The highest reflection intensity in a resolution shell

The Gumbel-Fisher-Tippett (GFT) extreme-value analysis is applied to evaluate the distribution, expectation value and standard deviation of the intensity J of the strongest reflection in a given resolution shell in the X-ray diffraction pattern of a crystal with many scattering atoms in the unit cell. For convenience, intensities are measured in units of the average reflection intensity in the resolution shell and, for simplicity, centric and acentric reflections are treated separately. For acentric reflections, the expectation value mu and standard deviation sigma of J are mu = ln n + gamma and sigma = pi/6(1/2), where n is the number of crystallographically independent reflections in the resolution shell and gamma approximately 0.577 is the Euler-Mascheroni constant. For centric reflections with expectation value 1 for the intensity, the corresponding expressions are mu = 2(ln n + gamma) - ln(pi ln n) and sigma = 2pi/6(1/2) - pi/(6(1/2) ln n). Extensive numerical simulations show that these formulas are excellent approximations for random atom configurations at all resolutions, and good approximations for real protein crystal structures in the resolution range between 2.5 and 1.0 A.