Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 3.3, pp. 413-414

Section 3.3.1. Crystal aggregates and intergrowths

Th. Hahna* and H. Klapperb

aInstitut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail:

3.3.1. Crystal aggregates and intergrowths

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Minerals in nature and synthetic solid materials display different kinds of aggregations, in mineralogy often called intergrowths. In this chapter, we consider only aggregates of crystal grains of the same species, i.e. of the same (or nearly the same) chemical composition and crystal structure (homophase aggregates). Intergrowths of grains of different species (heterophase aggregates), e.g. heterophase bicrystals, epitaxy (two-dimensional oriented intergrowth on a surface), topotaxy (three-dimensional oriented precipitation or exsolution) or the paragenesis of different minerals in a rock or in a technical product are not treated in this chapter.

  • (i) Arbitrary intergrowth: Aggregation of two or more crystal grains with arbitrary orientation, i.e. without any systematic regularity. Examples are irregular aggregates of quartz crystals (Bergkristall) in a geode and intergrown single crystals precipitated from a solution. To this category also belong untextured polycrystalline materials and ceramics, as well as sandstone and quartzite.

  • (ii) Parallel intergrowth: Combination of two or more crystals with parallel (or nearly parallel) orientation of all edges and faces. Examples are dendritic intergrowths as well as parallel intergrowths of spinel octahedra (Fig.[link]a) and of quartz prisms (Fig.[link]b). Parallel intergrowths frequently exhibit re-entrant angles and are, therefore, easily misinterpreted as twins.


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    Parallel intergrowth (a) of spinel octahedra and (b) of hexagonal quartz prisms. Part (a) after Phillips (1971[link], p. 172), part (b) after Tschermak & Becke (1915[link], p. 94).

    Two possible reasons for the formation of parallel intergrowths are mentioned:

    (a) A smaller crystal has set down in parallel orientation on a growth face of an already existing crystal of the same species and has further grown together with its host. Fig.[link] suggests such a mechanism.

    (b) The growth of one or several faces of a crystal is inhibited by a layer of impurities or by foreign particles. By a local `breaking down' of these obstacles, several parallel individual crystals may appear and grow together during further growth. This mechanism is suggested for Fig.[link].

    In this context the term mosaic crystal must be mentioned. It was introduced in the early years of X-ray diffraction in order to characterize the perfection of a crystal. A mosaic crystal consists of small blocks (size typically in the micron range) with orientations deviating only slightly from the average orientation of the crystal; the term `lineage structure' is also used for very small scale parallel intergrowths (Buerger, 1934[link], 1960a[link], pp. 69–73).

  • (iii) Bicrystals: This term is mainly used in metallurgy. It refers to the (usually synthetic) intergrowth of two single crystals with a well defined orientation relation. A bicrystal contains a grain boundary, which in general is also well defined. Usually, homophase bicrystals are synthesized in order to study the structure and properties of grain boundaries. An important tool for the theoretical treatment of bicrystals and their interfaces is the coincidence-site lattice (CSL). A brief survey of bicrystals is given in Section 3.2.2[link] ; a comparison with twins and domain structures is provided by Hahn et al. (1999[link]).

  • (iv) Growth sectors and optical anomaly: Crystals grown with planar faces (habit faces), e.g. from vapour, supercooled melt or solution, consist of regions crystallized on different growth faces (Fig.[link]). These growth sectors usually have the shapes of pyramids with their apices pointing toward the nucleus or the seed crystal. They are separated by growth-sector boundaries, which represent inner surfaces swept by the crystal edges during growth. In many cases, these boundaries are imperfections of the crystal.


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    (a) Optical anomaly of a cubic mixed (K,NH4)-alum crystal grown from aqueous solution, as revealed by polarized light between crossed polarizers: (110) plate, 1 mm thick, horizontal dimension about 4 cm. (b) Sketch of growth sectors and their boundaries of the crystal plate shown in (a). The {111} growth sectors are optically negative and approximately uniaxial with their optical axes parallel to their growth directions [\langle 111\rangle] [birefringence [\Delta n] up to [5 \times 10^{-5}]; Shtukenberg et al. (2001[link])]. The (001) growth sector is nearly isotropic ([\Delta n \,\lt\, 10^{-6}]). Along the boundaries A between {111} sectors a few small {110} growth sectors (resulting from small {110} facets) have formed during growth. S: seed crystal.

    Frequently, the various growth sectors of one crystal exhibit slightly different chemical and physical properties. Of particular interest is a different optical birefringence in different growth sectors (optical anomaly) because this may simulate twinning. A typical example of this optical anomaly is shown in Fig.[link]. Since this phenomenon has sometimes been misinterpreted as twinning, it is treated in detail in the Extended note below.

  • (v) Translation domains: Translation domains are homogeneous crystal regions that exhibit exact parallel orientations, but are displaced with respect to each other by a vector (frequently called a fault vector), which is a fraction of a lattice translation vector. The interface between adjoining translation domains is called the `translation boundary'. Often the terms antiphase domains and antiphase boundaries are used. Special cases of translation boundaries are stacking faults. Translation domains are defined on an atomic scale, whereas the term parallel intergrowth [see item (ii)[link] above] refers to macroscopic (morphological) phenomena; cf. Note (7)[link] in Section[link]

  • (vi) Twins: A frequently occurring intergrowth of two or more crystals of the same species with well defined crystallographic orientation relations is called a twin (German: Zwilling; French: macle). Twins form the subject of the present chapter. The closely related topic of Domain structures is treated in Chapter 3.4[link] .

In 1975, J. D. H. Donnay and H. Takeda even proposed a new name for the `science of twinning': geminography [as reported by Nespolo & Ferraris (2003)[link] and by Grimmer & Nespolo (2006)[link]]. A complete review of the history and the various theories of twinning, together with an extensive list of references, is provided in a recent monograph (in French) by Boulliard (2010)[link]. In addition, it contains an extraordinarily large set of beautiful colour photographs of many natural twins.

Extended note: Optical anomaly, Curie's principle and dis­symmetry

The phenomenon of optical anomaly [cf. Section 3.3.1(iv)[link] and Fig.[link]] can be explained as follows: as a rule, impurities (and dopants) present in the solution are incorporated into the crystal during growth. Usually, the impurity concentrations differ in symmetry-non-equivalent growth sectors (which belong to different crystal forms), leading to slightly changed lattice parameters and physical properties of these sectors. In mixed crystals these changes often arise from a partial ordering of the mixing components parallel to the growth face [example: (K,NH4)-alum, cf. Fig.[link]; Shtukenberg et al., 2001[link]]. Optical anomalies may occur also in symmetry-equivalent growth sectors (which belong to the same crystal form) owing to their different growth directions: as a consequence of growth fluctuations, layers of varying impurity content are formed parallel to the growth face of the sector (`growth striations'). This causes a slight change of the interplanar spacing normal to the growth face. For example, a cubic NaCl crystal grown on the {100} cube faces from an aqueous solution containing Mn ions consists of three pairs of (opposite) growth sectors exhibiting a slight tetragonal distortion with tetragonality 10−5 along their <100> growth directions and hence is optically uniaxial (Ikeno et al., 1968[link])1.

An analogous effect of optical anomaly may be observed in crystals grown from the melt on rounded interfaces with planar facets of prominent habit faces (e.g. of melt-grown synthetic garnets). Owing to different growth mechanisms on round and facet interfaces (rough growth and growth by supercooling, respectively), the incorporation of impurities or dopants is usually different on the two types of interfaces (e.g. Hurle & Rudolph, 2004[link]). The regions crystallized on the rounded faces and on the different facets correspond to different growth sectors and may exhibit optical anomalies.

Although the phenomenon of optical anomaly closely resembles all features of twinning, it does not belong to the category `twinning' because it is not an intrinsic property of the crystal species, but rather the result of different growth conditions (or growth mechanisms) on different faces of the same crystal (`growth anisotropy'). It is the consequence of the well known Curie principle2 (Curie, 1894[link]; Chalmers, 1970[link]) which describes (as an `effect') the reduction of the symmetry (`dissymmetry') of an object (crystal) under an external influence (`cause') which itself exhibits a symmetry. It says, in terms of group theory, that the point-group symmetry GCF of the crystal under the external influence (field) F is the intersection of the symmetry GC of the crystal without field and the symmetry GF of the influence without crystal:[G_{CF}=G_C \cap G_F,]i.e. GCF is a (proper or improper) subgroup of both groups GC and GF. In the example of the optical anomaly of the {111} growth sectors of (K,NH4)-alum (Fig.[link]) the crystal point group is [G_C=2/m\bar3] and the symmetry of the cause `growth in direction [111]' is [G_F=\infty m] (symmetry of a stationary cone, cf. ITA, Table[link] ). The intersection symmetry [i.e. the sym­metry of the (111) growth sector] is GCF = 3m (`dissymmetry') with the threefold axis along the growth direction [111] of this sector. This leads to a reduction of the isotropic optical birefringence of the `undisturbed' cubic alum crystal to an uniaxial birefringence of its {111} growth sectors.

A very early review of the optical anomaly of crystals with many examples was published in 1891 by von Brauns[link]. An actual review, treating the `historical' observations and various interpretations (starting with Brewster, 1818[link]) as well as the modern aspects of optical anomalies, is presented by Kahr & McBride (1992)[link]. A similar, very comprehensive review is contained in the monograph of Shtukenberg et al. (2007)[link].


Grimmer, H. & Nespolo, M. (2006). Geminography – the crystallography of twins. Z. Kristallogr. 221, 28–50.
Boulliard, J.-Cl. (2010). Le cristal et ses doubles. Paris: CNRS Èditions. (In French.) ISBN: 978–2–271–07049–4.
Brauns, R. von (1891). Die optischen Anomalien der Krystalle. Leipzig: S. Hirzel.
Brewster, D. (1818). On the optical properties of muriate of soda, fluate of lime, and the diamond, as exhibited in their action upon polarised light. Trans. R. Soc. Edinburgh, pp. 157–164.
Buerger, M. J. (1934). The lineage structure of crystals. Z. Kristallogr. 89, 195–220.
Buerger, M. J. (1960a). Crystal Structure Analysis, especially ch. 3. New York: Wiley.
Chalmers, A. F. (1970). Curie's principle. Brit. J. Philos. Sci. 21, 133–148.
Curie, P. (1894). Sur la symétrie des phénomènes physique: symétrie d'un champ électrique et d'un champ magnétique. J. Phys. 3, 393–415.
Hahn, Th., Janovec, V. & Klapper, H. (1999). Bicrystals, twins and domain structures – a comparison. Ferroelectrics, 222, 11–21.
Hurle, D. T. J. & Rudolph, P. (2004). A brief history of defect formation, segregation, faceting and twinning in melt-grown semiconductors. J. Cryst. Growth, 264, 550–564.
Ikeno, S., Maruyama, H. & Kato, N. (1968). X-ray topographic studies of NaCl crystals grown from aqueous solution with Mn ions. J. Cryst. Growth, 3/4, 683–693.
Kahr, B. & McBride, J. M. (1992). Optically anomalous crystals. Angew. Chem. Int. Ed. Engl. 31, 1–26.
Nespolo, M. & Ferraris, G. (2003). Geminography – The science of twinning applied to the early-stage derivation of non-merohedric twin laws. Z. Kristallogr. 218, 178–181.
Shtukenberg, A., Punin, Y. & Kahr, B. (2007). Optically Anomalous Crystals. Heidelberg: Springer.
Shtukenberg, A. G., Punin, Yu. O., Haegele, E. & Klapper, H. (2001). On the origin of inhomogeneity of anomalous birefringence in mixed crystals: an example of alums. Phys. Chem. Miner. 28, 665–674.

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