How does the experimentalist deal with Compton Scattering?

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Summary:

I want to understand how the experimentalist deals with Compton Scattering theory and see some examples.

Main Question or Discussion Point

I've been studying Compton Scattering (Mandl & Shaw, pages 142-147) and the derivation of the following formulas:

1) Differential cross section in the LAB frame

$$\Big( \frac{d \sigma}{d \Omega}\Big)_{LAB} = \frac{1}{(4 \pi)^2} \Big( \frac{\omega'}{\omega}\Big)^2 |\mathscr{M}|^2 \tag 1$$

2) The unpolarized differential cross-section in the LAB frame

$$\Big( \frac{d\sigma}{d \Omega} \Big)_{LAB} = \frac{\alpha^2}{2m^2} \Big(\frac{\omega'}{\omega}\Big)^2 \Big\{ \frac{\omega}{\omega'}+\frac{\omega'}{\omega}-\sin^2 \theta \Big\} \tag 2$$

Which in the low-energy limit (i.e. ##\omega' \sim \omega##) reduces to the differential cross-section for Thomson scattering

$$\Big( \frac{d\sigma}{d \Omega} \Big)_{LAB, \omega' \sim \omega} = \frac{\alpha^2}{2m^2} \Big\{ 2-\sin^2 \theta \Big\}=\frac{\alpha^2}{2m^2} \Big\{ 1+\cos^2 \theta \Big\} \tag 3$$

3) The polarized cross-section in the LAB frame (known as Klein-Nishina formula)

$$\Big( \frac{d\sigma}{d \Omega} \Big)_{LAB, pol} = \frac{\alpha^2}{4m^2} \Big(\frac{\omega'}{\omega}\Big)^2 \Big\{\frac{\omega}{\omega'}+\frac{\omega'}{\omega}+4(\epsilon \epsilon')^2 -2 \Big\} \tag 4$$

The unpolarized cross-section ##(2)## can be obtained due to

$$\frac 1 2 \sum_{pol} (\epsilon \epsilon')^2 = \frac 1 2 (1 + \cos^2 \theta) \tag 5$$

So my questions are:

Is ##(2)## the only important equation for the experimentalist? Based on what I've read most experiments involve unpolarized beams. Thus, the polarization of the particles produced in the collision are not observed. Are lasers an exception?

Why is Thomson approximation useful?

Thank you :smile:
 

Answers and Replies

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Simon Bridge
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That's three questions. I trained as an experimentalist, lets see if I understand the questions ... taken in order:
  • Is (2) the only important equation for the experimentalist?
No ... depends on what we are testing. However, in the usual teaching lab experiment, we are mainly interested in the lab-frame scattering distribution.

In practice you use the model that best fits the experiment you are doing. ie. If you are interested in polarization, then you will need (3).

We may build the model in the center of mass frame or something, because the maths is easier, but we'll have to convert that into whatever frame we are actually making measurements in (or convert the data, which is poor form).

  • Based on what I've read most experiments involve unpolarized beams. Thus, the polarization of the particles produced in the collision are not observed. Are lasers an exception?
In teaching experiments, we want to keep things as simple as possible. Lasers are normally used in my experience. Instead of basis your opinions on what you've read, alone, try also looking at the actual experimental setup being used.

  • Why is Thomson approximation useful?
You are not always doing the experiment at high energy. The maths is easier: why do harder maths than you have to?

When you have a new theory, to be successful, it has to show how the previous theory (in this case "Thompson scattering") could be so right, and yet not be good enough. That's what the approximation does -- it explains Thompson scattering in light of the new information.

Also: it is really common to use simple approximations... they often turn out to be good enough.
 
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