What is the hexagonal crystal family in crystallography?

In crystallography, the hexagonal crystal family is one of the six recognized crystal families. It is characterized by including two specific crystal systems: the hexagonal and the trigonal systems. Furthermore, it encompasses two lattice systems: the hexagonal and the rhombohedral lattice systems, representing all point groups whose space groups are based on the hexagonal lattice.

How does the hexagonal crystal family relate to the trigonal and hexagonal crystal systems?

The hexagonal crystal family is defined as the union of the trigonal crystal system and the hexagonal crystal system. It encompasses all point groups that have space groups assigned to either the hexagonal or the rhombohedral lattice system.

What are the two lattice systems that constitute the hexagonal crystal family?

The hexagonal crystal family is composed of two lattice systems: the hexagonal lattice system and the rhombohedral lattice system. Each of these lattice systems contains a single Bravais lattice.

What is the conventional description of a unit cell within the hexagonal crystal family?

Conventionally, a unit cell in the hexagonal crystal family is described as a right rhombic prism. This prism has two equal axes, typically denoted as 'a', which are separated by an angle of 120 degrees (gamma). A third axis, 'c', is perpendicular to the plane containing the 'a' axes and represents the height, which may differ in length from 'a'.

How is the rhombohedral lattice system represented using a hexagonal unit cell?

When described using a hexagonal unit cell, the rhombohedral lattice system is represented by an R-centered cell. This means that in addition to the corner lattice points, there are two extra lattice points located along a body diagonal of the unit cell, specifically at fractional coordinates like (2/3, 1/3, 1/3) and (1/3, 2/3, 2/3) in the obverse setting.

What are the two settings used to describe the rhombohedral lattice within a hexagonal unit cell?

The rhombohedral lattice can be described using two settings within a hexagonal unit cell: the obverse setting and the reverse setting. These settings differ in the fractional coordinates of the additional lattice points that define the rhombohedral centering.

Why is the hexagonal description of the rhombohedral lattice often preferred over the rhombohedral axes description?

The hexagonal description is generally preferred for practical calculations because its coordinate system includes two 90-degree angles, simplifying mathematical operations. However, the rhombohedral axes description is sometimes used because it more clearly illustrates the inherent 3m symmetry of the crystal lattice.

What are the defining symmetry requirements for the trigonal crystal system?

The trigonal crystal system is defined by point groups that possess at least one three-fold axis of rotation. This system includes 5 distinct point groups.

What are the defining symmetry requirements for the hexagonal crystal system?

The hexagonal crystal system is characterized by point groups that possess at least one six-fold axis of rotation. This system comprises 7 distinct point groups.

How many space groups are associated with the trigonal crystal system, and how are they distributed between lattice systems?

The trigonal crystal system is associated with a total of 25 space groups. Of these, 7 space groups are assigned to the rhombohedral lattice system, and 18 space groups are assigned to the hexagonal lattice system.

How many space groups are associated with the hexagonal crystal system, and where are they assigned?

The hexagonal crystal system is associated with 27 space groups. All of these 27 space groups are assigned exclusively to the hexagonal lattice system.

What is unique about the trigonal crystal system in terms of its associated lattice systems?

The trigonal crystal system is unique because its point groups are associated with more than one lattice system. Specifically, its space groups are assigned to both the rhombohedral and the hexagonal lattice systems, unlike other crystal systems which are typically tied to a single lattice system.

What are the five point groups that constitute the trigonal crystal system?

The five point groups within the trigonal crystal system are: Trigonal pyramidal (represented by 3), Rhombohedral (represented by 3-bar), Trigonal trapezohedral (represented by 32), Ditrigonal pyramidal (represented by 3m), and Ditrigonal scalenohedral (represented by 3-bar m).

Can you provide examples of minerals that belong to the trigonal crystal system?

Yes, several minerals exhibit symmetries of the trigonal crystal system. Examples include carlinite, jarosite, dolomite, ilmenite, alpha-quartz, cinnabar, schorl, cerite, tourmaline, alunite, lithium tantalate, antimony, hematite, corundum, and calcite.

What are the seven point groups that constitute the hexagonal crystal system?

The seven point groups within the hexagonal crystal system are: Hexagonal pyramidal (6), Trigonal dipyramidal (6-bar), Hexagonal dipyramidal (6/m), Hexagonal trapezohedral (622), Dihexagonal pyramidal (6mm), Ditrigonal dipyramidal (6-bar m2), and Dihexagonal dipyramidal (6/mmm).

Can you provide examples of minerals that belong to the hexagonal crystal system?

Minerals exhibiting the symmetries of the hexagonal crystal system include nepheline, cancrinite, cesanite, apatite, vanadinite, kalsilite, beta-quartz, greenockite, wurtzite, benitoite, and beryl.

What is hexagonal close-packed (hcp) structure?

Hexagonal close-packed (hcp) is a highly dense arrangement of atoms, similar to face-centered cubic (fcc). However, hcp is not classified as a Bravais lattice because it contains two distinct sets of lattice points, rather than just one. It can be constructed from the hexagonal Bravais lattice by adding a two-atom motif to each lattice point.

What is the relationship between the hexagonal close-packed (hcp) structure and lonsdaleite?

The atomic positions within the hexagonal close-packed (hcp) structure are identical to those found in lonsdaleite, which is known as the hexagonal form of diamond.

What are multi-element structures in the context of the hexagonal crystal family?

Multi-element structures, such as binary compounds, frequently adopt crystal structures based on the hexagonal crystal family. These structures can be conceptualized as multiple interpenetrating sublattices, where each sublattice occupies the interstitial spaces within the others.

Describe the Wurtzite crystal structure.

The Wurtzite crystal structure, designated B4 in the Strukturbericht system and hP4 by Pearson symbol, belongs to space group P6₃mc (No. 186). Its symmetry elements include a six-fold screw rotation axis along the c-axis, a mirror plane perpendicular to the {100} direction, and a glide plane parallel to the c-axis with a normal in the {120} direction.

What are some common examples of compounds that crystallize in the Wurtzite structure?

Numerous compounds adopt the Wurtzite structure, particularly semiconductors. Notable examples include wurtzite itself (a form of ZnS), silver iodide (AgI), zinc oxide (ZnO), cadmium sulfide (CdS), cadmium selenide (CdSe), silicon carbide (α-SiC), gallium nitride (GaN), aluminum nitride (AlN), and boron nitride (w-BN).

What are the key properties of Wurtzite crystals related to their symmetry?

Wurtzite crystals are classified as non-centrosymmetric, meaning they lack inversion symmetry. This characteristic symmetry deficiency enables them to exhibit important physical properties such as piezoelectricity and pyroelectricity, which are absent in crystals possessing centrosymmetry.

How are atoms coordinated in the Wurtzite structure?

In the Wurtzite structure, each atom is tetrahedrally coordinated. This means that each atom is surrounded by four neighboring atoms arranged in the shape of a tetrahedron, contributing to the overall structure's stability and properties.

What is the Nickel Arsenide structure?

The Nickel Arsenide structure is formed by two interpenetrating sublattices: a primitive hexagonal nickel sublattice and a hexagonal close-packed arsenic sublattice. In this arrangement, each nickel atom is octahedrally coordinated by six arsenic atoms, while each arsenic atom is trigonal prismatically coordinated by six nickel atoms.

What types of chemical compounds commonly adopt the Nickel Arsenide structure?

Compounds that typically exhibit the Nickel Arsenide structure are generally those formed between transition metals and elements like chalcogens, arsenic, antimony, and bismuth. Examples include transition metal sulfides, selenides, tellurides, arsenides, antimonides, and bismuthides.

What are the specific members of the nickeline group that adopt the Nickel Arsenide structure?

The nickeline group, which follows the Nickel Arsenide structure type, includes minerals such as Achavalite (FeSe), Breithauptite (NiSb), Freboldite (CoSe), Kotulskite (Pd(Te,Bi)), Langistite ((Co,Ni)As), Nickeline (NiAs), Sobolevskite (Pd(Bi,Te)), and Sudburyite ((Pd,Ni)Sb).

What is the distinction between the trigonal crystal system and the rhombohedral lattice system?

Although closely related and grouped within the hexagonal crystal family, the trigonal crystal system and the rhombohedral lattice system are not identical. A key difference is that some crystals, like alpha-quartz, possess trigonal symmetry but are classified under the hexagonal lattice system.

How many Bravais lattices are there in the hexagonal crystal family, and what are they?

The hexagonal crystal family contains two Bravais lattices: the hexagonal lattice and the rhombohedral lattice. These are often identified by their Pearson symbols, 'hP' for the hexagonal lattice and 'hR' for the rhombohedral lattice.

What is the formula for calculating the unit cell volume in the hexagonal crystal family?

The volume of a unit cell in the hexagonal crystal family is calculated using the formula V = a²c * sin(60°). This formula accounts for the dimensions of the hexagonal prism unit cell, where 'a' is the length of the base axes and 'c' is the height.

What is the characteristic symmetry element of the hexagonal crystal system?

The defining characteristic of the hexagonal crystal system is the presence of at least one six-fold axis of rotation. This symmetry element dictates the overall arrangement of atoms and the resulting crystal shape.

What is the characteristic symmetry element of the trigonal crystal system?

The defining characteristic of the trigonal crystal system is the presence of at least one three-fold axis of rotation. This symmetry element is fundamental to the classification of crystals within this system.

What is the unit cell description for the hexagonal lattice system?

The hexagonal lattice system is described using a hexagonal unit cell, which is a prism with a hexagonal base. This cell is defined by two equal axes (a) at a 120° angle to each other, and a third axis (c) perpendicular to the base, representing the height.

What is the unit cell description for the rhombohedral lattice system?

The rhombohedral lattice system is described by a rhombohedral unit cell. This unit cell has all three axes equal in length (a = b = c) and all interaxial angles equal, but not equal to 90° (α = β = γ ≠ 90°).

What is the Pearson symbol for the hexagonal Bravais lattice?

The Pearson symbol for the hexagonal Bravais lattice is 'hP'.

What is the Pearson symbol for the rhombohedral Bravais lattice?

The Pearson symbol for the rhombohedral Bravais lattice is 'hR'.

What are the two highest-density atomic packing types?

The two types of atomic packing that achieve the highest possible density are the hexagonal close-packed (hcp) structure and the face-centered cubic (fcc) structure. Both arrangements efficiently fill space with spheres.

What is the specific space group notation for the Wurtzite structure?

The Wurtzite crystal structure is associated with space group No. 186 according to the International Union of Crystallography classification. In the Hermann-Mauguin notation, this space group is designated as P6₃mc.

What does the Hermann-Mauguin symbol P6₃mc signify for the Wurtzite structure?

The Hermann-Mauguin symbol P6₃mc for the Wurtzite structure indicates: 'P' for a primitive lattice, '6₃' for a six-fold screw rotation axis along the c-axis, '.m.' for a mirror plane perpendicular to the {100} direction, and '..c' for a glide plane parallel to the c-axis with a normal in the {120} direction.

How are atoms coordinated in the Nickel Arsenide structure?

In the Nickel Arsenide structure, nickel atoms are octahedrally coordinated, meaning each nickel atom is surrounded by six arsenic atoms arranged in an octahedron. Conversely, each arsenic atom is trigonal prismatically coordinated by six nickel atoms.

How can the Nickel Arsenide structure be described in relation to close-packing?

The Nickel Arsenide structure can be understood as a hexagonal close-packed (HCP) arrangement of arsenic atoms, with nickel atoms occupying all the available octahedral voids within this packing.

What is the significance of the 'w-' prefix in chemical formulas for materials with the Wurtzite structure?

The prefix 'w-' is sometimes used in chemical formulas, such as 'w-BN' for boron nitride, to specifically denote that the material crystallizes in the Wurtzite structure. This helps distinguish it from other possible crystal structures the compound might adopt.

What is the coordination number and geometry for atoms in the Wurtzite structure?

In the Wurtzite structure, each atom is tetrahedrally coordinated. This means each atom is surrounded by four nearest neighbors arranged in a tetrahedral geometry, forming the basis of the structure's framework.

What is the coordination number and geometry for atoms in the Nickel Arsenide structure?

In the Nickel Arsenide structure, nickel atoms exhibit octahedral coordination, being surrounded by six arsenic atoms. Arsenic atoms, in turn, display trigonal prismatic coordination, being surrounded by six nickel atoms.

What are the two crystal systems that are part of the hexagonal crystal family?

The hexagonal crystal family comprises two distinct crystal systems: the trigonal crystal system and the hexagonal crystal system. These systems are differentiated by their characteristic symmetry elements, primarily the order of rotation axes.

What is the total number of space groups associated with the hexagonal crystal family?

The hexagonal crystal family is associated with a total of 52 space groups. These are precisely the space groups whose underlying Bravais lattice is either hexagonal or rhombohedral.

What is the distinction between a crystal family and a crystal system?

A crystal family is a classification that groups crystal systems based on shared lattice types and symmetry characteristics. A crystal system is a more specific classification based on the symmetry elements present in a point group and its assignment to a particular lattice system.

What are the two Bravais lattices that can be described using rhombohedral axes?

The two Bravais lattices that can be described using rhombohedral axes are the hexagonal Bravais lattice and the rhombohedral Bravais lattice. Both of these belong to the hexagonal crystal family.

What is the general characteristic of compounds that adopt the Wurtzite structure?

Many compounds that adopt the Wurtzite structure are semiconductors. Examples include zinc oxide (ZnO), cadmium sulfide (CdS), silicon carbide (SiC), and gallium nitride (GaN), among others.

What is the role of symmetry in classifying crystals into systems within the hexagonal family?

Symmetry is the primary basis for classifying crystals into systems. The trigonal system requires at least one three-fold rotation axis, while the hexagonal system requires at least one six-fold rotation axis. These rotational symmetries dictate the overall structure and properties of the crystals.

What are the common notations for the hexagonal and rhombohedral Bravais lattices?

The hexagonal Bravais lattice is commonly denoted by the Pearson symbol 'hP', while the rhombohedral Bravais lattice is denoted by 'hR'.

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Lattice Systems

Defining the Framework

In crystallography, the hexagonal crystal family encompasses two fundamental lattice systems: the hexagonal and the rhombohedral. Each system is characterized by a unique arrangement of points in space that repeats periodically, forming the underlying structure of crystals.

Hexagonal Lattice

The hexagonal lattice system is defined by a unit cell with two equal axes (a = b) that are 120° apart (γ = 120°), and a third axis (c) perpendicular to the plane of the other two. This system is represented by the Pearson symbol 'hP' for its primitive form.

Rhombohedral Lattice

The rhombohedral lattice system, denoted by 'hR', is characterized by a unit cell where all axes are equal in length (a = b = c) and all angles are equal but not 90° (α = β = γ ≠ 90°). While it can be described using rhombohedral axes, it is conventionally represented by a hexagonal unit cell for practical crystallographic descriptions.

Crystal Systems

Trigonal System

The trigonal crystal system is defined by the presence of at least one threefold rotation axis. It comprises 5 point groups and is associated with 7 space groups. Notably, these space groups are distributed across both the rhombohedral (7 groups) and hexagonal (18 groups) lattice systems, making it unique among crystal systems.

The 5 point groups within the trigonal system exhibit varying symmetries, from simple threefold rotation to more complex combinations involving mirror planes and inversion centers. These symmetries dictate the arrangement of atoms and the resulting macroscopic crystal forms.

Trigonal Crystal System Overview
Point Group Type	Intl Notation	Schoenflies Notation	Space Groups (Hexagonal Setting)	Space Groups (Rhombohedral Setting)	Examples
Trigonal pyramidal	3	C₃	P3, P3₁, P3₂	R3	Carlinite, Jarosite
Rhombohedral	3	C_3i (S₆)	P3	R3	Dolomite, Ilmenite
Trigonal trapezohedral	32	D₃	P312, P321, P3₁12, P3₁21, P3₂12, P3₂21	R32	Quartz, Cinnabar
Ditrigonal pyramidal	3m	C_3v	P3m1, P31m, P3c1, P31c	R3m, R3c	Tourmaline, Alunite
Ditrigonal scalenohedral	3m	D_3d	P31m, P31c, P3m1, P3c1	R3m, R3c	Corundum, Calcite

Hexagonal System

The hexagonal crystal system is characterized by the presence of a single sixfold rotation axis. It includes 7 point groups and is exclusively associated with the hexagonal lattice system, comprising 27 distinct space groups. This system represents a higher degree of symmetry compared to the trigonal system.

The 7 point groups in the hexagonal system exhibit symmetries based on a sixfold rotation axis, often combined with other symmetry elements like mirror planes or glide planes. These structures are fundamental in understanding materials with high rotational symmetry.

Hexagonal Crystal System Overview
Point Group Type	Intl Notation	Schoenflies Notation	Space Groups	Examples
Hexagonal pyramidal	6	C₆	P6, P6₁, P6₅, P6₂, P6₄, P6₃	Nepheline, Cancrinite
Trigonal dipyramidal	6	C_3h	P6	Cesanite, Laurelite
Hexagonal dipyramidal	6/m	C_6h	P6/m, P6₃/m	Apatite, Vanadinite
Hexagonal trapezohedral	622	D₆	P622, P6₁22, P6₅22, P6₂22, P6₄22, P6₃22	Kalsilite, Quartz (beta)
Dihexagonal pyramidal	6mm	C_6v	P6mm, P6cc, P6₃cm, P6₃mc	Greenockite, Wurtzite
Ditrigonal dipyramidal	6m2	D_3h	P6m2, P6c2, P62m, P62c	Benitoite
Dihexagonal dipyramidal	6/mmm	D_6h	P6/mmm, P6/mcc, P6₃/mcm, P6₃/mmc	Beryl

Atomic Packing

Hexagonal Close-Packed (HCP)

Hexagonal close-packed (HCP) is one of the most efficient ways to pack identical spheres, achieving a packing density of approximately 74%. Unlike the face-centered cubic (FCC) structure, HCP is not a Bravais lattice. It can be constructed from the hexagonal Bravais lattice by associating a two-atom motif with each lattice point, resulting in a non-primitive unit cell.

Symmetry and Properties

The HCP structure, while highly dense, lacks inversion symmetry. This characteristic is crucial as it enables materials exhibiting this packing to possess properties such as piezoelectricity and pyroelectricity, which are absent in centrosymmetric crystal structures.

Multi-Element Structures

Wurtzite Structure

The Wurtzite structure (Strukturbericht designation B4, Pearson symbol hP4) is a common arrangement for binary compounds. It is based on the hexagonal Bravais lattice with a two-atom motif. Each atom is tetrahedrally coordinated. The space group is P6₃mc (No. 186). This structure is non-centrosymmetric, leading to piezoelectric properties.

Numerous compounds adopt the Wurtzite structure, particularly semiconductors and related materials:

Zinc Sulfide (ZnS)
Zinc Oxide (ZnO)
Cadmium Sulfide (CdS)
Cadmium Selenide (CdSe)
Silicon Carbide (α-SiC)
Gallium Nitride (GaN)
Aluminum Nitride (AlN)
Boron Nitride (w-BN)

Nickel Arsenide Structure

The Nickel Arsenide (NiAs) structure involves two interpenetrating sublattices: a primitive hexagonal Ni sublattice and a hexagonal close-packed As sublattice. Each Ni atom is octahedrally coordinated by six As atoms, while each As atom is trigonal prismatically coordinated by six Ni atoms. This structure is typical for transition metal chalcogenides, arsenides, and antimonides.

Compounds adopting the NiAs structure include:

Achavalite (FeSe)
Breithauptite (NiSb)
Freboldite (CoSe)
Kotulskite (Pd(Te,Bi))
Langistite ((Co,Ni)As)
Nickeline (NiAs)
Sobolevskite (Pd(Bi,Te))
Sudburyite ((Pd,Ni)Sb)

Two-Dimensional Lattices

The Single 2D Hexagonal Lattice

In two dimensions, the hexagonal crystal family is represented by a single Bravais lattice: the hexagonal lattice. This lattice is characterized by a unit cell with two equal axes at a 120° angle, forming a perfectly symmetrical hexagonal arrangement of points in a plane.

Further Exploration

Related Concepts

Understanding the hexagonal crystal family naturally leads to exploring related crystallographic concepts. These include the fundamental principles of close-packing, the broader classification of crystal structures, and the specific characteristics of Bravais lattices and crystallographic point groups.

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References

Inorganic Chemistry by Duward Shriver and Peter Atkins, 3rd Edition, W.H. Freeman and Company, 1999, pp.47,48.

http://www.mindat.org/min-2901.html Mindat.org

A full list of references for this article are available at the Hexagonal crystal family Wikipedia page

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This content has been generated by an Artificial Intelligence, drawing upon established scientific literature and data, primarily sourced from Wikipedia. It is intended for educational and informational purposes at a postgraduate level. While efforts have been made to ensure accuracy and clarity, this material should not be considered a substitute for rigorous academic study or consultation with expert crystallographers or materials scientists.

This is not professional advice. The information provided herein is for academic enrichment and does not constitute expert consultation in crystallography, materials science, or any related field. Always refer to authoritative textbooks, peer-reviewed journals, and consult with qualified professionals for specific research or application needs. Never disregard professional advice or delay in seeking it because of information presented on this platform.

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Crystalline Architectures

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Lattice Systems 🔗

🌐 Defining the Framework

📐 Hexagonal Lattice

🧊 Rhombohedral Lattice

Crystal Systems ⚛️

3️⃣ Trigonal System

6️⃣ Hexagonal System

Atomic Packing 📦

✨ Hexagonal Close-Packed (HCP)

⚖️ Symmetry and Properties

Multi-Element Structures 🧪

💡 Wurtzite Structure

⚙️ Nickel Arsenide Structure

Two-Dimensional Lattices 📏

2️⃣ The Single 2D Hexagonal Lattice

Further Exploration 📚

➡️ Related Concepts

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Lattice Systems

Defining the Framework

Hexagonal Lattice

Rhombohedral Lattice

Crystal Systems

Trigonal System

Hexagonal System

Atomic Packing

Hexagonal Close-Packed (HCP)

Symmetry and Properties

Multi-Element Structures

Wurtzite Structure

Nickel Arsenide Structure

Two-Dimensional Lattices

The Single 2D Hexagonal Lattice

Further Exploration

Related Concepts

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