Scalar resonant Friedel-pair intensity difference ($I_{+}-I_{-}=4a{b}^{\prime \prime }\mathrm{Im}(A{B}^{\ast })$)

For positional structure factors $A,B$, a real scattering factor $a$, and a complex factor $b+i{b}^{\prime \prime }$, the amplitudes are $F_{+}=aA+(b+i{b}^{\prime \prime })B$ and $F_{-}=a{A}^{\ast }+(b+i{b}^{\prime \prime })B^{\ast }$. Expanding their squared moduli gives the displayed difference. Resonance and lack of inversion allow a difference but do not force one: proportional positional sums with a real proportionality factor give zero.