= Single isomorphous replacement
\b[Single isomorphous replacement] estimates phases in <X-ray crystallography> by comparing a native <crystal> with one <heavy-atom derivative>. Under the ideal <crystal isomorphism> assumption, a reflection's derivative <structure factor> is $F_{PH}=F_P+F_H$, where the heavy-atom model supplies the complex reference $F_H$. The measured magnitudes constrain $F_P$ to two circles in the <complex plane>: one centered at zero with radius $|F_P|$, the other centered at $-F_H$ with radius $|F_{PH}|$. A general intersection gives two phase choices.<Phasing diagrams in CCP4 Basic Maths for Protein Crystallographers>[SIR, two-circle construction]{ref}
An original exact example uses $|F_P|=5$, $F_H=3$ real and positive, and $|F_{PH}|=\sqrt{58}$. Writing $F_P=x+iy$ gives
$$
x^2+y^2=25,\qquad(x+3)^2+y^2=58.
$$
Subtracting the equations gives $6x+9=33$, hence $x=4$ and $y=\pm3$. The alternatives are
$$
F_P=4+3i\quad\text{or}\quad4-3i,\qquad
\phi_P=\pm\arctan(3/4)\simeq\pm36.87^\circ.
$$
Both alternatives predict exactly the same native and derivative intensities. Greater precision in these two magnitudes alone cannot select between them. The phase is the missing direction of a complex amplitude, not an unmeasured additive intensity.
<Multiple isomorphous replacement> can supply an additional circle from another derivative. The numerical example there resolves these particular candidates. In actual measurements, uncertainty, imperfect <crystal isomorphism> and an imperfect heavy-atom model broaden the constraints, so exact circle crossings are an explanatory idealization.
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