The Minimal Principle Solves Some Crystal Structures

Abstract

It is assumed throughout that a crystal structure S in the space group G and consisting of N identical atoms in the asymmetric unit is fixed, but unknown, that the magnitudes ∣E∣ of the normalized structure factors E are known, and that a sufficiently large base of phases, corresponding to the largest magnitudes ∣E∣, is specified. The magnitudes ∣E∣ determine a function R(I), called the minimal function and defined on the space of those structure invariants I which are generated by the specified base of phases: $$ R\left( I \right) = \frac{{\sum\limits_{H,K} {{A_{HK}}} {{\left\{ {\cos {\Phi _{HK}} - \frac{{{I_1}\left( {{A_{HK}}} \right)}} {{{I_0}({A_{HK}})}}} \right\}}^2} + \sum\limits_{L,M,N} {\left| {{B_{LMN}}} \right|{{\left\{ {\cos {\Phi _{LMN}} - \frac{{{I_1}\left( {{A_{HK}}} \right)}} {{{I_0}({A_{HK}})}}} \right\}}^2}} }} {{\sum\limits_{H,K} {{A_{HK}}} + \sum\limits_{L,M,N} {\left| {{B_{LMN}}} \right|} }}$$ ((1)) where $$ {A_{HK}} = \frac{2} {{{{\left( {nN} \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}}}\left| {{E_H}{E_K}{E_{H + K}}} \right| $$ ((2)) $$ {B_{LMN}} = \frac{2} {{nN}}\left| {{E_L}{E_M}{E_N}{E_{L + M + N}}} \right|\left( {{{\left| {{E_{L + M}}} \right|}^2} + {{\left| {{E_{M + N}}} \right|}^2} + {{\left| {{E_{N + L}}} \right|}^2} - 2} \right) $$ ((3)) $$ {\Phi _{HK}} = {\Phi _H} + {\Phi _K} + {\Phi _{ - H - K}} $$ ((4)) $$ {\Phi _{LMN}} = {\Phi _L} + {\Phi _M} + {\Phi _N} + {\Phi _{ - L - M - N}} $$ ((5)) n is the order of the space group and I1 and I0 are the Modified Bessel Functions. In view of (4) and (5), the ΦHK are seen to be triplets (three-phase structure invariants) and the ΦLMN are quartets (four-phase structure invariants). From (2), AHK > 0 but, from (3), BLMN may be greater than zero or less than zero depending on whether the three “cross-terms”, ∣EL+M∣, ∣EM+N∣, ∣EN+L∣ are all large or all small, respectively.