Contents

• Introduction
• Elements of crystallography of chiral minerals

  2.1

  The relation between symmetry and chirality

  2.1.1

  Enantiomorphism

  2.1.2

  Symmetry elements, point-group symmetries, and their relation to chirality

  2.2

  The lattice system

  2.3

  The unit-cell (UC)

  2.4

  The glide symmetry elements and the helical handedness labeling

  2.5

  Space-group symmetry

  2.5.1

  The space-group symmetry label of chiral crystals

  2.5.2

  Chiral crystals - Söhncke space-group symmetries

  2.6

  The asymmetric unit

  2.7

  Conglomerates and racemates

• The chiral minerals found in nature

  3.1

  The abundance of chiral minerals and the “missing glove” situation

  3.2

  Chiral minerals in the 22 enantiomorphic space-group symmetries

  3.3

  Chiral minerals in the 43 non-enantiomorphic Söhncke space-groups

  3.4

  Chiral organic and carbonate minerals

  3.5

  Chiral polymorphic phase-transitioned minerals

• Formation and transformations of chiral minerals

  4.1

  Crystallization from the melt

  4.2

  Chirality aspects of crystallization of minerals from aqueous solutions

  4.3

  Polymerizations leading to chiral minerals

  4.3.1

  The silicates

  4.3.2

  Opals

  4.3.3

  Ambers

  4.4

  Chemical transformations of chiral minerals

  4.4.1

  Hydrolyses

  4.4.2

  Carbonations

  4.4.3

  Oxygenations

• Chirality of the macroscopic mineral

  5.1

  Molecular level descriptors of the macroscopic crystal

  5.1.1

  The crystal class

  5.1.2

  The crystal forms

  5.2

  Chiral habits of minerals

  5.2.1

  Growth conditions and randomness as sources of habit chirality

  5.2.2

  Chiral twins

  5.2.3

  Chiral habits of biominerals

  5.3

  Chiral gemstone minerals

  5.4

  Handedness labeling of the chiral minerals

  5.4.1

  The problem of handedness labeling

  5.4.2

  The enantiomeric excess

• Physical, analytical and chemical properties of chiral minerals

  6.1

  The diastereomeric interactions of chiral minerals with polarized light

  6.2

  The interaction with x-rays and electrons: Absolute chiral configuration determination

  6.3

  Physical properties of non-centrosymmetric crystals

  6.4

  Surface chirality: the diastereomeric interactions of chiral minerals with chiral molecules

  6.5

  Property/chirality correlations: Quantifying the degree of chirality

1. Introduction

2. Elements of Crystallography of Chiral Minerals

2.1. The Relation between Symmetry and Chirality

The main structural characteristic of crystals is their symmetry on the molecular and sometimes on the macroscopic scales, and identifying chirality of a crystal requires knowledge of these symmetry features. This is so, because symmetry and chirality are structural properties which are intimately linked. The grounds for this relation are explained now:

2.1.1. Enantiomorphism

Chirality can be defined in several ways, and here is an elementary one: An object is chiral if it has left-handed and right-handed forms (either both real, or one real and the other virtual). These two related forms are generally termed enantiomorphs (enantiomorphic crystals, for instance). In the specific case of molecules and at the molecular level in general, this pair of forms is termed enantiomers (for instance, right- and left-handed amino acids). The main identifying property of a pair of enantiomorphs (or enantiomers) is that they are mirror images of each other, which cannot coincide with each other by their superposition (such as a pair of gloves). This non-superimposability is a simple test for detection of chirality, and it serves also as a more common definition of chirality: An object is chiral, if it cannot be superimposed with it mirror image. Thus, enantiomorphs are very similar and very different at the same time: They are similar because they are mirror reflections of each other; and they are different, because they cannot be superimposed. From the point of view of symmetry, the simplest manifestation of the non-superimposability of enantiomorphs is that the chiral object must not have mirror symmetry. This is so, because the reflection of a mirror-symmetric object coincides with the original and therefore both can be fully superimposed, leading to achirality. In fact, not only mirror symmetry renders an object achiral, but other symmetries do so as well. In this section we generalize the non-superimposability rule to include these other symmetries, and also see which symmetries chiral objects may have.

2.1.2. Symmetry Elements, Point-Group Symmetries, and Their Relation to Chirality

Symmetry, in its general geometrical definition, is the property of an object of being invariant to transformations, including translation, reflection, rotation and change of scale, or to combinations of these. A symmetry operation is a movement of an object into a new orientation in such a way that resulting moved object is indistinguishable from the original object. The movement maybe either real - such as rotation - termed proper symmetry, or imaginary – such as reflection through a mirror – termed improper symmetry. The symmetry operation movement is carried out with a symmetry element - with a line which is a rotation axis, with a plane which is reflection mirror, or with a point (in case of inversion symmetry – see below). Let us now detail the properties of the various symmetry operations and their relation to chirality:

* In rotational symmetry the symmetry element is an axis of rotation which passes through the center point of an object. The rotational symmetry operation brings the object symmetrically to a shape-identical self (for graphics of this and the other symmetry elements, see (Symmetry@Otterbein, https://symotter.org/tutorial/intro). Rotational symmetry is defined by the fraction of a full rotation, 360^o/n, which brings the object to its original shape. The integer n is the order or the rotation. For instance, for the rotational symmetric recycling symbol

the relevant symmetry rotational steps are of 120^o around a perpendicularly centered axis of rotation, that is n = 3, completing a full rotation after 3 such steps (120, 240, 360^o); each of these three steps is a symmetry operation.

The labeling of symmetry elements is done (unfortunately) mainly by two methods: The Schoenflies (Sch) notation convention, and the Hermann-Maguin (HM) notation convention. These are commonly used in crystallography, and appear in most mineral listings. To avoid confusion, Sch notation will appear is in italics, and HM in regular fonts. The rotational symmetry, it is denoted in the HM notation by the order number, n, or in the Sch notation by C_n. For instance, the recycling symbol has a 3 or C₃ threefold rotational symmetry, and the three symmetry operation steps are labeled with superscript, C₃¹, C₃² and C₃³, for 120, 240 and 360^o, respectively. A symmetric object, the symmetry of which is characterized only by rotational elements, is chiral, such as in a three-blades propeller of rotational 3 symmetry, for which a pair of enantiomorphs exits:

Thus, objects which are rotationally symmetric and only of these symmetries, are chiral.

For what follows it is important to note the symbol of rotational symmetry do not provide information about the enantiomorphism of that chiral object, namely weather it is left-handed or right-handed; this will require additional labeling, to which we return later on.

* Closely related to the rotational symmetry is the case where the object has no symmetry at all (asymmetric). This is so because a full 360^o rotation, brings the object back to itself. This is the identity symmetry operation, and using the rotational symmetry terminology, such objects are denoted by the number 1 in HM or C₁ (or E) in Sch. Objects which do not have any symmetry are chiral (because their reflection is not superimposable with the original).

* Mirror symmetry is characterized by exact and full (virtual) superposition of an object with its mirror image. To be a mirror-symmetric object requires that the object be of bilateral symmetry, that is, that it has at least one symmetry plane that bisects it into identical or mirror-symmetric halves. The symmetry element – the bisecting mirror plane – is denoted m in HM or C_s (also σ) in Sch. Objects with this symmetry (with or without additional rotational symmetries) are not chiral (achiral), because, as just mentioned, such objects superimpose with their mirror image. Most minerals have this and/or the next two symmetry elements, and are therefore achiral.

* The combination of reflection and rotation, roto-reflection, is denoted in HM with an even number numeral and with an upper bar (macron), $\overline{n}$ (also with a minus sign, -n), or S_n in Sch. For instance, $\overline{4}$ or -4 or S₄ denotes a fourfold rotation axis+reflection through a mirror perpendicular to it (Symmetry@Otterbein, https://symotter.org/tutorial/intro)). Since this combination contains reflection, objects with this symmetry are achiral.

* A special case of roto-reflection which has its own label (because it is very common) is the center point which is the element for inversion symmetry operation. It is a roto-reflection combined of a rotation of order 2 (180^o) and reflection, and is denoted $\overline{1}$ (also -1), or C_i (also S₂ or i). Since this symmetry is part of the roto-reflection family, minerals with inversion symmetry are achiral.

Symmetric objects may be characterized by more than one symmetry element, and the group of symmetry elements which describe such objects, is termed point-group symmetry (“point” because all of the symmetry elements gathered in that group pass through at least the same single point, which is left in place when the symmetry is operated). Point group symmetries which describe chiral objects contain only rotational elements and the identity. For instance, the point group which is marked D₄, is composed of one C₄, three C₂ and the identity C₁ rotational axes. On the other hand, the point group which describes the symmetry of an octahedron, O_h, (which we mention in the next Section), contains 24 planes of symmetry (as well as 24 rotational symmetry elements, and therefore describes achiral objects. Group-theory tells us which specific symmetry elements of those listed above can be gathered into point-groups, and there are infinite such combinations. However, crystals, because of their 3D symmetric space-filling restrictions, are quite limited in the allowed symmetry point groups – only 32 – and these are detailed in the next section and in Section 5.1.1 (The crystal class).

2.4. The Glide Symmetry Elements and the Helical Handedness Labeling

The point-group symmetry elements described in section 2.1 suffice for the description of the macroscopic symmetry of the crystal (Section 5) and for the description of many – but not all – of the symmetries of the molecular building blocks of the crystal. However, for a full description of the molecular level symmetry, two additional elements are defined, which involve the crystal translational symmetry (and therefore are not included in the point-group symmetries which require a fixed point) – these are the glide-plane and the roto-glide (screw axis) symmetry elements (Hoffmann, https://www.youtube.com/@FrankHoffmann1000) which relate to the translational symmetry properties within the unit-cell (UC). Most of this section is devoted for the roto-glide, which is a key symmetry element of chiral crystals; and we then briefly describe the glide plane, which characterizes achiral crystals only.

The roto-glide (or screw axis) symmetry element combines helical operation with translational glide in the direction of the long axis of the helix, contained inside the UC. It describes chiral crystals because the helix is chiral, the enantiomorphs/enantiomers of which are the left or right-handed versions of the helix. The roto-glide element is denoted (in practice, mainly by the HM notation system) as n_m (also appears as nm), where n describes the rotational symmetry component of the helix, that is, the rotational symmetry of the 2D-projection perpendicular to the helicity axis, z. The subscript m denotes the number of 360^o/n rotational steps taken along the helix, namely, the gliding distance inside the UC, which brings the object to an undistinguishable same (the symmetry operation) (Hoffmann, https://www.youtube.com/@FrankHoffmann1000). For instance, in the 3₁ (also appears as 31) notation, 3 is the rotation symmetry of the helix 2D projection, and the subscript 1 tell us that that the helical rotation of 360^o/3=120^o, is followed by a 1/3-UC glide along the long direction of the helix. Theory of crystallography tells us that only a limited set of rotational symmetries and their combinations can fulfil the space-filling restrictive condition of the lattices. These are the identity (1), the two-fold (2), three-fold (3), four-fold (4) and six-fold (6) rotational symmetries, and the associated helical roto-glides of 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, and 6₅.

Out of these 11 roto-glide elements, 8 represent 4 enantiomorphic pairs: 3₁ and 3₂, 4₁ and 4₃, 6₁ and 6₅, and 6₂ and 6₄. The agreed convention is that the smaller m in each pair represents a right-handed roto-glide helix (the regular clockwise motion for penetrating a screw), for instance 3₁, while the larger m - (3₂) - reparents the left-handed (counter-clockwise, unscrewing) roto-glide helix ^{Note 2}.

The 8 enantiomorphic roto-glide pairs 3₁, 3₂, 4₁, 4₃, 6₁, 6₂, 6₄, and 6₅, always indicate a chiral crystal and do not appear in achiral crystals. However, 2₁, 4₂, and 6₃, do not have enantiomorphs because each of them means a rotation of 180^o with a ½-UC glide, and that point is reached regardless of whether the rotation is to the left or to the right. Since these three elements are still rotation elements but they do not carry enantiomorphic information, they may describe both chiral (in conjunction with other rotational symmetries) or achiral crystals (in conjunction with other symmetry elements which are not rotations).

The second translational symmetry element is the glide-planeelement, which represents the combination of a glide (translation) with a reflection. Its HM notation uses a lower-case letter (a…e or n) depending on the glide direction. Most translational glides are ½ of the UC, and some are also ¼ of the UC. Crystals with this symmetry element are achiral, because they contain the reflection operation.

From the point of view of the chirality criterion, both the point group symmetry elements and the glide translational symmetry elements are put under a unified umbrella, divided into two types: Proper symmetry elements – which are the rotational, roto-glides and identity symmetries; and the improper symmetry elements – which contain mirror symmetry and its derivatives (roto-reflection, glide-plane). The chirality criterion is therefore:

Having that generalization for point groups and the glides, we can now move to the main identifier of a chiral crystal – its space group symmetry:

2.5. Space-Group Symmetry

2.5.1. The Space-Group Symmetry Label of Chiral Crystals

The main identifier of the molecular level symmetry/chirality of a crystalline mineral in all major mineral listings is its space-group symmetry (SGS, or “space-group”, in short)). The SGS combines into a single label the lattice-variety upper-case letter symbol (Section 2.2), and the symmetry elements within the UC including the glide-symmetries. The restriction imposed by the fact that the packing of the UCs must provide a space-filled construct, leads to a limit of the possible number of the SGS’s: 230 all in all.

As for the point-group symmetries, for space-group notations there are the Hermann-Maguin (HM) notation, and Schoenflies (Sch) notation and. However, in the case of the space groups, the Sch notation is rarely used, and so we proceed only with the HM notation ^{Note 3}. Let us first see how to read these notations, and we take for that purpose we take cristobalite, a chiral SiO₂ mineral. Its space group symmetry is described by the following notation:

P4₁2₁2.

The numbers in this notation are the symmetry elements of the UC molecular content. For cristobalite, these symmetry elements are three perpendicular rotational symmetry axes: two screw axes - 4₁ and 2₁, and a 2-fold rotation axis. The order of the axes follows a hierarchy – the first position is usually (but not always) assigned to the highest-order axis, 4 in our case ^{Note 4}. Being chiral, cristobalite appears in two different enantiomorphic crystals. Following the handedness notation explained in the previous section, P4₁2₁2 is the space group of the right-handed mineral, and P4₃2₁2 is of the left-handed one. As we shall see shortly, such notations of pairs of space group symmetries are possible only if they contain any of the 3₁ – 3₂, 4₁ - 4₃, 6₁ - 6₅, and 6₂ - 6₄ roto-glide symmetry enantiomorphic elements.

Any other symbol of symmetry elements in the SGS notation - lower case letters, numbers with a minus sign or a macron - indicates an achiral crystal, even if it comes in combination with rotational symmetries. For example, sphalerite ((Zn,Fe)S), an achiral mineral, has the SGS notation of P$\overline{4}$3m, that is, a combination of rotational symmetry (3) with the achiral elements of roto-reflection symmetry ($\overline{4}$) and mirror symmetry (m).

The first upper-case letter in the space-group notations refers to the lattice variety (primitive, P, for cristobalite). Mineral listings also routinely indicate the lattice-system - tetragonal in the case of cristobalite – but the practice is to attach the lattice variety (which together with the lattice system indicates one of the 14 Bravais lattices), to the list of symmetry elements. This distinguishes between space groups which contain the same symmetry elements but in different Bravais lattices. An example are the chiral crystals which have both 2-fold and 3-fold rotational symmetries (and only these symmetries), which appear in the space groups P23, F23, and I23 - all in the cubic lattice system - and in R32 in the trigonal lattice system. In any event and as already mentioned above, for a preliminary yes-no identification of chirality, the lattice-variety symbol can be ignored.

2.5.2. Chiral Crystals - Söhncke Space-Group Symmetries

Out of the crystallographic allowed 230 SGSs, 165 contain at least one improper symmetry element and are therefore describing achiral crystals (only) – these are the improper-elements space groups. The other 65 space groups contain only proper (rotational) symmetry elements - collectively known as Söhncke group symmetries (or proper or rotational space-groups). Following the above framed instruction of how to identify chirality, these 65 space groups are applicable for chiral crystals (only) – see Table 1, right column (the left columns is explained later on).

The 65 Söhncke group symmetries are further sub-divided into two distinctly different SGS’s: The first - Söhncke enantiomorphic space groups – are composed of the roto-glides 3₁-3₂, 4₁-4₃, 6₁-6₅ and 6₂-6₄, alone or in combination with the other allowed rotational elements, (the identity (1), the two-fold (2), three-fold (3), four-fold (4) and six-fold (6) rotational symmetries), and all of which are in the primitive (P) lattice. There are 22 Söhncke space groups of this kind, which are actually 11 pairs of right (the lower m index) left-handed (the higher one) space-group pairs (11x2=22), and these are bolded in Table 1. An example is again cristobalite, which appears in nature as two enantiomorphic minerals in either the space groups P4₁2₁2 or the space group P4₃2₁2. For these 11 enantiomorphic pairs, the handedness labeling rule is simple and follows the accepted convention for helixes, as expressed by the roto-glides and explained above: right-handed cristobalite is P4₁2₁2 while left-handed cristobalite is P4₃2₁2.

The other (65–22=) 43 Söhncke space groups – Table 1 – contain, alone or in combinations, all rotational elements including the roto-glides 2₁, 4₂, and 6₃, (but not 3₁-3₂, 4₃, 6₁-6₅ and 6₂-6₄ which appear only in the 11 enantiomorphic space groups). They also contain the roto-glide 4₁ but only in non-primitive lattices. Since rotational elements and the roto-glides 2₁, 4₂, and 6₃ do not provide enantiomorphic information (Section 2.4, and so is 4₁ in non-primitive lattices) this sub-group of 43 are termed Söhncke non-enantiomorphic space groups ^{Note 5}. The inability of this sub-group to indicate handedness creates aa problematic situation: For instance, although epsomite, MgSO₄·7H₂O, appears in nature as two distinctly different enantiomorphic minerals, the space group label of this mineral, P2₁2₁2₁, is blind to that fact – we return to that issue in Section 5.4.

We can now re-phrase the above criterion of chirality, and tailor it directly to crystals:

Having the same space group symmetry for two chemically different crystals may indicate a similar molecular structure, but this is not necessarily so. For instance, cinnabar (HgS) and quartz (SiO₂) are two chiral minerals in the enantiomorphic space groups P3₁21 and P3₂21, but their actual molecular structures – described in more detail below – are quite different: in HgS the 3₁ or 3₂ helixes are of -Hg-S- chains, while for quartz the helixes are of oxygen-linked SiO₄ tetrahedra. This, in fact, is an important fact to notice – a given symmetry never provides full information of the specific details of the actual molecular arrangement, but instead, provides a general shape descriptor. Yet, examples of crystals which share the same space group symmetry and also similar spatial arrangement of their different elements, exist and are referred to as isostructural crystals. Examples are berlinite (Al(PO₄)) and quartz, both of the enantiomorphic space groups P3₁21/P3₂21, and both with tetrahedral building units - tetrahedral phosphate and the tetrahedral Si(OSi)₄ units. Likewise, chemically related minerals may appear as either isostructural crystals or as crystals of different space groups. These two situations are exemplified with GeO₂ minerals: The element Ge is just below Si in the periodic table, and while pertoldite, a GeO₂ mineral, preserves the same space group and chirality of quartz (that is, quartz and pertoldite are isostructural), argutite, another GeO₂ mineral, packs in the completely different achiral space group of P4₂/mnm.

3. The Chiral Minerals Found in Nature

4. Formation and Transformations of Chiral Minerals

5. Chirality of the Macroscopic Mineral

5.1. Molecular Level Descriptors of the Macroscopic Crystal

5.1.1. The Crystal Class

The outcome of the various mineral formation routes described in Section 4, is the macroscopic mineral, form the microns to cm and even to meter scales. The molecular level should lead to some predicted ideal macroscopic shape, but in reality, the actual final shape – the habit – of the grown mineral, may have little resemblance, if at all, to that ideal shape. Nevertheless, in this Section and in the next Section 5.1.2, we briefly describe the idealized expectation, before moving to the real, found-in-field, shapes.

The combined collection of all molecular building blocks of a crystal, namely the molecular array of the whole crystal, has a symmetry of its own. That macroscopic symmetry carries with it the symmetry characteristics of the molecular assembly in the elementary UC, but without the translational glide elements (because translating the whole crystal is just shifting it to a different place); we are therefore back in the domain of point-group symmetries for the description of the macroscopic symmetrical shape. The crystallographic restriction of having symmetrical 3D space filling arrays, limits the number of possible whole crystal point-group symmetries to 32 (31+C₁). These are called the 32 crystal classes or crystallographic point groups, and each is labeled by its Hermann-Maguin (HM) or Schoenflies (Sch) point group notations (Hahn and Klapper, 2006). Crystal classes appear routinely in listings of minerals as one of their main characteristics, also because they are linked to a variety of chemical and their physical characteristics (Section 6). 11 of the 32 crystallographic point groups contain only rotational elements – see Table 1, first column – and describe the only 11 possible whole-crystal molecular assembly symmetries of chiral crystals. It is also seen in that table why these are termed “Classes”: Each class contains all space-group symmetries which share the same collection of rotational symmetry elements (roto-glide elements are treated with their rotational part only, without the gliding operation). For instance, the chiral 222-crystal class (D₂ point-group symmetry) encompasses all space groups which have three perpendicular 2 (C₂) rotational axes, including the axis of the 2₁ roto-glide. As this class is composed of rotational elements only, it describes, by definition, only chiral crystals. We have now therefore third version for a common chiral mineral identifier:

In addition to the symmetry point group label of the crystal class, the macroscopic crystal is also traditionally described by an ideal shape – open or closed - which has that point-group symmetry – see Table 1, first column: An example for an open shape there is the 2 (C₂) sphenoid – two faces related by a C₂ rotational symmetry; and an example for a closed shape is the 222 (D₂) rhombic-disphenoidal polyhedron, which is a tetrahedron with all faces being scalene triangles, rendering it chiral (Disphenoid, https://en.wikipedia.org/wiki/Disphenoid). This polyhedron as well as all other shapes associated with 11 chiral classes, appear as left – and right-handed versions (see (External Symmetry of Crystals, 32 Crystal Classes, https://www2.tulane.edu/~sanelson/eens211/32crystalclass.htm) for the shapes of all chiral crystal classes).

To put this classification in relevance, we already mentioned in this review quite a number of minerals which belong to the 222 crystal class, most of which are in the common space group P2₁2₁2₁, namely nabesite (Na₂BeSi₄O₁₀·4H₂O), epsomite, goslarite, and morenosite (MSO₄ · 7 H₂O, where M = Mg, Zn, Ni, respectively), wyartite (CaU⁵⁺(UO₂)₂(CO₃)O₄(OH)·7H₂O), chalcomenite (Cu(Se⁴⁺O₃)·2H₂O), austinite (CaZnAsO₄(OH)), leucophanite (NaCaBeSi₂O₆F), umbite (K₂(Zr,Ti)Si₃O₉·H₂O, and leucophanite ((Na,Ca)₂BeSi₂(O_.OH_.F)₇). We have already mentioned also minerals with other space groups belonging to the 222 class: tridymite (SiO₂, C222₁), edingtonite (Ba(Al₂Si₃O₁₀)·4H₂O, P2₁2₁2), gismondine-Sr (Sr₄(Si₈Al₈O₃₂)·9H₂O, C222₁), and tobermorite (Ca₅Si₆O₁₆(OH)₂·4H₂O) C222₁ (or P2₁).

The 32 crystal classes are traditionally divided into 7 crystal systems. While these do not provide information on the chirality/achirality of a crystal, we briefly describe it here, as this classification appears in most mineral listings. A crystal system groups the crystal classes (and the space groups they contain) that utilize the same lattice system. Thus, it divides the 32 crystal classes between, the cubic, hexagonal, trigonal, tetragonal, orthorhombic, monoclinic, and the triclinic lattices ^{Note 9}. For instance, the tetragonal crystal system groups seven crystal classes (out of the 32), two of which are chiral and appear in Table 1 (first column): The 4 (C₄, tetragonal-pyramidal) and 422 (D₄, tetragonal-trapezohedral) crystal classes; see Table 1 first column for the other crystal system-crystal class relations.

5.1.2. The Crystal Forms

Next, we explain the how the crystal classes just described, give rise to many possible macroscopic crystal forms. For that we first recall that the habit is often characterized by clearly observed planar faces. These faces indicate where the crystal develops grows, and where possible cleavage planes of the crystal exist, which are parallel to those external faces. The mutual orientations of these faces on the macroscopic habit are dictated by the symmetry and geometrical arrangement details of the atoms – specifically by their symmetry class - and by the lattice which packs them. These planes are termed crystallographic faces (or planes, which are defined by their Miller index labels ^{Note 10}. Out of the various possible growth/cleavage planes, one may build a variety of geometrical bodies, which are termed crystal forms. Crystallography considerations lead to of 47 (Hahn et al, 2016) or 48 (Perkins et al, 2023) possible crystal forms (depending on the methodology), the faces of which are those of the crystal classes. A given crystal class is associated in most cases with more than one crystal form, resulting in either achiral or chiral forms. For instance, the 222 (D₂) crystal class may give rise to crystal forms such pinacoid, rhombic disphenoid, and rhombic prism (Hahn and Klapper, 2006.). The opposite is also true: A specific crystal form may result from several different crystal classes. For instance, the tetragonal prism crystal form may emerge from the chiral 4 and 422 classes and from a number of achiral classes (4/m and more). For a full table of all general crystal forms and the specific classes, chiral and achiral that are associated with them, see (Perkins et al, 2023).

In reality crystals the situation is much richer from the shapes point of view, because crystals often grow (or are cleaved) along faces which come from a combination of several forms which are allowed by a specific crystal class. From that point of view, the term "crystal form" is somewhat misleading, and one should therefore regard the “crystal form” as a “form-unit building block”. A nice example is quartz, which is found in nature with as many as 40 different combinations of the crystal forms which belong to its crystal class 32 (D₃), some chiral, others achiral, including the most commonly cited chiral habit of quartz, which is in fact a combination of 5 crystal forms, preserving macroscopically the 32 symmetry classes – see (The Quartz Page, http://www.quartzpage.de/crs_forms.html) for details. The same is true for achiral minerals, an example being calcite, CaCO3, which may occur in either achiral or chiral (scalenohedral) forms (Hazen, 2004).

6. Physical, Analytical and Chemical Properties of Chiral Minerals

References

1. Addadi, L., Geva, M. 2003. Molecular recognition at the interface between crystals and biology: generation, manifestation and detection of chirality at crystal surfaces Cryst. Eng. Comm. 5, 140–146. [CrossRef]
2. Amber, https://www.mindat.org/min-188.html.
3. Amber, https://www.kremer-pigmente.com/elements/resources/products/files/60200e.pdf.
4. Armstrong, D.W. et al, 1996. Enantiomeric composition and prevalence of some bicyclic monoterpenoids in amber. Chirality 8, 39 – 48. [CrossRef]
5. Austinite, https://en.wikipedia.org/wiki/Austinite.
6. Avnir, D., Editor, 1989. The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers. Wiley, Chichester, 1989. 3rd printing: 1992.
7. Avnir, D., Huylebrouck, D. 2013. On Left and Right: Chirality in Architecture. Nexus Network Journal: Architecture and Mathematics 15, 171-182. [CrossRef]
8. Avnir, D., 2021. Critical review of chirality indicators of extraterrestrial life. New Astronomy Rev. 92, 101596. [CrossRef]
9. Avrahami, E.M., 2022. Complex morphologies of biogenic crystals emerge from anisotropic growth of symmetry-related facets. Science 376, 312–316. [CrossRef]
10. Bayliss, P., 1982. A further crystal structure refinement of gersdorffite. American Mineralogist 67, 1058-1064.
11. Beta-Tridymite, https://www.mindat.org/min-47886.html.
12. Bouropoulos, N. et al, 2001. Calcium oxalate crystals in tomato and tobacco plants: morphology and in vitro interactions of crystal-associated macromolecules. Chemistry A Europ. J. 7, 1881-1888. [CrossRef]
13. Bouska, V., et al, 1998. Hartite from Bilina. American Mineralogist 83, 1340–1346. [CrossRef]
14. Buchen, J. et al, 2019. Twins in YAl₃(BO₃)₄ and K₂Al₂B₂O₇ crystals as revealed by changes in optical activity. Crystals 9, 8. [CrossRef]
15. Cambridge Structural Database, https://www.ccdc.cam.ac.uk/media/CSD-Space-Group-Statistics-Space-Group-Number-Ordering-2024.pdf.
16. Carbon Mineral Challenge, https://en.wikipedia.org/wiki/Carbon_Mineral_Challenge.
17. Carretero-Genevrier, A. et al, 2015. Chiral habit selection on nanostructured.
18. epitaxial quartz films. Faraday Discuss., 179, 227-233. [CrossRef]
19. Chandrasekhar, S., 1953. The optical rotatory dispersion of cinnabar. Proc. Indian Acad. Sci. 37, 697–703. [CrossRef]
20. Chemicool Dictionary, https://www.chemicool.com/definition/lattice_crystal.html.
21. Claborn, K. et al, 2008. Optical Rotation of Achiral Compounds. Angew. Chem. Int. Ed. 47, 5706 – 5717. [CrossRef]
22. Cody, A.M., Cody, R.D., 1991. Chiral habit modifications of gypsum from epitaxial-like adsorption of stereospecific growth inhibitors. Journal of Crystal Growth 113, 508—519. [CrossRef]
23. Cooper, M. et al, 2000. Refinement of the crystal structure of cyrilovite from Cyrilov, Western Moravia, Czech Republic. Journal of the Czech Geological Society, 45, 95 - 100.
24. Crystalografia, https://www.xtal.iqf.csic.es/Cristalografia/parte_03_4-en.html.
25. Cyrilovite, https://gemstone.fandom.com/wiki/Cyrilovite.
26. Dapiaggi, M., 2015. The formation of silica high temperature polymorphs from quartz: Influence of grain size and mineralising agents. Journal of the European Ceramic Society 35, 4547–4555. [CrossRef]
27. Damien Daval, D., 2018. Carbon dioxide sequestration through silicate degradation and carbon mineralisation: promises and uncertainties. npj Materials Degradation 2, 11. [CrossRef]
28. Disphenoid, https://en.wikipedia.org/wiki/Disphenoid.
29. Dong, Z., Ma, Y., 2020. Atomic-level handedness determination of chiral crystals using aberration-corrected scanning transmission electron microscopy. Nature Commun. 11, 1588. [CrossRef]
30. Downs, R.T., Hazen, R.M., 2004. Chiral indices of crystalline surfaces as a measure of enantioselective potential. J. Mol. Catal. A Chem. 216, 273–285. [CrossRef]
31. Drewitt, J.W.E et al, 2022. From short to medium range order in glasses and melts by diffraction and Raman spectroscopy. Reviews in Mineralogy and Geochemistry 87, 55–103. [CrossRef]
32. Drees, L.R. et al, 1989. Silica in Soils: Quartz and Disordered Silica Polymorphs. Chapter 19 in Minerals in Soil Environments, Volume 1, Editors: Dixon, J.B., Weed, S.B. The Soil Science Society of America, Inc. [CrossRef]
33. Dryzun, C., Avnir, D., 2012. On the abundance of chiral crystals. Chem. Commun. 48, 5874–5876. [CrossRef]
34. Dryzun, C. et al., 2009. Chiral silicate zeolites. J. Mater. Chem. 19, 2062–2069. [CrossRef]
35. Erez, J., 2003. The source of ions for biomineralization in foraminifera and their implications for paleoceanographic proxies. Reviews in Mineralogy and Geochemistry 54, 115–149. https://doi.org/10.2113/0540115. [CrossRef]
36. External Symmetry of Crystals, 32 Crystal Classes, https://www2.tulane.edu/~sanelson/eens211/32crystalclass.htm.
37. Fábián, L., Brock, C.P., 2010. A list of organic kryptoracemates. Acta Crystallogr. B 66, 94-103. [CrossRef]
38. Flack H.D., 2003. Chiral and achiral crystal structures. Helvetica Chim. Acta 86, 905-921. [CrossRef]
39. Fortes, A.D. 2005. From Surrey to the moons of Jupiter (via Mars): The story of epsomite. Axis, 1, 1-28. https://mineralogicalrecord.com/wp-content/uploads/2020/10/pdfs/Epsomite-Article-Figs.pdf.
40. V. M. Fridkin, V.M. et al, 1982. The linear and circular bulk photovoltaic effect in piezoelectric crystals Er(HCOO)₃· 2H₂O and HgS. Ferroelectrics 44, 27- 31. [CrossRef]
41. Friis, H., Balic-Zunic, T., 2005. Complex twinning and incorporation of REE into leucophanite. In CER200 – Rare earth minerals, GFF, 127:1, 35. [CrossRef]
42. Gatta, G.D., et al, 2019. A single-crystal neutron diffraction study of wardite, NaAl₃(PO₄)₂(OH)₄·2H₂O. Phys Chem Minerals 46, 427–435. [CrossRef]
43. Gault, H.R., 1949. The frequency of twin types in quartz crystals. American Mineralogist 34, 142–162. http://www.minsocam.org/msa/collectors_corner/arc/qtztwin.htm.
44. Glassy igneous rock, https://www.mindat.org/min-50714.html.
45. Glazer, A.M., 2018. Confusion over the description of the quartz structure yet again. J. Appl. Cryst. 51, 915–918. [CrossRef]
46. Glazer, A.M., Stadnicka, K., 1986. On the origin of optical activity in crystal structures. J. Appl. Cryst. 19, 108-122. [CrossRef]
47. Glazer, A.M., Stadnicka, K., 1989. On the Use of the Term 'Absolute' in Crystallography. Acta Cryst. A 45, 234-238. [CrossRef]
48. Goodman, P., Secomb, T.W., 1977. Identification of enantiomorphously related space groups by electron diffraction. Acta Cryst. A33, 126-133. [CrossRef]
49. Goslarite Mineral Data, https://webmineral.com/data/Goslarite.shtml.
50. Götze, J. et al., 2021. Mineralogy and mineral chemistry of quartz: A review. Mineralogical Magazine 85, 639–664. [CrossRef]
51. Hahn,Th., Klapper, H., 2006. Tables 10.1.2.2 and 10.1.2.3 in International Tables for Crystallography Vol. A, Section 10.1.2, pp. 771 and 791. [CrossRef]
52. Hahn, Th. et al, 2016. Table 3.2.1.3 in International Tables for Crystallography (2016), Vol. A, p.724. https://onlinelibrary.wiley.com/iucr/itc/Ac/contents/.
53. Halasyamani, P.S., and Kenneth R. Poeppelmeier, K.R., 1998. Noncentrosymmetric oxides. Chem. Mater. 10, 2753-2769. [CrossRef]
54. Hatch, D.M., Ghose, S., 1991. The a-b Phase Transition in Cristobalite, SiO₂. Phys. Chem. Minerals 17, 554-562.
55. Hazen, R.M. et al, 2001. Selective adsorption of L- and D-amino acids on calcite: Implications for biochemical homochirality. PNAS 98, 5487–5490. www.pnas.orgycgiydoiy10.1073ypnas.101085998.
56. Hazen, R.M., 2004. Chiral crystal faces of common rock-forming minerals. Chapter 11 in Progress in Biological Chirality, Palyi, G. et al, Eds. Oxford: Elsevier, pp.137-151.
57. Hazen, R.M. et al, 2009. Evolution of uranium and thorium minerals. American Mineralogist 94, 1293–1311. [CrossRef]
58. Hazen, R.M. et al, 2016. Carbon mineral ecology: Predicting the undiscovered minerals of carbon. American Mineralogist 101, 889–906. [CrossRef]
59. Hazen, R.M., 2019. Symphony in C: Carbon and the Evolution of (Almost) Everything. W. W. Norton & Company.
60. Hazen, R.M. et al, 2022. On the paragenetic modes of minerals: A mineral evolution perspective. American Mineralogist 107, 1262–1287. [CrossRef]
61. Hejl, E., Finger, F. 2018. Chiral Proportions of Nepheline Originating from.
62. Low-Viscosity Alkaline Melts. A Pilot Study. Symmetry 10, 410. [CrossRef]
63. Helman, D.S., 2016. Symmetry-based electricity in minerals and rocks: A review with examples of centrosymmetric minerals that exhibit pyro and piezoelectricity. Periodico di Mineralogia 85, 201-248. [CrossRef]
64. Hoffmann, F., 2016. Introduction to Crystallography. Springer Fachmedien Wiesbaden GmbH. [CrossRef]
65. Hoffmann, F. Frank Hoffmann, https://www.youtube.com/@FrankHoffmann1000.
66. Hong, W. et al 1986. The piezoelectric and elastic properties of berlinite and the effect of defects on the physical properties. Journal of Crystal Growth 79, 227-231. [CrossRef]
67. Hongu, H., et al, 2019. Crystal structure and XANES investigation of petzite, Ag₃AuTe₂. Acta Cryst. B75, 273–278. [CrossRef]
68. Horwell, C.J. et al, 2013. The nature and formation of cristobalite at the Soufrière Hills volcano, Montserrat: implications for the petrology and stability of silicic lava domes. Bull Volcanol (2013) 75:696. [CrossRef]
69. Huang, T-Y., et al, 2024. Recent progress in chiral zeolites: Structure, synthesis, characterization and applications, Chinese Chemical Letters, in press. [CrossRef]
70. Hummer, D.R. et al, 2022. Evidence for the oxidation of Earth’s crust from the evolution of manganese minerals. Nature Commun. 13, 960. [CrossRef]
71. Iggland, M., Mazzotti, M., 2011. A population balance model for chiral resolution via Viedma ripening. Cryst. Growth Des. 11, 4611–4622. [CrossRef]
72. Inui, H. et al, 2007. Enantiomorph identification of crystals belonging to the point groups 321 and 312 by convergent-beam electron diffraction. J. Appl. Cryst. 40, 241–249. [CrossRef]
73. Jiang, W. et al, 2019. Homochirality in biomineral suprastructures induced by assembly of single-enantiomer amino acids from a nonracemic mixture. Nature Commun. 10, 2318. [CrossRef]
74. Kaminsky, W. et al, 2004. Polarimetric imaging of crystals. Chem. Soc. Rev. 33, 514–525. [CrossRef]
75. Kaolinite Subgroup, https://www.mindat.org/min-43755.html.
76. Katzenelson, O. et al. 1996. Chirality of large random supramolecular structures. Chemistry - A European J. 2, 174 – 181. [CrossRef]
77. Kavasmaneck, P.R., Bonner, W.A., 1977. Adsorption of amino acid derivatives by d- and l-quartz. Journal of the American Chemical Society 99, 44-50. [CrossRef]
78. I. B. Kobyakov, I.B. et al, 1982. On a peculiarity of the piezoeffect in carbonate-cancrinite crystals. Sov. Phys. JETP 56, 1112-1115.
79. Kozlovskaya, K.A. et al, 2023. Determination of the absolute cnfiguration of monoatomic chiral crystals using three-wave X-ray diffraction. Crystallography Reports 68, 374–379. [CrossRef]
80. Kremer, D. et al, 2022. Separation of reaction products from ex-situ mineral carbonation and utilization as a substitute in cement, paper, and rubber applications. Journal of CO₂ Utilization 62, 102067. [CrossRef]
81. Labéguerie, P. et al, 2010. Structural, electronic, elastic, and piezoelectric properties of α-quartz and MXO₄ (M=Al, Ga, Fe; X=P, As) isomorph compounds: A DFT study. Phys. Rev. B 81, 045107. [CrossRef]
82. Lee, C. et al, 2022. Chirality in organic and mineral Systems: A review of reactivity and alteration processes relevant to prebiotic chemistry and life detection missions. Symmetry 14, 460. [CrossRef]
83. List of gemstones by species, https://en.wikipedia.org/wiki/List_of_gemstones_by_species and other listings.
84. Loiacono, G.M. et al, 1982. Resolution of space group ambiguities in minerals. American Mineralogist 67, 846–847.
85. Liu, X-H et al, 2023. Genetic significance of trace elements in hydrothermal quartz from the Xiangzhong metallogenic province, South China. Ore Geology Reviews 152, 105229. [CrossRef]
86. Mace, H.A., Peterson, R.C., 1995. The crystal structure of fichtelite, a naturally occurring hydrocarbon. Can. Mineral., 33, 7–11.
87. Matsuura, T., Koshima, H., 2005. Introduction to chiral crystallization of achiral organic compounds. Spontaneous generation of chirality. Journal of Photochemistry and Photobiology C: Photochemistry Reviews 6, 7–24. [CrossRef]
88. McLaren, A.C., Phakey. P.P., 1969. Diffraction contrast from Dauphine twin boundries in quartz. Phys. Stat. Sol. 31, 723-737. [CrossRef]
89. Meierhenrich, U., 2008. Note 4, p.64 in Amino Acids and the Asymmetry of Life. Advances in Astrobiology and Biogeophysics. Springer-Verlag Berlin Heidelberg.
90. Nassau, K.,1978. The origins of color in minerals. American Mineralogist 63, 219-229.
91. Ng, H.N., C. Calvo, C., 1976. X-ray study of the α–β transformation of berlinite (AlPO₄). Canadian Journal of Physics 54, 638-647. [CrossRef]
92. Ok, K.M. et al, 2006. Bulk characterization methods for non-centrosymmetric materials: second harmonic generation, piezoelectricity, pyroelectricity, and ferroelectricity. Chem. Soc. Rev. 35, 710–717. [CrossRef]
93. Overview of Silica Polymorphs, http://www.quartzpage.de/gen_mod.html.
94. E. I. Parkhomenko, E.I., 1971. Piezoelectric and Pyroelectric Effects in Minerals, in Electrification Phenomena in Rocks, Chapter 2. Springer Science+Business Media New York.
95. Pažout, R. et al, 2018. Refikite from Krásno, Czech Republic: a crystal-and molecular-structure study Mineralogical Magazine 79, 59 – 70. [CrossRef]
96. Perkins, D., et al, 2023. Forms and Point Groups. Section 10.5.1 in Mineralogy, LibreTextsTM. https://geo.libretexts.org/Bookshelves/Geology/Mineralogy_(Perkins_et_al.).
97. Piezoelectric Gems, https://classicgems.net/info_Piezoelectricity.htm.
98. Petrícek, V. et al, 1990. Orientational disorder in phenanthrene. Structure determination at 248, 295, 339 and 344 K. Acta Cryst. B46, 830-832. [CrossRef]
99. Pinto, Y. et al, 1966. Continuous chirality analysis of interconversion pathways of the water-trimer enantiomers. J. Chem Soc. Faraday Trans. 92, 2523 – 2527.
100. Pinto Y, Avnir D., 2001. A generalized handedness labeling strategy: Addressing latent handedness in chiral structures. Enantiomer 6, 211-217.
101. Pogge von Strandmann, P.A.E., et al 2019.rRapid CO₂ mineralisation into calcite at the CarbFix storage site quantified using calcium isotopes. Nature Commun. 10, 1983. [CrossRef]
102. Poole, P.H., 1995. Amorphous polymorphism. Computational Materials Science 4, 373-382. [CrossRef]
103. Powder Diffraction on the WEB, http://pd.chem.ucl.ac.uk/pdnn/symm3/sgpfreq.htm.
104. Rekis, T., 2020. Crystallization of chiral molecular compounds: what can be learned from the Cambridge Structural Database? Acta Cryst. B76, 307–315. [CrossRef]
105. Ruby Dreamscape™ Cut, https://www.johndyergems.com/gemstones/ruby-7397.html.
106. Samotoin, N.D., 2011. Enantiomorphism of kaolinite: manifestation at the levels of elementary layer and microcrystals. Crystallography Reports 56, 327–334. [CrossRef]
107. Shindo, H. et al, 2013. Asymmetric autocatalysis induced by cinnabar: Observation of the enantioselective adsorption of a 5-pyrimidyl alkanol on the crystal Surface. Angew. Chem. Int. Ed. 52, 9135 –9138. See also the supporting information. [CrossRef]
108. Smith, P.P.K., 1979. The observation of enantiomorphous domains in a natural maghemite. Contr. Mineral. and Petrol. 69, 249–254. [CrossRef]
109. Kenso Soai, K., et al, 1999. d- and l-Quartz-Promoted Highly Enantioselective Synthesis of a Chiral Organic Compound. J. Am. Chem. Soc. 121, 11235-11236. [CrossRef]
110. Staples, L.W., 1935. Austinite, a new arsenate mineral, from Gold Hill, Utah. American Mineralogist 20, 112–119.
111. Strunz, H., 1962. Fichtelit. Dimethyl-isopropyl-perhydropenanthren. Naturwissen., 49, 9–10. [CrossRef]
112. Symmetry@Otterbein, https://symotter.org/tutorial/intro.
113. Támara, M., Preston, M.R., 2009. A statistical reassessment of the evidence for the racemic distribution of quartz enantiomorphs. American Mineralogist 94, 1556–1559. [CrossRef]
114. Tanaka, Y., et al, 2008. Right handed or left handed? Forbidden X-ray diffraction reveals chirality. Phys. Rev. Lett. 100, 145502. [CrossRef]
115. Tanaka, Y., et al, 2010. Determination of structural chirality of berlinite and quartz using resonant x-ray diffraction with circularly polarized x-rays. Phys. Rev. B 81, 144104. DOI: 10.1103/PhysRevB.81.144104. Erratum: 2011. Phys. Rev. B 84, 219905(E). [CrossRef]
116. The Mineral Cristobalite, https://www.minerals.net/mineral/cristobalite.aspx.
117. The Mineral Quartz, https://www.minerals.net/mineral/quartz.aspx;%20https:/www.mindat.org/min-3337.html.
118. The Quartz Page, http://www.quartzpage.de/.
119. The Quartz Page, http://www.quartzpage.de/crs_forms.html.
120. The Quartz Page, http://www.quartzpage.de/gen_struct.html.
121. The Structure of Materials, http://som.web.cmu.edu/structures/S092-wollastonite.html.
122. The Structure of Materials, http://som.web.cmu.edu/structures/S097-alpha-quartz.html.
123. Thomas, P.A. 1988. The crystal structure and absolute optical chirality of paratellurite, α-TeO₂. J. Phys. C: Solid State Phys. 21, 4611-4267. [CrossRef]
124. Thompson, A.L., Watkin, D.J., 2009. X-ray crystallography and chirality: understanding the limitations. Tetrahedron: Asymmetry 20, 712–717. [CrossRef]
125. Tridymite, https://en.wikipedia.org/wiki/Tridymite.
126. Tuvi-Arad, I. et al, 2024. CSM software: Continuous symmetry and chirality measures for quantitative structural analysis. J. Chem. Inf. Model. in press. [CrossRef]
127. Urusov, V.S., Nadezhina, T.N., 2009. Frequency distribution and selection of space groups in inorganic crystal chemistry. Journal of Structural Chemistry 50, 22-37. [CrossRef]
128. Valentín-Pérez, Á. et al, 2022. Chirality determination in crystals. Chirality 34, 163–181. [CrossRef]
129. Van Dyke, K.S., 1940. On the right- and left-handedness of quartz and its relation to elastic and other properties. Proceedings of the Institute of Radio Engineers (I.R.E.), September, 1940, 399-406.
130. Van der Biest, O., Thomas, G., 1975. Identification of Enantiomorphism in Crystals by Electron Microscopy. Acta Cryst. A31, 70-76. https://doi.org/10.1107/S0567739475000137. [CrossRef]
131. Vinegrad, E. et al, 2018. Circular Dichroism of Single Particles. ACS Photonics 5, 2151−2159. [CrossRef]
132. Ward, R.M. et al, 2010. Enantiomorphic symmetry breaking in crystallization of molten.
133. sodium chlorate. Chem. Commun. 46, 7634–7636. [CrossRef]
134. Watkin, D.J., and Richard Ian Cooper< R.I., 2020. Howard Flack and the Flack Parameter. Chemistry 2, 796–804. [CrossRef]
135. Webmineral, http://webmineral.com/crystal/Triclinic.shtml.
136. Wenk, H.-R., 2006. Neutron Diffraction Texture Analysis. Reviews in Mineralogy & Geochemistry 63, 399-426. [CrossRef]
137. Williams, S.A., de Azevedo, J., 1967. Austinite from Gold Hill, Utah. American Mineralogist 52, 1224–1226.
138. Wilson, P. 1979. Experimental investigation of etch pit formation on quartz sand grains. Geological Magazine 116, 477-482. [CrossRef]
139. Winkelmann, A., Nolze, G., 2014. Chirality determination of quartz crystals using Electron Backscatter Diffraction. Ultramicroscopy 149, 58-63. [CrossRef]
140. Christopher J. Winta, C.J. et al, 2018. Second-harmonic phonon spectroscopy of α-quartz. Phys. Rev. B 97, 094108. [CrossRef]
141. Wukovitz, S.W., Yeates, T.O., 1995.Why protein crystals favour some space-groups over others. Nat. Struct. Biol. 2,1062-7. [CrossRef]
142. Yogev-Einot, D., Avnir, D., 2003. Quantitative symmetry and chirality of the molecular building blocks of quartz. Chem. Mater. 15, 464 – 472. [CrossRef]
143. Yogev-Einot, D., Avnir, D., 2004. Pressure and temperature effects on the degree of symmetry and chirality of the molecular building blocks of low quartz, Acta Cryst. B60, 163–173. [CrossRef]
144. Yogev-Einot, D., Avnir, D., 2006. The temperature-dependent optical activity of quartz: from Le Chatelier to chirality measures. Tetrahedron: Asymmetry 17, 2723 – 2725. [CrossRef]
145. Yogev-Einot, D., Avnir, D., 2007. Left/right handedness assignment to chiral tetrahedral AB₄ structures. Tetrahedron: Asymmetry 18, 2295–2299. [CrossRef]
146. H. Zabrodsky, H., et al, 1992. Continuous Symmetry Measures, J. Am. Chem. Soc. 114, 7843-7851. [CrossRef]
147. H. Zabrodsky, H., Avnir, D., 1995. Continuous Symmetry Measures, IV: Chirality. J. Am. Chem. Soc. 117, 462-473. [CrossRef]