Isospin

What is Isospin

In particle physics and nuclear physics, isospin, I or I3,  is a quantum number related to the strong nuclear force. Isospin is associated with a conservation law which requires strong interaction decays to conserve isospin. This term was derived from isotopic spin, but physicists prefer the term isobaric spin, which is more precise in meaning.

Isospin was introduced by a German theoretical physicist and one of the key pioneers of quantum mechanics Werner Karl Heisenberg in 1932 as a way of distinguishing between protons and neutrons. It must be added, the concept of isospin was introduced before the development of the quark model, in the 1960s, which provides our modern understanding.

The observations have showed that the strong interaction does not distinguish between these nucleon. The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as neutrons or as protons. Instead of regarding protons and neutrons as totally different species, as far as strong interactions are concerned, they are regarded as being different isospin states of the same underlying nucleon particle. This particle is called the nucleon. Similarly, the three pions, π0,  π+ and π, seem to be only three different states of the same particle, when only a strong nuclear force interacts. Isospin is mathematically similar to spin, though it has nothing to do with angular momentum. The spin term is tacked on because the addition of the isospins follows the same rules as spin.

The proton has isotopic spin ½ as the neutron has isotopic spin ½, but in case of proton the spin is pointing upwards and the spin of neutron is pointing downwards.

In general, each multiplet is assigned a isospin number I that is a positive integer or half an odd positive integer. Isospin may be considered to be a vector not in coordinate space (x, y, z). The third component, T3, may take on any one of the values T, T − 1, T − 2, . . ., −T in a fashion similar to the values of the z component of angular momentum. Two other components, T1 and T2 can be ignored. Each of T3 values corresponds to a different member of the multiplet. There are 2T + 1 particles in the multiplet. This result follows from counting the possible values of T3. Thus singlets have T = 0, doublets have T = 1/2, and triplets have T = 1.

Total isospin for a collection of particles is computed in the same manner as for ordinary spin. , The maximum value of system isospin is the sum of the individual particle’s isospins. For example if one considers π+ – p scattering, Tmax = 3/2 and T3 = 3/2 so T can only have the value 3/2. For π − p scattering, Tmax = 3/2 and T3 = −1/2 so T can be either 3/2 or 1/2. In fact, measurement of T for π − -p sometimes produces T = 3/2 and sometimes T = 1/2 but always T3 = −1/2.

Conservation of Isospin
The benefit of introducing isospin is that it vastly simplifies the problems of strong interactions. Thus for a interaction of two nucleons, instead of dealing with four charge states (neutron-neutron, proton-proton, proton-neutron, neutron-proton) we are concerned with only two isospin states, I = 1 and I = 0. The isospin concept tells us why certain strong interactions are forbidden on account of violation of isospin I. Isospin is associated with a conservation law which requires strong interaction decays to conserve isospin. For weak interactions neither T3, nor T need be conserved.

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN: 978-0471805533
7. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
8. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
9. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.