Spectral Analysis of Gas Atoms: Temperature Calculation

[Dassinia]

Hello

Homework Statement

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
v=vo(1+vx/c)

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 ?

Homework Equations

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

tHANKS

 

[hjelmgart]

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!

 

[nasu]

What is Δ in this problem?