# Reference request: equivalence between formulas in fixed point and first-order logic

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in some sort of fixed-point logic to have an equivalent first-order formula.

Note that a recent post had a comment that introduced a textbook specifically on finite models. On the other hand I am interested in models that are not necessarily finite, and the specific relationship between first-order and fixed-point logics as stated above.