150
4 Two-Particle Systems in the Berggren Basis
interaction, simple characteristics of various nucleon-nucleon systems could be
experimentally discovered.
Two-nucleon systems have largest binding energy when the constituent nucleons
form a symmetric radial wave function, which maximizes overlap of their wave
functions and hence maximizes binding energy. When the two-nucleon wave
function is antisymmmetric, the two nucleons cannot form a bound system. This fact
is included in standard nucleon-nucleon interactions [2]. Indeed, due to the Pauli
principle, two protons and two neutrons have to be antisymmetric in coordinate
space, forming the 1 S 0 singlet state (S = 0 and T = 1). Consequently, the nucleonnucleon interaction in the S = 0 and T = 1 channel is constructed in such a way
that diproton or dineutron are unbound. On the contrary, a proton-neutron system is
not subject to this restriction, as proton and neutron are unlike particles, so that they
can be in the 3 S 1 triplet state (S = 1 and T = 0). The T = 0 nuclear interaction is
symmetric in space, whereas the T = 1 nuclear interaction is antisymmetric therein.
Therefore, the T = 0 proton-neutron interaction is stronger than that of the protonproton or neutron-neutron interaction. Consequently, the S = 1, T = 0 channel
possesses a bound state (deuteron ground state) close to the dissociation threshold,
of energy equal to −2.224 MeV.
A fundamental nontrivial characteristic of deuteron is the relative admixture of
S and D waves in the ground state, of respective weights close to 95% and 5% [2].
Indeed, a central and spin-orbit interaction would only generate S components, so
that the small but noticeable presence of D waves implies that the nuclear interaction
has also a tensor component.
The very different nature of bound deuteron compared to the unbound diproton
and dineutron reflects one of the most important aspects of the nuclear interaction,
which is the charge independence. Indeed, proton and neutron behave similarly
in the presence of the nuclear interaction. In particular, the singlet state S = 0,
T = 1 has almost the same binding energy in dineutron and deuteron. The charge
independence of nuclear interaction is, however, not exact. Proton and neutron
should have exactly the same mass in order to behave identically in the presence
of nucleon-nucleon interaction. However, the masses of the up (u) and down (d)
quarks are different, the down quark being about 2 MeV heavier than the up quark.
Consequently, neutron of udd structure is slightly heavier than proton of uud
structure and their masses equal 939.57 MeV/c 2 and 938.27 MeV/c 2 2, respectively.
While noticeable experimentally, this slight charge symmetry breaking is negligible
as compared to that which is generated by the Coulomb interaction.
The approximate charge independence of the nuclear interaction has lead Heisenberg to introduce the concept of nuclear isospin [3]. Obviously, isospin symmetry
is broken by the Coulomb interaction. Moreover, significant breaking of the isospin
symmetry is also expected in loosely bound or resonance states, as in general the
proton and neutron emission thresholds are different. All those effects can be clearly
visible when considering the corresponding spectroscopic factors and radial overlap
functions (see Chap. 7).
The Coulomb interaction strongly influences the nature of the singlet state of
the diproton compared to that of the dineutron and of the 1 S 0 singlet state of the
4 Two-Particle Systems in the Berggren Basis
interaction, simple characteristics of various nucleon-nucleon systems could be
experimentally discovered.
Two-nucleon systems have largest binding energy when the constituent nucleons
form a symmetric radial wave function, which maximizes overlap of their wave
functions and hence maximizes binding energy. When the two-nucleon wave
function is antisymmmetric, the two nucleons cannot form a bound system. This fact
is included in standard nucleon-nucleon interactions [2]. Indeed, due to the Pauli
principle, two protons and two neutrons have to be antisymmetric in coordinate
space, forming the 1 S 0 singlet state (S = 0 and T = 1). Consequently, the nucleonnucleon interaction in the S = 0 and T = 1 channel is constructed in such a way
that diproton or dineutron are unbound. On the contrary, a proton-neutron system is
not subject to this restriction, as proton and neutron are unlike particles, so that they
can be in the 3 S 1 triplet state (S = 1 and T = 0). The T = 0 nuclear interaction is
symmetric in space, whereas the T = 1 nuclear interaction is antisymmetric therein.
Therefore, the T = 0 proton-neutron interaction is stronger than that of the protonproton or neutron-neutron interaction. Consequently, the S = 1, T = 0 channel
possesses a bound state (deuteron ground state) close to the dissociation threshold,
of energy equal to −2.224 MeV.
A fundamental nontrivial characteristic of deuteron is the relative admixture of
S and D waves in the ground state, of respective weights close to 95% and 5% [2].
Indeed, a central and spin-orbit interaction would only generate S components, so
that the small but noticeable presence of D waves implies that the nuclear interaction
has also a tensor component.
The very different nature of bound deuteron compared to the unbound diproton
and dineutron reflects one of the most important aspects of the nuclear interaction,
which is the charge independence. Indeed, proton and neutron behave similarly
in the presence of the nuclear interaction. In particular, the singlet state S = 0,
T = 1 has almost the same binding energy in dineutron and deuteron. The charge
independence of nuclear interaction is, however, not exact. Proton and neutron
should have exactly the same mass in order to behave identically in the presence
of nucleon-nucleon interaction. However, the masses of the up (u) and down (d)
quarks are different, the down quark being about 2 MeV heavier than the up quark.
Consequently, neutron of udd structure is slightly heavier than proton of uud
structure and their masses equal 939.57 MeV/c 2 and 938.27 MeV/c 2 2, respectively.
While noticeable experimentally, this slight charge symmetry breaking is negligible
as compared to that which is generated by the Coulomb interaction.
The approximate charge independence of the nuclear interaction has lead Heisenberg to introduce the concept of nuclear isospin [3]. Obviously, isospin symmetry
is broken by the Coulomb interaction. Moreover, significant breaking of the isospin
symmetry is also expected in loosely bound or resonance states, as in general the
proton and neutron emission thresholds are different. All those effects can be clearly
visible when considering the corresponding spectroscopic factors and radial overlap
functions (see Chap. 7).
The Coulomb interaction strongly influences the nature of the singlet state of
the diproton compared to that of the dineutron and of the 1 S 0 singlet state of the