Show that differential cross section is independent of $θ$ for elastic scattering

Question

I have two hard sphere which took part in elastic scattering, I have attached a diagram below and I have assumed here that since its is elastic the angle of incidence $alpha$ is equal to the angle of reflection $alpha$
$b=(r_1+r_2 )sin⁡(pi/2-theta/2)$

AFG · Accepted Answer

Using $$sin(2x)=2sin (x) cos (x)$$
If $sinleft(frac{theta}{2}right)>0$:
$$cosleft(frac{theta}{2}right)Bigg|sinleft(frac{theta}{2}right)Bigg|=cosleft(frac{theta}{2}right)sinleft(frac{theta}{2}right)=frac{1}{2}sintheta,$$
so the $sintheta$ cancels with the one on the denominator. If $sinleft(frac{theta}{2}right)<0:$
$$cosleft(frac{theta}{2}right)Bigg|sinleft(frac{theta}{2}right)Bigg|=-cosleft(frac{theta}{2}right)sinleft(frac{theta}{2}right)=-frac{1}{2}sintheta,$$
the same applies. There is a change in the sign, but $sinleft(frac{theta}{2}right)$ is always positive in $[0,2pi]$, so $sigma$ is independent of $theta$.

Show that differential cross section is independent of $θ$ for elastic scattering

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