Atomic Paramagnetism: Understanding m_j Sublevels

[mooglue]

Hey everyone,

I've noticed that when paramagnetism is derived in the context of stat mech, we only consider the energy levels of the perturbation. Essentially, we take the fine structure of hydrogen, and we perturb it with a B-field, causing the weak zeeman effect to split the energy levels into m_j sublevels.

When we derive the partition function, we only take into account these levels. So, we weight the energies with bolzmann factors based on m_j.

My question is why don't we have to consider the n,j level distribtions. The energy levels in hydrogen as split by large gaps for various orbital quantum number n; yet, our partition function only accounts for the splitting of the j'th level into m_j sublevels. I've seen these derived many times, so I'm really not understanding why we don't consider the entire energy spectrum, and can only focus on the m_j sublevels.

Anyway help would be great.

 

[mooglue]

Actually, I think I may have a convincing argument for myself; however, if anyone still has anything to add, I'd of course still appreciate it.

 

[kanato]

I would guess the reason is that you're only interested in the low temperature magnetism, in which case you have the large splitting $$\Delta >> kT$$ so the effects of those excitations won't be apparent at the temperature scale you're interested in.

 

[manutdhk]

compared with the energy caused by n and j, the energy split caused by m_j is very small. In the partition function, the former one can be treated as a constant: exp(-E(n,j)/kT). However, the key of partition function is the energy distribution probability. If added a constant in front of this probability means nothing, due to normalize.