Past exam of the condensed matter physics course of the University of Oxford / 2015 / 2 / c / Solution

Replacing X-rays by neutrons of the same wavelength keeps the geometric Bragg positions, but replaces electron-density amplitudes by nuclear coherent scattering lengths. For Al all sites still carry the same nonzero amplitude, so its six reflections remain. For Si the identical two-site basis still has the same destructive-interference conditions, so its four reflections remain. Isotopic randomness can add diffuse/incoherent scattering but does not create a new periodic basis with different systematic Bragg absences.

K and Cl, however, have different nuclear amplitudes despite their isoelectronic ionic electron clouds. For natural isotopic composition, $b_K=3.67$ fm and $b_{\mathrm{Cl}}=9.577$ fm. The all-odd/all-even basis-intensity ratio becomeswhich is no longer the strong X-ray cancellation. The two odd rings should therefore be observable with suitable sensitivity, givingThe numerical contrast uses the NIST bound coherent lengths.^[ref] Counts here mean allowed and reasonably observable rings in the specified range, assuming sufficient angular coverage and resolution; no theory alone fixes an instrument's intensity threshold. These nonmagnetic examples use nuclear elastic scattering, not magnetic scattering.