Fall 2006 10 Misc

Nucleons (neutrons and protons) can be regarded as composed of three massless relativistic quarks freely moving inside a spherical volume with radius r and uniform energy per unit volume B inside the sphere. (The energy density outside the sphere is zero). Given that the experimentally measured rest energy of a nucleon is M ~ 1000 MeV and the radius of a nucleon is ~ 1 fm (10-15m), estimate the magnitude of B in MeV/m3.

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Answer

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The quarks are confined with radius r, so

(1)
\begin{align} E \sim 3 \left ( \frac{\hbar c }{r} \right ) + B r^3 \end{align}

$\partial E/ \partial r = 0$ implies that

(2)
\begin{align} \frac{\hbar c}{r^4} \sim B \end{align}

$\hbar c = 1.97 \times 10^{-13}$ MeV-m so

$B=\frac{1.97 \times 10^{-13}}{\left ( 10^{-15} \right )^4} = 1.97 \times 10^{47}$ MeV/m3

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