Isospin (pt. 1)
So something that I’m running into in my nuclear research that I’ve literally never heard of before is the concept of isospin, or isobaric spin.
The motivation for introducing this concept comes from noting that the strong nuclear interaction doesn’t distinguish between protons or neutrons - if we disregard the coulomb interaction, and the slight mass difference related to this, then the two particles behave identically.
As such, it can be convenient to think of the two particles as two states of a single particle, the nucleon. In the same way than an electron has a spin quantum number s = 1/2, with two possible states arising from the different z-projections of its spin vector (s_z = 1/2 or -1/2, in units of h-bar), a nucleon has an isospin quantum number of T = 1/2, and is characterized by the isospin vector’s orientation in an abstract isospin space. That is, if the isospin’s projection along the space’s third axis is T_3 = 1/2, the particle is a neutron, and if the projection is T_3 = -1/2, the particle is a proton:
For entire nuclei, the components T_3 add just as we’d expect of vector components, so for a particular nucleus with N neutrons and Z protons,
But that’s just the T_3 component - the isospin vector can take on different orientations in isospin space, meaning its magnitude can take a range of values as its direction varies while the T_3 component remains fixed. Just like angular momentum, it can range from a minimum to a maximum in integer steps:
For a given T within that range, T_3 can be within:
But note from how T_3 was defined that different values of T_3 correspond to different nuclei, with the same nucleon number but with certain protons and neutrons interchanged. Such nuclei are called isobars, hence the name isobaric spin or isospin for short. These isobars, with different proton numbers (and thus different chemical symbols) but the same isospin T belong to what’s called an isomultiplet, with 2T+1 members. For example, T = 1 gives an isotriplet consisting of nuclei with T_3 = -1, 0, or 1.
I’ll get more into the perks of this formalism when I get a better feel for it, in another post!
Here’s said other post (part 2)