- #1

- 9

- 0

It relates to the frequency of a photon emitted from an atom which then subsequently recoils. The atom has initial rest mass m0, and loses rest energy ‘e’ in the emission (“e is the difference between the rest energy of the final and initial atom”).

The question asks to prove the frequency of the emitted photon is:

[tex]f=\frac{e}{h}(1-\frac{e}{2m_0c^2})[/tex]

I really enjoy special relativity (we just had a lecture about how the existence of a magnetic field around a current carrying wire can be explained simply by considering SR, that’s so cool!), but I often find myself getting horrible confused. This is especially true when it comes to relativistic mass type questions like the above (and similarly nuclear reactions equations). I always thought the best thing to do with such problems was to just ensure both energy and momentum are conserved, but I still usually get mixed up.

Anyway, back to this problem, here’s what I tried.

For starters, I figured that since e is the difference in rest mass energies, this value must be equivalent to the gained kinetic energy of the atom plus the energy of the photon:

[tex]e = E_k + hf[/tex]

I then equated the initial and final energy of the system:

[tex]E_i=m_0c^2=E_f=hf+mc^2[/tex]

Where [tex]m[/tex] is the relativistic mass.

This is where the trouble starts… Is the above line correct? I get the feeling the final energy isn’t right. It has to be the energy of the photon plus the kinetic energy and rest mass energy of the new ‘lighter’ atom (one photon ‘lighter’), but how can this be expressed? Introduce a new rest mass: [tex]m_02[/tex]?

I get the feeling this is getting unnecessarily complicated…

Someone said the thing to do is use momentum rather than velocity, specifically the formula: [tex]E^2=p^2c^2+(m_0c^2)^2[/tex] but I really cant see how that helps.

I will be grateful for any help, tips and advice on this question (or even methods for solving such problems in general). The question is only worth 1 mark so really shouldn’t be so hard, I think I’m just doing something obviously wrong.

Thanks in advance.

(please excuse any tex errors, its my first time using it…)