# What do the subscripts 1/2 and 3/2 for the p-orbitals refer to?

What does $p_{1/2}$ and $p_{3/2}$ mean when referring to electron configuration?

I've seen sub shells and I'm assuming it has something to do with spin. I am asking this in regard to the $8p_{1/2}$ and $8p_{3/2}$ shells and the way they get filled at different energy levels?

• See this earlier Q & A, Shape of the P1/2 Orbital – ron May 10 '17 at 3:23
• We prefer to not use MathJax in the title field, see here for details. – Martin - マーチン May 10 '17 at 5:22
• Essentially, spin-orbit coupling leads to the degeneracy of the 8p orbitals being lifted. ron's question has all the details. – orthocresol May 10 '17 at 12:59

Essentially, this derives from Dirac’s relativistic quantum mechanics.

In nonrelativistic quantum mechanics—the Schrödinger equation that you are probably highly familiar with—p orbitals are triply degenerate while d orbitals are 5-way degenerate. However, nonrelativistic quantum mechanics does not include the electronic property known as spin unless it is introduced in a separate step.

Dirac’s relativistic quantum mechanics solves this problem by introducing $$c$$ as a finite constant. Almost magically, his equations directly result in a quantum number for the spin, $$\vec s$$. However, the result is that the azimuthal quantum number no longer behaves like a true quantum number (i.e. it no longer describes stationary states) but thankfully the sum $$\vec j = \vec l + \vec s$$ does. Therefore, instead of considering just one $$\vec l$$ as a descriptor to decide between s, p and d orbitals, we are stuck with $$\left |\vec l - \vec s\right |$$ and $$\left | \vec l + \vec s\right |$$. For $$\vec l = 1$$ (a p orbital) this gives us the values of $$\vec j = \frac12$$ and $$\vec j = \frac32$$. There are two p orbitals of the $$\mathrm p_\frac32$$ type and one of the $$\mathrm p_\frac12$$ type. These are no longer degenerate, i.e. $$\mathrm p_\frac12$$ has a lower energy than the $$\mathrm p_\frac32$$ ones.

The orbital of the $$\mathrm p_\frac12$$ type happens to be spherical in nature. For a more thorough background, I recommend Nicolau’s extensive answer which I mostly paraphrased in the above paragraphs and which contains links to further materials.

• Why $p_{3/2}$ has higher energy than $p_{1/2}$? Thanks in advance. – ado sar Oct 22 at 18:47