# Law of Conservation of Isospin

In particle and nuclear physics, isospin, I or I3,  is a quantum number related to the strong nuclear force. Isospin is associated with a conservation law that 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 to distinguish 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 shown that the strong interaction does not distinguish between these nucleons. 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 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 the case of a proton, the spin is pointing upwards, and the neutron’s spin is pointing downwards.

In general, each multiplet is assigned an isospin number I that is a positive integer or half an odd positive integer. Isospin may be considered 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 the 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, 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. 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 an interaction of two nucleons, we are concerned with only two isospin states instead of dealing with four charge states (neutron-neutron, proton-proton, proton-neutron, neutron-proton), 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 that requires strong interaction decays to conserve isospin. For weak interactions, neither T3 nor T need to be conserved.

See also: J. Christman,  (2001, December 11). ISOSPIN: CONSERVED IN STRONG INTERACTIONS. Retrieved from http://www.physnet.org/modules/pdf_modules/m278.pdf

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