Alpha particles with less kinetic energy are deflected through greater angles than more energetic ones.
When you calculate the scattering angle #θ# as a function of the kinetic energy #E# of the α particle, you get the equation
#(2bE)/(Z_1Z_2e^2) = cot(θ/2)#, where
#b# is the impact parameter — the distance of closest approach if the α particle were not deflected #Z_1# and #Z_2# are the charges on the α particle and the gold nucleus #e# is the electronic charge
For a fixed value of #b#, the equation says that
#E ∝ cot(θ/2)# or
#θ# ∝ arccot(#E#)
A plot of #y# =arccot(#x#) or #θ# = arccot(#E#) looks like this:
Since #E# is always positive, we need to look at only the right hand side of the plot.
We see that as the kinetic energy #E# increases, the deflection angle #θ# decreases. Also, as the kinetic energy #E# decreases, the deflection angle #θ# increases.
