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Hello

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 ?

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

## 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

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