Using the elastic scattering formulae In an experiment, particles of mass m and energy E are used to bombard stationary target particles of mass 2m.

Question

Using the elastic scattering formulae

In an experiment, particles of mass m and energy E are used to bombard stationary target particles of mass 2m.

Accepted Answer

Q. The experimenters wish to select particles that, after scattering, have energy E/3. At what scattering angle will they find such particles?

A. If [latex] E_{1} / E_{0}=1 / 3[/latex], then by energy conservation [latex] E_{2} / E_{0}=2 / 3 [/latex]. First use formula D to find ψ. Since the mass ratio γ = 1/2, this gives

[latex] \frac{2}{3}=\frac{8}{9} \sin ^{2}\left(\frac{1}{2} \psi ight),[/latex]

so that ψ = 120°. Now use formula A to find the scattering angle [latex] heta_{1} [/latex]. This gives tan [latex] heta_{1}=\infty ext { so that } heta_{1}=90^{\circ} [/latex]. Particles scattered with energy E/3 will therefore be found emerging at right angles to the incident beam.

Q. In one collision, the opening angle was measured to be 45°. What were the individual scattering and recoil angles?

A. First use formula C to find ψ. This gives

[latex] \cot \left(\frac{1}{2} \psi ight)=\frac{1}{3} [/latex],

so that [latex] \frac{1}{2} \psi=72^{\circ} [/latex], to the nearest degree. Now use formula B to find the recoil angle [latex] heta_{2} ext {. This gives } heta_{2}=90^{\circ}-72^{\circ}=18^{\circ} [/latex]. The scattering angle [latex] heta_{1} [/latex] must therefore be [latex] heta_{1}= heta- heta_{2}=45^{\circ}-18^{\circ}=27^{\circ} [/latex].

Q. In another collision, the scattering angle was measured to be 45°. What was the recoil angle?

A. First use formula A to find ψ. This shows that ψ satisfies the equation

[latex] 2 \cos \psi-2 \sin \psi=1,[/latex]

which can be written in the form*

[latex] \sqrt{8} \cos \left(\psi+45^{\circ} ight)=1.[/latex]

This gives ψ = 24°, to the nearest degree. Formula B now gives the recoil angle [latex] heta_{2} [/latex] to be [latex] heta_{2}=78^{\circ} [/latex], to the nearest degree.

Question 10.7: Using the elastic scattering formulae In an experiment, part...

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