The Pauli Exclusion Principle — Question 1

Question 1

A subshell capacity counts spin orbitals rather than spatial orbitals

For fixed n and l, the orbital projection has $2l+1$ values and each spatial orbital permits $m_s=\pm1/2$. Pauli exclusion allows at most one electron per complete spin orbital.

1. Derive the capacity of an l subshell.

2. Sum over $l=0,\ldots,n-1$ to find a complete shell capacity and evaluate it for $n=4$.

Question 1 - Solution

A subshell capacity counts spin orbitals rather than spatial orbitals

(a) Method and calculation

$$N_l=(2l+1)\times2=\boxed{4l+2}.$$$$\text{Spatial orbital count}=2l+1;\quad\text{spin-orbital count}=2(2l+1).$$

(b) Interpretation and further calculation

$$N_n=2\sum_{l=0}^{n-1}(2l+1)=2[n(n-1)+n]=\boxed{2n^2}.$$$$\boxed{N_4=32=2+6+10+14}.$$

Physical check

Shell capacity does not determine the energy ordering in a many-electron atom. Higher principal shells may begin filling before all lower-shell subshells are occupied.