Contribution of the second Landau level to the exchange energy of the three-dimensional electron gas in a high magnetic field

We derive a closed analytical expression for the exchange energy of the three-dimensional interacting electron gas in strong magnetic fields, which goes beyond the quantum limit (L=0) by explicitly including the effect of the second, L=1, Landau level and arbitrary spin polarization. The inclusion of the L=1 level brings the fields to which the formula applies closer to the laboratory range, as compared to previous expressions, valid only for L=0 and complete spin polarization. We identify, and explain, two distinct regimes, separated by a critical density $n_c$. Below $n_c$, the per-particle exchange energy is lowered by the contribution of L=1, whereas above $n_c$ it is increased. As special cases of our general equation we recover various known, more limited, results for higher fields, and identify and correct a few inconsistencies in some of these earlier expressions.