 1. Introduction
 2. On the number of face forms
 3. Are Miller indices always coprime integers?
 4. Laue indices of Bragg peaks versus Miller indices of families of lattice planes
 5. Geometric derivation of the reflection conditions
 6. Can the lack of symmetry impose restrictions?
 7. Conclusions
 References
 1. Introduction
 2. On the number of face forms
 3. Are Miller indices always coprime integers?
 4. Laue indices of Bragg peaks versus Miller indices of families of lattice planes
 5. Geometric derivation of the reflection conditions
 6. Can the lack of symmetry impose restrictions?
 7. Conclusions
 References
teaching and education
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The ash heap of crystallography: restoring forgotten basic knowledge
^{a}Materials and Structures Laboratory, Tokyo Institute of Technology, Midoriku, Yokohama City, Kanagawa 2268503, 4259 Nagatsuta, Japan
^{*}Correspondence email: massimo.nespolo@crm2.uhpnancy.fr
A critical analysis of some basic notions often overlooked in crystallographic education is presented to correct some common oversights occurring both in the literature and in textbooks. The crystal forms (face forms), defined in terms of their geometric
are 47 in number, not 48 as often found in the literature. The split of the dihedron into dome and sphenoid calls for the consideration of the physical properties of the faces building a form; in that case, however, the same criterion should be used for all forms. By taking the handedness of the faces as representative of the physical properties of the faces, the occurrence of 130 crystallographic face forms (97 affine face forms and 33 enantiomorphic pairs) is demonstrated. Next, the correct use of noncoprime when a centred is adopted is shown, and the inconsistent multiplication of in the Bravais–Friedel–Donnay–Harker law is pointed out. A geometric derivation of the is reviewed. Finally, the inconsistent presentation of metric restrictions imposed by the structural symmetry is pointed out and corrected.Keywords: crystal forms; systematic absences; Miller indices; symmetry restrictions; crystallographic education.
1. Introduction
In 1836, the Italian poet Giacomo Leopardi (1836) ironically challenged what he called the `magnificent and progressive fate of the human race', meaning the blind belief in an unlimited and extraordinary progress for all mankind, the optimistic legacy of the Age of Enlightenment. Two centuries later we realize how farsighted was his scepticism: the undeniable technological advances and the evolution of scientific knowledge have not been paralleled by a comparable social evolution in many parts of our world. Scientific advances have increased exponentially the number and complexity of notions and concepts that one is expected to master in any field, so that some domains are increasingly being overlooked and considered of lower priority, and the corresponding education is continually reduced, sometimes to a minimal set of practical notions. Particularly affected by this tendency are the most interdisciplinary subjects, which belong to everywhere but nowhere specifically. Crystallography is one perfect example: it is tightly connected to all domains dealing with the structure and properties of periodic matter (in whatever dimensions and whatever geometry), spanning thus mineralogy (from which it was born at the dawn of modern scientific thinking), materials science, metallurgy, physics, chemistry, biochemistry and mathematics, but one would have a tough task if today one wanted to find a fully crystallographic lecture within a standard university curriculum. Some aspects of crystallography, often reduced to a minimal set of indispensable concepts, are dispersed in lectures specific to the major subjects. As a consequence, approximate, imprecise and sometimes fundamentally wrong statements and definitions appear not only in lectures but also in textbooks, which go to press without passing through the necessary screening of a competent crystallographer, who may well become an endangered species. The adoption of less effective presentations of basic concepts is also frequent today, while older and too often forgotten textbooks contain more effective and convincing approaches. By paraphrasing Leopardi's poem, one could say that the `magnificent and progressive fate of crystallography', which one might expect as a legacy of the International Year of Crystallography celebrated last year, is jeopardized by the serious oversight and forgetting of many fundamental concepts which represent the roots of our discipline.
Without the ambition of an extensive analysis, a task which would require a much greater space and time frame, we dare to present in the following a few examples of the imprecisions, approximations and even mistakes often occurring in the literature, which we hope may awaken interest in filling some holes in the crystallographic background of those who have undertaken the noble yet hard mission of training future generations of knowledgeable crystallographers.
2. On the number of face forms
In crystallography, a form is defined as the set of all symmetrically equivalent `elements' with respect to the
P of the crystal. The definition of `element' depends on the space under consideration, namely(i) in crystal form or face form is a set of all symmetrically equivalent faces;
a(ii) in point form is a set of all symmetrically equivalent points.
aThe polyhedron (in threedimensional space) or polygon (in twodimensional space) of a point form is dual to the polyhedron or polygon of the corresponding face form, where `dual' means that they have the same number of edges but the number of faces and vertices is interchanged (ITA10; Hahn, 2005).^{1} The of a form is a E which either coincides with the generating P or is a of it. We treat here specifically face forms, which build the morphology of a crystal, to emphasize some inconsistencies occurring in the literature.
The number of possible forms of a hk0} – or three general – {hkl} – actually correspond to an infinite number of forms, for the various possible values of the In a real crystal only a limited number of forms are actually realized, those with small being typically of larger size. Forms are classified into a finite number of types, in terms of the Wyckoff positions of the point groups.
is infinite. In fact, symbols with two – such as {Face forms are classified on the basis of their symmetry properties and of their orientation with respect to the symmetry elements of the point groups in which they occur. Of particular relevance for our discussion is the classification on the basis of the characteristic if its E coincides with the generating P and noncharacteristic when E is a proper of P. Face forms bring a name dictated by their (Table 1) and in the following will be called geometric (face) forms because they are defined in terms of the of their stereographic poles. The number of forms is 47 and the minimal a geometric form may possess is mm2 or 222 (both of order 4).
of the form. A form is called
‡The special case of a ditrigonal scalenohedron where the dihedral angles are 60° is a hexagonal scalenohedron. The specialized dihedral angles do not increase the of the form, as would be the case for a pyramid or dipyramid instead. 
Frequently in the literature, 48 forms are quoted because the dihedron is split into `sphenoid' and `dome' depending on whether it occurs in m [concerning the difference between point groups, pointgroup types and see Nespolo & Souvignier (2009)]; the is, however, mm2 in both cases. The separation of the dihedron into a dome and a sphenoid implicitly means to take into account not only the of the geometric form but also some physical property of the faces building it. Quite obviously, if such an option is adopted, it should be applied to all 47 geometric forms, not only one of them. In that case, the number of forms becomes much larger, as we are going to show. [The following analysis was inspired by an email exchange (28 November 2005) with Theo Hahn, reporting the opinion of Ellen Novack on the purely geometric nature of the crystal forms typically used in crystallography.]
2 orAs a criterion to introduce the consideration of the symmetry of the physical properties of the faces building a form we can take inspiration from the dome/sphenoid distinction. The m) would be composed of one left and one right face. If the two faces of the form do not present a handedness, i.e. if they are achiral, one obtains the dihedron (crystal class mm2): the achirality comes from the fact that the of the faces contains a of the second kind (mirror reflection; the stereographic poles of the faces are on a mirror). Accordingly, the three forms that split from a dihedron when the handedness is taken into account can be symbolically indicated as L^{2}, LR and A^{2}, where L, R and A stand for left, right and achiral, respectively, and the superscript 2 indicates the number of faces with the indicated handedness. Obviously, L^{2} can equally be written as R^{2}, thus leading to an enantiomorphic pair of sphenoids.
relating the two faces is an operation of the first kind (rotation) in the case of a sphenoid but of the second kind (reflection) in the case of a dome. We can imagine assigning a handedness to each face, which is either preserved or reversed, depending on the nature of the Thus, the sphenoid (crystal class 2) would be composed of two left (or two right) faces, while the dome (crystal classThe same analysis applied to all 47 geometric crystal forms gives the results presented in Table 1, where they are split according to their handedness, represented by what we can call their handedness symbol, giving the sequence of the handedness of faces building the form. The handedness symbol is an immediate generalization of the symbols used for the analysis of the dihedron and is composed of sequences like [LR]^{n}, where n means n repetitions of the faces to which the superscript n is attached. For each form obtained in this way, the geometric crystal class(es) in which the form occurs is/are given in parentheses; bold indicates that the form is characteristic in that class, i.e. that the of the form coincides with the pointgroup type of that class. containing only operations of the first kind give rise to forms with faces of the same handedness, symbol L^{n}. For each of them the form of the opposite handedness, R^{n}, can also occur, giving birth to enantiomorphic pairs of forms.
Inspection of Table 1 reveals that the 47 geometric forms split into 97 affine forms. [The terms `affine forms' and `crystallographic forms' are introduced to follow the same criterion used to classify space groups into 219 affine types and 230 crystallographic types (ITA8).] Of these, 33 give rise to enantiomorphic pairs (L^{n} and R^{n}), to give a total of 130 crystallographic forms (no attempt to suggest specific names for each of them, like in the dome and sphenoid case, is made). It should thus be evident that the number of 48 found in some texts is spurious and finds its origin in the inconsistent consideration of the handedness of only one form.
Because the minimal
of a geometric form is orthorhombic, there are no characteristic forms in the triclinic and monoclinic crystal systems (ITA10). The morphology, at least if the crystal is does reveal the of the crystal, even in the lowersymmetry crystal systems. This comes from the fact that the morphological of the whole crystalline edifice corresponds to the intersection of the eigensymmetries of the different forms building the crystal:where E(i) is the of the ith form (geometric in bold in Table 1; the pedion and pinacoid are never characteristic and their eigensymmetries are ∞m and ∞m/m, respectively).
The inconsistent terminology occurring sometimes in the literature has to be emphasized. In particular, the use of `monoclinic prism' makes sense only with reference to an affine form. However, the morphological analysis is usually limited to the geometry of the sample, where no monoclinic prism exists, the corresponding geometric form having mmm.
If the term `monoclinic prism' is not intrinsically wrong but simply often used inconsistently, the term `triclinic prism' used by Li et al. (2010) is not correct, because a crystallographic prism is incompatible with the inversion centre as a unique symmetry element.
3. Are always coprime integers?
The i.e. reduced of all common factors. This is correct when the is indexed with respect to a However, to give the simplest possible matrix representation of the symmetry operations of a crystal it is customary to make reference to the socalled (ITA9), which is often a centred cell. In this case, the use of coprime integers to represent a family of planes, as done in many introductory texts, may become contradictory with respect to the definition itself of To show that this is the case, we have to recall that represent the orientation of a crystal face (whether actually developed in the crystal morphology or not) and thus also of all the planes parallel to it. The notation (hkl) indicates a family of planes, i.e. the whole set of planes having the same orientation with respect to the chosen axial setting; the last of these planes, if actually developed in the crystal morphology, is the crystal face with the same indices. The do not give the position of any of the planes of a family, which is instead represented by the equation
of a family of planes are usually presented as coprime integers,(see equation 1.1.2.3 in ITB1; Shmueli, 2008) which precisely fixes the position as a function of the parameter n. Now, n should take only integer values (ITB1) when the coordinates of a node are put as the x, y, z values in equation (2). In particular, it should give the first plane of the family after the origin, in the positive direction, when n = 1. This plane thus has intercepts 1/h, 1/k and 1/l on the a, b and c axes, respectively. When the is primitive or when the planes of the (hkl) family are parallel to the face of the containing the centring vector, the above condition is consistent with coprime Miller integers. This is no longer true in the other cases. The contradiction is shown in Fig. 1 for the case of an oP and an oC type of and the family of planes (h00). The intercept on the a axis of the first plane of the family is 1/h. If the is of type P then the first plane after the origin passes through the node 100, so that h = 1. In this case, the family of planes has (100). However, if the is of type C, the first plane after the origin passes through the node , so that h = 2. In this case, the family of planes has (200). To call this family (100) would mean to state that the plane passing through node 100 is the first of the family, while it is actually the second. In other words, it would mean denying the presence of the plane passing through , contradicting the choice of a Ccentred Also, putting the coordinates of node into equation (2) one obtains m = if the of this family are given as (100). Quite obviously, the same happens for the family (0k0) if the is B centred and for the family (00l) if the is A centred, and for all the three families simultaneously if the is F centred. This result is immediately extended to the families (hk0) for a Ccentred cell, (h0l) for a Bcentred cell and (0kl) for an Acentred cell: when the parity of the nonzero indices is the same, then the density of planes is the same as that of a primitive although the density of nodes on each plane of the family is doubled (Fig. 2).^{2} In contrast, when the parity of the nonzero indices is opposite, the density of planes is doubled with respect to that of a primitive but the density of nodes on each plane of the family is unchanged. Consequently, the first plane of the family does not correspond to coprime integers but to indices having 2 as common factor (Fig. 3).
These geometric considerations can be translated in an analytical way by simply putting the uvw coordinates of a node into equation (2). If the value n obtained is fractional, then the hkl indices chosen to represent the family of planes are not correct: if n is fractional with denominator p, with p integer, then there are p − 1 planes between the origin and what had been assumed to be the first plane of the family. Consequently, the hkl indices must be multiplied by a factor p. For example, in the case of Fig. 3, the family of planes indicated as (210) would lead to n = when the node is inserted into equation (2), and to n = when the node is used instead. If the family of planes is indicated as (420), then the two nodes above give n = 1 and 3, respectively, when used in equation (2). This analytical translation of the geometric considerations is useful to analyse cases which are more difficult to show in a simple twodimensional projection. For example, in the case of an Icentred the plane of the family (111) which passes through node would give n = in equation (2), meaning that the plane which intersects each axis on the first node is not the first of the family but the second. Thus, the correct of the family of planes are (222). In the same way, starting from hypothetical (100), (110) and (111) and inserting the coordinates of the node occurring in the obverse setting of a rhombohedral with respect to hexagonal axes, the results are n = , 1 and , showing that the corresponding planes are the third, first and third of the families, respectively. The thus have to be modified as (300), (110) and (333).
One immediately sees the relation between the and the in for which the same restrictions on the hkl value of the diffractions occur as on the of the family of planes.
correctly expressed to satisfy equation (2)3.1. When noncoprime become an artefact: the Bravais–Friedel–Donnay–Harker law
Friedel (1907) introduced the socalled which establishes a rough proportionality between the frequency of occurrence of a given face in the morphology of a crystal and the density of nodes on the planes corresponding to that face, i.e. having the same This geometric rule is an approximation introduced to predict the growth shapes of crystals, where the ranking of an {hkl} form is related to the thickness of its corresponding layer: the thicker the layer, the higher the morphological importance of the form. A number of exceptions remained unexplained, concerning in particular the absence or low frequency of occurrence of faces with small and thus high densities of nodes. To cope with those exceptions, Donnay & Harker (1937) introduced an extension of the today generally known as the Bravais–Friedel–Donnay–Harker law, which represents an effort to complement purely reticular considerations with some atomistic aspects. Unfortunately, this extension was performed in a rather spurious way, introducing confusion between the and the structure and leading to an unnecessary multiplication of Miller indices.
The idea behind the Donnay–Harker extension was to take into account the influence of screw axes and glide planes in nonsymmorphic space groups on the frequency of occurrence of the forms.
are generated, apart from the translations, by pointgroup operations; they may contain screw rotations and glide reflections too, but these are not generators, as is instead the case for nonsymmorphic space groups. On the other hand, screw axes and glide planes in nonsymmorphic space groups do represent a fundamental symmetry characteristic of the structure and do not depend on the axial setting chosen, although their representation obviously does: the of a family of planes cannot depend on the structural symmetry.Donnay & Harker (1937) made screw axes and glide planes act on nodes (termed `lattice equipoints') to produce a separate set of ghost points (termed `space group equipoints') which, although not being nodes, were treated as such, leading to an artificial multiplication of the We have seen that noncoprime come from the use of centred cells, which are by no means an intrinsic feature of the but a convenient choice for the axial setting. The Bravais–Friedel–Donnay–Harker law is a useful artefact in that it improves the predictive power of the on the relative importance of crystal faces, but it is still an artefact. The improvement in the `morphological aspect', as was called the decreasing sequence of forms, comes from the indirect consideration of the structural symmetry. Structural theories on the relative development of crystal faces, like the PBC (periodic bond chain) theory (Hartman & Perdok, 1955), were developed later, making this useful artefact outdated. For example, Sunagawa (2005) has shown that the PBC theory predicts the correct `morphological aspect' of pyrite, without the requirement of inconsistently multiplying the of the faces depending on the zonal and as was proposed by Donnay & Harker (1937) to explain the observed sequence of forms.
4. of Bragg peaks versus of families of planes
The indices of a Bragg reflection from the set of parallel equidistant planes (hkl) are called and are indicated as hkl without parentheses (ITA1). Bragg's law,
includes the order of diffraction n; it is sometimes neglected but it is of paramount importance. Let us remember that Bragg's law is a geometric interpretation, in terms of the reflection of Xrays by planes, of what is actually a diffraction phenomenon described by the It is the positive interference of the Xrays `reflected' by the various planes of the same family (hkl) that results in a Bragg peak, bringing the hkl. There is thus an obvious correspondence between the hkl and the (hkl). Bragg peaks along a direction of are characterized by a common factor n which increases with distance from the origin (position of the direct beam) and this is precisely the order of diffraction in equation (3). The expression `diffraction from the plane (nh nk nl)', which occurs more often than one might imagine, is at odds with both the definition of and Bragg's law. In fact, if the of a family of planes are (hkl), not necessarily coprime integers (as we have seen above), they represent a family whose first plane after the origin of the axes has intercepts a/h, b/k, c/l. A hypothetical family (nh nk nl) would have the same orientation but its first plane would have intercepts a/nh, b/nk, c/nl, which do not pass through any node. In other words, the hypothetical family (nh nk nl) would include n − 1 planes out of n which are not reticular planes (do not pass through any node). On the contrary, if these were reticular planes, then (hkl) would not be a family of planes, because it would be composed of only 1/n of the planes with the same orientation. Therefore, the nh nk nl do not represent the firstorder diffraction from a family (nh nk nl) but the nth order diffraction from the family (hkl).
5. Geometric derivation of the reflection conditions
Most modern textbooks introduce the hkl of the diffractions. While this is perfectly legitimate and absolutely correct, from the pedagogical viewpoint it is questionable as the best way to present the especially for beginners who may not yet have experience of diffraction experiments. A geometric approach, proposed a long time ago by Buerger (1942) but today often forgotten (although mentioned in ITA2), is preferable in that it gives a much more concrete image and allows an easy memorization of the without the need for an analytical derivation of the annihilation conditions for the This presentation makes use of notions and concepts in that the student has already (hopefully!) mastered and it is easier to understand, in the same way as the use of the is pedagogically useful when introducing the (Nespolo & Souvignier, 2010).
in an analytical way, showing that the is systematically zero for some relations between theIn §2 we have seen that a straightforward relation exists between the indexing of planes and the which are purely geometric in nature; they find their origin in the choice of the axial setting used to describe the and are common to all space groups, both symmorphic and nonsymmorphic. The specialization of the to two and onedimensional sections (i.e. obtained by zeroing one or two indices) has no structural origin, as shown by the fact that the use of a primitive nonconventional would completely annihilate the including those on these sections. On the other hand, nonsymmorphic space groups show zonal and/or coming from generating glide reflections and screw rotations, respectively; they do not disappear if a different is chosen, although their expression obviously depends on the axial setting chosen.
5.1. Zonal reflection conditions
shows the effect of an a glide perpendicular to the [001] direction, through the example of an orbit 2e in a of type Pmma. When the figure is observed in projection along this same direction, the distinction between atoms above and below the plane of the drawing is lost so that in this projection the periodicity along a is apparently halved. Because of the dual relation between the two spaces, the opposite is true in where the periodicity along a* is apparently doubled when a projection along [001] is observed. Accordingly, on the plane that is perpendicular to [001], i.e. the (001)* plane, one position out of two along a* seems to be missing. Expressed formally, this results in the well known reflection condition hk0: h = 2n. Obviously, by taking a b glide instead of an a glide, the become hk0: k = 2n, and by taking another direction of projection, the corresponding for h0l and 0kl are obtained.
come from the presence of glide planes. Fig. 4The case of a diagonal glide is shown in Fig. 5 for the case of an n glide perpendicular to [10], which occurs in cubic space groups (Pnn and Pmn, Nos. 222 and 223). The plane of the projection is (10) and contains the directions [110] and [001]. The glide component is half of the inplane horizontal direction, i.e. (a + b)/2, and half of the inplane vertical direction, i.e. c/2, so that the glide component is (a + b + c)/2. In the [10] projection, the periodicity along this direction is apparently halved. In the plane perpendicular to [10] is (10)* and in the projection on this plane the periodicity along the diagonal, which is h + h + l, is apparently doubled. The reflection condition for an n glide perpendicular to [10] is therefore hhl: 2h + l = 2n. However, because 2h is always an even number, the condition 2h + l = 2n gives the same result as l = 2n, and this is the way in which the are presented in ITA3 and ITA7. If the glide plane is of type d (which for the chosen orientation occurs in tetragonal and cubic space groups), the apparent periodicity in is reduced to onequarter, and in it is four times bigger, so that the reflection condition becomes hhl: 2h+l = 4n.
5.2. Serial reflection conditions
shows the effect of a 4_{1} screw axis parallel to the [001] direction. When the figure is observed in projection on this same direction, all the atoms around the axis are projected onto the axis, so that in this projection the periodicity along c is apparently reduced to onequarter. Because of the dual relation between the two spaces, the opposite is true in where the periodicity along c* is apparently four times larger when a projection on [001]* is observed. Accordingly, three positions out of four along c* seem to be missing. Expressed formally, this results in the well known reflection condition 00l: l = 4n. The same geometric reasoning applies to all the screw axes.
come from the presence of screw axes. Fig. 66. Can the lack of symmetry impose restrictions?
A number of recent (and not so recent) textbooks introduce the concept of crystalclass system) contains a complete set of of space groups (ITA8). The definition of a is thus based on the presence of a minimal set of operations that are common to all the crystals belonging to the same The symmetry of the conventional may, and often does, coincide with the of the crystal, but it may also be higher, in which case one speaks of accidental higher metric symmetry, which occurs more frequently than one might expect (Janner, 2004a,b). While the presence of a structural does impose restrictions on the metric of the the absence of it cannot impose any restriction. Thus, in a monoclinic crystal, the presence of a twofold axis, or of a mirror perpendicular to it (or of both, of course), imposes the metric restriction of two right angles, which are chosen as α and γ if the crystal is referred to the conventional bunique setting. However, these operations do not have any influence on the value of the third angle (β in the conventional bunique setting), which is symmetry unrestricted. It may take any value, and vary as a function of the external conditions (temperature T, pressure p), including 90° as a special case, at least in a certain interval of T and p and within the of the experimental determination. To define a monoclinic crystal as having β ≠ 90° is incorrect because no metric restriction can occur as a consequence of the lack of symmetry. Again, to state that in a monoclinic crystal a ≠ b ≠ c is incorrect for the same reasons.
on the basis of the metric relations of the conventional which is incorrect. A (shortened wording forThe situation is different if one makes reference to the lattice, thus making abstraction from the structure. A specialized metric does promote the to a higher symmetry, but without increasing the symmetry of the structure. Because of the possible divergence between symmetry and structural symmetry, the concept of a is used in ITA8 to assign space groups (which represent the structural symmetry) to the not affected by the accidental metric symmetry. However also in this case, the metric restrictions given in most textbooks are incorrect: they are too restrictive in some cases and not enough in others. The metric restrictions on the lattices are in fact much more complex than those on the structure, because one has to avoid all special cases, i.e. the specializations of the metric that promote the symmetry to a higher These specializations have recently been reviewed in great detail by Grimmer (2015); we give here just one example to emphasize how misleading are the metric restrictions frequently found in textbooks.
Let us consider the example of the metric restrictions for a monoclinic a ≠ b ≠ c; α = γ = 90°, β ≠ 90°, expressed in the common bunique setting. The condition on the b parameter is unnecessary. In fact, even in the case of a specialized metric with a = b or with b = c, the symmetry of the is not promoted to a higher unless further specializations exist too. On the other hand, to exclude only the 90° for β is not enough. If the cell parameters define a primitive and β = cos^{−1}(−a/2c), then the is actually oB (orthorhombic with a Bcentred cell, which can be transformed to oC or oA by a permutation of the axes). If the cell parameters define a bodycentred and β = cos^{−1}(−a/c), then the is actually oC; if a^{2} + b^{2} = c^{2} and a^{2} + accosβ = b^{2} or b^{2} + accosβ = a^{2}, then the is actually hR (rhombohedral); if c^{2} + 3b^{2} = 9a^{2} and c = −3acosβ or a^{2} + 3b^{2} = 9c^{2} and a = −3ccosβ, then the is again hR (rhombohedral) (data from Table 9.4.3.1 in ITA9; unit cells of type mI, mC, mA and mF are all easily converted into each other so that the above conditions for mI can be transformed into the setting of any of the other three types of see also Grimmer, 2015).
which are often given asIt should therefore be evident that, if one wants to give the metric restrictions for the lattices, the situation is much more complex than that given in any textbook. On the other hand, if the metric restrictions are those imposed by the structural symmetry, any accidentally specialized value which would increase the symmetry of the .
should not be excluded. Therefore, when giving the metric relations imposed by the symmetry of the crystal, the correct relations are those shown in Table 2

7. Conclusions
The time and space devoted to teaching fundamental crystallography in undergraduate programmes, especially in
has been reduced drastically during the past few decades and it is still shrinking. One of the consequences is that some basic aspects are no longer presented, while others are covered very quickly in an approximate way that often leads to imprecisions, axiomatic statements and less effective presentations. This also occurs in textbooks, where the authors sometimes seem to feel obliged to insert some basic topics without which the text could be considered incomplete, but cannot devote enough space to a convincing presentation, so that the result is quite often less satisfactory than that already available in older classical textbooks.We have presented a few examples of problems frequently and repeatedly seen in modern textbooks and class presentations and we have pointed out how a consistent and correct presentation can instead be realized by very simple, essentially geometric, considerations. We hope that this effort may be useful to avoid mistakes and increase the effectiveness of this basic part of the crystallography curriculum and contribute to restoring its charm, today unreasonably considered old fashioned.
Footnotes
‡On leave from Université de Lorraine, Faculté des Sciences et Technologies, Institut Jean Barriol FR 2843, CRM2 UMR CNRS 7036, BP 70239, Boulevard des Aiguillettes, F54506 VandoeuvrelèsNancy Cedex, France.
^{1}For the sake of brevity, the different chapters of International Tables for Crystallography will be quoted herein as ITAX for Vol. A, or ITBX for Vol. B, where X is the number of the chapter.
^{2}These geometric considerations were already present by Friedel (1926) but applied to the computation of the of twinned crystals [see details given by Nespolo & Ferraris (2007)], while the effects on the were not taken into account. The use of noncoprime integers for centred unit cells was proposed by Ungemach (1935) for the rhombohedral and its generalization was proposed by Donnay (1936) in a somewhat obscure way, so that the first clear occurrence of this idea can be traced back to a footnote at p. 450 of the paper by Donnay & Harker (1937).
Acknowledgements
This article was prepared during a stay at Tokyo Institute of Technology as Foreign Guest Professor. Comments from four anonymous reviewers, Professor Hans Grimmer and the Coeditor are gratefully acknowledged.
References
Buerger, M. J. (1942). Xray Crystallography: An Introduction to the Investigation of Crystals by Their Diffraction of Monochromatic XRadiation. New York: Wiley. Google Scholar
Donnay, J. D. H. (1936). Am. Mineral. 21 (No. 12, Part 2), 4–5. Google Scholar
Donnay, J. D. H. & Harker, D. (1937). Am. Mineral. 22, 446–467. CAS Google Scholar
Friedel, G. (1907). Bull. Soc. Franç. Mineral. 30, 326–455. Google Scholar
Friedel, G. (1926). Leçons de Cristallographie. Nancy, Paris, Strasbourg: BergerLevrault. Google Scholar
Grimmer, H. (2015). Acta Cryst. A71, 143–149. Web of Science CrossRef IUCr Journals Google Scholar
Hahn, Th. (2005). Editor. International Tables for Crystallography, Vol. A, SpaceGroup Symmetry, 5th ed. Heidelberg: Springer. Google Scholar
Hartman, P. & Perdok, W. G. (1955). Acta Cryst. 8, 49–52. CrossRef IUCr Journals Web of Science Google Scholar
Janner, A. (2004a). Acta Cryst. A60, 198–200. Web of Science CrossRef CAS IUCr Journals Google Scholar
Janner, A. (2004b). Acta Cryst. A60, 611–620. Web of Science CrossRef CAS IUCr Journals Google Scholar
Leopardi, G. (1836). La Ginestra, o il Fiore del Deserto. (The Broom, or the Flower of the Desert.) In Poems, translated by J. Galassi (2012). New York: Farrar, Straus and Giroux. Google Scholar
Li, L., Zhang, Y., Esling, C., Jiang, H., Zhao, Z., Zuo, Y. & Cui, J. (2010). J. Appl. Cryst. 43, 1108–1112. Web of Science CrossRef CAS IUCr Journals Google Scholar
Nespolo, M. & Ferraris, G. (2007). Acta Cryst. A63, 278–286. Web of Science CrossRef CAS IUCr Journals Google Scholar
Nespolo, M. & Souvignier, B. (2009). Z. Kristallogr. 224, 127–136. Web of Science CrossRef CAS Google Scholar
Nespolo, M. & Souvignier, B. (2010). J. Appl. Cryst. 43, 1144–1149. CrossRef CAS IUCr Journals Google Scholar
Schmueli, U. (2008). Editor. International Tables for Crystallography, Vol. B, Reciprocal Space, 3rd ed. Heidelberg: Springer. Google Scholar
Sunagawa, I. (2005). Crystals. Growth, Morphology and Perfection. Cambridge University Press. Google Scholar
Ungemach, H. (1935). Z. Kristallogr. 91, 97–113. CAS Google Scholar
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