4. Examples

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$\begin{displaymath} \langle{I}(hkl)_{l=2n}\rangle \mbox{ strong;} \qquad \langle{I}(hkl)_{l=2n+1}\rangle \mbox{ weak}.\end{displaymath}$

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 P2₁2₁2₁

$\begin{displaymath} \textstyle x, y, z; \quad \frac{1}{2} - x, \overline{y}, \fr... ...verline{z};\quad \overline{x}, \frac{1}{2} + y, \frac{1}{2} - z\end{displaymath}$

$\begin{displaymath} \textstyle (000,00\frac{1}{2}) + \frac{1}{4}, 0, z; \quad \frac{3}{4}, \frac{1}{2}, \overline{z}\end{displaymath}$

With $z^{\prime} = 2z$ this corresponds within the unit cell $a^{\prime} = a$ , $b^{\prime} = b$ , $c^{\prime} = c/2$ of the superposition structure, to the equipoints

$\begin{displaymath} \textstyle \frac{1}{4}, 0, z^{\prime}; \quad \frac{3}{4}, \frac{1}{2}, \overline{z}^{\prime}\end{displaymath}$

Obviously (Fig. 5), any of these two points lies on mirror planes perpendicular to $a^{\prime}$ and $b^{\prime}$ and are related by an n-glide plane perpendicular to $c^{\prime}$ . 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) 00z, $\frac{1}{2}\frac{1}{2}z$ would be found for Pmmn, which corresponds to the set found, after a shift of the origin by $a^{\prime}$ /4.

The space group for the superposition structure thus obtained may now be tested with the usual space group tests, and indeed, the hk0-reflections with h + k = 2n + 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.

$\begin{displaymath} \textstyle (x, y, z), \quad (x, y, \frac{1}{2} + z),\quad (\frac{1}{2} - x, y, z),\quad (\frac{1}{2} - x, y, \frac{1}{2} + z)\end{displaymath}$

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 = 2n + 1.

This indicated a deviation of x_j from $\frac{1}{4}$ , 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 P2₁2₁2₁.

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.

$\begin{displaymath} \langle{I}(hkl)_{h=2n}\rangle \mbox{ strong}; \quad \langle{I}(hkl)_{h=2n+1}\rangle \mbox{ weak}\end{displaymath}$

$\begin{displaymath} \textstyle a^{\prime} = \frac{a}{2},\quad b^{\prime} = b,\quad c^{\prime} = c.\end{displaymath}$

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 $F_{\mbox{obs}}(hkl)_{h=2n}$ but not of the $F_{\mbox{obs}}(hkl)_{h=2n+1}$ were determined.

With $F_{\mbox{obs}}(hkl)_{h=2n}$ a 3-dimensional Fourier synthesis of the superposition structure was calculated. This ${\rho}_{\mbox{sup}}(x, y, z)$ 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.

**Figure 6:** (y, z)-projection of the superposition structure with the pseudo-mirror-plane.
$\begin{figure} \includegraphics {fig6.ps} \end{figure}$

To preclude the spurious peaks in the Fourier synthesis a spatial minimum function M₄(x, y, z)¹⁰ was derived from the Patterson function by using the known bromine-bromine vectors.

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.

**Figure 7:** Peak coinciding in Fourier synthesis and minimum function.
$\begin{figure} \includegraphics {fig7.ps} \end{figure}$

monoclinic: P2₁/c, a = 6.51 Å, b = 8.64 Å, c = 8.24 Å, $\beta$ = 113.1 $^{\circ}$ ; Z = 2

$\begin{displaymath} \langle{I}(hkl)_{k+l=2n}\rangle \mbox{ strong,} \quad \langle{I}(hkl)_{k+l=2n+1}\rangle \mbox{ weak}.\end{displaymath}$

From Z = 2 and P2₁/c it follows that the bromine atom lies at the centre of symmetry forming, taken by themselves, an A-centred lattice. The $F_{\mbox{obs}}(hkl)_{k+l=2n}$ 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.

**Figure 8:** Electron density projection ${\rho} (x, z)$ of acetamide hemihydrobromide.
$\begin{figure} \includegraphics {fig8.ps} \end{figure}$

$\begin{displaymath} F(h1l)_{l=2n+1}=K\sum_j 4f_j \cos 2{\pi}(hx_j + lz_j) \cos 2{\pi}y_j\end{displaymath}$

$\begin{displaymath} F(h1l)_{l=2n}=-K\sum_j 4f_j \sin 2{\pi}(hx_j + lz_j) \sin 2{\pi}y_j\end{displaymath}$

where K is the scaling factor. The expressions $4f_j \cos 2{\pi}(hx_j + lz_j) = a_j$ and $-4f_j \sin 2{\pi}(hx_j + lz_j) = b_j$ may be calculated because x_j and z_j are known, whereas K cos 2 ${\pi}y_j$ = $C_j^{\prime}$ and K sin 2 ${\pi}y_j$ = $S_j^{\prime}$ are the unknown values. From the known position of the heavy atom (bromine) most of the signs of the F(h1l)_l=2n+1 could be determined and a system of equations (4.1) with a twelvefold overdetermination could be set up

$\begin{displaymath} F(h1l)_{l=2n+1} = \sum_j a_j(h1l)C_j^{\prime}.\end{displaymath}$

(4.1)

This system of equations was solved by least squares technique. Because the F(h1l) are on a relative scale the solutions $C_j^{\prime}$ were multiplied with a constant (1/K) so that (1/K) $C_{\mbox{Br}}^{\prime}$ = 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) $C^{\prime}$ (y_jo) obtained. To find out which of these two values is correct, equations of the type (4.2):

$\begin{displaymath} F(h1l)_{l=2n} = \sum_j b_j(h1l)S_j^{\prime}\end{displaymath}$

(4.2)

were used. The bromine atoms do not contribute to the F(h1l)_j=2n. The signs of these structure factors were unknown. Therefore the unobserved reflections F(h1l)_l=2n and one strong reflection F(h1l)_l=2n were used for setting up a system of inhomogeneous equations. The S_j = (1/k) $S_j^{\prime}$ 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.

**Figure 9:** The average structure ${\rho}(yz)$ of acetamide hemihydrobromide calculated only with $F_{\mbox{obs}}(0k1)$ for k + l = 2n. Atomic positions obtained with the help of *SFE* are marked by crosses, the refined positions are marked by squares.
$\begin{figure} \includegraphics {fig9.ps} \end{figure}$

monoclinic: Pc or P2/c, a = 5.86 Å, b = 6.93 Å, c = 7.78 Å, $\beta$ = 94 $^{\circ}$ ; Z = 2

$\begin{displaymath} \langle{I}(hkl)_{l=2n}\rangle \mbox{ strong}, \quad \langle{I}(hkl)_{l=2n+1}\rangle \mbox{ weak}\end{displaymath}$

The intensity statistic of Howells, Phillips and Rogers¹⁴ using the I(hkl) showed that the real structure has the centrosymmetric space group P2/c.

The Patterson function showed in agreement with Z = 2 and P2/c that the calcium atom occupies a special position on the twofold rotation axis with parameters $x_{\mbox{Ca}}$ = 0, $z_{\mbox{Ca}}$ = $\frac{1}{4}$ and $y_{\mbox{Ca}}$ 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(hk0). The Fourier projection ${\rho}(x, y)$ gave the positions of the oxygen and boron atoms. Because the signs of the F(hkl) with l = 2n + 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.

Two systems of equations were set up. For the first system F(hk1) and for the second F(hk2) 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
O₁	-	0.20	0.872	-	0.036	0.0359
O₂	1.00	0.51	0.703	0.756	0.070	0.0588
O₃	0.68	0.44	0.111	0.923	0.106	0.1042
O₄	-0.66	-0.74	-0.338	0.680	0.842	0.8193

monoclinic: P2₁, a = 11.37 Å, b = 8.62 Å, c = 7.48 Å, $\beta$ = 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 $y_{\mbox{Br}} = \frac{1}{4}$ . The 3-dimensional Fourier synthesis based on the phases of the $F_{\mbox{obs}}(hkl)$ derived from the bromine contributions is a superposition structure with the space group P2₁/m with an additional mirror plane at $y = \frac{1}{4}$ . For the calculation of this synthesis only F_o(hkl) for 0k4 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.

**Figure 10:** Comparison of the composite three dimensional electron density projected along 001 (only one full or dashed contour of the same height at arbitrary level is drawn) of atomic positions obtained by SFE (results from F(h21) are marked by open circles, from F(h31) by open squares, average values solid) and of the positions obtained by least-squares refinement marked by crosses.
$\begin{figure} \includegraphics {fig10.ps} \end{figure}$

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

$\begin{displaymath} A_o(h21) = K_1 \sum_j 2f_j \cos 2 {\pi}(hx_j + lz_j) \cos 2{\pi}2y_j\end{displaymath}$

$\begin{displaymath} A_o(h31) = K_2 \sum_j - 2f_j \sin 2{\pi}(hx_j + lz_j) \sin 2{\pi}3y_j\end{displaymath}$

where K₁ and K₂ are the scaling factors, two systems of equations were obtained taking A_o(hkl) equal to F_o(hkl). This may be done without creating large errors since the B_o(hkl), to which the bromine atoms do not contribute, are expected to be small. In these systems of equations only those A_o(hkl) were used whose signs could be deduced from the contributions $F_{\mbox{Br}}(hkl)$ of the bromine atoms.

The structure factors F_o(h21) and F_o(h31) are on a relative scale. The scaling factors K₁ and K₂ were given such values that (1/K₁) $C_{\mbox{Br}}^{(2)}$ = (1/K₂) $S_{\mbox{Br}}^{(3)}$ = 1. Four values for the coordinate y_j of any atom are in keeping with

$\begin{displaymath} \textstyle C_j^{(2)} = (1/K_1)C_j^{(2)}(y_{jo}) \quad \mbox{namely} \quad y_j = {\pm} y_{jo}, y_j = {\pm}y_{jo} + \frac{1}{2}.\end{displaymath}$

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⁾_j was improved using the results obtained from S⁽³⁾_j. The coordinates thus obtained were sufficiently accurate for a starting set for a least squares refinement of the real structure.