Multiple Electrons in Rectangular Traps — Question 5

Question 5

A cubic trap closes its second shell with eight electrons

A cube of side L has energies $E=\epsilon(n_x^2+n_y^2+n_z^2)$, with all indices positive. Each spatial orbital holds two spins. Eight independent electrons are present.

1. Enumerate the first two spatial energy shells and their electron capacities.

2. Calculate the ground energy and lowest one-electron promotion cost.

Question 5 - Solution

A cubic trap closes its second shell with eight electrons

(a) Method and calculation

$$(1,1,1):3\epsilon,\ g_{\mathrm{spatial}}=1\Rightarrow N_1=2.$$$$(2,1,1)\text{ and permutations}:6\epsilon,\ g_{\mathrm{spatial}}=3\Rightarrow\boxed{N_2=6}.$$

(b) Interpretation and further calculation

$$\boxed{E_g=2(3\epsilon)+6(6\epsilon)=42\epsilon}.$$$$(2,2,1)\text{ and permutations}:9\epsilon\Rightarrow\boxed{\Delta E_{\min}=3\epsilon}.$$

Physical check

The three permutations of the second shell are distinct. Zero quantum numbers are not allowed for hard-wall confinement, so states with a zero index must not be counted.