The crystal structure of realgar

The summations involved in the evaluation of the structure amplitudes are of the type ~r Zi cos 2zchxi cos 2 " cos 2zdz i • sm sin ~vYi sin 1 over the whole molecule (excepting the potassium ion and sulphur atom which are treated separately).The variation of scattering amplitude with 0 was introduced after these preliminary summations.For each atom the trigonometrical factors required are listed, and the sums ~cos 2z~hxi h sin 0 etc. are derived.These sums provide the checks according to the formulaeIf however xi _< 0.1 or x~_ 0.9 the magnitude of cot gx i is large and the rounding-off errors disguise any real error in the tabulated values.In such cases the following formulae are to be preferred: H ~v (_ 1)h cos 2z~hxi h o -~ ½ ~-½( ~ 1 )~ cos 2z~Hx i-½( --1 )~ sin 2x~Hx i tan z~xi, H (--1)~ sin 2zehx i h o -------½ tan ~xi+½(--1) a tan ~x~ cos 2z~Hx~ ~-½(--1) H sin 2z~Hxi.The various factors having been checked in this way, the products cos 2~hx i cos 27~kyi sm sin are formed and checked in a similar way, e.g.T a ~ ~ cos 2z~hx i cos 2z~kyi h 0 (k constant) z ----cos 2z~kyi ~v cos 2zehx i ~ T 1 cos 2z~ky i .