spectral line in a spectroscope. A stationary atom would emit light at the sharply
    de¯ned frequency vo. But because of the Doppler effect the frequency of the light emitted
    from an atom with horizontal velocity vx is not simply vo but rather

    Calculate the relative intensity distribution I(Δ) of the light measured in the spectroscope.

    The spectrum of a gas atom elitting at 638, nm follows a gauss distribution with σ=1.5 GHz
    What is the gas temperature ?
    2. Relevant equations

    3. The attempt at a solution
    So we have
    G(K)=Go exp(-(K-Ko)²/(2σ²))
    Go a constant and σ=Ko*√(k*T/(mc²))
    So I have to calculate
    I(Δ) = 1/2 ∫ Go exp(-(K-Ko)²/(2σ²)) * cos(KΔ) dK from 0 to infinity
    The result is given and we're supposed to find that
    I(Δ) = Io cos(Ko Δ) exp (-1/2 (σΔ)²)
    I tried integration by parts but I can't get to the result ..

    b/ T=σ²*m*c²/(Ko²*k)
    k=1/lambda and Ko=2pi*vo=2pic/lambda
    Replacing we obtain the temperature in fuction of the mass

    Last edited: Nov 27, 2013
  2. jcsd
  3. Nov 27, 2013 #2
    Well I think it is a general integral, you may be able to find it in a math handbook. I don't have mine at hand, but you can always try solving it at wolframalpha.com to see, what it gives.

    Also you don't need to solve any integral, if it is something with infinity, you can often find it in a handbook, which will be sufficient for most cases (unless you are a math student, ha ha).

    Also what you need to integrate looks like a Gaussian function to me! So it's definitely somewhere to be found!
    Last edited: Nov 28, 2013
  4. Nov 27, 2013 #3
    What is Δ in this problem?
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