In the constituent quark model each quark's magnetic moment is

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model · Symmetry, groups and the quark model

Symmetry and Flavour, Ch The constituent quark model: wave functions, magnetic moments and hyperfine masses

Problem

In the constituent quark model each quark's magnetic moment is μq=Qqe2mq\mu_q = Q_q\dfrac{e}{2m_q}, and the nuclear magneton is μN=e2mp\mu_N = \dfrac{e}{2m_p}, so μq=QqmpmqμN\mu_q = Q_q\dfrac{m_p}{m_q}\mu_N. For the proton's spin-flavour wave function this gives μp=13(4μu−μd)\mu_p = \tfrac13(4\mu_u - \mu_d). Exchanging u↔du \leftrightarrow d (isospin symmetry) gives the neutron's moment μn=13(4μd−μu)\mu_n = \tfrac13(4\mu_d - \mu_u). Take equal constituent masses mu=md=336 MeVm_u = m_d = 336\,\mathrm{MeV}, proton mass mp=938.3 MeVm_p = 938.3\,\mathrm{MeV}, and quark charges Qu=+23Q_u = +\tfrac23, Qd=−13Q_d = -\tfrac13. ((a)) Find μu\mu_u and μd\mu_d in units of μN\mu_N. ((b)) Hence find μn\mu_n in units of μN\mu_N, and compare it with the measured value −1.913 μN-1.913\,\mu_N.

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