Strong interaction and isospin

[stefano]

I have a problem in understanding the dependence by isospin of strong interaction.
In this potential one have a lot of operatorial terms which include central, spin-spin, tensor force, spin-orbit terms, and all this four terms multiply by a factor isospin-dependent (tau_i dot tau_j).
I don't understand the last term.
What does it mean (tau_i dot tau_j)? If I have two nucleon, they can have tau_z=1/2 or -1/2. Dot product involves tau_x, tau_y and tau_z and why there are a dependence of other coordinates?
Because strong interaction is invariant for a system composed by neutron-neutron or proton-proton, one would have the same interaction if tau_iz=tau_jz=1/2 (proton-proton) or tau_iz=tau_jz=-1/2 (neutron-neutron). So I don't understand why strong interaction contains (tau_i dot tau_j) terms, instead (tau_zi tau_zj).

Thanks a lot.

 

[Major]

Hello !

I will try to help you a bit, but i am not really sure of that.

The $$\tau_{i} \cdot \tau_{j}$$ is the product of two operators, and it's equal to :

$$\frac{1}{2} \left((\tau_{i} +\tau_{j})^{2} - (\tau_{i})^{2} - (\tau_{j})^{2}\right)$$

Each term of your potential appears once multiplied by 1, et once multiplied by $$\tau_{i} \cdot \tau_{j}$$. It permits to satisify the "charge independance". Under this form, you don't care about the isospin projections, but only of the quantum number of isospin (which is 1/2 for a nucleon).

Another thing, the nuclear potential should be +/- the same for p-p, n-n and n-p. In fact, n-p is a bit different from p-p or n-n, but not much (see experimental datas for "miror" or "isobare" nuclei for example)

Remark : you may sometimes have a $$Q_{12}$$ term in your nuclear potential (quadratic correction), with $$Q_{12} = 2 \left[(s_{1} \cdot L) (s_{2} \cdot L) + (s_{2} \cdot L) (s_{1} \cdot L)\right]$$

Cya :)

 

[stefano]

Ok, but in this case is tau=1/2 for protons and tau=-1/2 for neutrons?

 

[Major]

the quantum number associated to the projection on the 3rd axis in fact ...

 

[marlon]

Stefano...

isospin is the quantumnumber that distinguishes between a proton and a neutron...basically it works just like the electronspin : up and down...isospin up corresponds to a proton and isospin down corresponds to a neutron...


The strong force is characterized by the fact that isospin is invariant under this interaction. This means that the interaction does not change when isospin is changed from up to down...

This is the most simple picture...

regards
marlon