 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s19/2/d/solution

The stated conversion gives

Θ_{c} ≃ 0.10

eV. Hence the ideal longitudinal bath temperature is

Θ_{∥} ≃ \frac{( 0.10 eV ) ^{2}}{4 ( 5.0 \times 1 0 ^{4} eV )} = 5.0 \times 1 0^{- 8} eV, T_{p, eq} \sim 5.8 \times 1 0^{- 4} K .

This estimate equates the cooled proton thermal-energy scale with the effective electron bath scale, not the two particles' random speeds. The proton root-mean-square speed for a one-component convention

m_{p} σ_{v}^{2} = k_{B} T

would be about

2.2 m s^{- 1}

The result is an ideal floor under the hinted cooling model. It is not a promise of a sub-millikelvin stored beam in a real ring, where noise, beam self-fields, intrabeam scattering and other heat sources enter the balance.

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