**4.1. Nucleons and Nuclei**

Like the electronic structure of atoms, the basic structure
of neutrons and protons depend hardly at all on any of the parameters.
Ignoring for now the small effect of electric charge and
quark mass, proton and neutron structure are the same,
with labels related by isospin symmetry. Their internal
structure and mass are entirely determined by strong QCD SU(3) gauge
fields (gluons) interacting with each other and
with the quarks. There are no adjustable
parameters in the structure, not even a coupling constant, except for
the setting of the energy scale.
^{(8)}
Although these nucleon field
configurations are not really ``solved'', the equations which
govern them are known exactly and their structure
can be approximately solved
in lattice models of QCD which correctly estimate
for example the mass ratios of the proton and other hadrons. Basically, the
mass of the proton *m*_{proton} = 0.938 GeV
is some calculable dimensionless number (about 5) times the energy scale
_{QCD} fixed by the
strong interaction coupling constant.
The structure and mass of hadrons is as mathematically rigid as a Platonic
solid. Even so, because *n* and *p* are so similar,
the stability of the proton is very sensitive to the electromagnetic
effects and to the much smaller, and seemingly unrelated, up and down
quark masses, which break the symmetry.

Strong interactions not only create isolated
hadronic structures, but also bind them together into nuclei.
Although the individual hadrons are to first approximation
pure SU(3) solitons, nuclear structure is also directly influenced by
quark masses,
especially through their effect on the range of the nuclear potential.
The strong interactions of hadrons can
be thought of as being mediated by pions, which have relatively low mass
(*m*(_{0}) = 135 MeV)
and therefore a range which reaches significantly farther than
the hadronic radius.
The light quark masses determine the pion mass via breaking of
chiral symmetry,
*m*_{}
sqrt[*m*_{proton}(*m*_{u} +
*m*_{d})], and therefore
the details of nuclear energy levels are sensitive to
*u* and *d* masses.

The dependence of nuclear structures on quark masses and
electromagnetic forces is hard to compute exactly but we
can sketch the rough scalings.
The nuclear binding energy in a nucleus with *N* nucleons is about
*E*_{nuc}
*N*
*m*_{proton} where the specific binding energy per mass
is about
(*m*_{} /
*m*_{proton})^{2}
(*m*_{u} +
*m*_{d}) / *m*_{proton
} 10^{-2} and hence
the typical separation is
^{-1/2}
*m*_{proton}^{-1}.
The nuclear size therefore is typically
*R* *N*^{1/3}
^{-1/2}
*m*_{proton}^{-1}. Larger
nuclei develop increasing electromagnetic repulsion, scaling like
*E*_{em}
*N*^{2}/*R*. They become unstable above a
maximum charge where the nuclear and electrostatic energies match,

The basic reason for the number of stable nuclei is that the electromagnetic coupling is weak, but not extremely weak, compared to the strong interactions.

^{8} Ironically,
the nucleon rest mass (which of course includes most of the rest mass of
ordinary matter) is 99% dominated by the kinetic
energy of the constituents, including roughly
equal contributions from very light quarks and massless gluons.
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