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**A. Demissidine hydroiodide ^{8}**

Crystal data: C_{27}H_{45}NO.HI.C_{2}H_{5}OH

orthorhombic: *P*2_{1}2_{1}2_{1}; *a* = 23.0 Å, *b* = 7.6 Å, *c* = 16.0
Å; *Z* = 4

Observed systematic intensity distribution:

According to the chemical formula and number of molecules per unit cell there
are 4 heavy atoms per unit cell, i.e. one per asymmetric unit. Thus the 2 heavy
atoms related by a shift of *c*/2 (on account of systematic intensity
distribution) must necessarily belong to the same set of equipoints. This
results if and only if the atoms lie on screw dyads parallel to *c* (see Fig.
5), thus the set of equipoints in *P*2_{1}2_{1}2_{1}

specializes to

With this corresponds within the unit cell , , of the superposition structure, to the equipoints

Obviously (Fig. 5), any of these two points lies on mirror planes perpendicular
to and and are related by an *n*-glide plane
perpendicular to . Thus the space group of the superposition
structure is *Pmmn*.

The same result could have been obtained by scanning the orthorhombic higher
symmetry group for such equipoints. Then the set of special positions (a)
00*z*, would be found for *Pmmn*, which corresponds to
the set found, after a shift of the origin by /4.

The space group for the superposition structure thus obtained may now be tested
with the usual space group tests, and indeed, the *hk*0-reflections with *h* +
*k* = 2*n* + 1 are weak (corresponding to the *n*-glide plane). The
superposition structure thus contains for each of 4 symmetry related atoms (*x*,
*y*, *z*) etc., of the real structure the following sets of 4 atoms.

i.e. 4 atoms to any atom of the real structures. This superposition structure would be obtained, if the usual heavy atom technique could be applied, and would certainly be difficult to interpret.

The Patterson function gave, however, not only the *z*-coordinate of the heavy
atom but also hinted that it may not lie exactly on the dyad screw, but only
approximately so; this was confirmed by the Patterson of the complementary
structure, obtained from the reflections with *l* = 2*n* + 1.

This indicated a deviation of *x*_{j} from , and this deviation
results in contribution of reflections with high values of *h* which even
determine their phases.

The iodine parameters were refined and with the resulting phases a first Fourier
synthesis of the complementary structure was obtained in space group
*P*2_{1}2_{1}2_{1}.

This result was compared with the known part of the model and thus a part of the structure deduced and used as a starting point for the final determination of the real structure.

**B. Piperidino-acet-m-bromo-anilide ^{9}**

Crystal data: C_{13}H_{17}N_{2}OBr

orthorhombic: *Pbca*; *a* = 23.65 Å, *b* = 12.66 Å, *c* = 9.37
Å; *Z* = 8

Observed systematic intensity distribution:

Space group of : *Pbcm* with lattice parameters

The 3-dimensional Patterson function explained the systematic distribution in
the intensities by the particular position of the bromine-atom on the *a*-glide
plane with the fractional coordinates *x* = 0.159, *y* = 0.193, *z* = 0.25.
With the known position of the bromine atom (refined by least squares methods)
the signs of most of the but not of the
were determined.

With a 3-dimensional Fourier synthesis of the
superposition structure was calculated. This involves
perpendicular to *c* an additional mirror plane, not existing in the real
structure, through the bromine atom. That is why each maximum in the synthesis
has a corresponding reflected one (Fig. 6). But only one of these pairs
corresponds to an atom in the real structure. In addition many maxima occurred
in the Fourier synthesis which do not refer to atoms. Therefore the
interpretation of the synthesis by the model of the molecule failed.

To preclude the spurious peaks in the Fourier synthesis a spatial minimum
function *M _{4}*(

The comparison of Fourier synthesis and minimum function revealed to which of
the pairs of peaks connected by the mirror plane atoms could be assigned. These
peaks of the superposition structure are shown in Fig. 7. Its
(*x*, *y*)-projection is identical with the projection of the real structure.
A model of the molecule enabled us to determine the *z*-coordinates of
the atoms by eliminating the ambiguity in the Fourier synthesis.

**C. Acetamide hemihydrobromide ^{11,12}**

Crystal data: (CH_{3}CONH_{2})_{2}.HBr

monoclinic: *P*2_{1}/*c*, *a* = 6.51 Å, *b* = 8.64 Å, *c* = 8.24 Å,
= 113.1; *Z* = 2

From *Z* = 2 and *P*2_{1}/*c* it follows that the bromine atom lies at the centre of
symmetry forming, taken by themselves, an *A*-centred lattice. The
correspond to a superposition structure with the
space group *A* 2/*m*, which has a mirror plane perpendicular *b* in
addition to the space group of the real structure.

The (

and

where *K* is the scaling factor. The expressions
and
may be calculated
because *x*_{j} and *z*_{j} are known, whereas *K* cos 2 = and *K* sin 2 = are the unknown values. From the known
position of the heavy atom (bromine) most of the signs of the *F*(*h*1*l*)_{l=2n+1}
could be determined and a system of equations (4.1) with a twelvefold
overdetermination could be set up

(4.1) |

This system of equations was solved by least squares technique. Because the
*F*(*h*1*l*) are on a relative scale the solutions were multiplied
with a constant (1/*K*) so that (1/*K*) = 1 is valid.

Two values *y*_{j} = *y*_{jo} and *y*_{j} = 1 - *y*_{jo} are in keeping with the
solutions *C*_{j} = (1/*K*)(*y*_{jo}) obtained. To find out which of
these two values is correct, equations of the type (4.2):

(4.2) |

were used. The bromine atoms do not contribute to the *F*(*h*1*l*)_{j=2n}. The
signs of these structure factors were unknown. Therefore the unobserved
reflections *F*(*h*1*l*)_{l=2n} and one strong reflection *F*(*h*1*l*)_{l=2n} were used
for setting up a system of inhomogeneous equations. The *S*_{j} =
(1/*k*) were less accurate than the *C*_{j} because this system of
equations had only a twofold overdetermination. That is why the absolute values
of the *y*-coordinates were calculated from the *C*_{j}, but the ambiguity was
eliminated by the *S*_{j}. The results are shown in Fig. 9. Structure refinement
proved these approximate values to be correct.

**D. -Calcium tetraborate hydrate ^{13}**

Crystal data: CaB_{2}O_{4}.4H_{2}O

monoclinic: *Pc* or *P*2/*c*, *a* = 5.86 Å, *b* = 6.93 Å, *c* = 7.78
Å, = 94; *Z* = 2

Observed systematic intensity distribution:

The intensity statistic of Howells, Phillips and Rogers^{14} using the
*I*(*hkl*) showed that the real structure has the centrosymmetric space group
*P*2/*c*.

The Patterson function showed in agreement with *Z* = 2 and *P*2/*c* that the
calcium atom occupies a special position on the twofold rotation axis with
parameters = 0, = and approximately
zero. This position is near the *c*-glide plane and thus explains why the
reflections *I*(*hkl*)_{l=2n+1} are systematically weak.

On the other hand the calcium atom determined most of the signs of the *F*(*hk*0).
The Fourier projection gave the positions of the oxygen and boron
atoms. Because the signs of the *F*(*hkl*) with *l* = 2*n* + 1 were not determined
by the contribution of the calcium atom the *z*-parameters of the atoms could
not be derived from a Fourier synthesis based on the contributions of the
calcium atom to the sign of the *F*(*hkl*). But with the SFE-method the
approximate *z*-coordinates were easily obtained.

From the structure factor formula follows

Two systems of equations were set up. For the first system *F*(*hk*1) and for the
second *F*(*hk*2) were used. In each case the unobserved structure factors and
one strong structure factor with arbitrary sign was used. These systems of
equations gave approximate values for *C*^{(L)}_{j} and *S*^{(L)}_{j}, by which
*F*_{c}(*hkl*) were calculated.

By comparing the *F*_{c}(*hkl*) with the *F*_{o}(*hkl*) the signs of more structure
factors could be determined. The corresponding equations were added to the
previous systems of equations. In this way the overdetermination of the systems
of equations was increased and the accuracy of the results improved. The final
results obtained after several cycles are listed in the Table. The last column
contains the refined parameter for comparison.

Atom | C^{(1)}_{j} |
S^{(1)}_{j} |
C^{(2)}_{j} |
S^{(2)}_{j} |
z_{j} |
z_{j}
refined |

Ca | - | 1.00 | -1.000 | - | 0.250 | 0.25 |

O_{1} |
- | 0.20 | 0.872 | - | 0.036 | 0.0359 |

O_{2} |
1.00 | 0.51 | 0.703 | 0.756 | 0.070 | 0.0588 |

O_{3} |
0.68 | 0.44 | 0.111 | 0.923 | 0.106 | 0.1042 |

O_{4} |
-0.66 | -0.74 | -0.338 | 0.680 | 0.842 | 0.8193 |

**E. Dimethylaminomethylpinene ^{12,15}**

Crystal data: C_{13}H_{24}NBr

monoclinic: *P*2_{1}, *a* = 11.37 Å, *b* = 8.62 Å, *c* = 7.48 Å,
= 97.4; *Z* = 2

The *x*- and *z*-parameters of the bromine atom were determined from the
Patterson synthesis and refined by Fourier methods. The *y* coordinate was
chosen arbitrarily as . The 3-dimensional Fourier
synthesis based on the phases of the derived from the bromine
contributions is a superposition structure with the space group *P*2_{1}/*m* with an
additional mirror plane at . For the calculation of this
synthesis only *F*_{o}(*hkl*) for 0*k*4 were available because the crystals were
very small. The Fourier synthesis revealed the positions of all non-hydrogen
atoms, most of them resolved in the *x* and *z* directions. Nearly all these
atoms, however, located so closely to the pseudo mirror plane that the peak
corresponding to one atom and its mirror image were not separated but formed an
elliptical maximum with its peak on the mirror plane (Fig. 10).

The main problem of the structure determination was to determine the small deviations of the light atoms from this pseudo mirror plane.

Analysis of the peak shape resulted in rather inaccurate values of the
deviations from the mirror plane. Better values were obtained by the SFE
method. To start with, the positions of all atoms in (*x*, *z*)-projection were
refined by difference Fourier synthesis to an *R* value of 0.16. Using the
formulae

and

where *K _{1}* and

With the abbreviations

the systems of the equations have the form

The structure factors *F*_{o}(*h*21) and *F*_{o}(*h*31) are on a relative scale. The
scaling factors *K _{1}* and

Two of these values for any atom could be excluded by comparison with the
Fourier synthesis of the superposition structure (see above). From the two
remaining values one could be precluded for most of the atoms by using a model
of the molecule (Fig. 10). The accuracy of the *y* coordinates obtained from
the *C ^{(2)}*

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