What defines a crystal or crystalline solid at a microscopic level?

A crystal or crystalline solid is defined as a solid material where its constituent parts, such as atoms, molecules, or ions, are arranged in a highly ordered microscopic structure. This ordered arrangement forms a crystal lattice that extends uniformly in all directions.

Beyond the microscopic structure, how are macroscopic single crystals often identifiable?

Macroscopic single crystals are often identifiable by their distinct geometrical shape, which typically consists of flat faces with specific, characteristic orientations. These external shapes are a macroscopic manifestation of the underlying ordered microscopic structure.

What is the scientific discipline dedicated to the study of crystals and their formation?

The scientific study of crystals and how they form is known as crystallography. The process by which crystals grow is called crystallization or solidification.

What is the etymological origin of the word 'crystal'?

The word 'crystal' originates from the Ancient Greek word 'krustallos', which meant both 'ice' and 'rock crystal'. This term itself derived from 'kruos', meaning 'icy cold' or 'frost'.

Can you provide examples of common substances that exist as crystals?

Common examples of large crystals include snowflakes, diamonds, and table salt (sodium chloride). These are all solids where the atoms or molecules are arranged in a highly ordered, repeating pattern.

How do most inorganic solids, like metals and rocks, differ from ideal single crystals?

Most inorganic solids are not single crystals but are instead polycrystals. A polycrystal is composed of many microscopic crystals, called crystallites or grains, that are fused together into a single solid mass. While each crystallite has an ordered internal structure, the overall structure of the polycrystal lacks a single, continuous periodic arrangement due to the boundaries between these grains.

What are amorphous solids, and how do they contrast with crystalline solids?

Amorphous solids, such as glass, wax, and many plastics, are distinct from crystalline solids because they lack any periodic arrangement of atoms, even at the microscopic level. A key difference in their formation is that the process of forming a glass (an amorphous solid) does not release the latent heat of fusion, whereas the formation of a crystal does.

Are products like 'crystal glass' or 'lead crystal' actual crystals?

Despite their names, products like lead crystal and crystal glass are not true crystals. They are types of glass, which are classified as amorphous solids because their atomic structure lacks the ordered, repeating pattern characteristic of crystalline materials.

What is the microscopic definition of a crystal structure?

Scientifically, a crystal structure refers to the periodic arrangement of atoms within a crystal. A crystal is defined as a solid where atoms form this repeating, ordered pattern, although quasicrystals represent an exception to strict periodicity.

How does the formation of a polycrystalline structure, like in ice, differ from forming an amorphous solid like glass?

When liquid water freezes, it often forms a polycrystalline structure where small ice crystals (crystallites) grow and fuse together. In contrast, amorphous solids like glass lack this microscopic periodic arrangement entirely. A key thermodynamic difference is that crystal formation releases latent heat of fusion, while glass formation does not.

What is a 'unit cell' in the context of crystal structure?

A unit cell is a fundamental concept in describing a crystal structure. It is a small, imaginary box that contains one or more atoms arranged in a specific spatial configuration. These unit cells are then conceptually stacked in three-dimensional space to build up the entire crystal.

What limits the possible symmetries found in crystals?

The symmetry of a crystal is constrained by the requirement that its unit cells must stack perfectly without any gaps. This geometric constraint leads to a finite number of possible crystal symmetries, known as crystallographic space groups.

How many crystallographic space groups exist, and how are they categorized?

There are 219 possible crystallographic space groups, which describe the unique symmetries a crystal can possess. These groups are further organized into 7 broader categories called crystal systems, such as the cubic crystal system and the hexagonal crystal system.

What are the defining characteristics of the cubic crystal system?

Crystals belonging to the cubic crystal system exhibit a high degree of symmetry. Their unit cells can form cubes or rectangular boxes, and this symmetry is reflected in their macroscopic shapes, as seen in examples like halite (table salt).

How are the macroscopic shapes of crystals, like flat faces, related to their microscopic structure?

The flat faces, or facets, of a euhedral (well-formed) crystal are oriented in specific ways relative to the underlying atomic arrangement. These faces correspond to crystallographic planes with relatively low Miller indices, which are surfaces with lower surface energy, making them more stable and prone to growth.

What is a 'crystallographic form' in relation to a crystal's faces?

A crystallographic form refers to a set of crystal faces that are related to each other through the symmetries of the crystal's structure. For instance, the six faces of a cube or the eight faces of an octahedron can each represent a single crystallographic form.

How are crystallographic forms denoted?

A crystallographic form is typically described using Miller indices enclosed in curly braces, such as {111}. This notation implies all faces related by the crystal's symmetry, even if only one face's indices are explicitly shown.

What is the difference between closed and open crystallographic forms?

A closed crystallographic form is one that can completely enclose a volume of space, while an open form cannot. All forms within the isometric crystal system are closed, whereas forms in the monoclinic and triclinic systems are open.

What factors influence a crystal's external shape, known as its habit?

A crystal's habit, or its visible external shape, is determined by a combination of factors: the underlying crystal structure (which dictates possible facet orientations), the specific chemical bonding within the crystal (which can favor certain facets), and the environmental conditions under which the crystal formed.

Where are the largest concentrations of crystals found on Earth by volume and weight?

The largest concentrations of crystals on Earth, by both volume and weight, are found within its solid bedrock. These crystalline structures form the basis of many rocks.

What was the largest known naturally occurring crystal as of 1999, and where was it found?

As of 1999, the world's largest known naturally occurring crystal was a crystal of beryl located in Malakialina, Madagascar. It measured 18 meters (59 ft) in length, 3.5 meters (11 ft) in diameter, and weighed an estimated 380,000 kg (840,000 lb).

How do igneous rocks develop their crystalline structure?

Igneous rocks form from molten magma. Their degree of crystallization depends on how quickly they cool and the pressures involved. Rocks like granite, which cool very slowly under high pressure, are completely crystallized, while lavas that cool rapidly at the surface often contain some glassy or amorphous material.

What are metamorphic rocks, and how do they relate to crystal formation?

Metamorphic rocks, such as marbles, mica-schists, and quartzites, are formed when existing rocks like limestone, shale, or sandstone are subjected to high temperatures and pressures. These conditions cause recrystallization in the solid state, erasing original structures and forming new crystalline arrangements without the rock ever becoming molten.

How do crystals like halite and gypsum form naturally?

Crystals such as halite (table salt) and gypsum are often formed through a process called evaporation. They deposit from aqueous solutions, primarily due to the evaporation of water in arid climates, leading to the precipitation of these minerals.

What is the crystalline nature of common forms of water like snow and ice cubes?

Water-based ice, found in forms like snow, sea ice, and glaciers, exists as crystalline or polycrystalline structures. A single snowflake is typically a single crystal or a collection of crystals, while an ice cube is a polycrystal composed of many fused microscopic ice crystals.

Besides cooling liquid water, how else can ice crystals form?

Ice crystals, such as those found in frost or snowflakes, can also form directly from water vapor in the air. This occurs when the air becomes supersaturated with water vapor and the temperature drops below the dew point, causing the vapor to transition directly into ice without passing through a liquid state.

What is unique about water's expansion behavior during crystallization?

Water exhibits an unusual property when it crystallizes: it expands rather than contracts. This is why ice floats on liquid water and why frozen pipes can burst.

What are 'organigenic crystals', and can you provide examples?

Organigenic crystals are crystals produced by living organisms. Examples include calcite and aragonite crystals found in most mollusks, and hydroxylapatite crystals that form the structure of bones and teeth in vertebrates.

What is polymorphism in the context of solid materials?

Polymorphism is the ability of a solid substance to exist in more than one distinct crystal form. These different forms, also known as phases, can have different physical properties even though they are composed of the same atoms or molecules.

What is allotropy, and how does it relate to polymorphism?

Allotropy is a specific term for polymorphism that applies to pure chemical elements. For example, carbon exhibits allotropy, existing in crystalline forms like diamond and graphite, as well as amorphous forms.

How do different polymorphs of the same substance, like carbon, exhibit vastly different properties?

Polymorphs can have dramatically different properties due to their distinct atomic arrangements. For instance, diamond, a crystalline form of carbon, is the hardest known substance, while graphite, another crystalline form of carbon, is very soft and used as a lubricant. This difference arises from the different ways carbon atoms are bonded and arranged in each structure.

What is crystallization, and what factors influence its outcome?

Crystallization is the process by which a solid crystalline structure forms from a fluid or from dissolved substances. The final form of the crystal (e.g., single crystal vs. polycrystal, specific phase, defects) is determined by various conditions during solidification, including the fluid's chemistry, ambient pressure, temperature, and the rate at which these parameters change.

What are some industrial methods used to produce large single crystals?

Industrial techniques for growing large single crystals, often referred to as 'boules', include the Czochralski process and the Bridgman technique. Other methods like hydrothermal synthesis, sublimation, or solvent-based crystallization may also be employed depending on the substance's properties.

Can geological processes create exceptionally large crystals, and if so, where?

Yes, geological processes can create very large crystals. For example, selenite crystals exceeding 10 meters in length have been found in the Cave of the Crystals in Naica, Mexico.

What are crystallographic defects?

Crystallographic defects are imperfections or disruptions in the perfect, repeating pattern of atoms within a crystal lattice. While an ideal crystal has every atom in an exact, repeating arrangement, real crystals contain these defects, which can significantly influence their properties.

What are some common types of crystallographic defects?

Common types of crystallographic defects include vacancy defects (an empty spot where an atom should be), interstitial defects (an extra atom squeezed into a space where it doesn't fit), and dislocations (line defects that disrupt the atomic planes). Dislocations are particularly important as they affect the mechanical strength of materials.

How do impurities function as crystallographic defects, and what effects can they have?

An impurity is a type of crystallographic defect where the 'wrong' type of atom is present within the crystal lattice. These impurities can alter the crystal's properties, such as its color. For example, boron impurities can make diamonds appear blue, and the specific impurities present in corundum determine whether it is classified as ruby or sapphire.

What is crystal twinning?

Crystal twinning is a phenomenon where two or more crystals of the same substance grow together in a specific, symmetrical orientation relative to each other. It's considered intermediate between a crystallographic defect and a grain boundary, as the orientation changes across a twin boundary in a mirror-image fashion.

What is mosaicity in the context of crystals?

Mosaicity refers to a spread or misalignment in the orientation of crystal planes within a material. A mosaic crystal is composed of smaller crystalline units that are slightly tilted relative to one another.

What types of chemical bonds can hold solids together, and how do they relate to crystallinity?

Solids can be held together by metallic, ionic, covalent, or van der Waals bonds. While none of these bond types exclusively dictate whether a solid is crystalline or amorphous, there are general trends. For example, metals typically crystallize rapidly into polycrystalline forms, while ionic and covalently bonded solids are often crystalline.

Why are single-crystal metals sometimes preferred over polycrystalline metals, and how are they produced?

Single-crystal metals, such as specialized titanium alloys, are sometimes grown synthetically because they exhibit increased strength and higher melting points compared to their polycrystalline counterparts. For instance, fighter-jet turbine blades are often made from single crystals to withstand extreme conditions.

How can very slow cooling lead to large single crystals in metals?

Very slow cooling, such as experienced by iron meteorites in the vacuum of space, can allow atoms to arrange themselves into large, single crystals or very large crystalline grains over extended periods. This slow process facilitates the formation of ordered structures.

What are quasicrystals, and how do they differ from traditional crystals?

Quasicrystals are solids composed of atoms arranged in an ordered, but not strictly periodic, manner. Unlike traditional crystals, which have repeating patterns, quasicrystals can exhibit symmetries, such as five-fold symmetry, that are mathematically impossible in periodic structures.

How has the definition of 'crystal' evolved to include quasicrystals?

The International Union of Crystallography has broadened the definition of a crystal to encompass both ordinary periodic crystals and quasicrystals. The updated definition includes 'any solid having an essentially discrete diffraction diagram', recognizing the ordered, albeit non-periodic, nature of quasicrystals.

What unique symmetry property do quasicrystals possess that ordinary crystals do not?

Quasicrystals are notably known for their ability to display five-fold symmetry. This type of symmetry is forbidden in traditional periodic crystals due to the crystallographic restriction theorem, which states that only 2, 3, 4, and 6-fold rotational symmetries are compatible with a periodic lattice.

What are some special properties that crystals can exhibit due to anisotropy?

Crystals can possess unique electrical, optical, and mechanical properties stemming from their anisotropy, meaning their properties vary with direction. Examples include the piezoelectric effect (generating voltage under stress) and birefringence (splitting light into two rays), which are typically not found in isotropic materials like glass or polycrystals.

What is anisotropy in a crystal, and how does it relate to its properties?

Anisotropy in a crystal refers to the lack of rotational symmetry in its atomic arrangement, causing its physical properties to differ depending on the direction. For instance, graphite is mechanically much stronger along the plane of its sheets than perpendicular to them due to its anisotropic structure.

Can non-crystalline materials like glass or polycrystals exhibit anisotropy?

While anisotropy is a natural property of most crystals, it can also be induced in glasses or polycrystals. This can happen through processes like mechanical working or applying stress, which can align their internal structures or create directional internal stresses, leading to anisotropic behavior like stress-induced birefringence.

What is crystallography, and what is a primary technique used within this science?

Crystallography is the scientific field dedicated to determining the crystal structure, or the precise arrangement of atoms, within a crystal. A widely employed technique in crystallography is X-ray diffraction, which uses the scattering of X-rays to deduce the atomic arrangement.

Where are known crystal structures cataloged?

Extensive collections of known crystal structures are maintained and stored in specialized crystallographic databases, making this information accessible for research and reference.

What does the image caption 'Crystals of amethyst quartz' describe?

The image caption indicates that the photograph displays crystals of amethyst, which is a variety of quartz. Amethyst is known for its characteristic purple color.

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The Essence of Crystals

Ordered Microstructure

A crystal, or crystalline solid, is a solid material characterized by a highly ordered microscopic structure. Its constituent atoms, molecules, or ions are arranged in a repeating pattern that extends in all three dimensions, forming a crystal lattice. This ordered arrangement is the defining scientific characteristic of a crystal.

Etymology and Forms

The term "crystal" originates from the Ancient Greek word "krustallos," meaning both "ice" and "rock crystal." While macroscopic crystals often exhibit distinct geometric shapes with flat faces, this external form is not a strict requirement; the defining feature is the internal atomic order.

Beyond Crystals: Polycrystals and Amorphous Solids

Not all solids are true crystals. Most inorganic solids, like metals and rocks, are polycrystalline, composed of many microscopic crystals (crystallites or grains) fused together. Solids lacking any periodic atomic arrangement are termed amorphous solids, such as glass or wax. The formation of crystals releases latent heat, a characteristic absent in the formation of amorphous solids.

Crystal Structure: The Atomic Blueprint

The Unit Cell

The fundamental building block of a crystal structure is the unit cell. This is a small, repeating three-dimensional box containing atoms arranged in a specific spatial pattern. By stacking these unit cells infinitely, the entire crystal lattice is constructed.

Symmetry and Systems

The requirement for perfect stacking without gaps imposes constraints on crystal symmetry. There are 219 possible crystallographic space groups, which are categorized into 7 distinct crystal systems. Notable examples include the cubic crystal system, exemplified by halite (table salt), and the hexagonal crystal system, characteristic of water ice.

Atomic Arrangement

The precise arrangement of atoms within the unit cell dictates the crystal's overall symmetry and properties. Understanding this microscopic order is the domain of crystallography, a field that utilizes techniques like X-ray diffraction to map these atomic positions.

Crystal Faces and Forms

Euhedral vs. Anhedral

Crystals can be classified by their external appearance. Euhedral crystals possess well-defined, flat faces, reflecting their underlying atomic structure. Anhedral crystals lack these distinct faces, typically occurring as grains within a polycrystalline solid.

Facet Formation

The flat faces of euhedral crystals are planes of relatively low Miller index, corresponding to orientations with lower surface energy. As a crystal grows, atoms preferentially attach to rougher surface areas, allowing stable, flat faces to enlarge and become dominant.

During crystal growth, new atoms readily attach to surface areas with higher energy and fewer bonds. This leads to rapid growth in these regions. Conversely, areas with lower surface energy and more stable bonding form smooth, flat faces. Eventually, the entire crystal surface may consist of these stable facets.

Crystallographic Forms

Crystallographic forms represent sets of crystal faces related by symmetry operations. These forms are described using Miller indices. For instance, the cubic and octahedral forms are closed forms capable of enclosing a volume, while other forms may be open. A crystal's overall habit is its visible external shape, determined by its structure, chemistry, and growth conditions.

Crystals in Nature

Rocks and Minerals

Crystals form the bedrock of our planet. They are found in vast crystalline rock masses formed through magmatic and metamorphic processes. Examples include granite, marble, and quartzite. Crystals also precipitate from fluids, forming veins and druses, or through evaporation in arid environments, yielding minerals like halite and gypsum.

Ice and Water

Water is a common substance that readily forms crystals. Snowflakes, sea ice, and glaciers are all manifestations of crystalline or polycrystalline ice. Ice crystals can form directly from water vapor or from freezing liquid water. Notably, water expands upon crystallization, a unique property.

Biological Crystallization

Many living organisms produce crystals. Mollusks create calcite and aragonite, while vertebrates form hydroxylapatite crystals in bones and teeth. These biological processes demonstrate nature's ability to control crystallization for structural and functional purposes.

Polymorphism and Allotropy

Multiple Forms

Polymorphism refers to a solid's ability to exist in multiple distinct crystal forms. For example, water ice exhibits various phases (Ice I_h, Ice I_c, Ice II, etc.). When this phenomenon occurs in pure chemical elements, it is called allotropy, as seen with the different crystalline forms of carbon: diamond and graphite.

Property Differences

These different crystalline structures, despite being composed of the same atoms, can possess vastly different physical properties. Diamond, for instance, is exceptionally hard, while graphite is soft and used as a lubricant. Similarly, the various crystal forms of chocolate affect its texture and melting point.

Beyond Crystals

Substances can also exist in amorphous states, like fused silica versus crystalline quartz. The ability to transition between these forms, known as polymorphism, is crucial in materials science, influencing properties and applications, such as the heat treatment of steel.

The Process of Crystallization

From Fluid to Solid

Crystallization is the process by which a crystalline structure forms from a fluid (liquid or gas) or from dissolved materials. The final form of the solid—whether a single crystal, a polycrystal, or amorphous material—is dictated by factors such as fluid chemistry, pressure, temperature, and the rate of change of these parameters.

Industrial Crystal Production

Techniques like the Czochralski and Bridgman methods are employed to grow large, high-quality single crystals, often referred to as boules. These are essential for applications in electronics and optics. Other methods include hydrothermal synthesis and sublimation.

Natural and Biological Growth

Crystals form naturally through geological processes, such as the large selenite crystals found in Mexico's Cave of Crystals. Biological processes also yield crystals, as noted previously. Conversely, some organisms utilize antifreeze proteins to prevent detrimental ice crystallization.

Imperfections: Defects, Impurities, and Twinning

Deviations from Perfection

An ideal crystal possesses a perfectly repeating atomic pattern. However, real crystals contain crystallographic defects—interruptions in this order. These include vacancy defects (missing atoms), interstitial defects (extra atoms), and dislocations, which significantly influence a material's mechanical properties.

The Role of Impurities

Impurities, where foreign atoms are present within the crystal lattice, dramatically affect properties. For example, trace amounts of boron impart a blue color to diamond, while chromium ions create the red hue in ruby. In semiconductors, controlled impurities (dopants) are fundamental to creating transistors and other electronic devices.

Twinning and Mosaicity

Twinning occurs when a crystal consists of two or more intergrown crystal individuals with specific, mirror-image orientations relative to each other. Mosaicity describes a crystal whose internal structure is composed of smaller crystalline domains that are slightly misaligned.

Chemical Bonds in Crystals

Metallic Bonding

Metals typically form polycrystalline solids due to rapid crystallization. However, single-crystal metals can be grown for enhanced properties, such as in jet engine turbines. The slow cooling of meteorites can result in large single crystals with unique patterns.

Ionic Bonding

Ionic compounds, like salts (e.g., sodium chloride), often form brittle crystals that cleave easily. They are typically crystalline or polycrystalline. Hard, gemstone-quality crystals like ruby and synthetic sapphire are examples of hard, covalently bonded oxides.

Covalent and Van der Waals Bonds

Covalently bonded solids, such as diamond and quartz, are known for their hardness and rigidity. Crystals held together by weaker van der Waals forces, like molecular solids (e.g., ice, dry ice, certain fats), tend to be softer and more easily deformed.

Quasicrystals: Order Without Periodicity

Unique Symmetry

Quasicrystals represent a fascinating class of solids where atoms are ordered but lack strict periodicity. Unlike conventional crystals, they can exhibit symmetries forbidden by the crystallographic restriction theorem, such as five-fold symmetry, as seen in materials like holmium-magnesium-zinc.

Discovery and Definition

First discovered in 1982, quasicrystals are relatively rare compared to periodic crystals. The International Union of Crystallography now defines a crystal broadly to include both periodic crystals and quasicrystals, recognizing any solid with a discrete diffraction pattern.

Anisotropy and Special Properties

Directional Dependence

The inherent anisotropy of crystal structures—the lack of rotational symmetry in atomic arrangements—can lead to unique properties not found in amorphous solids or typical polycrystals. These directional properties can include electrical conductivity, optical behavior, and mechanical response.

Notable Phenomena

Crystals can exhibit phenomena like the piezoelectric effect (generating voltage under stress) and birefringence (splitting light into two rays). Properties such as electrical permittivity and mechanical modulus can vary significantly depending on the direction within the crystal, as exemplified by the layered structure of graphite.

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References

The surface science of metal oxides, by Victor E. Henrich, P. A. Cox, page 28, google books link

Nucleation of Water: From Fundamental Science to Atmospheric and Additional Applications by Ari Laaksonen, Jussi Malila -- Elsevier 2022 Page 239--240

Encyclopedia of the Solar System by Tilman Spohn, Doris Breuer, Torrence V. Johnson -- Elsevier 2014 Page 632

Science for Conservators, Volume 3: Adhesives and Coatings by Museum and Galleries Commission -- Museum and Galleries Commission 2005 Page 57

A full list of references for this article are available at the Crystal Wikipedia page

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The Crystalline Universe

💡 Dive in with Flashcard Learning! 💡

The Essence of Crystals 💎

⚛️ Ordered Microstructure

🧊 Etymology and Forms

⚖️ Beyond Crystals: Polycrystals and Amorphous Solids

Crystal Structure: The Atomic Blueprint 🏗️

📦 The Unit Cell

📐 Symmetry and Systems

🔗 Atomic Arrangement

Crystal Faces and Forms ✨

🌟 Euhedral vs. Anhedral

💡 Facet Formation

🔢 Crystallographic Forms

Crystals in Nature 🌍

⛰️ Rocks and Minerals

💧 Ice and Water

🌱 Biological Crystallization

Polymorphism and Allotropy 🔄

🔠 Multiple Forms

⚖️ Property Differences

💡 Beyond Crystals

The Process of Crystallization 🌱

💧➡️💎 From Fluid to Solid

🏭 Industrial Crystal Production

❄️ Natural and Biological Growth

Imperfections: Defects, Impurities, and Twinning ⚠️

🧩 Deviations from Perfection

✨➡️🌈 The Role of Impurities

👯 Twinning and Mosaicity

Chemical Bonds in Crystals 🔗

⚡ Metallic Bonding

⚡↔️⚡ Ionic Bonding

🤝 Covalent and Van der Waals Bonds

Quasicrystals: Order Without Periodicity 🌌

💫 Unique Symmetry

🔬 Discovery and Definition

Anisotropy and Special Properties 💡

📐↔️⚡ Directional Dependence

💡 Notable Phenomena

Teacher's Corner 🧑‍🏫

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📜 Important Notice

Dive in with Flashcard Learning!

The Essence of Crystals

Ordered Microstructure

Etymology and Forms

Beyond Crystals: Polycrystals and Amorphous Solids

Crystal Structure: The Atomic Blueprint

The Unit Cell

Symmetry and Systems

Atomic Arrangement

Crystal Faces and Forms

Euhedral vs. Anhedral

Facet Formation

Crystallographic Forms

Crystals in Nature

Rocks and Minerals

Ice and Water

Biological Crystallization

Polymorphism and Allotropy

Multiple Forms

Property Differences

Beyond Crystals

The Process of Crystallization

From Fluid to Solid

Industrial Crystal Production

Natural and Biological Growth

Imperfections: Defects, Impurities, and Twinning

Deviations from Perfection

The Role of Impurities

Twinning and Mosaicity

Chemical Bonds in Crystals

Metallic Bonding

Ionic Bonding

Covalent and Van der Waals Bonds

Quasicrystals: Order Without Periodicity

Unique Symmetry

Discovery and Definition

Anisotropy and Special Properties

Directional Dependence

Notable Phenomena

Teacher's Corner

True or False?

Test Your Knowledge!

Gamer's Corner

Discover other topics to study!

References

Feedback & Support

Disclaimer

Important Notice