research papers
Analysis of the structural continuity in twinned crystals in terms of pseudoeigensymmetry of crystallographic orbits
^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands, and ^{b}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
^{*}Correspondence email: massimo.nespolo@crm2.uhpnancy.fr
The reticular theory of
gives the necessary conditions on the level for the formation of twins. The latter are based on the continuation, more or less approximate, of a through the composition surface. The analysis of this structural continuity can be performed in terms of the of the crystallographic orbits corresponding to occupied Wyckoff positions in the structure. If is the of the individual and a which fixes the obtained as an intersection of the space groups of the individuals in their respective orientations, then a structural continuity is obtained if (1) the of an orbit under contains the (2) the of a union of orbits under contains the (3) the of a split orbit under contains the or (4) the of a union of split orbits under contains the The case of the twins in melilite is analysed: the (approximate) restoration of some of the orbits explains the formation of these twins.Keywords: melilite; structural continuity; twinned crystals.
1. Symbols
(a, b, c): basis vectors of the unit cell.
a, b, c: length of basis vectors.
r_{i} = : coordinates of the ith crystallographically independent atom A_{i}.
:
of the individual, = {, , …}, with = 1 the identity element of .O_{i}: orbit of r_{i} under , O_{i} = {r_{i}, r_{i}, …} = {r_{i}^{1}, r_{i}^{2}, …} with r_{i}^{k} = for ∈ .
O_{ij}: splitting of the orbit O_{i} under the action of a of .
m(O_{i}): multiplicity of the orbit O_{i}, defined as the number of equivalent points in the conventional of .
(P, p): matrixcolumn pair representing a change of basis; composed of a 3 × 3 matrix P and a 3 × 1 column p.
: matrix representation of the
in the basis of the twin.: sitesymmetry group of r_{i}.
:
associated with the structure of the twin.: O_{i}.
of the orbit2. Introduction
A twin is a heterogeneous crystalline edifice composed of two or more homogeneous crystals of the same phase with different orientation related by a i.e. a crystallographic operation mapping the orientation of one individual onto that of the other(s) (Friedel, 1904, 1926, 1933). A is the in (plane, line, centre) about which the is performed.
Twins can be classified from the genetic viewpoint in three categories:
(1) Transformation twins, which form during a
leading to a loss of point symmetry.(2) Mechanical twins, which form as the result of a mechanical action (typically, an oriented pressure) on the crystal.
(3) Growth twins, which form during crystal growth, either at the nucleation stage or by oriented attachment (for a review, see Nespolo & Ferraris, 2004a).
For cases (1) and (2), the cause of the formation of the twin is known. For the growth twins the formation can be a response to a mistake in the normal crystal growth of the individual or the random association of two or more crystals with different orientation (nonequivalent under the symmetry group of the crystal). This category of twins appears not only during the formation of a natural crystal but also during the synthesis of artificial crystals.
The interface that separates the individuals represents a discontinuity for at least a substructure. This heterogeneity gives rise to serious problems in the structural study of materials and biomaterials and it represents an obstacle for structural investigations as well as for crystal engineering and material design. For example:
(a) The potential technological applications are hindered by the presence of (e.g. the piezoelectric effect is reduced or annihilated).
(b) The presence of reduces the amount of details that can be obtained from a structural study, especially for samples with large unit cells (for example, macromolecules) for which the resolution that can be achieved is already limited by the size of the unit cell.
From the viewpoint of the material scientist and of the crystal grower, the development of a synthesis protocol capable of reducing, if not suppressing, the formation of twins is an important goal. To reach this aim a detailed understanding of the formation mechanism of twins is of paramount importance.
A prerequisite for the formation of a twin is a partial structural continuity through the interface. In fact, without any structural continuity the edifice built by the individual crystals would be unstable or simply not form at all; a complete structural continuity is the feature of a single crystal; in a twin a part of the structure has to continue, more or less unperturbed, across the interface. This atomic continuity implies the continuity of a sublattice. In fact, the structurally necessary for the formation of twinned crystals. A general theory on this has not been developed yet.
represents the periodicity of the and the continuity of a sublattice is a necessary condition for the continuity of a substructure. The reticular approach abstracts from the structure and estimates the restoration by the in terms of the and the obliquity. A good restoration of the is a necessary but not sufficient condition to obtain a good structure restoration. The latter would enhance the reticular theory to conditions which areExtensive research from the ), Mallard (1885) and Friedel (1904, 1926), based on the existence of a common (sub)lattice in the three dimensions of the crystallographic (note however the special case of monoperiodic twins reported by Friedel, 1933). The common (sub)lattice, called the (Donnay, 1940), is based on the (twin plane or twin axis) and the element (line or plane) that are mutally (quasi)perpendicular. The L_{T} is defined by these two elements (hkl)_{T} and [uvw]_{T}. When the two elements are reciprocally perpendicular one speaks of symmetry (TLS: Donnay & Donnay, 1974) and the two elements are symmetry elements for L_{T}. Otherwise one speaks of quasi symmetry (TLQS: Donnay & Donnay, 1974); the two elements are only pseudosymmetry elements for L_{T}. The degree of pseudosymmetry corresponds to the deviation from the perpendicularity condition and is measured by the angle ω called the obliquity.^{1} The n is the inverse of the fraction of nodes restored by the and corresponds to the ratio between the volumes of the primitive cells of the twin and the individual, n = V(L_{T})/V(L_{ind}). Friedel gave as empirical limits for the occurrence of twins n ≤ 6 and ω ≤ 6. Twins falling within these limits are called Friedelian twins (Nespolo & Ferraris, 2005). The frequency of occurrence of a twin depends on the degree of restoration: the lower the and the obliquity, the better is the restoration and the higher is the probability that the twin actually occurs. This relation between the occurrence frequency of twins and the values of n and ω is an empirical observation, based, however, on the extensive study of twins over more than a century. It shows the necessary (not sufficient) character of the restoration. Nevertheless some twins with higher index are known that violate the empirical limits: they are called nonFriedelian twins (Nespolo & Ferraris, 2005). These twins seem to contradict the general conclusion that a high degree of restoration is a necessary condition for a twin to form. However, in most cases they can be explained by the fact that two or more sublattices contribute to the quasirestoration. When all the concurrent sublattices are taken into account, the necessary conditions are no longer contradicted. The interpretation of the occurrence of this kind of twins is the object of the hybrid theory of (Nespolo & Ferraris, 2005), which represents an extension of the reticular theory and measures the quasirestoration in terms of an effective twin index n_{E} (Nespolo & Ferraris, 2006), a real number defined as the ratio between the nodes of the individual and the nodes belonging to any of the quasirestored sublattices. In the case of a single quasirestored this coincides with the classical otherwise it is lower. In the few examples which are neither explained by the classical reticular nor by the hybrid theory of the possibility of a wrong choice of the has to be considered (reflection twins in place of rotation twins or vice versa). This indeed resolves the apparent contradiction of a higher frequency of twins with higher index than twins with a lower index observed in some cases like the staurolite twins. The Saint Andrews cross twin of staurolite, with index n = 12, is more frequent than the Greek cross twin with index n = 6 (Nespolo & Ferraris, 2007). These twins are often reported as reflection twins on (031) and (231), respectively, but experimental studies have shown (Hurst et al., 1956) that this interpretation is incorrect and that they actually are rotation twins. For the Saint Andrews cross twin (n = 12), the correct choice of the as a line shows the existence of two planes quasiperpendicular to it and correspondingly two sublattices are quasirestored by the This gives an effective index n_{E} = 6.0 and as a consequence the Saint Andrews twin is brought back into the Friedelian limits. The occurrence frequency no longer contradicts the necessary condition of a good restoration (Nespolo & Ferraris, 2009).^{1}
viewpoint during more than a century led to the reticular theory developed by Bravais (1851The reticular theory of a posteriori study of known twins. Our purpose is to develop a general structural theory of to predict the structurally necessary conditions for the formation of twins in a general way through an algebraic algorithm. A twin fulfilling these conditions can form (and may even be likely to form), but does not necessarily have to form. Indeed, a growth twin is a `mistake' originated by defects or perturbation of growth conditions and does not correspond to the thermodynamically most stable situation (Buerger, 1945). Donnay & Curien (1960) were the first to suggest the application of the analysis of the of crystallographic orbits, in the case of pyrite and digenite, which led to the introduction of a restoration index for a subset of atoms (Takeda et al., 1967). This subset must be quasicontinuous across the interface, otherwise the interface would be incoherent, the contact between the individuals would be unstable and the twin would not form. Under the action of the , each atom in a crystal is repeated in space to form a O, i.e. O is the set of all atoms obtained under the symmetry operations of the . The (O) of the orbit may be a of or coincide with it; accordingly, crystallographic orbits are classified in three types according to the relation between and :
can only provide partial prerequisites for the formation of twins, which are governed by the structure. More conclusive conditions can only be obtained by the analysis of the structural coherence at the interface, but such an analysis reduces to a casebycaseCharacteristic orbit: = .
Noncharacteristic orbit: but = .
Extraordinary orbit: , a special case of noncharacteristic orbit defining a (smaller unit cell) with respect to .
Here and are the translation subgroups of and , respectively. When , an operation t belonging to but not to may map the orientation of crystal 1 onto that of crystal 2 and may thus serve as twin operation.
3. approach to the analysis of structural continuity in twins
Depending on the nature of the
twins can be classified into three categories:(1) twins by reflection;
(2) twins by rotation;
(3) twins by inversion.
An i.e. it corresponds to n = 1 and does not give rise to a because the whole of the individual is (quasi)restored. For a twin with index n > 1, the is not about a direction, which makes its matrix representation nonintegral with respect to the basis of the individual. By expressing the in the basis of the twin, its representation becomes integral again.
is always by (pseudo)merohedry,The reticular theory of i.e. the radius: ionic, covalent or atomic depending on the type of bond) seems a reasonable figure.
shows that an exact restoration of the is not an absolute condition for the twin to form, a limited departure from the restoration, measured by the obliquity or the twin misfit, being the rule rather than the exception. In the same way, we can expect that a limited departure from structural continuity at the interface does not represent a hindrance to twin formation. In the following, all the occurrences of `restoration' should thus be read as `restoration or quasirestoration'. As a consequence, the of an orbit has to be taken with some degree of tolerance: a pseudoeigensymmetry will result in quasirestoration. The choice of this tolerance has clearly important consequences on the conclusions one may draw about the structural quasicontinuity. Choosing a too small tolerance may lead to a relatively good coherence at the interface being overlooked; a too large tolerance would have no real physical meaning. Clearly, the tolerance has to be chosen keeping in mind the atomic size: it is greater for a large atom than that for a small one. As a rule of the thumb, about 50% of the atomic diameter (Let (hkl)_{T} and [uvw]_{T} be the mutually (quasi)perpendicular plane and direction which define the cell of the Let v_{1} and v_{2} be two vectors defining a twodimensional in (hkl)_{T}. The three linearly independent vectors v_{1}, v_{2} and [uvw]_{T} form the twin basis, denoted by (abc)_{T}, which is related to the basis (abc)_{I} of the individual by the basis transformation P:
L_{ind} and L_{T} have a common origin: there is thus no vector part in the relation between the two references. Given the coordinates (xyz)_{I} of an atom in the individual basis, the new coordinates (xyz)_{T} of this atom in the twin basis are obtained by the relation:
Each atom with coordinates r_{i} generates a O_{i} with under the action of the symmetry operations of the . If the orbit is noncharacteristic, its group may contain the t, in which case the orbit is restored by the This cannot be true for all the orbits, otherwise t would belong to the of the individual and the structure would be a single individual and not a twin. When the orbit is not fully restored, a subset of atoms belonging to the orbit can instead be restored. This subset is defined by a of obtained by intersecting the space groups of the individuals. Since the is n > 1, is a proper of , the translation of is a of index n in the translation of .
Let be the t maps the first individual to the second individual (assuming, for ease of description, the case of a twofold twin) and the of the second individual is the conjugate group . In addition, the t maps the L of the first individual to the tL of the other individual and the intersection is the Since tL_{T} = = = L_{T}, the fixes the The compatible with the is the intersection of the space groups of the two individuals, written with respect to the twin basis, i.e. = . The is uniquely determined; it consists of those isometries which fix both individuals separately. In particular, its translation consists of the translations by vectors from the L_{T}. The above relation is easily generalized to twin operations higher than twofold by replacing L_{1} ∩ L_{2} = L ∩ tL with ∩_{i}L_{i} = ∩_{i}t_{i}L_{1}.
of one of the individuals of a twinned crystal. TheTo find the elements of , let W_{i}, w_{i} be the linear and translation parts of a of the first individual, written with respect to the twin basis, i.e. (W_{i}, w_{i}) ∈ P^{−1}P. Since the linear parts of a act on its translation the elements belonging to necessarily have an integral linear part W_{i}. Moreover, if (W_{i}, w_{i}) belongs to the intersection, the conjugate (W_{j}, w_{j}) = (W_{i}, w_{i}) must be an element of the form (, ) ∈ P^{−1}P. Choosing an element (, ) with = W_{j}, one finally has to check whether w_{j} − ∈ L_{T}. Since the translations in are by vectors in L_{T}, two elements (W_{i}, w_{i}) and (W_{i}, ) with the same linear part can only belong to if w_{i} − ∈ L_{T}. This means that for a given element (W_{i}, w_{i}) of P^{−1} P one has to check elements of the form (W_{i}, w_{i} + v) for representatives v of L with respect to L_{T}.
The study of the orbit behaviour in the twin basis is characterized by the P. Considering the group–subgroup related space groups , atoms which are symmetrically equivalent under , i.e. belong to the same orbit of , may become non equivalent under (splitting of crystallographic orbits), and/or their sitesymmetry group can be reduced (Wondratschek, 1993). Let O_{i} be an orbit under , [, m(O_{i})] the group and the multiplicity of the orbit with respect to the of , and let [, m(O_{ij})] be defined correspondingly for a split orbit O_{ij} under , the double index indicating the original orbit under (index i) as well as the number of split orbits stemming from it under restriction to (index j).
and the matrixIn the case of splitting, the orbit O_{i} = {r_{i}, ∈ } is divided into two or more orbits of , with the same/or reduced group and a multiplicity equal or lower than m(O_{i}). The atoms belonging to O_{i} have as coordinates in the twin basis. The possibilities of the splitting of the orbit O_{i} are described by the following relations:
where [i] is the finite index of in , R_{j} is the ratio of the order of the sitesymmetry groups of the orbits O_{i} and O_{ij} in and in , respectively, and k is the number of orbits in stemming from O_{i} in (Wondratschek, 1993).
The atomic restoration by the
can finally be realised in four cases.(1) The orbit O_{i} is noncharacteristic and its contains the t. In this case, P = I, where I is the identity matrix.
(2) The union of two or more orbits has an t belongs to where = ∪_{i}O_{i} and i spans the orbits which are not restored by t and are occupied by atoms with similar structural role. Here again, P = I.
which is higher than that of any of the orbits of the union. This may in particular happen in presence of a specialized metric corresponding, exactly or approximately, to a higher In this case, if the is included in this higher the set of atoms belonging to the union is restored although each orbit, taken separately, is not. The union can obviously be formed only from atoms with interchangable roles in the structure. For example, the union of orbits defined by crystallographically different types of oxygen, or of atoms having the same coordination environment although a different chemical species. Clearly, the fact that a different atom occurs in the same coordination on the opposite sides of the interface does not affect the structural continuity, especially if the atomic size is not extremely different. The choice of the orbits to be considered in the union must thus rely on the analysis of the structural roles of these orbits. From a formal viewpoint, the restoration occurs if(3) When neither the orbits O_{i} nor their union is restored, a split orbit O_{ij} under may be restored by the t if its contains t.
(4) As in case (2) above, for orbits O_{ij} whose does not contain the t, the union = ∪_{ij}O_{ij}, defined on the same criteria as , has to be considered. The restoration of a union of orbits under may in particular happen when the fixed by has a specialized metric corresponding, exactly or approximately, to a higher crystal family.
Cases (1) and (3) could of course be subsumed under cases (2) and (4) as unions of a single orbit or split orbit, but we emphasize the importance of these cases by discussing them separately.
The actual analysis performed is exactly the same no matter whether the group considered is or and whether we work on a single orbit or a union of orbits. Let be a general notation for either or and O a general notation for one of O_{i}, O_{ij}, or . If O is restored by the t, then the is a of containing t. Such an orbit which belongs to the continuing across the interface of the twin structure that is invariant under the explains (in part) the formation of the twin.
Because the d_{min} be the minimal distance between the position to which a chosen atom in O is mapped under the t and the atoms in O. If t ∈ , then d_{min} = 0 for all atoms in O. If t is only a pseudosymmetry of O, then d_{min} > 0 and its value is a measure for the degree of quasirestoration.
of (split) orbits or unions thereof is often approximate and as a consequence the restoration is imperfect, we need a quantitative measure for the degree of restoration. LetThe advantage of dealing with split orbits under the intersection group = ∩ tt^{−1} is that the value of d_{min} is the same for all atoms in a split orbit under , as is shown by the theorem in the Appendix A.
Let O^{1} be an orbit O in the first individual, O^{2} the corresponding orbit generated by the t in the second individual. The application of the t to O^{1} generates O^{2}. For a fixed orientation of the the formation of a twin may result in a variable degree of atomic restoration depending on the position of the in the i.e. depending on which atoms are exposed to the surface or close to it. Since is a phenomenon that occurs at a macroscopic level, the orientation of a only determines the linear part of the but not its translational part, corresponding to the position of the On the other hand, the operation which restores an orbit acts on the structure, at the microscopic (atomic) level and may well also contain an intrinsic translational part (glide or screw component). In other words, the one observes macroscopically as well as in the diffraction pattern as the overlap of differently oriented reciprocal lattices, can be realised at the atomic levels at different locations and with or without an intrinsic translation. This realisation of the is hereafter called a restoration operation. In order to find the possible restoration operations, one starts with the intersection group and determines its minimal supergroups which contain an operation with the required linear part. However, dealing with split orbits for the intersection simplifies the analysis drastically. For a single split orbit and pairs of split orbits one simply checks whether the (pseudo) contains an operation of the same type as the and with its parallel to that of the The analysis then provides the location of the and the nature of the restoration operation.
O^{1} is restored if t ∈ (O^{1}) or if d_{min} is lower than a certain threshold which depends on the atomic size (being smaller for smaller atoms). When comparable degrees of restoration are obtained for different locations of the the probability of twin formation is higher because the twin can form at different stages of crystal growth, corresponding to different atomic surfaces exposed when the twin formation starts. In the opposite case, a higher probability of formation corresponds to the occurrence of a stacking defect, during crystal growth, on a surface corresponding to more restricted, possibly unique, locations of the twin element.
4. Case study: the melilite twins
Melilite is a group of sorosilicate minerals with general formula X_{2}YZ_{2}O_{7} with X = Ca, Na, Sr, K in octahedral coordination, Y = Mg, Al, Fe, B in tetrahedral coordination and Z = Si, Al again in tetrahedral coordination. These minerals crystallize in space groups of type with X and Z in Wyckoff positions 4e, Y in 2a and oxygen atoms distributed over three different Wyckoff positions, 2c, 4e and 8f, respectively. We have analysed the structure reported by Bindi & Bonazzi (2005) for which a = 7.826 (1), c = 5.004 (1) Å. The atomic coordinates are given in Table 1, together with an analysis of the quasirestoration of each orbit. This analysis has been performed with the PSEUDO program (Capillas et al., 2011) at the Bilbao Crystallographic Server (Aroyo et al., 2006). Given the difference in the dimensions of the cations and the anions, a tolerance of 1 Å for the former and 1.5 Å for the latter has been used to evaluate the pseudoeigensymmetry.
Two twins in melilite are reported by Deer et al. (1997), with reflections in {001} and {100} as twin operations: both are twins by so that L_{T} coincides with L_{ind}. The analysis has to be performed on planes, not on forms, and for this reason in the following the planes (001) and (100) are used; the result is obviously exactly the same if another plane from the same form is used. Since the twins are by the intersection group = ∩ tt^{−1} coincides with the group of the individual which is of type (No. 113). The minimal supergroups containing symmetry operations with the required linear parts are (all symmetry operations are expressed with respect to the standard setting of ):
(1) P4/mbm (No. 127), with the m x,y,0 for the (001) twin and b ¼,y,z for the (100) twin;
(2) P4/nmm (No. 129), with n(½,½,0) x,y,0 for the (001) twin and m 0,y,z for the (100) twin;
(3) P4_{2}/mnm (No. 136), with m x,y,¼ for the (001) twin and n(0,½,½) ¼,y,z for the (100) twin;
(4) P4_{2}/ncm (No. 138), with n(½,½,0) x,y,¼ for the (001) twin and c 0,y,z for the (100) twin.
The last two columns in Table 1 give the respective restoration operations contained in the of the different orbits.
Both (001) and (100) twins are by bglide reflection shifted ¼ from the origin for the (100) twin, with displacements ranging from 0 (perfect restoration) to 0.6415 Å. On the other hand, all anions are quasirestored by a reflection shifted ¼ from the origin for the (001) twin and by an nglide reflection shifted ¼ from the origin for the (100) twin, with displacements between 0.0580 and 0.6956 Å. The two further possible restorations for O_{3} correspond to different pseudoeigensymmetries but the much higher value of d_{min} makes their contribution hardly significant.
with the whole restored by the twin operations. The degree of structural restoration is the same for both twins, since the minimal supergroups of containing a restoration operation for one of the twins also contain one for the other twin. All cation orbits are approximately restored by a reflection located at the origin for the (001) twin and by aMore recently, a further et al. (2003). The restoration under the action of the has to be checked in = for each orbit O_{i} [this is easily done by inspecting Table 1: never contains ] as well as for the union of atoms with similar structural role, i.e. Y and Z, which are both in tetrahedral coordination, and the three types of oxygen atoms (Table 2). Neither nor contain as a proper or pseudosymmetry which therefore does not restore any orbit or union of orbits under . The next step is to check for the restoration of split orbits under .
on (), has been reported in melilite by Bindi

In a tetragonal hk0) is exactly perpendicular to the direction with the same indices [hk0]; the direction is therefore exactly perpendicular to the twin plane, which can thus also be indicated as . This perpendicularity imposed by the metric of the is known as intrinsic TLS or iTLS (Nespolo & Ferraris, 2006). is by reticular polyholohedry, with n = 5 (for details, see Nespolo & Ferraris, 2004b). The two shortest inplane directions are [210] and [001] so that the transformation from the basis of the individual to that of the twin, see equation (1), is immediately obtained as follows:
a plane (Applying the inverse transformation, the twin plane in the basis of the m_{[100]}, equation (1′):
becomes (100) orso that the matrix representation T of the t in the twin basis is simply:
In our case, = ∩ tt^{−1} = , a = 17.4995, c = 5.0040 Å: in fact, neither the 2fold axis nor the reflection plane contained in fix the whereas the axis does fix it and is common to and tt^{−1}.
Let m(O_{i}) be the multiplicity of each orbit O_{i} in , i ∈{1, 2, ..6}, and let n_{i} be the number of the atoms of the orbit O_{i} in the of the Then:
where P is the determinant of the transformation matrix P. The number of atoms n_{i}, equivalent under , is divided in the twin basis on s nonequivalent subsets of atoms under the : each subset corresponds to a split orbit O_{ij} indexed by s and such that:
The restoration of a split orbit O_{ij} is realised when contains a restoration operation with linear part m_{[100]T}. The extensions of (No. 81) containing such an operation are (No. 115), (No. 116), (No. 117) and (No. 118); the corresponding restoration operations are m 0,y,z, c 0,y,z, b ¼,y,z and n(0,½,½) ¼,y,z, respectively. To evaluate whether a split orbit under = is quasirestored by the operation in , one checks whether one of these four operations maps a split orbit either to itself or to another split orbit of the same type (within the accepted tolerance). This is what is displayed in Tables 3–8. It turns out that the reflection located in the origin gives by far the best restoration results, therefore we will only discuss the restoration by the operation m 0,y,z.






The atoms of type X in 4e for = fall under the action of the into five split orbits in 4h for = , each having four atoms in the of the The split orbit X_{1} is almost perfectly restored (with a deviation of 0.03764 Å), X_{4} and X_{5} are also quasirestored with a much larger but still acceptable deviation (0.8617 Å).
The atoms of type Y in 2a fall into four split orbits, two of which have four atoms in the twin cell and the other two a single atom. The two split orbits with a single atom in the twin cell are perfectly restored; the split orbit Y_{4} is quasirestored to the split orbit Z_{3} with a deviation of 0.6493 Å. This is an admissible replacement, since both the Y and the Z atoms are in tetrahedral coordination.
The atoms of type Z in 4e fall again into five split orbits each having four atoms in the twin cell. Besides the split orbit Z_{3} which is interchanged with Y_{4}, three more split orbits are approximately restored (with deviations between 0.5621 and 0.9793 Å).
The oxygen atoms in c fall into two orbits with four atoms in the twin cell and one orbit with two atoms in the twin cell. The split orbit with two atoms is exactly restored, the other two split orbits are only quasirestored when the threshold for anions is relaxed to 1.5 Å (deviations 1.1740 and 1.3402 Å) and one may doubt whether these are still meaningful for the formation of the twin. The oxygen atoms in 4e fall into five split orbits (each having four atoms in the twin cell). The split orbits O_{25} and O_{22} are approximately restored to themselves (with deviations of 0.5432 and 0.9856 Å), the orbit O_{24} is quasirestored to the split orbit O_{34} belonging to the oxygen atoms in 8f (with deviation 0.4103 Å) and the remaining two orbits are quasirestored to different split orbits with deviations between 1 and 1.5 Å. Finally, the oxygen atoms in 8f fall into ten split orbits with four atoms each. Besides the split orbit O_{34} that is interchanged with O_{24}, the two orbits O_{310} and O_{38} are quasirestored to themselves with low deviations (0.1946 and 0.3283 Å). Six more of these split orbits are quasirestored with higher deviations (between 1 and 1.5 Å).
2Table 9 shows a summary of the above analysis, where we see that the percentage of atoms quasirestored by the reflection is much better than for the three glide reflections. The fact that 68% of the cations and 37% of the anions are restored within 1 Å is a strong justification for the occurrence of this twin.

In Figs. 1 and 2 we display views of the twin cell. Figs. 1(a) and 2(a) show all atoms, and Figs. 1(b) and 2(b) the quasirestored atoms. Fig. 1 is a view along the c axis, i.e. the direction of the fourfold rotoinversion axis contained in the ; Fig. 2 is along the normal of the (111) plane.
5. Conclusions
The reticular theory of a priori predictive tool is limited: while a low restoration clearly indicates low probability of formation, a high restoration is indicative, but not conclusive, of a probable occurrence.
represents an elegant and general approach for estimating the probability of the occurrence of a twin. However, because it provides a necessary condition only on the level, its application as anThe analysis of the
of the crystallographic orbits corresponding to occupied Wyckoff positions is the key for obtaining a quantitative estimation of the structural restoration realised by the twin operation(s) and for obtaining structurally necessary conditions enhancing the reticular conditions for the twin formation. The example of melilite is particularly instructive. The (001) and (100) twins are both twins by and from the reticular viewpoint both twins should have a high probability of occurrence. As a matter of fact, the structural restoration is also fairly good, although the cations and anions require different locations of the The twin, despite a of 5, also leads to a relatively high degree of atomic restoration, which explains the occurrence of this twin.The approach we have developed in this article opens new perspectives in the study of twins and is currently being applied to other known examples.
APPENDIX A
Theorem. Assume^{2} that t is the such that t^{2} is an element of . Let O_{ij} be a split orbit under the intersection group = ∩ tt^{−1} and let x be the position of an atom in O_{ij}. Let x′ be the position of the atom in the structure closest to the mapped position t(x) of x under the thus d_{min} = ∥t(x) − x′∥. Then the value of d_{min} is the same for every atom in O_{ij}, i.e. the distance of the image of any atom in O_{ij} under t to the closest atom position in the structure is always d_{min}.
Moreover, if the position x′ belongs to the split orbit O_{i′j′}, then the closest atoms to the mapped split orbit t(O_{ij}) all belong to O_{i′j′}. In particular, if one atom of O_{ij} is exactly restored to an atom in O_{i′j′}, then the full split orbit O_{ij} is mapped to the full split orbit O_{i′j′} under the twin operation.
Proof: Let x be the position of an atom in O_{ij}, let x′ be the position of the atom in the structure closest to t(x) and let the split orbit to which x′ belongs be O_{i′j′}. If y is the position of another atom in O_{ij}, then there is a h in mapping x to y. Since t is a twofold one has tht^{−1} ∈ tt^{−1} ∩ t^{2}t^{−2} = tt^{−1} ∩ = and hence tht^{−1} = h′ ∈ . This means that th = h′t and thus mapping y = h(x) by the t gives t(y) = th(x) = h′t(x). If one defines y′ = h′(x′), then from the fact that h′ is an and thus preserves distances, it follows that ∥t(y) − y′∥ = ∥h′t(x) − h′(x′)∥ = ∥h′[t(x) − x′]∥ = ∥t(x) − x′∥ = d_{min}. Since h′ is an element of , it follows that O_{i′j′} contains an atom with distance d_{min} to y. The same argument applied with the roles of O_{ij} and O_{i′j′} interchanged now shows that the structure can not contain an atom closer to t(y) than y′, because that would result in an atom with distance less than d_{min} to t(x).
Remark: The above proof is easily generalized to the case of a kfold twin. In this case, the intersection has to be chosen as = ∩ tt^{−1} ∩ t^{2}t^{−2} ∩ … ∩ t^{k − 1}t^{−(k − 1)}. Then the crucial argument in the proof that tht^{−1} = h′ ∈ remains valid.
Footnotes
^{1}For manifold twins (i.e. twins in which the is higher than twofold), a zeroobliquity TLQS may occur. In this case, a different parameter is necessary to measure the deviation from the exact restoration of nodes, like the twin misfit introduced by Nespolo & Ferraris (2007).
^{2}This includes the of a twofold twin as well as twin operations of higher order about symmetry elements for the individual, like a fourfold rotation about a twofold symmetry axis or a sixfold rotation about a threefold symmetry axis. For details, see Nespolo (2004).
Acknowledgements
We would like to thank two anonymous referees for their valuable remarks which helped to significantly improve the manuscript.
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