How is the threshold for Cerenkov counters calculated?

[erwinscat]

Hi everyone,

I have a problem with the following equation that is related to Cerenkov counter. You can find it at page 56 of "Introduction of High Energy Physics" by Perkins.

The equation is the following:

$$sin ^{2} (\theta _1)=1-\frac{1}{\beta ^{2}_{1}n^2}\approx\frac{m^{2}_{2}-m^{2}_1}{p^2}$$

I do know why : $$sin ^{2}(\theta _1})= 1-\frac{1}{\beta ^{2}_{1}n^{2}}$$

since it comes out from $$cos(\theta _{1})=\frac{1}{\beta n}$$

but I don't know where : $$1-\frac{1}{\beta ^{2}_{1}n^2}\approx \frac{m^{2}_{2}-m^{2}_1}{p^2}$$

comes from...

Could anyone explain this to me ?
Thank you very much in advance for any help !

Erwin

 

[Vanadium 50]

You'll have better luck if you explain what the problem is, rather that assuming everyone here has the same edition of Perkins that you do.