Abstract
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 we identify and correct a few inconsistencies in some of these earlier expressions.
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