What defines the cubic crystal system in crystallography?

The cubic crystal system, also known as the isometric system, is defined by its unit cell being in the shape of a cube. This is one of the most common and simplest shapes observed in crystals and minerals.

What are the three main varieties of cubic crystals?

The three main varieties of cubic crystals are primitive cubic (cP), body-centered cubic (cI), and face-centered cubic (cF).

What are the three Bravais lattices found in the cubic crystal system, and what are their abbreviations?

The three Bravais lattices in the cubic crystal system are primitive cubic (cP), body-centered cubic (cI), and face-centered cubic (cF).

Describe the arrangement of lattice points in a primitive cubic (cP) Bravais lattice and its total lattice points per unit cell.

A primitive cubic (cP) lattice consists of one lattice point at each corner of the cube. Since each corner atom is shared by eight adjacent cubes, the unit cell contains a total of one lattice point (1/8 contribution from each of the 8 corners).

Explain the structure of the body-centered cubic (cI) Bravais lattice and its net total of lattice points per unit cell.

The body-centered cubic (cI) lattice includes lattice points at each corner of the cube, plus an additional lattice point located at the exact center of the unit cell. This results in a net total of two lattice points per unit cell (1/8 from corners + 1 from the center).

Detail the composition of the face-centered cubic (cF) Bravais lattice in terms of lattice points per unit cell.

The face-centered cubic (cF) lattice has lattice points at each corner and also at the center of each of the six faces of the cube. Each corner point contributes 1/8, and each face-centered point contributes 1/2, leading to a total of four lattice points per unit cell (1/8 * 8 corners + 1/2 * 6 faces = 1 + 3 = 4).

What are the coordination number and atomic packing factor (APF) for the primitive cubic (cP) lattice?

The primitive cubic (cP) lattice has a coordination number of 6, meaning each atom has six nearest neighbors. Its atomic packing factor (APF) is approximately 0.524, indicating that about 52.4% of the unit cell volume is occupied by atoms.

What are the coordination number and atomic packing factor (APF) for the body-centered cubic (cI) lattice?

The body-centered cubic (cI) lattice has a coordination number of 8, meaning each atom is surrounded by eight nearest neighbors. Its atomic packing factor (APF) is approximately 0.680, meaning about 68.0% of the unit cell volume is occupied by atoms.

What are the coordination number and atomic packing factor (APF) for the face-centered cubic (cF) lattice?

The face-centered cubic (cF) lattice has a coordination number of 12, meaning each atom has twelve nearest neighbors. Its atomic packing factor (APF) is approximately 0.740, which is the maximum possible for spheres of equal size, meaning about 74.0% of the unit cell volume is occupied by atoms.

How do the atomic packing factors of the primitive cubic, body-centered cubic, and face-centered cubic lattices compare?

The atomic packing factors (APF) increase with the density of packing: primitive cubic (cP) has an APF of about 0.524, body-centered cubic (cI) has an APF of about 0.680, and face-centered cubic (cF) has the highest APF at about 0.740.

How is the face-centered cubic (cF) lattice related to the hexagonal close-packed (hcp) system?

The face-centered cubic (cF) lattice is closely related to the hexagonal close-packed (hcp) system. The primary difference lies in the relative stacking sequence of their layers, and the [111] plane of a face-centered cubic lattice forms a hexagonal grid.

What happens if one attempts to create a base-centered cubic lattice?

Attempting to create a base-centered cubic lattice, by adding lattice points to the center of each horizontal face, results in a structure that is equivalent to a simple tetragonal Bravais lattice.

How many cubic space groups are there in total?

There are a total of 36 distinct cubic space groups.

What are the main types of crystal classes within the isometric crystal system, and what are their general characteristics?

The isometric crystal system includes classes like Tetartoidal (enantiomorphic, tetrahedral symmetry), Diploidal (centrosymmetric, pyritohedral symmetry), Gyroidal (enantiomorphic, octahedral symmetry), Hextetrahedral (tetrahedral symmetry), and Hexoctahedral (highest symmetry, octahedral symmetry, centrosymmetric).

Which cubic crystal class is described as enantiomorphic and includes examples like Ullmannite and Sodium chlorate?

The Tetartoidal crystal class is described as enantiomorphic and includes examples such as Ullmannite and Sodium chlorate.

What is the common name for the crystal structure that includes Pyrite and has a specific space group designation?

The Diploidal crystal class, with a space group designation of Pm3m (or 23 in Hermann-Mauguin notation), includes Pyrite as an example.

Which crystal class is characterized by tetrahedral symmetry and includes examples like Sphalerite?

The Hextetrahedral crystal class, which exhibits tetrahedral symmetry, includes Sphalerite as an example.

What is the common name for the highest symmetry cubic crystal class, and what are some of its examples?

The Hexoctahedral crystal class represents the highest symmetry within the cubic system and is also known as the normal or holohedral class. Examples include Galena and Halite.

Why is the primitive cubic structure, with its low atomic packing factor, considered rare in nature?

The primitive cubic structure is rare in nature because, with its low atomic packing factor (around 0.524), it is less tightly packed compared to other cubic structures like BCC and FCC. Tightly packed arrangements are generally favored in solids due to atomic attraction, although specific bonding requirements can lead to less packed structures.

Provide examples of elements that commonly crystallize in body-centered cubic (bcc) and face-centered cubic (fcc) structures.

Elements commonly found in the body-centered cubic (bcc) structure include iron, chromium, tungsten, and niobium. Examples of elements in the face-centered cubic (fcc) structure are aluminum, copper, gold, and silver.

What is the diamond cubic structure, and which elements are known to exhibit it?

The diamond cubic structure is a significant cubic crystal structure that is not a Bravais lattice itself, as it contains multiple atoms within its primitive cell. Elements such as carbon (in its diamond form), silicon, germanium, and tin exhibit this structure.

How does the diamond cubic structure differ from a Bravais lattice?

The diamond cubic structure differs from a Bravais lattice because it contains multiple atoms within its primitive cell, whereas a Bravais lattice is defined by a single lattice point per primitive cell.

What is the A15 structure, and which element is mentioned as having it?

The A15 structure is another type of cubic elemental structure. Tungsten is mentioned as an element that exhibits the A15 structure.

How can compounds with multiple elements often have crystal structures based on the cubic system?

Multi-element compounds often form cubic crystal structures by having two or more interpenetrating sublattices, where each sublattice occupies the interstitial sites of the others. This arrangement allows for complex structures to be built within the cubic framework.

What is the alternative name for the 'interpenetrating primitive cubic' structure, and what is its Strukturbericht designation?

The 'interpenetrating primitive cubic' structure is also known as the caesium chloride (CsCl) or B2 structure. Its Strukturbericht designation is B2.

How does the caesium chloride (CsCl) structure differ from a body-centered cubic (bcc) structure, despite their similar atomic arrangements?

While the CsCl structure appears similar to bcc, it differs because it has a basis composed of two different atomic species. In a true bcc structure, there would be translational symmetry along the [111] direction, which results in a change of species in the CsCl structure.

What is the coordination number for ions in the caesium chloride structure?

In the caesium chloride structure, each ion is at the center of a cube formed by ions of the opposite kind, resulting in a coordination number of eight for each ion.

What is the space group for the caesium chloride (CsCl) structure?

The space group of the caesium chloride (CsCl) structure is designated as Pm3m in Hermann-Mauguin notation, and it is numbered 221 in the International Tables for Crystallography.

What is the radius ratio guideline for forming the rock-salt structure?

The rock-salt structure is more likely to form when the cation is somewhat smaller than the anion, specifically within a cation/anion radius ratio range of approximately 0.414 to 0.732.

What is the space group and Strukturbericht designation for the rock-salt (halite) structure?

The rock-salt, or halite (sodium chloride) structure, has the space group Fm3m in Hermann-Mauguin notation (number 225) and the Strukturbericht designation B1.

How is the rock-salt structure described in terms of interpenetrating lattices?

The rock-salt structure can be described as two interpenetrating face-centered cubic lattices, one for each type of atom, arranged in a three-dimensional checkerboard pattern. Alternatively, it can be viewed as one FCC lattice with the other atom type occupying all the octahedral voids.

What is the coordination number and geometry for atoms in the rock-salt structure?

In the rock-salt structure, each atom has six nearest neighbors of the opposite type, arranged in an octahedral geometry, similar to the vertices of a regular octahedron.

List some common examples of compounds that adopt the rock-salt structure.

Common examples of compounds with the rock-salt structure include sodium chloride itself, most other alkali halides, and many divalent metal oxides, sulfides, selenides, and tellurides.

How does the fluorite structure compare to the rock-salt structure in terms of space group and ion ratio?

The fluorite structure (AB2) shares the same Fm3m space group as the rock-salt structure, but it differs in its ion ratio, having a 1:2 ratio of ions instead of the 1:1 ratio found in rock-salt.

What are the Wyckoff positions for the fluorite and rock-salt structures?

In the fluorite structure, the ions occupy Wyckoff positions 4a and 8c, whereas in the rock-salt structure, the positions are 4a and 4b.

What is the space group and Strukturbericht designation for the Zincblende structure?

The Zincblende structure has the space group F43m in Hermann-Mauguin notation (number 216) and is designated as B3 in the Strukturbericht system.

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

In the zincblende structure, each atom is tetrahedrally coordinated, meaning its nearest neighbors consist of four atoms of the opposite type, positioned at the vertices of a regular tetrahedron.

How does the arrangement of atoms in the zincblende structure relate to the diamond cubic structure?

The arrangement of atoms in the zincblende structure is fundamentally the same as the diamond cubic structure, but with alternating types of atoms occupying the different lattice sites.

Provide examples of compounds that commonly exhibit the zincblende structure, including semiconductor materials.

Examples of compounds with the zincblende structure include zinc sulfide (sphalerite), and many compound semiconductors such as gallium arsenide and cadmium telluride. Other binary compounds also adopt this structure.

What is the common name for the group of compounds like Gallium arsenide and Cadmium telluride?

Gallium arsenide and cadmium telluride are examples of compounds belonging to the III-V family, which often exhibit the zincblende structure.

What is the Heusler structure, its space group, and what types of compounds typically adopt it?

The Heusler structure, based on Cu2MnAl, is common for ternary compounds involving transition metals. It has the space group Fm3m (No. 225) and the Strukturbericht designation L21.

Describe the iron monosilicide structure, including its space group and potential associated properties like chirality.

The iron monosilicide structure has the space group P213 (No. 198) and the Strukturbericht designation B20. It is notable for being a chiral structure, which can be associated with helimagnetic properties.

What is the symmetry of the Weaire-Phelan structure, and in what context is it often known in chemistry?

The Weaire-Phelan structure possesses Pm3n symmetry (No. 223). In chemistry, it is often recognized as a 'type I clathrate structure'.

Describe the pyrite crystal shown in the first image.

The first image displays a rock containing three crystals of pyrite (FeS2), which exhibit cubic symmetry in their natural facets, reflecting their primitive cubic crystal structure.

What does the second image illustrate regarding cubic crystal systems?

The second image provides a network model illustrating the primitive cubic system, showing the arrangement of points in a cubic lattice.

What do the unit cells depicted in the third image represent?

The third image shows a comparison between the primitive cubic and cubic close-packed (face-centered cubic) unit cells, illustrating their structural differences.

What does the image of the diamond cubic unit cell visualize?

The image of the diamond cubic unit cell visualizes its components, a single unit cell, and a lattice composed of three by three by three unit cells.

What does the image of the caesium chloride unit cell show?

The image of the caesium chloride unit cell depicts the arrangement of two different types of atoms, representing the structure of caesium chloride.

What does the graphic of cesium chloride illustrate about its structure?

The graphic of cesium chloride illustrates the interlocking simple cubic lattices of cesium and chlorine atoms, showing how they combine to form an arrangement that resembles a body-centered cubic structure.

What does the image of the rock-salt crystal structure depict regarding coordination?

The image of the rock-salt crystal structure shows that each atom has six nearest neighbors, arranged with octahedral geometry.

What does the image of the zincblende unit cell depict?

The image of the zincblende unit cell illustrates the arrangement of atoms within this specific crystal structure.

Cubic Crystal System Wiki: The Cubic Crystal System: A Foundation in Geometric Crystallography

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Introduction

The Cubic Form

In the field of crystallography, the cubic, or isometric, crystal system is defined by a unit cell that assumes the shape of a cube. This geometric configuration represents one of the most prevalent and straightforward forms observed in crystals and minerals, serving as a fundamental building block in understanding solid-state structures.

Simplicity and Symmetry

The cubic system is characterized by its high degree of symmetry, possessing three equivalent crystallographic axes that are mutually perpendicular and of equal length. This inherent symmetry simplifies the description and analysis of crystal structures, making it a cornerstone in materials science and solid-state physics.

Ubiquity in Nature

Many common minerals and elements exhibit cubic structures due to the energetic favorability of close-packed arrangements. Understanding these structures is crucial for predicting and explaining the physical properties of materials, from metals to ionic compounds.

Bravais Lattices

Lattice Points

The cubic crystal system encompasses three distinct Bravais lattices, which are unique arrangements of points in space that describe the translational symmetry of a crystal. These are:

Primitive Cubic (cP): Features lattice points only at the corners of the cube.
Body-Centered Cubic (cI): Includes lattice points at the corners and one in the exact center of the cube.
Face-Centered Cubic (cF): Possesses lattice points at the corners and in the center of each face of the cube.

While the unit cells are cubic, the primitive unit cells for these lattices may not always be cubic themselves.

Lattice Characteristics

These lattices differ in their atomic packing efficiency and the number of lattice points per unit cell:

cP: Coordination Number (CN) = 6, Atomic Packing Factor (APF) ≈ 0.524.
cI: Coordination Number (CN) = 8, Atomic Packing Factor (APF) ≈ 0.680.
cF: Coordination Number (CN) = 12, Atomic Packing Factor (APF) ≈ 0.740.

The face-centered cubic (cF) structure represents the densest possible packing of spheres.

Bravais Lattice Table

The following table summarizes the key characteristics of the cubic Bravais lattices:

Bravais lattice	Primitive cubic	Body-centered cubic	Face-centered cubic
Pearson symbol	cP	cI	cF
Unit Cell (Conceptual)	⬢	⬣	⬡
Lattice Points per Unit Cell	1	2	4
Coordination Number (CN)	6	8	12
Atomic Packing Factor (APF)	~0.524	~0.680	~0.740

Crystal Classes

Symmetry Groups

The cubic crystal system is further classified into 36 distinct space groups, which describe the full symmetry of the crystal structure, including translational symmetry. These groups are organized into several crystal classes based on their point group symmetry. The highest symmetry class, the hexoctahedral class, possesses the full symmetry of a cube.

Classification Framework

Crystallographic point groups define the rotational and reflectional symmetries of a crystal's lattice. In the cubic system, these range from lower symmetry groups like tetrahedral (T) to the highest octahedral (O_h) symmetry. Each point group corresponds to specific crystal classes and is associated with a set of space groups.

Crystal Classes Overview

The following table outlines the major crystal classes within the cubic system, their associated point group notations, and examples:

No.	Point group					Type	Example	Space groups
No.	Name	Schön.	Intl	Orb.	Cox.	Type	Example	Primitive	Face-centered	Body-centered
195–197	Tetartoidal	T	23	332	[3,3]⁺	enantiomorphic	Ullmannite, Sodium chlorate	P23	F23	I23
198–199	Tetartoidal	T	23	332	[3,3]⁺	enantiomorphic	Ullmannite, Sodium chlorate	P2₁3		I2₁3
200–204	Diploidal	T_h	2/m3 (m3)	3*2	[3⁺,4]	centrosymmetric	Pyrite	Pm3, Pn3	Fm3, Fd3	I3
205–206	Diploidal	T_h	2/m3 (m3)	3*2	[3⁺,4]	centrosymmetric	Pyrite	P3m		Ia3
207–211	Gyroidal	O	432	432	[3,4]⁺	enantiomorphic	Petzite	P432, P4₂32	F432, F4₁32	I432
212–214	Gyroidal	O	432	432	[3,4]⁺	enantiomorphic	Petzite	P4₃32, P4₁32		I4₁32
215–217	Hextetrahedral	T_d	43m	*332	[3,3]		F43m	I43m
218–220	Hextetrahedral	T_d	43m	*332	[3,3]	P43m		I43d
221–230	Hexoctahedral	O_h	4/m32/m (m3m)	*432	[3,4]	centrosymmetric	Galena, Halite	Pm3m, Pn3n, Pm3n	Fm3m, Fm3c, Fd3m, Fd3c	Im3m, Ia3d

Single Element Structures

Packing Efficiency

The arrangement of atoms in elemental solids is often dictated by the principle of maximizing packing efficiency. Structures with higher atomic packing factors (APFs) are generally more stable.

Primitive Cubic (cP): Rare due to its low APF (~0.524). Found in Polonium.
Body-Centered Cubic (cI): Common, with an APF of ~0.680. Examples include Iron, Chromium, Tungsten, and Niobium.
Face-Centered Cubic (cF): Also very common and represents the densest packing (APF ~0.740). Examples include Aluminum, Copper, Gold, and Silver.

Diamond Cubic Structure

A notable cubic structure is the diamond cubic arrangement, which is not a simple Bravais lattice but rather a lattice with a basis. This structure is found in elements like Carbon (diamond allotrope), Silicon, Germanium, and Tin. It is characterized by tetrahedral bonding, resulting from specific orbital hybridization.

Multi-Element Structures

Interpenetrating Lattices

Many compounds, particularly binary and ternary compounds, adopt crystal structures based on the cubic system. These structures can often be visualized as two or more interpenetrating Bravais lattices, where atoms occupy the interstitial sites of each other.

Common Cubic Structures

Several fundamental structures are prevalent in multi-element compounds:

Caesium Chloride (CsCl) Structure: A simple cubic arrangement of one ion type with the other ion type occupying the cubic interstitial site. It exhibits 8-fold coordination.
Rock-Salt (NaCl) Structure: Two interpenetrating face-centered cubic lattices, resulting in octahedral coordination (6-fold) for each atom.
Zincblende (ZnS) Structure: Similar to the rock-salt structure but with tetrahedral coordination (4-fold). It is derived from the diamond cubic structure by alternating atom types.

Structure Examples

The following tables list examples of compounds exhibiting these and other cubic structures:

Caesium Chloride Structure (B2)

Characterized by 8-fold coordination. Space group: Pm3m (No. 221).

Alkali Metal Halides & Hydrides
Element	Hydrides	Fluorides	Chlorides	Bromides	Iodides
Lithium	LiH	LiF	LiCl	LiBr	LiI
Sodium	NaH	NaF	NaCl	NaBr	NaI
Potassium	KH	KF	KCl	KBr	KI
Rubidium	RbH	RbF	RbCl	RbBr	RbI
Caesium	CsH	CsF	(CsCl structure)

Alkaline Earth Metal Chalcogenides
Element	Oxides	Sulfides	Selenides	Tellurides	Polonides
Magnesium	MgO	MgS	MgSe	MgTe	MgPo
Calcium	CaO	CaS	CaSe	CaTe	CaPo
Strontium	SrO	SrS	SrSe	SrTe	SrPo
Barium	BaO	BaS	BaSe	BaTe	BaPo

Rock-Salt Structure (B1)

Features 6-fold octahedral coordination. Space group: Fm3m (No. 225).

Alkali Metal Halides & Hydrides (Common)
Element	Hydrides	Fluorides	Chlorides	Bromides	Iodides
Lithium	LiH	LiF	LiCl	LiBr	LiI
Sodium	NaH	NaF	NaCl	NaBr	NaI
Potassium	KH	KF	KCl	KBr	KI
Rubidium	RbH	RbF	RbCl	RbBr	RbI
Caesium	CsH	CsF	(CsCl structure)

Transition Metal Monoxides
Element	Oxides
Titanium	TiO
Vanadium	VO
Chromium	CrO
Manganese	MnO
Iron	FeO
Cobalt	CoO
Nickel	NiO
Cadmium	CdO

Zincblende Structure (B3)

Characterized by 4-fold tetrahedral coordination. Space group: F43m (No. 216).

II-VI Compounds
Element	Sulfides	Selenides	Tellurides	Polonides
Beryllium	BeS	BeSe	BeTe	BePo
Zinc	ZnS	ZnSe	ZnTe	ZnPo
Cadmium	CdS	CdSe	CdTe	CdPo
Mercury	HgS	HgSe	HgTe	-

III-V Compounds
Element	Nitrides	Phosphides	Arsenides	Antimonides
Boron	BN*	BP	BAs	BSb
Aluminium	AlN*	AlP	AlAs	AlSb
Gallium	GaN*	GaP	GaAs	GaSb
Indium	InN*	InP	InAs	InSb

*Some nitrides also exist in the hexagonal wurtzite structure.

Heusler Structure

Common for ternary compounds, often involving transition metals. Space group: Fm3m (No. 225), Strukturbericht designation L2₁.

Heusler Compounds (e.g., Co₂MnSi)
Example Compound	Structure Type
Cu₂MnAl	Heusler
Co₂FeSi	Heusler
Ni₂MnGa	Heusler

Iron Monosilicide Structure (B20)

A chiral structure found in transition metal silicides and germanides. Space group: P2₁3 (No. 198).

Transition Metal Silicides/Germanides
Element	Silicides	Germanides
Manganese	MnSi	MnGe
Iron	FeSi	FeGe
Cobalt	CoSi	CoGe

Weaire–Phelan Structure

A complex structure related to clathrates. Space group: Pm3n (No. 223).

Clathrate Hydrate Structures
Molecule	Structure Type
Water (with Methane)	Type I Clathrate (Weaire–Phelan)
Water (with Propane)	Type I Clathrate (Weaire–Phelan)

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References

Crystallography and Minerals Arranged by Crystal Form, Webmineral

Birkbeck College, University of London

The Zincblende (B3) Structure. Naval Research Laboratory, U.S.

A full list of references for this article are available at the Cubic crystal system Wikipedia page

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This content has been generated by an Artificial Intelligence model and is intended for educational and informational purposes only. The information presented is derived from publicly available data, primarily Wikipedia, and may not represent the most current or complete understanding of the subject matter.

This is not a substitute for expert consultation. The details provided herein should not be considered professional advice in crystallography, materials science, or any related field. Always consult with qualified experts and refer to authoritative sources for critical applications or decision-making.

The creators of this page are not liable for any errors, omissions, or consequences arising from the use of this information.

The Cubic Crystal System

💡 Dive in with Flashcard Learning! 💡

Introduction ℹ️

⬢ The Cubic Form

💎 Simplicity and Symmetry

🌐 Ubiquity in Nature

Bravais Lattices 📊

🔗 Lattice Points

🔢 Lattice Characteristics

🗂️ Bravais Lattice Table

Crystal Classes 🗂️

⚖️ Symmetry Groups

🔬 Classification Framework

📋 Crystal Classes Overview

Single Element Structures ⚛️

⚡ Packing Efficiency

💎 Diamond Cubic Structure

Multi-Element Structures 🔗

🧪 Interpenetrating Lattices

🏠 Common Cubic Structures

🗄️ Structure Examples

Caesium Chloride Structure (B2)

Rock-Salt Structure (B1)

Zincblende Structure (B3)

Heusler Structure

Iron Monosilicide Structure (B20)

Weaire–Phelan Structure

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Disclaimer ⚠️

📜 Important Notice

Dive in with Flashcard Learning!

Introduction

The Cubic Form

Simplicity and Symmetry

Ubiquity in Nature

Bravais Lattices

Lattice Points

Lattice Characteristics

Bravais Lattice Table

Crystal Classes

Symmetry Groups

Classification Framework

Crystal Classes Overview

Single Element Structures

Packing Efficiency

Diamond Cubic Structure

Multi-Element Structures

Interpenetrating Lattices

Common Cubic Structures

Structure Examples

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References

Feedback & Support

Disclaimer

Important Notice