What is the difference between parallel and antiparallel spins for a pair of nucleons?
It simply means that nucleons can be parallel in spin or opposite nature of spin- example of two nucleons
( up , up ) or ( down , down ) or ( up , down ) or (down, up)
if they are parallel total spin S = s(1) +s(2) = 1 a triplet state
if they are anti -parallel
total spin S = s(1) +s(2) = 0 a singlet state
the spin-spin or spin-orbit interactions will be different for two states.
which may affect the nuclear force.
the scattering cross sections for two states will give different results.
As nucleons are added in a nucleus in a specific state the nucleons get paired and the final spin state of the nucleus is decided by the odd nucleon (its spin). In case of even number nucleus the total spin is zero.
A characteristic of the collection of protons and neutrons (which are fermions) is that a nucleus of odd mass number A will have a half-integer spin and a nucleus of even A will have integer spin. The suggestion that the angular momenta of nucleons tend to form pairs is supported by the fact that all nuclei with even Z and even N have nuclear spin I=0.
Ref.-Reference-http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/nspin.html
Spin-spin coupling is the coupling of the intrinsic angular momentum (spin) of different particles.
Examples are-
Such coupling between pairs of nuclear spins is an important feature of (NMR) spectroscopy as it can provide detailed information about the structure and conformation of molecules.
Spin-spin coupling between nuclear spin and electronic spin is responsible for hyperfine structure in atomic spectral transitions.
Am I understanding this correctly? - I'm thinking I might be missing something in regards to the Pauli Exclusion Principle. How can two nucleons with the same spin state pair?
Due to the similarity in mass and nuclear properties between the proton and neutron, they are sometimes considered as two symmetric types of the same object, a nucleon.
The symmetry relating the proton and neutron is known as isospin and denoted I (or sometimes T).
Isospin is an SU(2) symmetry, like ordinary spin, so is completely analogous to it.
The proton and neutron form doublet, with a downstate (↓) being a neutron, and an up (↑) state is a proton.
A pair of nucleons can either be in an antisymmetric state of isospin called singlet or in a symmetric state called triplet.
An example-
A nucleus with one proton and one neutron, i.e. a deuterium nucleus.
and thus consists of three types of nuclei, which are supposed to be symmetric: a deuterium nucleus (actually a highly excited state of it), a nucleus with two protons, and a nucleus with two neutrons. The latter two nuclei are not stable or nearly stable, and therefore so is this type of deuterium (meaning that it is indeed a highly excited state of deuterium).
The deuteron wavefunction must be antisymmetric if the isospin representation is used (since a proton and a neutron are not identical particles, the wavefunction need not be antisymmetric in general). Apart from their isospin, the two nucleons also have spin and spatial distributions of their wavefunction. The latter is symmetric if the deuteron is symmetric under parity (i.e. have an "even" or "positive" parity), and antisymmetric if the deuteron is antisymmetric under parity (i.e. have an "odd" or "negative" parity). The parity is fully determined by the total orbital angular momentum of the two nucleons: if it is even then the parity is even (positive), and if it is odd then the parity is odd (negative).
The deuteron, being an isospin singlet, is antisymmetric under nucleons exchange due to isospin, and therefore must be symmetric under the double exchange of their spin and location.
Ref. < https://en.wikipedia.org/wiki/Deuterium#Spin_and_energy
Therefore one may try constructing the wave function of deuteron holding the Pauli principle and the ground state of deuteron will come to a triplet state of nucleons with Binding of about 2.2 MeV.
Best Answer
In general that is a difficult question from first principles. Simulation can often answer it, but the problem can be pretty involved and demanding.
However, as a practical matter the configuration and energy levels of many nuclei are known from extensive experiments. For example an online level diagram and a table of level data for Gd-157 from http://www.nndc.bnl.gov/nudat2/ .