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In many cases two coordinates (e.g. *x*_{j}, *y*_{j}) of any atom in the real
structure are the same as those of the superposition structure. There arises
the task of determining the third atomic coordinate (*z*_{j}). The method of
linear structure factor equations (SFE) by Kutschabsky^{2} and by Kutschabsky
and Höhne^{3} allows us to calculate these atomic coordinates (*z*_{j}) directly,
using the reflection of the first level of the reciprocal lattice (*F*(*hkl*)).

The basic relation follows directly from the formula for the structure factor

(2.1) |

where *P* is the number of equipoints in the unit cell, *N* is the number of
symmetrically independent atoms, *f*_{sj} is the atomic scattering factor and
is the radius vector of the centre of atom *s*, *j*.

Vector is defined by * + * + *, where (*, *, *) are the basic reciprocal vectors.

Using the symmetry matrices *R*_{s} and the translation *t*_{s} we obtain:

(2.2) |

The separation into the components leads to

(2.3) |

(2.4) |

In the most important cases the factor depends only on *l* (not *h*
or *k*). In those cases we obtain for structure factors *P*(*hkl*) with constant
*L*

(2.5) |

where the unknown variables cos 2 and sin 2 have been
denoted by *C*^{(L)}_{j} and *S*^{(L)}_{j}, respectively, and their known
coefficients by *a*_{j}, *b*_{j}, *c*_{j} and *d*_{j}.

*a*_{j} = *a*_{j}(*h*, *k*, *f*_{j}, *x*_{j}, *y*_{j}) etc. where the exact form of dependence on *h*,
*k*, *f*_{j}, *x*_{j} and *y*_{j} may be obtained from Table 4 of
*International Tables for X-ray Crystallography*, Vol. 1.

The 2*N* unknown variables *C*^{(L)}_{j} and *S*^{(L)}_{j} may be determined by a
system of linear equations using )for ).

If the phases of the are unknown the unobserved
reflections may be used to obtain a system of homogeneous linear equations.
Often it is of advantage to use in addition to the homogeneous equations one
equation belonging to a strong structure factor whose phase may be fixed
arbitrarily in centrosymmetrical space groups, and in the non-centrosymmetrical
space groups in which the origin may have any position in the *z*-direction.

Because the coefficients of these equations are inaccurate and, moreover, the
structure factors are zero only approximately a more accurate solution for the
values *C*^{(L)}_{j} and *S*^{(L)}_{j} may be obtained by using more equations than
there are variables and by minimizing the sum of the squares of the deviations
,where stands
for the right side of equation (2.5) and *F*(*H*) is to be replaced by
in this relation. The *C*^{(L)}_{j} and *S*^{(L)}_{j} from
the first calculation may be used to determine the phases of further structure
factors. Taking these equations in addition to those used already, the number
of equations increases and thus the accuracy of *C*^{(L)}_{j} and *S*^{(L)}_{j} is
improved.

If the *F*(*hkl*) with *L* = 1 are used, the atomic parameters *z*_{j} of all atoms
resolved in the (*x*, *y*)-projection follow from *C ^{(1)}*

**Copyright © 1981, 1998 International Union of
Crystallography**