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As a basis for direct-methods phasing at very low resolution for macromolecular crystal structures, normalized structure-factor algebra is presented for few-atoms structure models with N = 1, 2, 3, ... equal atoms or polyatomic globs per unit cell. Main results include: For 
,
For 
,
For 
, 
is obtained in an exact closed form as a rational function of nine magnitudes 
, 
, 
, 
, 
, 
, 
, 
, 
. For 
,
Triplet discriminant 
and triplet weight 
parameters, 
and 
, respectively, were determined empirically in numerical error analyses. Tests with phases calculated for few-atoms `super-glob' models of the protein apo-D-glyceraldehyde-3-phosphate dehydrogenase (~10000 non-H atoms) showed that low-resolution phases from the new few-atoms tangent formula were much better than conventional tangent formula phases for and 3; phases from the two formulae were essentially the same for 
.
, 
, 
, 
is obtained in an exact closed form as a rational function of nine magnitudes 
, 
, 
, 
, 
, 
, 
, 
, 
. For 
, ![\displaylines{ \quad\tan ({\varphi _{\bf h} - \alpha _{\bf h} } ) \approx {{ - \textstyle\sum\limits_{\bf k} {W_{{\bf hk}} } \sin ({\varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } )} \Big/ {\textstyle\sum\limits_{\bf k} {W_{{\bf hk}} } \cos ({\varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } ),}} \hfill\cr\quad \alpha _{\bf h} = \left\{ \matrix{ 0\quad{\rm if }\quad\textstyle\sum\limits_{\bf k} {\Delta _{{\bf hk}} | {\Delta _{{\bf hk}} } |} \,\gt\, 0 ,\hfill \cr \pi \quad{\rm if }\quad\textstyle\sum\limits_{\bf k} {\Delta _{{\bf hk}} | {\Delta _{{\bf hk}} } |}\, \lt\, 0 ,\hfill \cr} \right. \quad \Delta _{{\bf hk}} = | {E_{\bf h} }|{}^2 +| {E_{\bf k} } |{}^2 + | {E_{ - {\bf h} - {\bf k}} } |{}^2 - ({a /N}), \hfill\cr \quad W_{{\bf hk}} = | {\Delta _{{\bf hk}} E_{\bf h} E_{\bf k} E_{ - {\bf h} - {\bf k}} } | [{1 - ({1 - | {\Delta _{{\bf hk}} } |{}^{ - 1} } )\tanh ({N - b} )} ]. \hfill}](http://journals.iucr.org/a/issues/2002/04/00/dr0021//teximages/dr0021fd3.gif)
and triplet weight 
parameters, 
and 
, respectively, were determined empirically in numerical error analyses. Tests with phases calculated for few-atoms `super-glob' models of the protein apo-D-glyceraldehyde-3-phosphate dehydrogenase (~10000 non-H atoms) showed that low-resolution phases from the new few-atoms tangent formula were much better than conventional tangent formula phases for and 3; phases from the two formulae were essentially the same for 
.Supporting information
 | Portable Document Format (PDF) file https://doi.org/10.1107/S0108767302005597/dr0021sup1.pdf |
