Multiplet Structure - Isospin and Hypercharges

As far as strong interactions are concerned, the neutron and the proton are the two states of equal mass of a nucleon doublet. A glance at Tables 27.2B, C shows that particles can be grouped in multiplets of equal mass but different charges. Examples are shown below:

An important outcome of the multiplicity number M of an equal mass group is the concept of Isospin I. This is not a true mechanical spin but its quantum-mechanical derivation follows similar lines to that of electron spin in spectroscopy and obeys similar rules so that we put M = 2I + 1 for the charge multiplicity,

In the case of the nucleon doublet it is reasonable to assume that they are two states of the same nuclear particle. They are distinguished only by their charge and thus by the interaction of the proton with the electromagnetic field. The Isospin I = 1/2 is assigned to all nucleons but with the component I 3 = 1/2 for the proton and I 3 = - 1/2 for the neutron. Here I 3 = ±1/2 is the z component of I. Thus the proton and neutron form a doublet with the same Isospin. Similarly for the pion triplet we have M = 3 and 1 = 1 giving

The kaons are grouped into two pairs with I 3 = ± 1/2 for each pair. These are Finally, the delta particle, has four states, viz., for which I =3/2 and the I 3 values are 3/2, 1/2, -1/2and -3/2 respectively. In general there are 2I+1 isospin states for a particle of given I Using these data we now obtain isospins as follows:

Since Y and B are each conserved in strong and electromagnetic interactions S must also be so conserved. As S is a function of the quantum numbers Y and B it becomes redundant if Y and B are used, although it is still frequently used.

Classification of Elementary Particles By inspection of the list of particles now available and the multiplet structures to which they conform it is possible to regroup the mesons and baryons (all the strongly interacting particles), using only three of the conserved quantities just discussed. These are B, Y and I and we can now refer to these as conserved quantum numbers. This gives only four basic meson groups eta, pi, and the two kaon groups according to their B, Y and I values. These are shown in Table 27.4, where M is the charge multiplicity.

These are the well-established resonances. There are many more which are not fully confirmed but which would still fit the above patterns. They are being actively researched. It is evident then that the large number of particles can be reduced to simpler descriptions of families or regularities by the application of the conservation laws. These regularities or symmetries are not fortuitous. Is it possible therefore to devise a physical or mathematical model which would enable us to explain the above properties of all the known particles, and so help us in our search for new particles by predicting their properties in much the same way as searching for unknown elements in the periodic table?