From Current Literature

Direct Determination of the Phases of Fourier Coefficients from X-ray Measurement Data*)

D. Harker and J. S. Kasper

The intensity of an X-ray beam reflected from a crystal face determines the absolute value of the structural (“crystal-structural”) factor \( |F_{hkl}| \), but does not give the phase of the latter. It is possible, however, to establish a sufficiently narrow limit for the magnitude of the phase, or, if \(F_{hkl}\) is a real quantity, to determine its sign by combining intensity data for several diffraction reflections. In the following lines the general principles of such a method are set forth; a more detailed exposition will be given by us later.

Although this is not absolutely necessary, it is very useful for clarifying the method to assume that the atomic (“atomic-structural”) factors \(f_j\) of the atoms composing the crystal are related to one another by the ratio \(f_j = z_j \hat f\), where \(z_j\) is the atomic number of the \(j\)-th atom, and \(\hat f\) is a function of \(\dfrac{\sin \theta}{\lambda}\), which we call the unit (normalized) atomic-structural factor. If the number of atoms in the unit cell of the crystal is \(N\), and if \(z = \sum_{j=1}^{N} z_j\) is equal to the total number of electrons in the unit cell, then \(\sum_{j=1}^{N} f_j = z \hat f\). Since

\[ F_{hkl} = \sum_{j=1}^{N} f_j \exp[-2\pi i (h x_j + k y_j + l z_j)], \]

the assumption for \(\hat f\) makes it possible to write

\[ \hat F_{hkl} = \sum_{j=1}^{N} n_j \exp[-2\pi i (h x_j + k y_j + l z_j)]. \]

*) D. Harker and J. S. Kasper, Phases of Fourier Coefficients Directly from Crystal Diffraction Data. Journ. of Chem. Physics 15, 882–884 (1947).

where \(\widehat{F}_{hkl}=F_{hkl}/z\hat f\) and \(n_j=z_j/z\). We have called the quantity \(\widehat{F}_{hkl}\) the unit crystal-structure factor, or the unit (normalized) Fourier coefficient. This manner of writing implies that all atoms in the crystal have the same “shape.”

To the expression for \(\widehat{F}_{hkl}\) we may apply the well-known Cauchy inequality, namely:

\[ \left|\sum_{j=1}^{N} a_j b_j\right|^2 \le \left(\sum_{j=1}^{N}|a_j|^2\right) \left(\sum_{j=1}^{N}|b_j|^2\right). \]

Putting

\[ a_j=\sqrt{n_j} \quad\text{and}\quad b_j=\sqrt{n_j}\exp[-2\pi i(hx_j+ky_j+lz_j)], \]

we obtain

\[ |\widehat{F}_{hkl}|^2 \le \left\{\sum_{j=1}^{N} n_j\right\} \left\{\sum_{j=1}^{N} n_j \exp[-2\pi i(2hx_j+2ky_j+2lz_j)]\right\}, \]

or \(|\widehat{F}_{hkl}|^2 \le 1\), since

\[ \sum_{j=1}^{N} n_j=1. \]

This result explains the term “unit” (normalized) in the designation \(\widehat{F}_{hkl}\).

If the crystal possesses symmetry elements, then Cauchy’s inequality leads to more interesting relations. Thus, in particular, if there is a center of inversion, then we write

\[ \widehat{F}_{hkl} = 2\sum_{j=1}^{N/2} n_j \cos 2\pi(hx_j+ky_j+lz_j) \]

and, noting that in this case \(\widehat{F}_{hkl}\) is a real quantity, on the basis of Cauchy’s inequality we have

\[ \widehat{F}_{hkl}^{\,2} \le 4\left\{\sum_{j=1}^{N/2} n_j\right\} \left\{\sum_{j=1}^{N/2} n_j \cos^2 2\pi(hx_j+ky_j+lz_j)\right\}. \]

Applying the elementary formula \(2\cos^2 a=1+\cos 2a\) and making a simple regrouping, we obtain:

\[ \widehat{F}_{hkl}^{\,2} \le \frac{1}{2}+\frac{1}{2}\widehat{F}_{2h\,2k\,2l}. \]

This important relation requires, for example, that \(\widehat{F}_{2h\,2k\,2l}\) be positive or equal to zero if \(\widehat{F}_{hkl}\ge \frac{1}{2}\). If, however, \(|\widehat{F}_{2h\,2k\,2l}|>\frac{1}{2}\), then \(\widehat{F}_{2h\,2k\,2l}\) must be positive when \(\widehat{F}_{hkl}^{\,2}\ge \frac{1}{4}\), etc. Conversely, if this inequality is not satisfied, then the crystal cannot have a center of symmetry (this is not in contradiction with Friedel’s law, since the latter states only that \(|F_{hkl}|=|F_{\bar h\,\bar k\,\bar l}|\)).

The presence of other symmetry elements leads to other inequalities, which are obtained from the corresponding expressions for \(\widehat{F}_{hkl}\), using Cauchy’s inequality in exactly the same way as was done above. The table collects such inequalities for crystals which

DETERMINATION OF PHASES OF FOURIER COEFFICIENTS

Cauchy inequalities for various types of symmetry axes

*) See next page.

Continuation

\[ {}^{*})\ \widehat F_{hkl}\ \text{may be expressed in the form}\ \exp(-2\pi i\alpha_{hkl})\left|\widehat F_{hkl}\right|, \]

where \(\alpha_{hkl}\) is the “phase” of \(\widehat F_{hkl}\). \(HKL\) are indices in the hexagonal system.

are characterized by various axes of symmetry: rotational, screw, and also mirror-rotational.

The sign of the structure factor can also be obtained for such \(\widehat F_{hkl}\) as are not determined by inequalities involving only a single \(\left|\widehat F_{hkl}\right|^2\). Thus, for example, when a center of inversion is present, consideration of the quantity \(\left|\widehat F_{hkl}\pm\widehat F_{h'k'l'}\right|^2\) by the methods just described gives the relation

\[ \left|2\widehat F_{hkl}\widehat F_{h'k'l'}-\widehat F_{h+h',\,k+k',\,l+l'}-\widehat F_{h-h',\,k-k',\,l-l'}\right|\le \]

\[ \le 1+\frac{1}{2}\left(\widehat F_{2h\,2k\,2l}+\widehat F_{2h'\,2k'\,2l'}\right) -\widehat F^{\,2}_{hkl}-\widehat F^{\,2}_{h'k'l'} . \]

We hope that the further development of this method will make it possible to determine the sign or, more generally speaking, the phase for the greater part of the \(\widehat F_{hkl}\) of a crystal structure. The fact that not all signs (phases) can be determined by these methods follows from those fundamental ambiguities in the determination of crystal structure that were revealed by Patterson*).

The application of these inequalities, of course, essentially requires that the experimental values \(F_{hkl}\) be expressed in absolute units. If X-ray methods are used, then the number of electrons in the unit cell must serve as these units.