Abstract
Over the past 2–3 years, a number of works have appeared that could not be included in existing monographs and are of fundamental importance for the theory of X-ray structural analysis. It seems of interest to summarize some results of this rapid development and to consider, from general points of view, the possibilities and prospects for the development of X-ray structural analysis of crystals.
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Advances in X-ray Structural Analysis of Crystals
A. I. Kitaigorodskii
In the last 2–3 years a number of works have appeared which have not yet been included in the existing monographs and which are of fundamental importance for the theory of X-ray structural analysis. It seems of interest to summarize certain results of this rapid development and to consider, from general points of view, the possibilities and prospects for the development of X-ray structural analysis of crystals.
1. Representation of the Result of an Experimental Study of a Crystal by a Set of Interatomic Vectors
As is known,^1 the diffraction of X-rays by a crystal may be regarded as selective reflection from systems of planes of the crystal lattice. The diffraction indices, which number the arising rays, are the integers \(h, k, l\). The common divisor of the indices is the order of reflection \(n\) in the Wulff–Bragg formula \(n\lambda = 2d\sin\theta\), where \(\lambda\) is the wavelength of the X-ray beam, \(\theta\) is the glancing angle at which the beam falls on a system of planes with interplanar spacing \(d\) and crystallographic indices \(h/n, k/n, l/n\).
It is convenient to represent the diffraction conditions in the space of the so-called reciprocal lattice, whose nodes are defined by the vector \(\mathbf{H} = h\mathbf{a}^{*} + k\mathbf{b}^{*} + l\mathbf{c}^{*}\), where \((\mathbf{a}\mathbf{a}^{*}) = (\mathbf{b}\mathbf{b}^{*}) = (\mathbf{c}\mathbf{c}^{*}) = 1\). The diffracted ray \(hkl\) arises in the case when a sphere of radius \(1/\lambda\), passing through the node 000, passes through the node \(hkl\), and the radius of the sphere passing through 000 coincides with the direction of the primary beam.
For a given wavelength, effective in the sense of diffraction are all nodes of the reciprocal lattice that can be intersected by a sphere of radius \(1/\lambda\) in some position with respect to the axes. It is obvious that this region will be a sphere drawn with radius \(2/\lambda\) from the node 000.
The intensities of reflections from various planes \(I(hkl)\) are related to the structure of the crystal. If the unit cell of the crystal contains \(N\) atoms with relative (with respect to the lengths of the cell edges) coordinates \(x_k y_k z_k\), then the amplitude of the diffracted beam with indices \(hkl\), due to scattering by the atoms of one cell—the structural amplitude—is equal to
\[ F(hkl)=\sum_{k=1}^{N} f_k e^{2\pi i(hx_k+ky_k+lz_k)}, \]
where \(f_k\) is the scattering amplitude of the \(k\)-th atom.
The dependence of \(I(hkl)\) on \(F(hkl)\) is established by the theory of diffraction. It proves possible to calculate the modulus of the structural amplitude from the intensity values.
The dependence of \(I(hkl)\) on \(|F(hkl)|\) is determined by the geometry of the experiment and by the properties of the radiation. These factors can be taken into account quite exactly. In addition, in passing from intensity to \(|F|\), in principle it is necessary to know the absorption of the specimen. Allowance for absorption requires calculation of the integral \(\int e^{-\mu(s_1+s_2)}\,dv\), where \(\mu\) is the absorption coefficient, and \(s_1\) and \(s_2\) are the paths of the incident and scattered beams to the volume \(dv\) of the crystal. Determination of the absorption factor presents no difficulties in principle, but is very complicated in practice. The most difficult part is taking into account the mosaicity of the crystal, i.e., the size, shape, and angular dispersion of the coherently scattering blocks. Theory shows that in the case where the whole crystal is a single large block, the diffraction intensity is directly proportional to the first power of \(|F|\). If, on the contrary, the crystal consists of many blocks of size on the order of \(10^{-4}\)—\(10^{-5}\) cm, then the scattering intensity is proportional to \(|F^2|\). Experience has shown that in the overwhelming majority of cases the intensity of the diffracted radiation is proportional to \(|F^2|\).
If it has been possible to take absorption into account and there is no doubt as to the mosaic structure of the crystal, then the quantities \(|F^2|\) are known to the investigator with the same accuracy with which the intensities have been measured. As for the latter, when special measures are taken, an accuracy of up to 1% can be achieved. If, however, absorption has not been estimated (and its magnitude is significant), or the scattering intensity by the crystal cannot be regarded as the sum of the scattering intensities of the individual mosaic blocks, then the errors in determining \(|F^2|\) may amount to many tens and even hundreds of percent of the measured quantity.
Of extraordinary importance for the practice of X-ray structural analysis is the circumstance that a rough determination of the values of the structural amplitudes has little effect on the accuracy
determination of interatomic distances. As was shown by the author1, an accuracy of 50% in the determination of \(F^2\) is quite good. Moreover, there is no point in striving for greater accuracy, since the forced truncation of Fourier series, by means of which we depict the crystal structure (see below), introduces a more significant error into the determination of interatomic distances. This circumstance greatly simplifies the experimental technique of X-ray structural analysis, makes it possible in most cases to neglect absorption by the crystal, and permits one not to be concerned with its mosaic structure. In accordance with what has been said, the determination of the intensities of diffraction rays when working by the photographic method is carried out by visually estimating the density of blackening of the film.
Thus, by a simple calculation from the direct experimental data one can obtain the values of \(F^2(hkl)\) for all measured diffraction rays.
The number of these rays depends on the degree of perfection of the order in the arrangement of atoms within the crystal. The causes of imperfection of order are, moreover, immaterial (thermal vibrations, impurities, lattice distortions, etc.).
The higher the order of diffraction, the smaller the effective interplanar distance of that system of planes, reflection from which in the first order is equivalent to the diffraction \(hkl\). If the mean displacements of atoms from the positions occupied by them in the ideal lattice, in the direction of the normal to the reflecting planes, are equal to half the interplanar distance, then diffraction in the corresponding direction does not occur. In this direction the atoms scatter rays with continuously distributed phases, i.e., as an amorphous, and not a crystalline, substance. Practically, diffraction rays standing out above the background will not be detected even in the case when the normal displacements reach one quarter of the corresponding interplanar distance.
The amplitudes of zero-point vibrations of atoms are quantities of the order of \(0.1\,\text{Å}\). Consequently, interplanar distances of the order of \(0.4\,\text{Å}\) are limiting small ones. Thus, a sphere of radius \(2.5\,\text{Å}^{-1}\) in reciprocal-lattice space bounds all nodes of the reciprocal lattice capable of giving diffraction rays.
If the X-ray investigation is carried out with the aid of the \(K_{\alpha}\) radiation of molybdenum \((\lambda = 0.7\,\text{Å})\), then the radius of the effective region will already be greater than \(2.5\,\text{Å}^{-1}\).
Thus, molybdenum radiation has the shortest wavelength needed in X-ray structural analysis. It should be noted that for most substances the region of nodes,
effective with respect to diffraction at room temperature are considerably smaller, namely of the order of \(1\)—\(1.5\ \text{\AA}^{-1}\).
The smallest elementary cells have linear cell dimensions of the order of \(3.5\)—\(4\ \text{\AA}\), the largest (globular proteins) \(350\)—\(400\ \text{\AA}\).
Thus, along the radius of the sphere \(2/\lambda\) there can fit from 10 to 1000 nodes. It is true that, as the size of the cell increases, as a rule, the degree of perfection of the order in the arrangement of atoms decreases. In the same globular proteins, diffraction of the highest order takes place from planes with interplanar spacings of \(2\ \text{\AA}\) (and not \(0.4\ \text{\AA}\)).
Experience shows that well-formed crystals can give, at room temperature, \(300\)—\(3000\) diffraction maxima. Probably these numbers may be increased on passing to investigations at low temperatures.
Thus, the result of an x-ray experiment is several hundred or several thousand quantities \(F^2(hkl)\). What information can be obtained about the structure of a crystal on the basis of this enormous numerical material?
A direct relation is given by the Fourier series
\[ A(u, v, w)=\sum_{h=0}^{\infty}\sum_{k=-\infty}^{+\infty}\sum_{l=-\infty}^{+\infty} F^2(hkl)\cos 2\pi(hu+kv+lw), \]
which is called the series of interatomic vectors, or, briefly, the \(F^2\)-series.
It can be shown\(^1\) that the function \(A(u, v, w)\), with periods equal to the lengths of the edges (\(u, v, w\) are relative coordinates along the edges \(a, b, c\)), can be represented in the form of the following sum:
\[ A(u, v, w)=\sum_{p=1}^{N}\sum_{s=1}^{N}\Phi_{p,s}, \]
where
\[ \Phi_{p,s}=\sum_{h,k,l=-\infty}^{+\infty} f_p f_s e^{2\pi i\left[h(u-u_{sp})+k(v-v_{sp})+l(w-w_{sp})\right]}. \]
Here \(N\) is the number of atoms in the cell, and \(u_{sp}, v_{sp}\), and \(w_{sp}\) are the projections of the interatomic vector joining atoms \(s\) and \(p\). Thus, the function \(\Phi_{sp}\) may be called the function of the interatomic vector. This function possesses spherical symmetry; its maximum lies at the point \(u=u_{sp},\ v=v_{sp}\), and \(w=w_{sp}\). The height
the maximum is equal to \(\sum_{h,k,l} f_p f_s\); it will always be proportional to the product of the atomic numbers \(Z_p Z_s\) of atoms \(p\) and \(s\).
If the atomic scattering is represented in the form \(f_p = k_p e^{-\alpha_p H}\) and \(f_s = k_s e^{-\alpha_s H}\), then, replacing summation by integration, we easily see that the height of the maximum of the function of the interatomic vector \(\Phi_{ps}\) is proportional to \(Z_p Z_s\) and inversely proportional to \((\alpha_p + \alpha_s)^3\). If one assumes that the scattering amplitudes of atoms of different kinds entering the cell differ only by a constant factor (i.e. \(\alpha_p = \alpha_s\)), then the ratio of the heights of the maxima of two functions \(\Phi_{p_1s_1}\) and \(\Phi_{p_2s_2}\) is equal to \(Z_{p_1}Z_{s_1} : Z_{p_2}Z_{s_2}\). Obviously, this relation will hold, at least approximately, in real cases.
Thus, the experimentally found function \(A(u, v, w)\) is a sum of functions of interatomic vectors that can be drawn within the elementary cell.
In a cell containing \(N\) atoms, \(N^2\) interatomic vectors can be drawn; of these, \(N\) are trivial \((p = s)\), i.e. connecting an atom “with itself.” These \(N\) trivial functions \(\Phi_{ss}\) will have a maximum at the point 000. The remaining \(N(N - 1)\) functions \(\Phi_{p,s}\), if their maxima are resolved, will give us information about all interatomic vectors present in the structure.
The function \(A(u, v, w)\) has maxima at those points whose coordinates are equal to the projections of the interatomic vectors. The heights of the maxima are determined by the atomic numbers of the atoms that form the interatomic vector. The number of nontrivial maxima in the space of the \(A\)-function is equal to \(N(N - 1)\).
In a number of cases the symmetry of the crystal may lead to the strict superposition of certain maxima. In order for maxima to coincide, the crystal must contain parallel or antiparallel interatomic vectors.
The only symmetry operation that transforms an arbitrary vector into an antiparallel one is the operation of a center of inversion. An arbitrary vector is transformed into a parallel one only by a translation.
Thus, when a center of inversion is present in the crystal, every interatomic vector not passing through the center of symmetry is transformed into an antiparallel one. In the case of a centrosymmetric crystal, the interatomic vectors are divided into two groups—\(N\), passing through the center of symmetry, and \(\dfrac{N^2}{2} - N\) pairs of antiparallel ones. Thus, in the space of the \(A\)-function there will be present \(N\) single and \(\dfrac{N^2}{2} - N\) double maxima.
Let us emphasize that this reasoning applies to an arbitrary case. It is obvious that a vector which, either accidentally or by virtue of symmetry, is parallel to an axis or plane of symmetry will be transformed into one parallel to itself.
2. A SERIES OF INTERATOMIC VECTORS AND THE SYMMETRY OF A CRYSTAL
The function \(A(u, v, w)\) gives an exhaustive knowledge of the structure of crystals (if one does not count the periods of the cell and the angles between the axes of the cell). It determines the symmetry and structure of the crystal.
a) Symmetry of the \(F^2\)-series and symmetry of the crystal
The coefficients in the expansion into a series of the function \(A(u, v, w)\) are real and positive quantities. In accordance with this, the symmetry of the space of the function \(A(u, v, w)\) depends on the point group of the crystal. Indeed, equality or inequality of the intensities of reflection from the planes \(hkl\), forming one form \(\{hkl\}\), characterizes both the point group of the crystal and the symmetry of the space of the function \(A(u, v, w)\).
It is not difficult to see that through the point 000 of the space of the function \(A(u, v, w)\) pass the symmetry elements of the point group of the crystal, with their position relative to the origin of coordinates preserved. However, since the reflection intensities \(hkl\) and \(\bar{h}\bar{k}\bar{l}\) are equal, a center of symmetry is added to the aggregate of symmetry elements of the crystal.
The function \(A(u, v, w)\) is a periodic function. Consequently, the group of symmetry elements passing through 000 is translated in three dimensions.
As for the translational group of the space \(A(u, v, w)\), it coincides with the translational group of the crystal.
From what has been said follows the list, given in Table 1, of the Fedorov groups of the space of the function \(A(u, v, w)\). The origin of coordinates of the space is always located at the center of inversion.
b) Other features of the \(F^2\)-series and the symmetry of the crystal
At first glance it may seem that the \(F^2\)-series does not allow one to judge the space group of the crystal, since 230 Fedorov groups of the crystal are represented by 23 symmetry groups of the space of the \(A\)-function.
This, however, is not so. The point is that not only the symmetry of the \(A\)-function depends on the symmetry of the crystal, but also certain features in the distribution of maxima. Each symmetry element, except the center of inversion, has in the space of the \(A\)-function a corresponding geometric image. The presence in the crystal
Table 1
| Crystal system | Fedorov groups of the crystal | Fedorov group of the space \(A(u,v,w)\)-function |
|---|---|---|
| Triclinic | Both triclinic groups | \(P\,1\) |
| Monoclinic | Translation group \(P\) | \(P\,2/m\) |
| Monoclinic | Translation group \(C\) | \(C\,2/m\) |
| Orthorhombic | Translation group \(P\) | \(P\,mmm\) |
| Orthorhombic | Translation group \(C\) | \(C\,mmm\) |
| Orthorhombic | Translation group \(F\) | \(F\,mmm\) |
| Orthorhombic | Translation group \(J\) | \(J\,mmm\) |
| Tetragonal | Translation group \(P\) of the classes \(4,\ \bar{4}\) and \(4/m\) | \(P\,4/m\) |
| Tetragonal | Translation group \(J\) of the classes \(4,\ \bar{4}\) and \(4/m\) | \(J\,4/m\) |
| Tetragonal | Translation group \(P\) of the classes \(\bar{4}2m,\ 422,\ 4mm\) and \(4/mmm\) | \(P\,4/mmm\) |
| Tetragonal | Translation group \(J\) of the classes \(\bar{4}2m,\ 422,\ 4mm,\ 4/mmm\) | \(J\,4/mmm\) |
| Hexagonal | Translation group \(C\) of the classes \(3\) and \(\bar{3}\) | \(C\,\bar{3}\) |
| Hexagonal | Translation group \(R\) of the classes \(3\) and \(\bar{3}\) | \(R\,\bar{3}\) |
| Hexagonal | Translation group \(C\) of the classes \(3m,\ 32,\ \bar{3}m\) | \(C\,\bar{3}m\) |
| Hexagonal | Translation group \(R\) of the classes \(3m,\ 32,\ \bar{3}m\) | \(R\,\bar{3}m\) |
| Hexagonal | Translation group \(C\) of the classes \(6,\ \bar{6},\ 6/m\) | \(C\,6/m\) |
| Hexagonal | Translation group \(C\) of the classes \(\bar{6}m2,\ 6mm,\ 622\) and \(6/mmm\) | \(C\,6/mmm\) |
| Cubic | Translation group \(P\) of the classes \(23\) and \(m3\) | \(P\,m3\) |
| Cubic | Translation group \(F\) of the classes \(23\) and \(m3\) | \(F\,m3\) |
| Cubic | Translation group \(J\) of the classes \(23\) and \(m3\) | \(J\,m3\) |
| Cubic | Translation group \(P\) of the classes \(\bar{4}3m,\ 432\) and \(m3m\) | \(P\,m3m\) |
| Cubic | Translation group \(F\) of the classes \(\bar{4}3m,\ 432\) and \(m3m\) | \(F\,m3m\) |
| Cubic | Translation group \(J\) of the classes \(\bar{4}3m,\ 432\) and \(m3m\) | \(J\,m3m\) |
of a rotation axis means the presence of interatomic vectors whose projection onto the direction of the axis is zero.
Thus, rotation axes are represented in the space of the \(A\)-function by maxima lying in the coordinate plane perpendicular to the axis. The presence in a crystal of a screw axis means the presence of interatomic vectors whose projection onto the direction of the axis is equal to the screw displacement \(\left(\frac{1}{2}\right.\) for the axis \(2_1\), \(\frac{1}{3}\) for the axes \(3_1\) and \(3_2\), etc.). Thus, screw axes are represented in the space of the \(A\)-function by maxima lying in planes perpendicular to the axis and displaced by \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{6}\) from the coordinate plane.
Planes of symmetry are represented in the space of the \(A\)-function by maxima lying on a line. The presence of a plane of mirror symmetry means the presence of interatomic vectors for which both projections onto this plane are equal to zero. Thus, a plane of mirror symmetry is represented in the space of the \(A\)-function by maxima lying along the coordinate line perpendicular to the plane of symmetry. The presence of a glide plane means the presence of interatomic vectors with components \(\frac{1}{2}\) (or \(\frac{1}{4}\) in the case of \(d\)-planes) along the glide axis.
Thus, a glide plane is represented in the space of the \(A\)-function by maxima situated on a line perpendicular to the glide plane and passing through the middle (or one quarter, for \(d\)) of the segment of the glide line in the cell; if the case of the \(d\)-plane is excluded, then the lines representing glide planes may have, in the glide plane, coordinates \(0,\frac{1}{2}\); \(\frac{1}{2},0\), or \(\frac{1}{2},\frac{1}{2}\).
It follows from the foregoing that, despite the centrosymmetry of the \(F^2\)-series, by considering it one can prove the absence of a center of symmetry in the crystal.
Until recently it was assumed that an X-ray experiment does not allow one to establish the absence or presence of a center of symmetry. It was shown that, for this reason, the 230 Fedorov groups (it should be remembered that 11 pairs among them are enantiomorphic) correspond to 120 X-ray diffraction groups[^3].
However, with the aid of the \(F^2\)-series the number of objectively distinguishable Fedorov groups can be brought up to 192. The remaining 38 groups are indistinguishable in pairs. These include the 11 enantiomorphic pairs and the following 8 pairs: \(J23\) and \(J2_1 3\), \(P3/m\) and \(P6/m\), \(R3\) and \(R\bar{3}\), \(P3\) and \(P\bar{3}\), \(J4\) and \(J\bar{4}\), \(P4\) and \(P\bar{4}\), and \(J222\) and \(J2_1 2_1 2_1\), \(P1\) and \(P\bar{1}\).
If, for example, one compares the groups \(Pa\) and \(P2/a\) \((C_s^4\) and \(C_{2h}^4)\), which differ from one another only by a center of inversion (and, consequently, are indistinguishable by the extinctions of interferences), it turns out that the spaces of the \(A\)-function must look different. In the case of the first of the groups mentioned, the \(F^2\)-series will lead to the line
\[ \frac{1}{2},\, v,\, 0, \]
along which the maxima must be distributed. In the case of the group \(P2/a\), in addition to the special line, there also arises a special plane \(u,0,w\), along which the maxima must be distributed.
A similar consideration of all space groups makes it possible to characterize them by the following table\(^4\) (Table II):
It would be the opposite extreme to suppose that, when a crystal belongs to one of the 192 groups, the form of the \(F^2\)-series makes it possible to determine the symmetry of the crystal uniquely.
Two limitations apply:
1) It is possible to prove the absence, but not the presence, of a center of inversion\(^5\).
2) It is impossible to distinguish a true symmetry element from the corresponding pseudosymmetry element\(^6\).
Let us consider these propositions in somewhat more detail. If we know that in the elementary cell of the crystal there is only one atom, situated in a general position, i.e. not on a symmetry element, then determination of the space group is indeed unambiguous. The finding of a center of inversion in this case does not even require consideration of the \(F^2\)-series. Indeed, a center of inversion doubles the multiplicity of the space group. If it is known that there are 2 atoms in the cell, then the choice between the groups \(Pa\) and \(P2/a\) is unambiguously decided in favor of the first.
Let us assume, however, continuing the consideration of the same example, that there are 4 atoms in the cell. Then either one atom is in a general position and the symmetry of the crystal is \(P2/a\), or two atoms are in general positions and the symmetry of the crystal is \(Pa\).
As was indicated above, the number of maxima in the case of a centrosymmetric crystal is equal to \(N\) single and
\[ \frac{N^2}{2}-N \]
double ones, i.e. in our example \(4+4\). In the absence of a center of symmetry all maxima are single, and their number is \(N(N-1)\), i.e. 12.
If, therefore, upon consideration of the \(F^2\)-series we find 12 maxima, then the absence of a center of inversion leaves no doubt. But if the number of maxima is 8, then one can say only that the crystal structure is either centrosymmetric or differs only slightly from a centrosymmetric one. Indeed, the \(F^2\)-series has a finite resolving power, which may be estimated at approximately \(0.5\,\text{Å}\). Maxima that are closer to one another will merge into one.
Characteristic of the space of \(A\)-functions of 230 Fedorov groups
Table 11
| Crystal system | Fedorov group of the crystal | Fedorov space group of the \(A\)-function | Images of planes of symmetry in the space of the \(A\)-function | Images of planes of symmetry in the space of the \(A\)-function | Images of planes of symmetry in the space of the \(A\)-function | Images of axes of symmetry in the space of the \(A\)-function | Images of axes of symmetry in the space of the \(A\)-function | Images of axes of symmetry in the space of the \(A\)-function | Images of axes of symmetry in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Triclinic | \(P1\) | \(P1\) | — | — | — | — | — | — | — |
| Triclinic | \(P1\) | \(P1\) | — | — | — | — | — | — | — |
| Monoclinic | \(P2\) | \(P2/m\) | — | — | — | — | — | \(x0z\) | — |
| Monoclinic | \(P2_1\) | \(P2/m\) | — | — | — | — | — | \(x\frac{1}{2}z\) | — |
| Monoclinic | \(Pm\) | \(P2/m\) | — | — | \(0y0\) | — | — | — | — |
| Monoclinic | \(Pc\) | \(P2/m\) | — | — | \(0y\frac{1}{2}\) | — | — | — | — |
| Monoclinic | — | \(P2/m\) | — | — | \(0y0\) | — | — | \(x0z\) | — |
| Monoclinic | \(P2/c\) | \(P2/m\) | — | — | \(0y\frac{1}{2}\) | — | — | \(x0z\) | — |
| Monoclinic | \(P2_1/c\) | \(P2/m\) | — | — | \(0y0\) | — | — | \(x\frac{1}{2}z\) | — |
| Monoclinic | \(P2_1/c\) | \(P2/m\) | — | — | \(0y\frac{1}{2}\) | — | — | \(x\frac{1}{2}z\) | — |
| Monoclinic | \(C2\) | \(C2/m\) | — | — | — | — | — | \(x0z\) | — |
| Monoclinic | \(Cm\) | \(C2/m\) | — | — | \(0y0\) | — | — | — | — |
| Monoclinic | \(Cc\) | \(C2/m\) | — | — | \(0y\frac{1}{2}\) | — | — | — | — |
| Monoclinic | \(C2/m\) | \(C2/m\) | — | — | \(0y0\) | — | — | \(x0z\) | — |
| Monoclinic | \(C2/c\) | \(C2/m\) | — | — | \(0y\frac{1}{2}\) | — | — | \(x0z\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov space group of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Rhombic | \(P222\) | \(Pmmm\) | — | — | — | \(0yz\) | \(x0z\) | \(xy0\) | — |
| Rhombic | \(P222_1\) | — | — | — | — | \(0yz\) | \(x0z\) | \(xy\frac12\) | — |
| Rhombic | \(P2_12_12\) | — | — | — | — | \(\frac12 yz\) | \(x\frac12 z\) | \(xy0\) | — |
| Rhombic | \(P2_12_12_1\) | — | — | — | — | \(\frac12 yz\) | \(x\frac12 z\) | \(xy\frac12\) | — |
| Rhombic | \(Pmm2\) | — | \(x00_1\) | \(0y0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Pmc2_1\) | — | \(x00\) | \(0y\frac12_1\) | — | — | — | \(xy\frac12_1\) | — |
| Rhombic | \(Pcc2\) | — | \(x0\frac12\) | \(0y\frac12\) | — | — | — | \(xy0\) | — |
| Rhombic | \(Pma2\) | — | \(x00_1\) | \(\frac12 y0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Pca2_1\) | — | \(x0\frac12_1\) | \(\frac12 y0\) | — | — | — | \(xy\frac12_1\) | — |
| Rhombic | \(Pnc2\) | — | \(x\frac12\frac12\) | \(0y\frac12\) | — | — | — | \(xy0\) | — |
| Rhombic | \(Pmn2_1\) | — | \(x00\) | \(\frac12 y\frac12_1\) | — | — | — | \(xy\frac12_1\) | — |
| Rhombic | \(Pba2\) | — | \(x\frac12 0_1\) | \(\frac12 y0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Pna2_1\) | — | \(x\frac12\frac12_1\) | \(\frac12 y0\) | — | — | — | \(xy\frac12_1\) | — |
| Rhombic | \(Pnn2\) | — | \(x\frac12\frac12\) | \(\frac12 y\frac12\) | — | — | — | \(xy0\) | — |
| Rhombic | \(Pmmm\) | — | \(x00_{1,2}\) | \(0y0_{3,4}\) | \(00z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — |
| Rhombic | \(Pnnn\) | — | \(x\frac12\frac12\) | \(\frac12 y\frac12\) | \(\frac12\frac12 z\) | \(0yz\) | \(x0z\) | \(xy0\) | — |
Continuation of Table II
| Crystal system | Fedorov crystal group | Fedorov group of the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Orthorhombic | \(Pccm\) | — | \(x0\frac12_1\) | \(0y\frac12_2\) | \(00z_{3,4}\) | \(0yz_{2,3}\) | \(x0z_{1,4}\) | \(xy0\) | — |
| Orthorhombic | \(Pbam\) | — | \(x\frac12 0_1\) | \(\frac12 y0_2\) | \(\frac12\frac12 z\) | \(0yz\) | \(x0z\) | \(xy0_{1,2}\) | — |
| Orthorhombic | \(Pmma\) | — | \(x00_{1,2}\) | \(0y0_3\) | \(\frac12 0z_{4,5}\) | \(\frac12 yz_4\) | \(x0z_{1,5}\) | \(xy0_{2,3}\) | — |
| Orthorhombic | \(Pnna\) | — | \(x\frac12\frac12_1\) | \(\frac12 y\frac12\) | \(\frac12 0z\) | \(0yz\) | \(x\frac12 z_1\) | \(xy0\) | — |
| Orthorhombic | \(Pmna\) | — | \(x00_1\) | \(\frac12 y\frac12_2\) | \(\frac12 0z_3\) | \(0yz\) | \(x0z_{1,3}\) | \(xy\frac12_2\) | — |
| Orthorhombic | \(Pcca\) | — | \(x0\frac12_1\) | \(0y\frac12\) | \(\frac12 0z_{2,3}\) | \(\frac12 yz\) | \(x0z_{1,3}\) | \(xy0\) | — |
| Orthorhombic | \(Pbam\) | — | \(x\frac12 0_1\) | \(\frac12 y0_{2,3}\) | \(00z_4\) | \(\frac12 yz_2\) | \(x0z_4\) | \(xy0_{1,3}\) | — |
| Orthorhombic | \(Pccn\) | — | \(x0\frac12\) | \(0y\frac12\) | \(\frac12\frac12 z_{1,2}\) | \(\frac12 yz_1\) | \(x\frac12 z_2\) | \(xy0\) | — |
| Orthorhombic | \(Pbcm\) | — | \(x\frac12 0_1\) | \(0y\frac12_{2,3}\) | \(00z_4\) | \(0yz_{2,4}\) | \(x\frac12 z_1\) | \(xy\frac12_3\) | — |
| Orthorhombic | \(Pnnm\) | — | \(x\frac12\frac12_1\) | \(\frac12 y\frac12_2\) | \(00z\) | \(\frac12 yz_2\) | \(x\frac12 z_1\) | \(xy0\) | — |
| Orthorhombic | \(Pmnn\) | — | \(x00_1\) | \(0y0_2\) | \(\frac12\frac12 z_{3,4}\) | \(\frac12 yz_2\) | \(x\frac12 z_4\) | \(xy0_{1,2}\) | — |
| Orthorhombic | \(Pbcn\) | — | \(x\frac12 0\) | \(0y\frac12_1\) | \(\frac12\frac12 z_2\) | \(\frac12 yz_2\) | \(x0z\) | \(xy\frac12_1\) | — |
| Orthorhombic | \(Pbca\) | — | \(x\frac12 0_1\) | \(0y\frac12_2\) | \(\frac12 0z_3\) | \(\frac12 yz_3\) | \(x\frac12 z_1\) | \(x0\frac12_2\) | — |
| Orthorhombic | \(Pnma\) | — | \(x\frac12\frac12_{1,2}\) | \(0y0\) | \(\frac12 0z_3\) | \(\frac12 yz_3\) | \(x\frac12 z_1\) | \(x0\frac12_2\) | — |
| Orthorhombic | \(C222_1\) | \(C\) | — | — | — | \(0yz\) | \(x0z\) | \(xy\frac12\) | — |
| Orthorhombic | \(C222\) | — | — | — | — | \(0yz\) | \(x0z\) | \(xy0\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov group of the \(A\)-function space | Images of symmetry planes in the \(A\)-function space | Images of symmetry planes in the \(A\)-function space | Images of symmetry planes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space |
|---|---|---|---|---|---|---|---|---|---|---|
| Orthorhombic | \(Cmm2\) | — | \(x00_1\) | \(0y0_2\) | — | — | — | — | \(xy0_{1,2}\) | — |
| Orthorhombic | \(Cmc2_1\) | — | \(x00\) | \(0y\frac12_1\) | — | — | — | — | \(xy\frac12_1\) | — |
| Orthorhombic | \(Ccc2\) | — | \(x0\frac12\) | \(0y\frac12\) | — | — | — | — | \(xy0\) | — |
| Orthorhombic | — | — | — | \(0y0_1\) | \(00z_2\) | \(0yz_{1,2}\) | — | — | — | — |
| Orthorhombic | — | — | — | \(0y0_1\) | \(0\frac12 z_2\) | \(0yz_{1,2}\) | — | — | — | — |
| Orthorhombic | — | — | — | \(0y\frac12_1\) | \(00z_2\) | \(0yz_{1,2}\) | — | — | — | — |
| Orthorhombic | — | — | — | \(0y\frac12_1\) | \(0\frac12 z_2\) | \(0yz_{1,2}\) | — | — | — | — |
| Orthorhombic | \(Cmcm\) | — | \(x00_1\) | \(0y\frac12_{2,3}\) | \(00z_{4,5}\) | \(0yz_{2,4}\) | \(x0z_{1,5}\) | \(xy\frac12_3\) | — | — |
| Orthorhombic | \(Cmca\) | — | \(x00_1\) | \(0y\frac12_{2,3}\) | \(\frac12 0z_{4,5}\) | \(0yz_{2,4}\) | \(x0\frac12_{1,5}\) | \(xy\frac12_3\) | — | — |
| Orthorhombic | \(Cmmm\) | — | \(x00_{1,2}\) | \(0y0_{3,4}\) | \(00z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — | — |
| Orthorhombic | \(Cccm\) | — | \(x0\frac12_1\) | \(0y\frac12_2\) | \(00z_{3,4}\) | \(0yz_{2,3}\) | \(x0z_{1,4}\) | \(xy0\) | — | — |
| Orthorhombic | \(Cmma\) | — | \(x00_{1,2}\) | \(0y0_{3,4}\) | \(\frac12 0z_5\) | \(0y\frac12_{3,5}\) | \(x0\frac12_1\) | \(xy0_{2,4}\) | — | — |
| Orthorhombic | \(Ccca\) | — | \(x0\frac12_1\) | \(0y\frac12_2\) | \(\frac12 0z_{3,5}\) | \(0yz_{2,5}\) | \(x0z_{1,3}\) | \(xy0\) | — | — |
| Orthorhombic | \(I222\) | \(Jmmm\) | \(\}\) | — | — | — | \(0yz\) | \(x0z\) | \(xy0\) | — |
| Orthorhombic | \(I2_12_12_1\) | \(Jmmm\) | \(\}\) | — | — | — | \(0yz\) | \(x0z\) | \(xy0\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov group of the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Rhombic | \(Imm2\) | — | \(x00_{1}\) | \(0y0_{2}\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Iba2\) | — | \(x\tfrac12 0_{1}\) | \(\tfrac12 y0_{2}\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Ima2\) | — | \(x00_{1}\) | \(\tfrac12 y0_{2}\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Immm\) | — | \(x00_{1,2}\) | \(0y0_{3,4}\) | \(00z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — |
| Rhombic | \(Ibam\) | — | \(x\tfrac12 0_{1,2}\) | \(\tfrac12 y0_{3,4}\) | \(00z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — |
| Rhombic | \(Ibca\) | — | \(x\tfrac12 0_{1,2}\) | \(0y\tfrac12_{3,4}\) | \(\tfrac12 0z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — |
| Rhombic | \(Imma\) | — | \(x00_{1,2}\) | \(0y0_{3,4}\) | \(\tfrac12 0z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — |
| Rhombic | \(F222\) | \(Fmmm\) | — | — | — | \(0yz\) | \(x0z\) | \(xy0\) | — |
| Rhombic | \(Fmm2\) | — | \(x00_{1}\) | \(0y0_{2}\) | — | — | — | \(xy0_{1,2}\) | — |
| Rhombic | \(Fdd2\) | — | \(x\tfrac14\tfrac14\) | \(\tfrac14 y\tfrac14\) | — | — | — | \(xy0\) | — |
| Rhombic | \(Fmmm\) | — | \(x00_{1,2}\) | \(0y0_{3,4}\) | \(00z_{5,6}\) | \(0yz_{3,5}\) | \(x0z_{1,6}\) | \(xy0_{2,4}\) | — |
| Rhombic | \(Fddd\) | — | \(x\tfrac14\tfrac14\) | \(\tfrac14 y\tfrac14\) | \(\tfrac14\tfrac14 z\) | \(0yz\) | \(x0z\) | \(xy0\) | — |
Continuation of Table 11
| Crystal system | Fedorov group of the crystal | Fedorov group of the space of the \(A\)-function | Image of symmetry planes in the space of the \(A\)-function | Image of symmetry planes in the space of the \(A\)-function | Image of symmetry planes in the space of the \(A\)-function | Image of symmetry axes in the space of the \(A\)-function | Image of symmetry axes in the space of the \(A\)-function | Image of symmetry axes in the space of the \(A\)-function | Image of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Tetragonal | \(P\bar{4}\) | \(P4/m\) | — | — | — | — | — | \(xy0\) | — |
| Tetragonal | \(P4\) | — | — | — | — | — | — | — | — |
| Tetragonal | \(P4_1\) | — | — | — | — | — | — | — | — |
| Tetragonal | \(P4_3\) | — | — | — | — | — | — | \(xy^{1/4}\) | \(xy^{1/8}\) |
| Tetragonal | \(P4_2\) | — | — | — | — | — | — | \(xy0\) | \(xy^{1/2}\) |
| Tetragonal | \(P4/m\) | — | — | — | \(00z\) | — | — | \(xy0\) | — |
| Tetragonal | \(P4_2/m\) | — | — | — | \(00z\) | — | — | \(xy0\) | \(xy^{1/2}\) |
| Tetragonal | \(P4/n\) | — | — | — | \(\tfrac12\,\tfrac12\,z\) | — | — | \(xy0\) | — |
| Tetragonal | \(P4_2/n\) | — | — | — | \(\tfrac12\,\tfrac12\,z\) | — | — | \(xy0\) | \(xy^{1/2}\) |
| Tetragonal | \(I\bar{4}\) | \(I4/m\) | — | — | — | — | — | \(xy0\) | — |
| Tetragonal | \(I4\) | — | — | — | — | — | — | — | — |
| Tetragonal | \(I4_1\) | — | — | — | — | — | — | \(xy0\) | \(xy^{1/4}\) |
| Tetragonal | \(I4/m\) | — | — | — | \(00z\) | — | — | \(xy0\) | — |
| Tetragonal | \(I4_1/a\) | — | — | — | \(\tfrac12\,0\,z\) | — | — | \(xy0\) | \(xy^{1/4}\) |
Continuation of Table 11
| Crystal system | Fedorov group of the crystal | Fedorov group of the \(A\)-function space | Examples of symmetry planes in the \(A\)-function space | Examples of symmetry planes in the \(A\)-function space | Examples of symmetry planes in the \(A\)-function space | Examples of symmetry axes in the \(A\)-function space | Examples of symmetry axes in the \(A\)-function space | Examples of symmetry axes in the \(A\)-function space | Examples of symmetry axes in the \(A\)-function space |
|---|---|---|---|---|---|---|---|---|---|
| Tetragonal | \(P422\) | \(P4/mmm\) | — | — | — | \(x0z\) | \(xxz\) | \(xy0\) | — |
| Tetragonal | \(P42_12\) | — | — | — | — | \(x\frac12 z\) | \(xxz\) | \(xy0\) | — |
| Tetragonal | \(P4_122\) | — | — | — | — | — | — | — | — |
| Tetragonal | \(P4_322\) | — | — | — | — | \(x0z\) | \(xxz\) | \(xy\frac14\) | \(xy\frac12\) |
| Tetragonal | \(P4_12_12\) | — | — | — | — | — | — | — | — |
| Tetragonal | \(P4_32_12\) | — | — | — | — | \(x\frac12 z\) | \(xxz\) | \(xy\frac14\) | \(xy\frac12\) |
| Tetragonal | \(P4_122\) | — | — | — | — | \(x0z\) | \(xxz\) | \(xy0\) | \(xy\frac12\) |
| Tetragonal | \(P4_212\) | — | — | — | — | \(x\frac12 z\) | \(xxz\) | \(xy0\) | \(xy\frac12\) |
| Tetragonal | \(P4mm\) | — | \(x00_1\) | \(xx0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Tetragonal | \(P4bm\) | — | \(x\frac12 0_1\) | \(xx0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Tetragonal | \(P4_2cm\) | — | \(x0\frac12_1\) | \(xx0_2\) | — | — | — | \(xy0_2\) | \(xy\frac12_1\) |
| Tetragonal | \(P4_2nm\) | — | \(x\frac12\frac12_1\) | \(xx0\) | — | — | — | \(xy0_2\) | \(xy\frac12_1\) |
| Tetragonal | \(P4cc\) | — | \(x0\frac12\) | \(xx\frac12\) | — | — | — | \(xy0\) | — |
| Tetragonal | \(P4nc\) | — | \(x\frac12\frac12\) | \(xx\frac12\) | — | — | — | \(xy0\) | — |
| Tetragonal | \(P4_2mc\) | — | \(x00_1\) | \(xx\frac12_3\) | — | — | — | \(xy0_1\) | \(xy\frac12_2\) |
| Tetragonal | \(P4_2bc\) | — | \(x\frac12 0_1\) | \(xx\frac12_2\) | — | — | — | \(xy0_1\) | \(xy\frac12_3\) |
Continuation of Table 1
| Crystal system | Fedorov group of the crystal | Fedorov space group of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Tetragonal | \(P\bar{4}2m\) | — | — | \(xx0_1\) | — | \(x0z\) | — | \(xy0_1\) | — |
| Tetragonal | \(P\bar{4}2c\) | — | — | \(xx\tfrac12\) | — | \(x0z\) | — | \(xy0\) | — |
| Tetragonal | \(P\bar{4}2_1m\) | — | — | \(xx0_1\) | — | \(x\tfrac12 z\) | — | \(xy0_1\) | — |
| Tetragonal | \(P\bar{4}2_1c\) | — | — | \(xx\tfrac12\) | — | \(x\tfrac12 z\) | — | \(xy0\) | — |
| Tetragonal | \(P\bar{4}m2\) | — | \(x00_1\) | — | — | — | \(xxz\) | \(xy0_1\) | — |
| Tetragonal | \(P\bar{4}c2\) | — | \(x0\tfrac12\) | — | — | — | \(xxz\) | \(xy0\) | — |
| Tetragonal | \(P\bar{4}b2\) | — | \(x\tfrac12 0_1\) | — | — | — | \(xxz\) | \(xy0_1\) | — |
| Tetragonal | \(P\bar{4}n2\) | — | \(x\tfrac12\,\tfrac12\) | — | — | — | \(xxz\) | \(xy0\) | — |
| Tetragonal | \(P4/mmm\) | — | \(x00_{1,2}\) | \(xx0_{3,4}\) | \(00z_{5,6}\) | \(x0z_{1,5}\) | \(xxz_{3,6}\) | \(xy0_{2,4}\) | — |
| Tetragonal | \(P4/mcc\) | — | \(x0\tfrac12_1\) | \(xx\tfrac12_2\) | \(00z_{3,4}\) | \(x0z_{1,3}\) | \(xxz_{2,4}\) | \(xy0\) | — |
| Tetragonal | \(P4/nbm\) | — | \(x\tfrac12 0_1\) | \(xx0_{2,3}\) | \(\tfrac12\,\tfrac12 z_4\) | \(x0z\) | \(xxz_{2,4}\) | \(xy0_{1,3}\) | — |
| Tetragonal | \(P4/nnc\) | — | \(x\tfrac12\,\tfrac12\) | \(xx\tfrac12_1\) | \(\tfrac12\,\tfrac12 z_2\) | \(x0z\) | \(xxz_{1,2}\) | \(xy0\) | — |
| Tetragonal | \(P4/mbm\) | — | \(x\tfrac12 0_{1,2}\) | \(xx0_{3,4}\) | \(00z_5\) | \(x\tfrac12 z_1\) | \(xxz_{3,5}\) | \(xy0_{2,4}\) | — |
| Tetragonal | \(P4/mnc\) | — | \(x\tfrac12\,\tfrac12_1\) | \(xx\tfrac12_2\) | \(00z_3\) | \(x\tfrac12 z_1\) | \(xxz_{2,3}\) | \(xy0\) | — |
| Tetragonal | \(P4/nmm\) | — | \(x00_1\) | \(xx0_{2,3}\) | \(\tfrac12\,\tfrac12 z_{4,5}\) | \(x\tfrac12 z_4\) | \(xxz_{2,5}\) | \(xy0_{1,3}\) | — |
| Tetragonal | \(P4/ncc\) | — | \(x0\tfrac12\) | \(xx\tfrac12_1\) | \(\tfrac12\,\tfrac12 z_{2,3}\) | \(x\tfrac12 z_2\) | \(xxz_{1,3}\) | \(xy0\) | — |
| Tetragonal | \(P4_2/mmc\) | — | \(x00_{1,2}\) | \(xx0_{3,4}\) | \(00z_{5,6}\) | \(x0z_{1,5}\) | \(xxz_{3,6}\) | \(xy0_{2,4}\) | \(xy\tfrac12\) |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov space group of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Tetragonal | \(P4_2/mcm\) | — | \(x0\frac12_1\) | \(xx0_{2,3}\) | \(00z_{4,\bar5}\) | \(x0z_4\) | \(xxz_{2,\bar5}\) | \(xy0_{\bar3}\) | \(xy\frac12_1\) |
| Tetragonal | \(P4_2/nbc\) | — | \(x\frac12 0_1\) | \(xx\frac12_{2,3}\) | \(\frac12\,\frac12\,z_4\) | \(x0z\) | \(xxz_{2,4}\) | \(xy0_1\) | \(xy\frac12_3\) |
| Tetragonal | \(P4_2/nnm\) | — | \(x\frac12\,\frac12_1\) | \(xx0_{2,3}\) | \(\frac12\,\frac12\,z_4\) | \(x0z\) | \(xxz_{2,4}\) | \(xy0_8\) | \(xy\frac12_1\) |
| Tetragonal | \(P4_2/mbc\) | — | \(x\frac12 0_{1,2}\) | \(xx\frac12_{3,4}\) | \(00z_5\) | \(x\frac12 z_1\) | \(xxz_{3,\bar5}\) | \(xy0_2\) | \(xy\frac12_4\) |
| Tetragonal | \(P4_2/mnm\) | — | \(x\frac12\,\frac12_{1,2}\) | \(xx0_{3,4}\) | \(00z_5\) | \(x\frac12 z_1\) | \(xxz_{3,5}\) | \(xy0_4\) | \(xy\frac12_2\) |
| Tetragonal | \(P4_2/nmc\) | — | \(x00_1\) | \(xx\frac12_{2,3}\) | \(\frac12\,\frac12\,z_{5,6}\) | \(x\frac12 z_5\) | \(xxz_{2,6}\) | \(xy0_1\) | \(xy\frac12_3\) |
| Tetragonal | \(P4_2/ncm\) | — | \(x0\frac12_1\) | \(xx0_{2,3}\) | \(\frac12\,\frac12\,z_{5,6}\) | \(x\frac12 z_5\) | \(xxz_{2,6}\) | \(xy0_8\) | \(xy\frac12_1\) |
| Tetragonal | \(J422\) | \(J4/mmm\) | — | — | — | \(x0z\) | \(xxz\) | \(xy0\) | — |
| Tetragonal | \(J4_1 22\) | — | — | — | — | \(x0z\) | \(xxz\) | \(xy0\) | \(xy\frac14\) |
| Tetragonal | \(J4mm\) | — | \(x00_1\) | \(xx0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Tetragonal | \(J4cm\) | — | \(x0\frac12_1\) | \(xx0_2\) | — | — | — | \(xy0_{1,2}\) | — |
| Tetragonal | \(J4_1md\) | — | \(x00_1\) | \(x/x+\frac12\ [[unclear: subscript]]\) | — | — | — | \(xy0_1\) | \(xy\frac14_2\) |
| Tetragonal | \(J4_1cd\) | — | \(x0\frac12_1\) | \(x/x+\frac12\ [[unclear: subscript]]\) | — | — | — | \(xy0_1\) | \(xy\frac14_2\) |
| Tetragonal | \(J\bar4m2\) | — | \(x00_1\) | — | — | — | \(xxz\) | \(xy0_1\) | — |
| Tetragonal | \(J\bar4c2\) | — | \(x0\frac12_1\) | — | — | — | \(xxz\) | \(xy0_1\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov group of the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|
| Tetragonal | \(I\bar{4}2m\) | — | — | \(xx0_1\) | — | \(x0z\) | — | \(xy0_1\) | — |
| Tetragonal | \(I\bar{4}2d\) | — | — | \(x/x+\frac12\, [[unclear]]\) | — | \(x0z\) | — | \(xy0\) | — |
| Tetragonal | \(I4_1/mmm\) | — | \(x00_{1,2}\) | \(xx0_{3,4}\) | \(00z_{5,6}\) | \(x0z_{1,5}\) | \(xxz_{3,6}\) | \(xy0_{2,4}\) | — |
| Tetragonal | \(I4_1/mcm\) | — | \(x0\frac12_{1,2}\) | \(xx0_{3,4}\) | \(00z_{5,6}\) | \(x0z_{1,5}\) | \(xxz_{4,6}\) | \(xy0_{2,4}\) | — |
| Tetragonal | \(I4_1/amd\) | — | \(x00_{1,2}\) | \(x/x+\frac12\, [[unclear]]\) | \(\frac12 0z_4\) | \(x0z_{1,4}\) | \(xxz\) | \(xy0_2\) | \(xy'_{[[unclear]]}\) |
| Tetragonal | \(I4_1/acd\) | — | \(x0\frac12_{1,2}\) | \(x/x+\frac12\, [[unclear]]\) | \(\frac12 0z_4\) | \(x0z_{1,4}\) | \(xxz\) | \(xy0_2\) | \(xy'_{[[unclear]]}\) |
| Trigonal | \(P\bar{3}\) | \(P\bar{3}\) | \(\}\) | — | — | — | — | \(xy0\) | — |
| Trigonal | \(P3\) | — | \(\}\) | — | — | — | — | ||
| Trigonal | \(P3_1\) | — | \(\}\) | — | — | — | — | \(xy1'_3\) | |
| Trigonal | \(P3_1\) | — | \(\}\) | — | — | — | — | ||
| Trigonal | \(R\bar{3}\) | \(R\bar{3}\) | \(\}\) | — | — | — | — | \(xy0\) | |
| Trigonal | \(R3\) | — | |||||||
| Trigonal | \(P321\) | \(P\bar{3}m1\) | \(x2xz\) | \(xy0\) | |||||
| Trigonal | \(P3_121\) | — | \(\}\) | — | \(x2xz\) | \(xy1'_3\) | |||
| Trigonal | \(P3_221\) | — | \(\}\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov space group of the \(A\)-function | Images of symmetry planes in the space of \(A\)-functions | Images of symmetry planes in the space of \(A\)-functions | Images of symmetry planes in the space of \(A\)-functions | Images of symmetry planes in the space of \(A\)-functions | Images of symmetry axes in the space of \(A\)-functions | Images of symmetry axes in the space of \(A\)-functions | Images of symmetry axes in the space of \(A\)-functions | Images of symmetry axes in the space of \(A\)-functions | Images of symmetry axes in the space of \(A\)-functions |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Hexagonal | \(P3m1\) | — | — | \(xx0_1\) | — | — | — | — | \(xy0_1\) | — | — |
| Hexagonal | \(P3c1\) | — | — | \(xx\frac12\) | — | — | — | — | \(xy0\) | — | — |
| Hexagonal | \(P\overline{3}m1\) | — | — | \(xx0_1\) | — | — | \(x2xz\) | — | \(xy0_1\) | — | — |
| Hexagonal | \(P\overline{3}c1\) | — | — | \(xx\frac12\) | — | — | \(x2xz\) | — | \(xy0\) | — | — |
| Hexagonal | \(P312\) | \(P\overline{3}1m\) | — | — | — | — | — | \(xxz\) | \(xy0\) | — | — |
| Hexagonal | \(P3_4 12\) | — | — | — | — | — | — | — | — | — | — |
| Hexagonal | \(P3_2 12\) | — | — | — | — | — | — | \(xxz\) | \(xy\frac12\) | — | — |
| Hexagonal | \(P31m\) | — | \(x2x0_1\) | — | — | — | — | — | \(xy0_1\) | — | — |
| Hexagonal | \(P31c\) | — | \(x2x\frac12\) | — | — | — | — | — | \(xy0\) | — | — |
| Hexagonal | \(P\overline{3}1m\) | — | \(x2x0_1\) | — | — | — | — | \(xxz\) | \(xy0_1\) | — | — |
| Hexagonal | \(P\overline{3}1c\) | — | \(x2x\frac12\) | — | — | — | — | \(xxz\) | \(xy0\) | — | — |
| Hexagonal | \(R32\) | \(R\overline{3}m\) | — | — | — | — | \(x2xz\) | — | \(xy0\) | — | — |
| Hexagonal | \(R3m\) | — | — | \(xx0_1\) | — | — | — | — | \(xy0_1\) | — | — |
| Hexagonal | \(R3c\) | — | — | \(xx\frac12\) | — | — | — | — | \(xy0\) | — | — |
Continuation of Table 1
| Crystal system | Fedorov group of the crystal | Fedorov group of the \(A\)-function space | Image of symmetry planes in the \(A\)-function space | Image of symmetry planes in the \(A\)-function space | Image of symmetry planes in the \(A\)-function space | Image of symmetry axes in the \(A\)-function space | Image of symmetry axes in the \(A\)-function space | Image of symmetry axes in the \(A\)-function space | Image of symmetry axes in the \(A\)-function space | Image of symmetry axes in the \(A\)-function space |
|---|---|---|---|---|---|---|---|---|---|---|
| \(R\overline{3}m\) | — | — | \(xx0_1\) | — | \(x2xz\) | — | \(xy0_1\) | — | — | |
| \(R\overline{3}c\) | — | — | \(xx^{1/2}\) | — | \(x2xz\) | — | \(xy0\) | — | — | |
| Hexagonal | \(P6\) | \(P6/m\) | — | — | — | — | — | \(xy0\) | — | — |
| Hexagonal | \(P6_1\) | — | — | — | — | — | — | \(xy^{1/6}\) | \(xy^{1/3}\) | \(xy^{1/2}\) |
| Hexagonal | \(P6_5\) | — | — | — | — | — | — | — | — | — |
| Hexagonal | \(P6_2\) | — | — | — | — | — | — | \(xy0\) | \(xy^{1/3}\) | — |
| Hexagonal | \(P6_4\) | — | — | — | — | — | — | — | — | — |
| Hexagonal | \(P6_3\) | — | — | — | — | — | — | \(xy0\) | — | \(xy^{1/2}\) |
| Hexagonal | \(P6=P3/m\) | — | — | — | \(00z\) | — | — | \(xy0\) | — | — |
| Hexagonal | \(P6/m\) | — | — | — | — | — | — | — | — | — |
| Hexagonal | \(P6_3/m\) | — | — | — | \(00z\) | — | — | \(xy0\) | — | \(xy^{1/2}\) |
| Hexagonal | \(P622\) | \(P6/mmm\) | — | — | — | \(x2xz\) | \(xxz\) | \(xy0\) | — | — |
| Hexagonal | \(P6_122\) | [[unclear: entry obscured]] | — | — | — | — | — | — | — | — |
| Hexagonal | \(P6_522\) | [[unclear: entry obscured]] | — | — | — | \(x2xz\) | \(xxz\) | \(xy^{1/6}\) | \(xy^{1/3}\) | \(xy^{1/2}\) |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov group of the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|---|
| Hexagonal | \(P6_2 22\) | — | — | \(x^2xz\) | \(xxz\) | \(xy0\) | \(xy^1\) | |||
| Hexagonal | \(P6_4 22\) | — | — | \(x^2xz\) | \(xxz\) | \(xy0\) | \(xy^{1/2}\) | |||
| Hexagonal | \(P6_3 22\) | — | — | |||||||
| Hexagonal | \(P6mm\) | — | \(x2x0_1\) | \(xx0_2\) | — | — | — | \(xy0_{1,2}\) | ||
| Hexagonal | \(P6cc\) | — | \(x2x^{1/2}\) | \(xx^{1/2}\) | — | — | — | \(xy0\) | — | — |
| Hexagonal | \(P6_3cm\) | — | \(x2x0_1\) | \(xx^{1/2}_2\) | — | — | — | \(xy0_1\) | — | \(xy^{1/2}_2\) |
| Hexagonal | \(P6_3mc\) | — | \(x2x^{1/2}_1\) | \(xx0_2\) | — | — | — | \(xy0_2\) | — | \(xy^{1/2}_1\) |
| Hexagonal | \(P6m2\) | — | — | \(xx0_{1,2}\) | — | — | \(xxz_1\) | \(xy0_2\) | — | — |
| Hexagonal | \(P\bar{6}c2\) | — | — | \(xx^{1/2}_1\) | — | — | \(xxz_1\) | \(xy0\) | — | — |
| Hexagonal | \(P\bar{6}2m\) | — | \(x2x0_{1,2}\) | — | — | \(x2xz_1\) | — | \(xy0_2\) | — | — |
| Hexagonal | \(P\bar{6}2c\) | — | \(x2x^{1/2}_1\) | — | — | \(x2xz_1\) | — | \(xy0\) | — | — |
| Hexagonal | \(P6/mmm\) | \(x2x0_{1,2}\) | \(xx0_{3,4}\) | \(00z_{5,6}\) | \(x^2xz_{1,5}\) | \(xxz_{3,6}\) | \(xy0_{2,4}\) | — | — | |
| Hexagonal | \(P6/mcc\) | \(x2x^{1/2}_1\) | \(xx^{1/2}_2\) | \(00z_{3,4}\) | \(x2xz_{1,3}\) | \(xxz_{2,4}\) | \(xy0\) | — | — | |
| Hexagonal | \(P6_3/mcm\) | \(x2x0_{1,2}\) | \(xx^{1/2}_{3,4}\) | \(00z_{5,6}\) | \(x2xz_{1,5}\) | \(xxz_{3,6}\) | \(xy0_2\) | \(xy^{1/2}_4\) | ||
| Hexagonal | \(P6_3/mmc\) | \(x2x^{1/2}_{1,2}\) | \(xx0_{3,4}\) | \(00z_{5,6}\) | \(x2xz_{1,5}\) | \(xxz_{3,6}\) | \(xy0_4\) | \(xy^{1/2}_2\) |
Continuation of Table 11
| Crystal system | Fedorov group of the crystal | Fedorov group of the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cubic | \(P23\) | \(Pm3\) | — | — | — | — | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(P2_13\) | — | — | — | — | — | — | — | \(xy^{1/2}\) | — | \(xy(x+y)\) | — |
| Cubic | \(Pm3\) | — | \(x00_1\) | — | — | — | — | — | \(xy0_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(Pn3\) | — | \(x^{1/2}\,{}^{1/2}\) | — | — | — | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(Pa3\) | — | \(x^{1/2}0_1\) | — | — | — | — | — | \(xy^{1/2}_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(I23\) | \(Im3\) | — | — | — | — | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(I2_13\) | — | — | — | — | — | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(Im3\) | — | \(x00_1\) | — | — | — | — | — | \(xy0_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(Ia3\) | — | \(x^{1/2}0_1\) | — | — | — | — | — | \(xy0_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(F23\) | \(Fm3\) | — | — | — | — | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(Fm3\) | — | \(x00_1\) | — | — | — | — | — | \(xy0_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(Fd3\) | — | \(x^{1/4}\,{}^{1/4}\) | — | — | — | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(P432\) | \(Pm3m\) | — | — | — | \(xxz\) | \(xy0\) | — | \(xy(x+y)\) | — | ||
| Cubic | \(P4_232\) | — | — | — | — | \(xxz\) | \(xy0\) | \(xy^{1/2}\) | \(xy(x+y)\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov group of the \(A\)-function space | Images of symmetry planes in the \(A\)-function space | Images of symmetry planes in the \(A\)-function space | Images of symmetry planes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space | Images of symmetry axes in the \(A\)-function space |
|---|---|---|---|---|---|---|---|---|---|---|
| Cubic | \(P4_3 32\) | — | — | — | — | \(xxz\) | \(xy^{1/4}\) | \(xy^{1/2}\) | \(xy(x+y)\) | — |
| Cubic | \(P4_1 32\) | — | — | — | — | \(xxz\) | \(xy^{1/4}\) | \(xy^{1/2}\) | \(xy(x+y)\) | — |
| Cubic | \(P\bar{4}3m\) | — | — | \(xx0_1\) | — | — | \(xy0_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(P\bar{4}3n\) | — | — | \(xx^{1/2}\) | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(Pm\bar{3}m\) | — | \(x00_{1,2}\) | \(xx0_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_{2,4}\) | — | \(xy(x+y)\) | — |
| Cubic | \(Pn\bar{3}n\) | — | \(x^{1/2}\,{}^{1/2}_{1}\) | \(xx^{1/2}_{2}\) | — | \(xxz_{1,2}\) | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(Pn\bar{3}n\) | — | \(x00_{1,2}\) | \(xx^{1/2}_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_2\) | \(xy^{1/2}_4\) | \(xy(x+y)\) | — |
| Cubic | \(Pn\bar{3}m\) | — | \(x^{1/2}\,{}^{1/2}_{1,2}\) | \(xx0_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_4\) | \(xy^{1/2}_2\) | \(xy(x+y)\) | — |
| Cubic | \(I432\) | \(Im\bar{3}m\) | — | — | — | \(xxz\) | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(I4_1 32\) | — | — | — | — | \(xxz\) | \(xy0\) | \(xy^{1/4}\) | \(xy(x+y)\) | — |
| Cubic | \(I\bar{4}3m\) | — | — | \(xx0_1\) | — | — | \(xy0_1\) | — | \(xy(x+y)\) | — |
| Cubic | \(I\bar{4}3d\) | — | — | \(x(x+\tfrac{1}{2})^{1/4}\) | — | — | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(Im\bar{3}m\) | — | \(x00_{1,2}\) | \(xx0_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_{2,4}\) | — | \(xy(x+y)\) | — |
| Cubic | \(Ia\bar{3}d\) | — | \(x^{1/2}0_1\) | \(x(x+\tfrac{1}{2})^{1/2}\) | — | \(xxz\) | \(xy0_1\) | \(xy^{1/4}_2\) | \(xy(x+y)\) | — |
Continuation of Table II
| Crystal system | Fedorov group of the crystal | Fedorov space group of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry planes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function | Images of symmetry axes in the space of the \(A\)-function |
|---|---|---|---|---|---|---|---|---|---|---|
| Cubic | \(F\bar{4}32\) | \(Fm\bar{3}m\) | — | — | — | \(xxz\) | \(xy0\) | — | \(xy(x+y)\) | — |
| Cubic | \(F4_132\) | — | — | — | — | \(xxz\) | \(xy0\) | \(xy^{1/4}\) | \(xy(x+y)\) | — |
| Cubic | \(F\bar{4}3m\) | — | — | \(xx0_{1,2}\) | — | \(xxz\) | \(xy0_2\) | — | \(xy(x+y)\) | — |
| Cubic | \(F\bar{4}3c\) | — | — | \(xx\frac{1}{2}_{1,2}\) | — | \(xxz_1\) | \(xy0_2\) | — | \(xy(x+y)\) | — |
| Cubic | \(Fm\bar{3}m\) | — | \(x00_{1,2}\) | \(xx0_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_{2,4}\) | — | \(xy(x+y)\) | — |
| Cubic | \(Fm\bar{3}c\) | — | \(x00_{1,2}\) | \(xx\frac{1}{2}_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_{2,4}\) | — | \(xy(x+y)\) | — |
| Cubic | \(Fd\bar{3}m\) | — | \(x\frac{1}{4}\frac{1}{4}_{1,2}\) | \(xx0_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_4\) | \(xy^{1/2}\) | \(xy(x+y)\) | — |
| Cubic | \(Fd\bar{3}c\) | — | \(x\frac{1}{4}\frac{1}{4}_{1,2}\) | \(xx\frac{1}{2}_{3,4}\) | — | \(xxz_{1,3}\) | \(xy0_4\) | \(xy^{1/4}_2\) | \(xy(x+y)\) | — |
Note to Table II. The table has been compiled on the basis of the principles set forth in the text. In the first column the name of the crystal system is given. In the second column are the symbol and setting of the Fedorov group. In the third column the Fedorov space group of the \(A\)-function is given.
In the next column are given the maxima of the \(F^1\)-series lying on lines parallel to the coordinate axes. These maxima are the images of symmetry planes perpendicular, respectively, to the axes \(x\), \(y\), and \(z\) in the accepted setting of the Fedorov group. Only independent maxima are given; the rest may be obtained from those listed by applying the symmetry elements of the Fedorov space group of the \(A\)-function.
In the last column are given the images of symmetry axes. Here also, of course, only independent maxima of the \(F^2\)-series are given. Space groups indistinguishable by the distribution of maxima in the \(F^2\)-series are marked with a paragraph sign (see the text).
End of the note to Table 11
The indices at the coordinate triples are given in order to show which two-dimensional sections of the last graph contain identical sections of the preceding graph. The totality of the data given in the last two columns is nothing other than a list of the symmetrically independent maxima in the space of the \(A\)-function produced by a single atom situated in a general position. Thus, for example, if in the Fedorov group \(Pbam\) there is one atom in a general position (i.e., since the multiplicity of the group is 8, a total of 8 atoms in the cell), then the \(F^2\)-row will give maxima on the lines \(x \tfrac12 0\), \(\tfrac12 y0\), and \(00z\), on the planes—\(\tfrac12 yy\), \(x0z\), and \(xy0\), and also, of course, a maximum in the general position.
Relative heights of maxima are not difficult to calculate. In all, there are \(8 \times 7 = 56\) interatomic vectors, and the independent part of the space of the \(A\)-function contains 7 maxima. Of these, one is in the general position, corresponding to the vector connecting centrosymmetric atoms. We shall take the height of this maximum as unity. Atoms related by symmetry planes give an interatomic vector parallel to two other symmetry planes. These vectors occur in sets of 4 parallel ones. Consequently, the maxima on the lines will have a height of 4 units; they constitute, as it were, a fourfold merger of maxima. The independent fraction of the space of the \(A\)-function contains \(\tfrac14\) of each such merged maximum, i.e., one maximum.
The maxima on planes have a height of 2 units. Each maximum is formed by two interatomic vectors.
The description of the \(F^2\)-rows given in the table is not complete. For a special publication it would be necessary to recommend such a method of describing the space of the \(A\)-function corresponding to a crystal of a Fedorov group. A Fedorov group is characterized by several arrangements of points (several special and one general) \(a, b, c, d, \ldots\). Each position, if it is occupied by atoms, gives in the space of the \(A\)-function the typical combination of maxima \(aa\), if only this one position is occupied—and, in addition, \(ab, ac, ad, \ldots\), if other positions are also occupied by atoms.
Consequently, the space of the \(A\)-function corresponding to a Fedorov group, for example with four possible positions \(a, b, c, d\), must be characterized by specifying 10 types of distributions of maxima
\[ \begin{array}{cccc} aa & ab & ac & ad\\ & bb & bc & bd\\ & & cc & cd\\ & & & dd \end{array} \]
The table given also lacks the relative heights of the maxima.
Thus, as in other physical methods for determining an inversion center, by the X-ray method one can establish its absence, but one cannot assert the presence of an inversion center.
Let us proceed to consider the second limitation of the X-ray diffraction method in establishing the symmetry of a crystal.
We have said that an axis of symmetry is reflected in the space of the \(A\)-function by maxima lying in a plane, and a plane of symmetry by maxima lying on an axis. However, maxima lying in planes arise in the space of the \(A\)-function also in the case where the crystal contains atoms not related by symmetry, located in one and the same plane (a pseudo-rotation axis), or in planes displaced by \(\frac{1}{2}\) of a period (a pseudo-axis \(2_1\)), and so on. In exactly the same way, the \(F^2\)-series will imitate planes of symmetry if the crystal contains atoms located along one line (a mirror pseudo-plane of symmetry) or on two parallel lines displaced by half a translation (a pseudo-glide plane).
It follows from what has been said that here too, as in the case of the inversion center, X-ray diffraction can strictly prove the absence, but not the presence, of elements of symmetry. The absence of maxima along lines or planes in the space of the \(A\)-function is strict proof of the absence of the corresponding elements of symmetry.
The phenomenon of pseudosymmetries is not so very widespread. Nevertheless, one should keep in mind that the determination of the space group in a number of cases may be impossible without invoking crystal-chemical rules or without carrying the structural analysis “to the end,” i.e. up to the finding of the coordinates of the atoms in the cell.
c) Gaussian distribution of the values of \(F\)
In determining the symmetry of a crystal, and also for some other purposes to be discussed below, finding the mean value \(|F^2|\) for all measured diffraction orders \(hkl\) may help. We write the expression \(|F^2|\) in the form
\[ |F^2|=\left(\sum_{j=1}^{N} f_j \cos \alpha_j\right)^2+ \left(\sum_{j=1}^{N} f_j \sin \alpha_j\right)^2, \]
where \(f_j\) is the atomic factor of atom \(j\), and \(\alpha_j=2\pi(hx_j+ky_j+lz_j)\), \(x_j, y_j, z_j\) are the coordinates of atom \(j\) in the cell, \(N\) is the number of atoms in the cell.
Suppose that the sums \(A=\sum f_j\cos\alpha_j\) and \(B=\sum f_j\sin\alpha_j\)—the real and imaginary parts of the structural amplitude—are variables (for different \(hkl\)) that assume with equal probability ...
with equal probability any values and, moreover, values independent of one another.
To what extent this initial assumption is justified we shall discuss below.
The mean values of \(A\) and \(B\) are equal to zero. The mean values of \(A^2\) and \(B^2\) we find without difficulty:
\[ \overline{A^2} = \sum_{j=1}^{N} f_j^2 \frac{1}{\pi}\int_{0}^{\pi}\cos^2\alpha_j\,d\alpha_j + \sum_{j\ne k}^{N}\sum f_j f_k \frac{1}{2\pi} \int_{0}^{2\pi}\int_{0}^{2\pi} \cos\alpha_j\cos\alpha_k\,d\alpha_j\,d\alpha_k . \]
The double sum is equal to zero and, consequently,
\[ \overline{A^2}=\frac{1}{2}\sum_{j=1}^{N} f_j^2; \]
similarly,
\[ \overline{B^2}=\frac{1}{2}\sum_{j=1}^{N} f_j^2. \]
Thus,
\[ \overline{F^2}=\sum_{j=1}^{N} f_j^2. \]
If, as we have assumed, the quantities \(A\) and \(B\) are randomly distributed among different orders of reflections, then for these quantities a Gaussian distribution holds.
If by \(P(A)\,dA\) we denote the fraction of structural amplitudes whose real parts lie between \(A\) and \(A+dA\), then, as is known,
\[ P(A)=\frac{1}{m\sqrt{2\pi}}\,e^{-\frac{A^2}{2m^2}}, \]
where \(m=\sqrt{\overline{A^2}}\). An analogous equality holds for the imaginary parts of the structural amplitudes.
The fraction of reflections \(hkl\) for which the value \(F^2\) lies within the limits from \(F^2\) to \(F^2+d(F^2)\) is related to \(P(A)\) and \(P(B)\) in the following way. The fraction of reflections \(hkl\) for which \(A\) lies within the limits \(A, A+dA\) and \(B\) simultaneously within the limits from \(B\) to \(B+dB\), is equal to
\[ P(A)P(B)\,dA\,dB=(\pi\Delta)^{-1}e^{-\frac{A^2+B^2}{\Delta}}\,dA\,dB, \]
where
\[ \Delta=\sum_{j=1}^{N} f_j^2 . \]
In order to find \(P(F^2)\,d(F^2)\), this expression must be integrated over the ring lying in the plane of the variables \(A, B\), with radii from \(F^2\) to \(F^2+d(F^2)\). The integration reduces to multiplication by the area of the ring. Thus,
\[ P(F^2)=\Delta^{-1} e^{-\frac{F^2}{\Delta}} . \]
We have carried out our reasoning for a crystal that does not possess a center of symmetry. Let us show that the function \(P(F^2)\) changes its form in the case when the crystal has a center of inversion. Indeed, in this case
\[ |F^2|=4\left(\sum_{j=1}^{N/2} f_j \cos \alpha_j\right)^2 , \]
since the atoms are pairwise related by a center of symmetry.
\[ \overline{|F^2|}=4\cdot \frac{1}{2}\sum_{j=1}^{N/2} f_j^2 =\sum_{j=1}^{N} f_j^2=\Delta, \]
i.e., the mean value of \(|F^2|\) is the same as in the general case.
Here too we assume that \(F\) (equal to \(A\), since the imaginary part of the structure amplitude is now absent) has a Gaussian distribution over the different diffraction orders \(hkl\), i.e.,
\[ P_c(F)\,dF=\frac{1}{m_c\sqrt{2\pi}}\, e^{-\frac{F^2}{2m_c^2}}\,dF, \]
where
\[ m_c=\sqrt{\overline{|F^2|}}=\Delta^{1/2}. \]
Replacing \(dF\) by \(d(F^2)\), we obtain:
\[ P_c(F^2)=(2\pi \Delta F^2)^{-1/2}\, e^{-\frac{F^2}{2\Delta}} . \]
We arrive at an interesting result\(^8\), namely that the distribution of \(F^2\) by magnitude among all \(hkl\) has a different form for crystals possessing and not possessing a center of inversion. In crystals with a center of inversion the fraction of weak reflections is considerably larger.
Instead of constructing a Gaussian distribution, one may be satisfied with calculating the ratios
\[ \frac{\overline{|F|}}{|F|}=\varepsilon . \]
For a crystal without a center of symmetry,
\[ |\overline{F}|=\Delta^{-1}\int_{0}^{\infty} F e^{-\frac{F^{2}}{\Delta}}\,d(F^{2}) =\frac{1}{2}(\pi\Delta)^{\frac{1}{2}}; \]
for a centrosymmetric crystal,
\[ |\overline{F}|=(2\pi\Delta)^{-\frac{1}{2}}\int_{0}^{\infty}e^{-\frac{F^{2}}{2\Delta}}\,d(F^{2}) =(2\Delta/\pi)^{\frac{1}{2}}, \]
whence
for a crystal without a center of symmetry
\[ \varepsilon=\frac{\pi}{4}=0.785 \]
and
for a crystal with a center of symmetry
\[ \varepsilon=\frac{2}{\pi}=0.637. \]
At first sight we have at our disposal a brilliant method for finding the center of symmetry in a crystal. However, on closer examination it becomes clear that this method is in any case no better\(^5\) than the existing ones.
Indeed, in calculating the Gaussian curve for a non-centrosymmetric crystal it was assumed that the distribution of the quantities \(A\) and \(B\) in the plane of the variables \(AB\) is circular in character. Of course, such a situation will not occur for crystals in which the arrangement of the atoms differs little from a centrosymmetric one. Let us suppose, for example, that some fraction of the atoms in the cell are in positions related to one another by a center of symmetry. In this case the mean value of \(A^{2}\) will be greater than the mean value of \(B^{2}\). The distribution of the quantities \(A\) and \(B\) will possess not circular but elliptical symmetry. One may safely assert that elliptical distributions are possible with an axial ratio from unity down to zero (the line being the case of a centrosymmetric crystal). Therefore the quantities \(\varepsilon\) may take any values within the limits from 0.785 to 0.637.
One may suppose the existence of “ideally non-centrosymmetric” crystals, in which the quantities are distributed in the complex plane in a circle. Such crystals probably possess distinct piezoelectric, pyroelectric, etc., effects. Therefore it is natural that, for a number of optically active substances, the method described led to values of \(\varepsilon\) of the order of 0.64—0.70. It is likewise natural that crystals known to be centrosymmetric gave a ratio of the order of 0.76—0.79\(^9\).
However, as was to be expected, the method fails precisely where the solution of the problem is nontrivial. Triphenylphosphor, containing 4 molecules in the monoclinic cell, was assigned, on the basis of an analysis of the values of \(\varepsilon\), to the space group \(P2/a\).
This must be incorrect: as in many analogous cases, the packing of molecules in the space group \(P2_1/a\), which is fundamental for organic substances, is difficult. Then a loss of symmetry occurs—the axis \(2_1\) (or, equivalently, the center of inversion) is lost and the group \(Pa\) is formed with four molecules (2 nonequivalent) in the cell. The molecules in such a crystal are very densely packed, and their arrangement differs little from packing with an inversion center.
The rules of organic crystal chemistry \(^{10}\), which exclude the group \(C^4_{2h}\;(=P2/a)\) from the number suitable for an organic crystal, are in this case a more serious means of judging the absence of a center of symmetry. It is difficult to state a criterion for the “degree” of centrosymmetry of a crystal, if one bears in mind the arbitrariness of the initial assumption of the theory—uniform and independent distribution of the quantities \(A\) and \(B\). Inevitable experimental errors can also distort the value \(\varepsilon\) to a significant degree. An error of the order of 10% is sufficient for determination of the inversion center by the described method to become impossible.
Be that as it may, here, as in other methods, with accurate work one can refute the presence of an inversion center. However, no experiment can make it possible to prove the presence of a center of symmetry. The structure can always be close to centrosymmetric.
On the basis of an analysis of the distribution of intensities for the reflections \(hkl\), zonal reflections, and different orders of reflection from the same plane, one can make judgments not only about centers of symmetry, but also with respect to other symmetry elements. A similar method has been developed \(^{11}\); however, in our opinion, it is of little interest.
It is nevertheless essential that determination of \(\varepsilon\) can help in determining the Fedorov group where the method of \(F^2\)-series fails. Of the 8 indistinguishable pairs, 6 can in principle be distinguished by the method described. These are the pairs of groups that differ by a center of inversion. It is impossible to distinguish only the following two pairs: \(J23\) and \(J2_1 3\), as well as \(J222\) and \(J2_1 2_1 2_1\).
г) Conclusion
Those 8 pairs of Fedorov groups which are characterized by the same distribution of maxima along lines and planes of the space of the \(A\)-function and, generally speaking, are indistinguishable by the method of \(F^2\)-series, will, of course, give \(A\)-functions that differ from one another. Here, however, determination of the Fedorov group will require knowledge of certain elements of the structure. The same also applies to other limitations of the method described.
Thus, to 219 Fedorov groups (excluding 11 enantiomorphic ones) there correspond 217 X-ray groups (the pairs \(J222\) and \(J2_1 2_1 2_1\), and also \(J23\) and \(J2_1 3\), constitute two X-ray groups).
However, an unambiguous determination of the Fedorov group from the X-ray group, because of the limited resolving power of the \(F^2\)-series, may not always be possible. In any case, it is possible to prove the absence of an element of symmetry; but one can be confident of the presence of an element of symmetry only when the arrangement of the atoms in the cell has been determined.
Cases of pseudosymmetry appear to occur rather rarely; therefore judgments about the presence of screw axes and glide planes from extinctions, or from accumulations of maxima on lines and planes of the space of the \(A\)-function, are sufficiently justified in the overwhelming majority of cases.
Nothing can be said about how often noncentrosymmetric crystals close to centrosymmetric ones occur in nature—too small a number of noncentrosymmetric crystals have been studied by the method of \(F^2\)-series.
From what has been said, the great significance is clear of the general ideas of crystallochemistry and, above all, of the idea of close packing in the determination of space groups. In the case, for example, of organic crystals, one may confidently reject, on the basis of the rules of organic crystallochemistry, the existence in the crystal of mirror-symmetry planes of the packing molecules. A more complicated characteristic example of triphenylphosphine was given above.
3. SERIES OF INTERATOMIC VECTORS AND THE STRUCTURE OF THE CRYSTAL
With an arbitrarily chosen origin of coordinates, the structure of a crystal is characterized by the set of vectors \(\mathbf r_i\) \((i=1,\ldots,N)\), where \(N\) is the number of atoms in the unit cell, and \(\mathbf r_i\) is the vector connecting the origin of coordinates with atom \(i\) of the unit cell.
The series of interatomic vectors is characterized by the set of interatomic vectors \(\mathbf r_{ik}=\mathbf r_k-\mathbf r_i\). Not counting the trivial zero solution, the space of the \(A\)-function has \(N(N-1)\) maxima, at which the vectors \(\mathbf r_{ik}\) drawn from the origin of coordinates terminate.
The maxima of the \(F^2\)-series split into \(N\) groups of \(N-1\) maxima (not counting the trivial \(\mathbf r_{11}, \mathbf r_{22}\), etc.):
\[ \begin{array}{ccccc} * & \mathbf r_{12} & \mathbf r_{13} & \ldots & \mathbf r_{1N},\\ \mathbf r_{21} & * & \mathbf r_{23} & \ldots & \mathbf r_{2N},\\ \mathbf r_{31} & \mathbf r_{32} & * & \ldots & \mathbf r_{3N},\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ \mathbf r_{N1} & \mathbf r_{N2} & \mathbf r_{N3} & & * . \end{array} \]
It is not difficult to see that each of the groups gives the arrangement of the atoms in the crystal with the origin of coordinates at the first, second, third, etc., atoms. Indeed,
\[ \mathbf r_{12}=\mathbf r_2-\mathbf r_1,\quad \mathbf r_{13}=\mathbf r_3-\mathbf r_1,\ldots,\quad \mathbf r_{1N}=\mathbf r_N-\mathbf r_1. \]
Thus the following, rather obvious theorem has been proved: the maxima of the space of the \(A\)-function can be divided into \(N\) groups, each of which gives the arrangement of the atoms of the crystal\(^2\). These \(N\) groups of points are, consequently, related by translation.
Thus, the problem of determining the atomic configuration in a crystal from the \(F^2\)-series would be solved in the case where it were possible to give a method for selecting maxima belonging to one group in the sense indicated above.
The division into \(N\) groups may take place both by rows and by columns. The \(N\) groups \(\mathbf r_{1k}, \mathbf r_{2k},\ldots\) give groups of points passing into one another by a parallel translation. For the groups \(\mathbf r_{1k}\) and \(\mathbf r_{2k}\) this translation vector is equal to \(\mathbf r_{12}=\mathbf r_2-\mathbf r_1\). The same is true also for the groups \(\mathbf r_{k1}, \mathbf r_{k2}\ldots\). Groups of the first type, generally speaking, do not pass into groups of the second type by a translation. However, they are related. Indeed, the groups \(\mathbf r_{1k}\) and \(\mathbf r_{k1}\), \(\mathbf r_{2k}\) and \(\mathbf r_{k2}\) pass into one another by inversion at the origin of the coordinates of the \(A\)-function space. For brevity, we shall call the groups \(\mathbf r_{1k}, \mathbf r_{2k},\ldots\) direct, and \(\mathbf r_{k1}, \mathbf r_{k2},\ldots\)—inverse.
Thus there exist in all \(2N\) methods of selecting the structure of the crystal from the maxima of the \(F^2\)-series. If there is a center of symmetry in the crystal, the groups of type \(\mathbf r_{1k}\) and \(\mathbf r_{k1}\) are translationally identical, since the operation of the center of symmetry, applied to centrosymmetric groups, is equivalent to a translation.
In the \(F^2\)-series there are \(N(N-1)\) maxima. The number of all combinations of \(N(N-1)\) taken \(N-1\) at a time is \((N-1)!\) Of this number, only \(2N\) combinations lead to the required result. What, then, is the method for selecting the required combination, i.e. for finding the group of atoms of the crystal among the maxima in the space of the \(A\)-function?
Let us consider a group of \(N-1\) peaks \(\mathbf r_{1k}\). Let us call \(\mathbf r_{12}\) the fundamental vector. Obviously,
\[ \mathbf r_{12}+\mathbf r_{k1}+\mathbf r_{2k}=0 \]
for any \(k\). Or
\[ \mathbf r_{12}=\mathbf r_{1k}+\mathbf r_{k2}. \]
In other words, the fundamental vector is the diagonal of a parallelogram constructed on two vectors, one of which belongs to the first row (the first direct one), and the other to the second column (the second inverse one). Obviously, the midpoint of the fundamental vector is a center of symmetry for each pair \(\mathbf r_{1k}\) and \(\mathbf r_{k2}\).
Thus, the following has been proved: the maxima of the space of the \(A\)-function that are inverted in the center of the basic vector belong only to two groups of points—one direct and one inverse. Inversion will be carried out not only with respect to the coordinates, but also with respect to the weight of the maxima, only when the atomic numbers of atoms 1 and 2, forming the basic vector, are equal. Moreover, a violation of symmetry in the magnitude of a maximum is possible because of the superposition of maxima.
Of course, any vector may be taken as the basic vector. Therefore the theorem proved can also be formulated as follows: if from the matrix \(r_{ik}\) an arbitrary column and row are selected, then the selected vectors will correspond to two groups of points inverted at the midpoint of the vector that is the common element of the row and column.\(^{12}\)
For example, the selected row and column
\[ \begin{array}{ccccc} * & r_{12} & r_{13} & r_{14} & r_{15}\ldots\\ r_{21} & * & r_{23} & r_{24} & r_{25}\ldots\\ r_{31} & r_{32} & * & r_{34} & r_{35}\ldots\\ \hline r_{41} & r_{42} & r_{43} & * & r_{45}\ldots\\ \hline r_{51} & r_{52} & r_{53} & r_{54} & *\ldots\\ \multicolumn{5}{c}{\cdots\ \cdots\ \cdots\ \cdots\ \cdots} \end{array} \]
are inverted at the midpoint of the vector \(r_{43}\). In this case the vectors \(r_{4k}\) and \(r_{k3}\) are related by the center of inversion, i.e. \(r_{41}\) and \(r_{13}\), \(r_{42}\) and \(r_{23}\), etc.
The proposition proved sharply restricts the number of points from which the structure of the crystal must be composed: instead of \(N(N-1)\) maxima, \(2(N-1)\) maxima are to be considered.
Let us carry out the further selection of the atomic configuration first for a centrosymmetric crystal. As indicated above, in this case in the space of the \(A\)-function there will be \(N\) single and
\[ \frac{N^2}{2}-N \]
double maxima. If the indices \(k\) and \(k'\) denote atoms related in the crystal by the center of inversion, then the maxima \(r_{kk'}\) will be single.
Suppose that one of the maxima can be assigned with certainty to the single ones. We choose the vector of this maximum as the basic one and select the maxima of the \(F^2\)-series that are inverted at its center. But the center of this basic vector is the center of symmetry of the group of atoms (since a vector passing through the center of symmetry connects only atoms related by this operation). Consequently, the direct group coincides in this case with the inverse one, and the selection of maxima will lead us to an unambiguous solution of the problem.
This is illustrated by the following table:
| \(*\) | \(\mathbf r_{12}\) | \(\mathbf r_{13}\) | \(\mathbf r_{14}\ldots\) | \(\mathbf r_{11'}\) | \(\mathbf r_{12'}\) | \(\mathbf r_{13'}\) | \(\mathbf r_{14'}\ldots\) | ||
| \(\mathbf r_{21}\) | \(*\) | \(\mathbf r_{23}\) | \(\mathbf r_{24}\ldots\) | \(\mathbf r_{21'}\) | \(\mathbf r_{22'}\) | \(\mathbf r_{23'}\) | \(\mathbf r_{24'}\ldots\) | ||
| \(\mathbf r_{31}\) | \(\mathbf r_{32}\) | \(*\) | \(\mathbf r_{34}\ldots\) | \(\mathbf r_{31'}\) | \(\mathbf r_{32'}\) | \(\mathbf r_{33'}\) | \(\mathbf r_{34'}\ldots\) | ||
| \(\mathbf r_{41}\) | \(\mathbf r_{42}\) | \(\mathbf r_{43}\) | \(*\ldots\) | \(\mathbf r_{41'}\) | \(\mathbf r_{42'}\) | \(\mathbf r_{43'}\) | \(\mathbf r_{44'}\ldots\) | ||
| \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) | \(\cdot\) |
| \(\mathbf r_{1'1}\) | \(\mathbf r_{1'2}\) | \(\mathbf r_{1'3}\) | \(\mathbf r_{1'4}\ldots\) | \(*\) | \(\mathbf r_{1'2'}\) | \(\mathbf r_{1'3'}\) | \(\mathbf r_{1'4'}\ldots\) | ||
| \(\mathbf r_{2'1}\) | \(\mathbf r_{2'2}\) | \(\mathbf r_{2'3}\) | \(\mathbf r_{2'4}\ldots\) | \(\mathbf r_{2'1'}\) | \(*\) | \(\mathbf r_{2'3'}\) | \(\mathbf r_{2'4'}\ldots\) | ||
| \(\mathbf r_{3'1}\) | \(\mathbf r_{3'2}\) | \(\mathbf r_{3'3}\) | \(\mathbf r_{3'4}\ldots\) | \(\mathbf r_{3'1'}\) | \(\mathbf r_{3'2'}\) | \(*\) | \(\mathbf r_{3'4'}\ldots\) | ||
| \(\mathbf r_{4'1}\) | \(\mathbf r_{4'2}\) | \(\mathbf r_{4'3}\) | \(\mathbf r_{4'4}\ldots\) | \(\mathbf r_{4'1'}\) | \(\mathbf r_{4'2'}\) | \(\mathbf r_{4'3'}\) | \(*\ldots\) |
The single vector \(\mathbf r_{11'}\) has been taken as the principal vector. All maxima of the selected row and column, except \(\mathbf r_{11'}\), merge in pairs and become double. The maxima \(\mathbf r_{21'}\) and \(\mathbf r_{12'}\), \(\mathbf r_{31'}\) and \(\mathbf r_{13'}\), etc., coincide, since the vectors \(\mathbf r_{1'}-\mathbf r_2\) and \(\mathbf r_{2'}-\mathbf r_1\), \(\mathbf r_{1'}-\mathbf r_3\) and \(\mathbf r_{3'}-\mathbf r_1\) are parallel. The maxima \(\mathbf r_{1k}\) and \(\mathbf r_{k1'}\), i.e. \(\mathbf r_{12'}\) and \(\mathbf r_{2'1'}\), \(\mathbf r_{13'}\) and \(\mathbf r_{3'1'}\), etc., are inverted at the center of the vector \(\mathbf r_{11'}\).
Let us now assume that the single maxima are poorly expressed and that the principal vector has been drawn through a double maximum. Then, by inversion at the center of the principal vector, two groups of points, \(\mathbf r_{1k}\) and \(\mathbf r_{k2'}\), will be selected; among them, along with double maxima, there will also be single ones. Since the crystal has a center of symmetry, both groups of points are related by a translation. Thus, after selection we have two “crystal structures” superposed on one another with some parallel displacement. How can these groups of points be separated from one another?
The most direct method is to search for the displacement vector of the two “structures.” Obviously, the whole pattern can be divided into \(N-1\) pairs of maxima connected by the vector \(\rho\), and, consequently, the group \(\mathbf r_{1k}\) will automatically separate from the group \(\mathbf r_{k2'}\).
We proceed to discuss the case of a non-centrosymmetric crystal. Following the previous scheme, by inversion at the middle of the principal vector \(\mathbf r_{12}\), we select two groups \(\mathbf r_{1k}\) and \(\mathbf r_{k2}\). These two groups are not parallel to one another. To isolate the points of one group, we proceed as follows.
In an arbitrary manner we select the first three points of the group. These will be: the origin of coordinates of the space of the \(A\)-function, the end of the principal vector \(\mathbf r_{12}\), and any of the selected maxima, which we shall call \(\mathbf r_{13}\). Let us now consider two points \(A\) and \(B\), inverted at the middle of \(\mathbf r_{12}\); let these be the ends of the vectors \(\mathbf r_{14}\) and \(\mathbf r_{42}\).
However, we do not know which of these vectors corresponds to point \(A\) or \(B\). The group that we began to construct contains \(\mathbf r_{14}\), but not \(\mathbf r_{42}\). Let us join point \(A\) with the end of vector \(\mathbf r_{13}\). If \(A\) is \(\mathbf r_{14}\), then the segment drawn is equal to \(\mathbf r_{14} - \mathbf r_{13} = \mathbf r_{43}\); this is an interatomic vector and, consequently, there exists a vector drawn from the origin, equal and parallel to the segment drawn from \(A\) to the end of \(\mathbf r_{13}\). If \(A\) is \(\mathbf r_{42}\), then the segment drawn is equal to \(\mathbf r_{42} - \mathbf r_{13} = \mathbf r_2 - \mathbf r_4 - \mathbf r_1 + \mathbf r_3\); this is not an interatomic vector, and in this case there is no vector drawn from the origin that is equal and parallel to the one drawn.
In the manner indicated, we can test all pairs of maxima \(\mathbf r_{1k}\) and \(\mathbf r_{k2}\) and select from them the maxima \(\mathbf r_{1k}\).
What difficulties may arise in the practical implementation of this method? It is obvious that the method is doomed to failure if a large number of maxima is not resolved. Projections of the \(F^2\)-series may therefore be used to carry out this method only in the simplest cases. The possibilities of the three-dimensional series, however, are quite great, as follows from the following approximate estimate. In the case of 4 molecules of 8 atoms in the unit cell, the total number of peaks of the \(F^2\)-series is almost 1000. With the usual division of the cell, the calculation of the series is carried out at approximately \(50 \times 50 \times 50 = 125\,000\) points. Thus, on average, one peak corresponds to 100 points. In the case of 32 atoms in the cell, its volume will be of the order of \(1000\ \text{\AA}^3\); consequently, one peak will correspond to a volume of the order of \(1\ \text{\AA}^3\). With a resolving power of the series of the order of \(0.5\ \text{\AA}\), in such a volume there may be not one, but 25 resolved maxima. Thus, merging of maxima, owing to their accidental closeness, will be relatively rare, and a developed method has chances of success in the majority of quite complicated cases.
Up to the present time, structural studies by the method of \(F^2\)-series have been carried out only for structures with one or two heavy atoms. Suppose, for example, that there are two heavy atoms and some number of light ones in the cell. Then \(\mathbf r_{12}\) gives one strongest maximum in the space of the \(A\)-function. The column and row of the matrix \(\mathbf r_{ik}\) that intersect at \(\mathbf r_{12}\) will give maxima of intermediate strength; all the remaining \(\mathbf r_{ik}\) will give weak maxima. Thus, the heavy atom automatically performs that selection of groups, the artificial carrying out of which we discussed above. It was precisely in such a case that Sanadze V. V. and Zhdanov G. S.\(^{13}\) first established the connection of a group of points of the \(F^2\)-series with an inversion center in the middle of an interatomic vector.
The chief interest, however, is the use of the relation between the \(F^2\)-series and the structure of the crystal in the case of crystals built of atoms with nearly equal scattering power. Experimental investigations based on this method will undoubtedly begin to appear in the near future.
The indicated method is of principal interest in the case of structures without heavy atoms. A structure without a center of symmetry, with atoms of equal scattering power, is precisely where it is especially interesting to test the possibility of directly determining the structure from a three-dimensional series of interatomic vectors.
If further experience shows that a structural problem can be solved without the aid of electron-density series, then the simplification of the method of crystal-structure analysis will be exceptionally great. The most laborious stage of the analysis—the calculation of structural amplitudes—will become unnecessary.
The arguments presented show that there is not only a fundamental but also a practical route to finding the structure directly from the \(F^2\)-series.
It should also not be forgotten that symmetry provides a number of known, important simplifications. We have already emphasized that for a lattice with a center of symmetry the \(F^2\)-series contains \(N\) single maxima (all the others being double), which give the structure directly. If all these maxima can be clearly identified, then finding the structure is trivial. Axes and planes of symmetry bring maxima into special positions, etc.
The method considered here should lead to a positive result in all cases. Of great importance is the possibility of repeated checking within the framework of the method itself (any vector may be chosen as the fundamental one). We do not dwell here on the use of \(F^2\)-series in structural analysis when the cell contains a small number of heavy atoms, since these methods are well known and described in the monograph \({}^{1}\). For the same reason we do not discuss here methods for calculating Fourier series.
4. DETERMINATION OF THE PHASES OF STRUCTURAL AMPLITUDES AND CONSTRUCTION OF ELECTRON-DENSITY SERIES
The Fourier series of the electron density in a crystal
\[ \rho(xyz)=\frac{1}{v}\sum_h \sum_k \sum_l F_{hkl} e^{2\pi i(hx+ky+lz)} \]
can be constructed only if the phases of the complex quantities \(F_{hkl}\) are known. Experiment gives only \(|F_{hkl}|\); at first sight the phases cannot be found by an objective method.
All (with the exception of isolated) structure determinations carried out up to now have been made as follows. From \(F^2\)-series or from crystal-chemical considerations, a “rough” structure was found, i.e. an approximate determination of the arrangement
atoms in the cell. From the values found for the coordinates of the atoms, the phases of the structural amplitudes were calculated. After this, a series \(\rho(xyz)\) was constructed, in which the phases \(F_{hkl}\) were taken from calculation, and the absolute values from experiment. If the series constructed in this way gives maxima of the function \(\rho(xyz)\) at points not quite coinciding with the coordinates of the atoms used for the calculation of the phases, then from the constructed series a second set of atomic coordinates is determined, and from it a second set of phases of the structural amplitudes is calculated. The procedure is then repeated and, if the initial rough structure was correct, leads to the final result. The phases \(F_{hkl}\) calculated from the last series \(\rho(xyz)\) must coincide with the corresponding phases used for its construction.
The technique of working by this method and the accuracy of the results obtained are discussed elsewhere\(^{1}\), and we shall not dwell on them here.
In 1948 Harker and Kasper drew attention to the possibility of an objective determination of phases, in any case for centrosymmetric structures, where the phases have only the values \(0^\circ\) and \(180^\circ\) (the phases reduce to signs). In 1950, by this method, the very complex structure of decaborane was determined\(^{14}\). At the present time there is no reason to doubt the possibility of applying this method to the solution of structural problems. It is therefore of interest to consider the generalizations of Harker and Kasper’s work, published in 1950\(^{15,16,17}\).
The structural amplitude \(F_{hkl}\) can be expressed through the electron-density distribution function as the coefficient of the Fourier series
\[ F_{hkl}=v\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\rho(xyz)e^{2\pi i(hx+ky+lz)}\,dx\,dy\,dz. \]
Starting from this definition of \(F_{hkl}\) and from the requirement that \(\rho(xyz)\) be a positive quantity for all values of \(xyz\), it is not hard to show that the expression
\[ \sum_{hkl}^{m}\sum_{h'k'l'}^{m} X_{hkl}X^{*}_{h'k'l'}F_{h-h',\,k-k',\,l-l'} \geq 0 \quad (m=1,2,\ldots). \]
The sextuple sum is taken over diffraction orders (or, what is the same thing, over nodes of the reciprocal lattice); \(X\) and \(X^{*}\) are an arbitrary quantity and its complex conjugate.
The validity of this condition is easily verified by substituting the integral expression for \(F_{hkl}\) into the inequality. It can also be proved that the condition written above is not only necessary, but also sufficient for \(\rho(xyz)\) not to have negative values.
On the basis of general theorems of higher algebra it is proved that, in order for the inequality written above to hold, it is in turn necessary that the following series of inequalities be satisfied:
\[ \left|F_{\xi_n\eta_n\xi_n}-\delta\right|\leq r \quad (n=0,1,2,\ldots), \]
where
\[ \varepsilon_n=h_1+h_2+\cdots+h_n,\quad \eta_n=k_1+k_2+\cdots+k_n, \]
\[ \xi_n=l_1+l_2+\cdots+l_n; \]
\[ \delta=\frac{\Delta'}{\Delta},\quad r=\frac{\Delta_1^{1/2}\Delta_2^{1/2}}{\Delta}; \]
\(\Delta,\Delta',\Delta_1\) and \(\Delta_2\) are the following determinants:
\[ \Delta= \begin{vmatrix} F_{\varepsilon_1-\varepsilon_1}, & F_{\varepsilon_1-\varepsilon_2}, \ldots, & F_{\varepsilon_1-\varepsilon_n-1}\\ F_{\varepsilon_2-\varepsilon_1}, & F_{\varepsilon_2-\varepsilon_2}, \ldots, & F_{\varepsilon_2-\varepsilon_n-1}\\ \cdot & \cdot & \cdots & \cdot\\ F_{\varepsilon_n-1-\varepsilon_1}, & F_{\varepsilon_n-1-\varepsilon_2}, \ldots, & F_{\varepsilon_n-1-\varepsilon_n-1} \end{vmatrix}, \]
\[ \Delta'=(-1)^n \begin{vmatrix} F_{\varepsilon_1}, & F_{\varepsilon_1-\varepsilon_1}, & F_{\varepsilon_1-\varepsilon_2}, \ldots, & F_{\varepsilon_1-\varepsilon_n-1}\\ F_{\varepsilon_2}, & F_{\varepsilon_2-\varepsilon_1}, & F_{\varepsilon_2-\varepsilon_2}, \ldots, & F_{\varepsilon_2-\varepsilon_n-1}\\ \cdot & \cdot & \cdot & \cdots & \cdot\\ F_{\varepsilon_n-1}, & F_{\varepsilon_n-1-\varepsilon_1}, & F_{\varepsilon_n-1-\varepsilon_2}, \ldots, & F_{\varepsilon_n-1-\varepsilon_n-1}\\ 0, & F_{\varepsilon_n-\varepsilon_1}, & F_{\varepsilon_n-\varepsilon_2}, \ldots, & F_{\varepsilon_n-\varepsilon_n-1} \end{vmatrix} \]
\[ \Delta_1= \begin{vmatrix} F_0, & F_{-\varepsilon_1}, \ldots, & F_{-\varepsilon_n-1}\\ F_{\varepsilon_1}, & F_{\varepsilon_1-\varepsilon_1}, \ldots, & F_{\varepsilon_1-\varepsilon_n-1}\\ \cdot & \cdot & \cdots & \cdot\\ F_{\varepsilon_n-1}, & F_{\varepsilon_n-1-\varepsilon_1}, \ldots, & F_{\varepsilon_n-1-\varepsilon_n-1} \end{vmatrix}, \]
\[ \Delta_2= \begin{vmatrix} F_{\varepsilon_1-\varepsilon_1}, & F_{\varepsilon_1-\varepsilon_2}, \ldots, & F_{\varepsilon_1-\varepsilon_n}\\ F_{\varepsilon_2-\varepsilon_1}, & F_{\varepsilon_2-\varepsilon_2}, \ldots, & F_{\varepsilon_2-\varepsilon_n}\\ \cdot & \cdot & \cdots & \cdot\\ F_{\varepsilon_n-\varepsilon_1}, & F_{\varepsilon_n-\varepsilon_2}, \ldots, & F_{\varepsilon_n-\varepsilon_n} \end{vmatrix}. \]
In order that the notation of the determinants not be too cumbersome, the indices of \(F\) were written in abbreviated form: instead of \(-\xi_1\), \(\eta_1\), \(\zeta_1\), etc.; instead of \(\varepsilon_n-\varepsilon_1\), one should read \(\varepsilon_n-\varepsilon_1\), \(\eta_n-\eta_1\), \(\zeta_n-\zeta_1\), etc.*)
Thus, a general rule has been obtained for deriving all possible inequalities that exist between structure amplitudes. Naturally, the larger \(n\) is, the more cumbersome the inequality becomes. Probably \(n=3\), and in any case \(4\), will serve as the practical limit for the use of the inequalities; \(n=0\) and \(n=1\) give the trivial results \(F_{000} \geqslant 0\) and \(|F_{hkl}| \leqslant F_{000}\). The inequality for \(n=2\) has the form:
\[ \left| F_{h_1+h_2,\,k_1+k_2,\,l_1+l_2} - \frac{F_{h_1k_1l_1}F_{h_2k_2l_2}}{F_{000}} \right| \leqslant \frac{ \left| \begin{matrix} F_{000} & F_{\bar h_1 \bar k_1 \bar l_1}\\ F_{h_1k_1l_1} & F_{000} \end{matrix} \right|^{\frac12} \left| \begin{matrix} F_{000} & F_{\bar h_2 \bar k_2 \bar l_2}\\ F_{h_2k_2l_2} & F_{000} \end{matrix} \right|^{\frac12} }{F_{000}}. \]
From this general equation one can obtain a number of particular ones, taking into account the symmetry relation between reflections of various orders, and also taking different types of relations between \(h_1k_1l_1\) and \(h_2k_2l_2\). For example, for crystals with a center of symmetry \((F_{hkl}=F_{\bar h \bar k \bar l})\), and putting \(h_1=h_2=h\), etc., we obtain one of the simplest inequalities, which appeared in the first paper,
\[ \left(\frac{F_{hkl}}{F_{000}}\right)^2 \leqslant \frac{1}{2} + \frac{1}{2}\, \frac{F_{2h,\,2k,\,2l}}{F_{000}}. \]
Thus, the most recent works have constructed a theory that permits the systematic derivation of all possible inequalities between structure amplitudes.
What possibilities does this method offer for carrying out structure analysis? So far, as was mentioned above, only one structure determination carried out by the described method has been published—the investigation of decaborane. By the method of inequalities it proved possible to find the signs of a sufficiently large number of structure amplitudes for constructing the first rows of the electron density. The structure of decaborane crystals is very complex, and the result of applying the method must be regarded as very encouraging.
It is unlikely that the method of inequalities will prove applicable to determining phases of structure amplitudes different from \(0\) and \(180^\circ\). This method will probably serve only for determining the signs of structure amplitudes.
*) The idea of the calculation carried out is based on the work of H. Harker and M. Kasper.\(^{13}\)
In a number of cases (it is now difficult to say what percentage of such cases) the method of inequalities may fail. It is quite obvious that the inequalities will be effective only when among the structural amplitudes there is a sufficient number of “strong” ones, i.e., those for which \(\dfrac{F_{hkl}}{F_{000}}\) is greater than \(0.4\text{—}0.6\).
As was shown above,
\[ \overline{F^2}=\sum_{j=1}^{N} f_j^2 . \]
Suppose that the crystal consists of atoms of a single kind. Then
\[ \overline{F^2}=Nf^2 . \]
Since
\[ \left(\frac{\overline{F}}{F_{000}}\right)^2=\frac{\overline{F^2}}{N^2Z^2}, \]
where \(Z\) is the atomic number, it follows that
\[ \left(\frac{\overline{F}}{F_{000}}\right)^2=\frac{f^2}{NZ^2}<\frac{1}{N}. \]
Thus, as \(N\) increases, the mean value of the unit structural amplitude decreases. In other words, the more complex the structure, the “weaker” the inequalities become.
It was shown by a specific example that, in the presence of 9 atoms in a general position, the method of inequalities is already inapplicable. For most structures of such a degree of complexity there will very often be cases in which all reflections give
\[ \frac{F_{hkl}}{F_{000}}<0.3. \]
In such cases the application of the method of inequalities (in any event, of this method alone) is doomed to failure.
5. REFINEMENT OF ATOMIC COORDINATES BY THE METHOD OF LEAST SQUARES AND BY ANALYSIS OF ELECTRON-DENSITY SERIES
In the last two or three years, the comparative merits and shortcomings of refining the values of atomic coordinates in the cell by the method of least squares and by electron-density series have repeatedly been discussed.\(^{19}\)
Let us denote by \(F_{\mathrm{n}}\) the observed values of the structural amplitudes and by \(F_{\mathrm{p}}\) the values calculated from the atomic coordinates of the “rough” structure obtained by some one of the methods discussed above. One may assume that the structure closest to the true one will be that for which the sum
\[ \sum w(F_{\mathrm{n}}-F_{\mathrm{p}})^2, \]
taken over all reflections, will be minimal. Here \(w\) is the “weight” of a single observation. Any coordinate of an atom, say the coordinate along the \(x\)-axis of the \(r\)-th atom, can then be found from the equation
\[ \sum w(F_{\mathrm{n}}-F_{\mathrm{p}})\frac{\partial F_{\mathrm{p}}}{\partial x_r}=0 \]
or, since \(x_r\) enters into \(F_{\mathrm{p}}\) in the term \(f_r e^{2\pi i(hx_r+ky_r+lz_r)} = f_r e^{i\alpha_r}\) (atoms symmetrically related to the given ones may be disregarded—they will give the same expressions), then
\[ 2\pi i \sum wh f_r(F_{\mathrm{n}}-F_{\mathrm{p}})e^{i\alpha_r}=0. \]
Hence
\[ 2\pi i \sum wh f_r F_{\mathrm{n}} e^{i\alpha_r} = 2\pi i \sum wh f_r F_{\mathrm{p}} e^{i\alpha_r}. \]
It is not difficult to see that each part of the equality represents the partial derivative with respect to \(x\) of the functions
\[ \varphi_{\mathrm{n}}=\sum w f_r F_{\mathrm{n}} e^{i\alpha_r}, \]
\[ \varphi_{\mathrm{p}}=\sum w f_r F_{\mathrm{p}} e^{i\alpha_r}. \]
These sums, like the preceding ones, are taken over all reflections \(hkl\).
If the weight of the observations \(w\) is set equal to \(\dfrac{1}{f_r}\), then \(\varphi_{\mathrm{n}}=\rho_{\mathrm{n}}\) and \(\varphi_{\mathrm{p}}=\rho_{\mathrm{p}}\), i.e., the functions \(\varphi\) correspond to the measured and calculated electron densities. Then determining the atomic coordinates from the equation
\[ \sum w(F_{\mathrm{n}}-F_{\mathrm{p}})\frac{\partial F_{\mathrm{p}}}{\partial x_r}=0 \]
acquires the meaning of finding the maximum of the electron density.
We have shown by the calculation carried out that the least-squares method and the method of electron-density series will give identical results if, in calculating the coordinate of the \(r\)-th atom, the reciprocal of the atomic factor of the \(r\)-th atom is taken as the weight of an individual observation.
However, the methods will give identical results only in the case where the atomic factors of the atoms taken for the calculation are capable of correctly conveying the picture of the electron-density distribution in the crystal under investigation. But this may not be the case. Indeed, in order for the idea of bringing \(F_{\mathrm{n}}\) and \(F_{\mathrm{p}}\) closer together to be correct in its basis, it is necessary that the atoms be approximated with sufficient accuracy by a spherical distribution
electron density, \(\rho(xyz)\), of the crystal must be the sum of the electron densities \(\rho_r\) of the individual free atoms. Then and only then will the atomic factor
\[ f_r=\int 4\pi r^2\rho_r\,\frac{\sin sr}{sr}\,dr \]
(where
\[ s=\frac{4\pi\sin\theta}{\lambda} \]
) of the free atom make it possible to calculate exactly the structure of the crystal, i.e. the function \(\rho(xyz)\). The anisotropy of thermal vibrations and the chemical interactions between atoms do not allow one to regard the least-squares calculation as fully exact.
The method of electron-density series is free of the indicated shortcomings. It was said that the least-squares method is free of the principal shortcoming of the electron-density series—the distortions introduced by truncation of the series. However, these distortions can easily be taken into account if, along with the principal series \(\rho_{\mathrm{n}}(xyz)\), one constructs the series \(\rho_{\mathrm{p}}(xyz)\).
The second series, constructed from the values \(F_{\mathrm{p}}\), can be used to determine the character of the distortions arising when the series is truncated. With some justification it may be assumed that the shifts of the maxima of the series \(\rho_{\mathrm{p}}(xyz)\) from the coordinate values taken as the basis for calculating \(\rho_{\mathrm{p}}(xyz)\) correspond to the shifts of the maxima of the series \(\rho_{\mathrm{n}}(xyz)\) from the sought coordinates of the atoms.
We believe that the least-squares method has no advantages whatever over the method of electron-density series.
A quite specific case of the least-squares method is its application to the determination of one coordinate of the atoms when the other two coordinates are known.
The expression for the structural amplitude, or for sums and differences of structural amplitudes, can always be reduced to linear forms of the type
\[ \sum_i A_i\cos ky_i=|F| \quad\text{and}\quad \sum_i B_i\sin ky_i=|F|, \]
where \(A_i\) and \(B_i\) are known quantities computed from the known \(xz\) coordinates for the given reflection. The index \(i\) numbers the independent atoms in the cell. The number of linear equations for each \(k\) considerably exceeds the number of unknown quantities \(y_i\). Usually it is possible to determine approximate values of \(y_i\), and together with them the signs of all \(F_{hkl}\), using the equations for accidentally extinguished or very weak reflections. Having determined the signs of \(F_{hkl}\), we find the optimal values of the coordinates \(y_i\) by forming, for each \(k\), the polynomials
\[ \Phi_{h,l}=\left(\sum_i A_i a_i-F_{h,l}\right)^2, \]
where \(a_i=\cos ky_i\), adding them into the sum
\[ \sum_{h,l}\Phi_{h,l} \]
(in this case, into the sum
may include any number of terms, provided that it does not exceed the number of unknowns), taking the partial derivatives of this sum and setting them equal to zero. The number of equations obtained in this way is, of course, equal to the number of unknowns.
6. POSSIBILITIES OF X-RAY STRUCTURAL ANALYSIS OF CRYSTALS
The exceptionally great advances in the theory and methodology of X-ray structural analysis made over the last several years, as well as a number of structural investigations of very complex chemical compounds carried out with the utmost thoroughness and reliability, make it possible to express a definite opinion on the possibilities of X-ray structural analysis.
The principal difficulty in determining the electron density \(\rho(xyz)\) in a crystal lies in the inevitable truncation of the series representing \(\rho(xyz)\).
The true electron density is
\[ \rho(xyz)'=\sum_{hkl=0}^{\infty} F_{hkl} e^{2\pi i(hx+ky+lz)}. \]
In experiment, however, a series consisting of a finite number of terms is measured:
\[ \rho(xyz)=\sum_{hkl=0}^{HKL} F_{hkl} e^{2\pi i(hx+ky+lz)}. \]
The remainder of the series \(\rho(xyz)' - \rho(xyz) = \Delta \rho\) may be very significant.
As was shown above, the mean value \(\overline{F^2}\) decreases with increasing scattering angle in the same way as the square of the atomic factor. In the very best investigations the most distant reflections corresponded to \(0.03\)—\(0.05\) of the maximum value of the atomic factor. In ordinary investigations this figure is \(0.1\). At the same time the author has shown that, in order to obtain accurate (to within \(1\%\)) values of the electron density, the \(f\)-curve must be extended experimentally down to \(0.001\) of the maximum value of \(f\).
Thus, the experimentally measured function \(\rho\) and the function \(\rho'\) may differ considerably. It would be proper to introduce a special term for \(\rho\), for example, to speak of a conditional electron density.
As was shown by the author, knowledge of the conditional electron density is quite sufficient for an extremely accurate investigation of interatomic vectors. At the same time, knowledge of the conditional electron—
of the density characterizes to a very small degree the true distribution of electrons in the crystal.
Nor should one forget the need for an incomparably more accurate estimate of the \(F\)-quantities for determining the values of the electron density than is required for the investigation of interatomic distances.
Determinations of interatomic distances by the method of \(F^2\)-series, by the least-squares method, and by constructing electron-density series may lead to somewhat different figures. Experience shows that this difference, as a rule, lies within the limits of experimental errors. Such good agreement of these three methods of calculation is apparently explained by the fact that, for determining interatomic distances, the deviation of the atom from a spherical form is immaterial.
Indeed, in the theory of \(F^2\)-series and in the least-squares method it is assumed that the electron density in the crystal
\[ \rho(xyz)=\sum_{k=1}^{N}\rho_k(\mathbf r-\mathbf r_k), \]
where \(\mathbf r_k\) is the radius vector of the “center” of the atom; the atomic functions \(\rho_k(\mathbf r-\mathbf r_k)\) are assumed to be spherically symmetric.
It is quite obvious that the presence of a chemical bond and the anisotropy of thermal vibrations allow this equality to be regarded only as approximate. However, the degree of this approximation is very high; as for the chemical bond, recent studies have shown that the redistribution of electron density which violates the spherical symmetry of the atom is very small. The structure of diamond differs from a structure composed of a sum of spherically symmetric functions only by an amount of 0.1–0.2 electron distributed along the valence bond.^20
If, at atomic number 6, the role of the valence electrons is so insignificant, it will be still smaller in heavier substances. The influence of thermal vibrations is more serious. On the basis, however, of the still rather limited experimental material, it may be assumed that optical vibrations do not violate the spherical symmetry of the atom.
By contrast, vibrations of the molecule as a whole lead to substantial violations of the spherical symmetry of the atom.
Be that as it may, this, apparently the principal phenomenon that violates the sphericity of the atomic function, is not so large as to render the methods of \(F^2\)-series, least squares, and electron-density series nonequivalent in the determination of interatomic distances.
The problem of accuracy in the determination of interatomic distances has been discussed more than once. Theory and experiment show that the principal
the factor determining the accuracy of the result is the number of measured reflections and, to an utterly negligible degree, the accuracy of measuring intensities1. In this connection, the only way to increase the accuracy of X-ray structural analysis is to study the structure of crystals at low temperatures. In this case, owing to the reduction of thermal vibrations, the number of diffraction orders that can be measured sharply increases.
Since the characteristic temperature of crystals may differ greatly, at room temperature the accuracy of structural analysis with respect to interatomic distances may vary. Referring the reader to the error formulas derived in the original papers[^21][^22], we shall confine ourselves merely to pointing out that, in the worst case, the experimental material makes it possible to determine interatomic distances with an accuracy of \(\pm 0.1\ \text{Å}\), and in the best case \(\pm 0.01\ \text{Å}\). Of course, not all interatomic distances in a structure are determined with the same accuracy. If the cell contains atoms of different weight, then the distances between the lightest atoms will be determined least accurately.
The detectability of light atoms in the presence of heavy ones also depends on the volume of the experiment, i.e., on the number of measured reflections. Because of the termination of the series, the atomic functions \(\rho_k(\mathbf r-\mathbf r_k)\) will differ somewhat from the monotone, essentially positive functions that they would be in the ideal case of an infinite series. So-called “termination waves” arise, distorting the form of \(\rho_k(\mathbf r-\mathbf r_k)\); the function \(\rho_k(\mathbf r-\mathbf r_k)\) becomes similar to the intensity distribution diffracted by an aperture. The amplitude of the spurious oscillations of the functions \(\rho_k\) is proportional to the atomic number of the element. If the magnitude of this amplitude for a heavy atom is of the order of the maximum value of the electron density of a light atom, then the detectability of the latter becomes problematic. The greater the number of heavy atoms in the cell, the smaller the probability of detecting a light atom, which may “drown” in the termination wave of one or another heavy atom.
The individual features of the crystal structure do not allow one to predict with complete certainty the possibility of detecting a light atom in the presence of heavy ones. Apparently the record case is the finding of a carbon atom in iodoform (one C atom per three iodine atoms).
The determination of interatomic distances and the detection of light atoms—these problems confront the investigator only when a rough structure has been found. For this it is necessary to decode the \(F^2\)-series, or to find in some way the phases of the structural amplitudes (at least of the strongest ones, in a number sufficient for constructing an electron-density series). Is it always possible to do this?
Undoubtedly, not always. If a large number of maxima in the \(F^2\)-series have merged, then its interpretation becomes impossible. This is the case, for example, in crystals of globular proteins. The criterion for the fundamental possibility of solving a structural problem is the ratio of the number of maxima of the \(F^2\)-series to the number of cells of the space of the \(A\)-function, if by a cell one understands the volume \(\Delta^3\), where \(\Delta\) is the resolving power of the series, depending on the size of the smallest observed interplanar spacings \(\left(\Delta \simeq 0.6 \cdot d_{\min}\right)\). As was indicated at the beginning of the article, one can hardly expect values \(d_{\min} < 0.4\), and consequently, in the best case \(\Delta = 0.25 \text{ Å}\). As a rule, \(\Delta\) will be twice as large. If the size of the cell is \(a\), then the number of cells per period will be \(\frac{a}{\Delta}\), and the number of cells per elementary volume of the space of the \(A\)-function will be \(\left(\frac{a}{\Delta}\right)^3\). If the number of atoms in the cell is denoted by \(N\), then the condition for solvability of the structural problem can be written in the form
\[ \left(\frac{a}{\Delta}\right)^3 > N(N-1). \]
For a rough estimate the following approximation is also of interest: on average, one atom accounts for a volume
\[ \frac{4}{3}\pi (1.5)^3 \text{ Å}^3 \simeq 10—20 \text{ Å}^3. \]
Instead of \(a^3\) one may substitute \(20N\). Then we obtain
\[ \frac{20}{\Delta^3} > (N-1). \]
The value \(\Delta\) in most cases is of the order of magnitude \(0.5 \text{ Å}\). The condition takes the form
\[ N < 200. \]
Thus, when there are more than 200 atoms in the cell, the solution of the structural problem by existing methods becomes problematic and in any case exceedingly difficult. In these cases X-ray structural analysis will serve no longer to find the structure, but to confirm or refute one or another model of the structure.
In conclusion it is necessary to make several further remarks about the possibilities of other methods for determining structure. The mutual arrangement of atoms can be found with certainty only for a crystal. Only diffraction methods are of significance. In specific cases, alongside the diffraction of X-rays, electronographic and neutronographic methods may be of considerable benefit. These specific cases are mainly two problems: 1) determination
coordinates of hydrogen atoms (which scatter X-rays to a very small extent in comparison with other atoms)—successes in applying electron diffraction for this purpose have been obtained by Z. G. Pinsker and B. K. Vainshtein^23—and 2) the distinction of atoms with close atomic numbers. Such atoms scatter X-rays practically identically and at the same time may have different scattering cross sections for neutrons. The prospects of neutron diffraction and electron diffraction as methods for analyzing the structure of crystals are not yet entirely clear at the present time. However, one may confidently predict that in the future these methods of investigation will become a valuable support for the method of X-ray structural analysis.
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