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Home , Cubic crystal system, Diamond cubic, Triclinic crystal system

ASSIGNMENT

PRESENTED BY :- SOLOMON JOHN TIRKEY 2020UGCS002 MOHIT RAJ 2020UGCS062 RADHA KUMARI 2020UGCS092 PALLAVI PUSHPAM 2020UGCS122

SUBMITTED TO:-

DR. RANJITH PRASAD

CRYSTAL STRUCTURE

CRYSTALLINE AND NON-CRYSTALLINE MATERIALS .

Introduction

We see a lot of materials around us on the earth. If we study them , we found few of them having regularity in their structure . These types of materials are crystalline materials or solids.

A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances; that is, long-range order exists, such that upon solidification, the atoms will position themselves in a repetitive three-dimensional pattern, in which each atom is bonded to its nearest-neighbor atoms.

X-ray diffraction photograph [or Laue photograph for a single of magnesium.

Crystalline solids have well-defined edges and faces, diffract x-rays, and tend to have sharp melting points.

What is meant by and why to study the structure of crystalline solids? Crystallography is the experimental science of determining the arrangement of atoms in the crystalline solids. The properties of some materials are directly related to their crystal structures. For example, pure and undeformed magnesium and , having one , are much more brittle (i.e., fracture at lower degrees of deformation) than pure and undeformed such as and that have yet another crystal structure.

Furthermore, significant property differences exist between crystalline and non-crystalline materials having the same composition. For example, non-crystalline ceramics and polymers normally are optically transparent; the same materials in crystalline (or semi-crystalline) forms tend to be opaque or, at best, translucent.

An important reason to have an understanding of interatomic bonding in solids is that, in some instances, the type of bond allows us to explain a material’s properties. For example, consider , which may exist as both graphite and . Whereas graphite is relatively soft and has a ―Greasy‖ feels to it, diamond is the hardest known material. This dramatic disparity in properties is directly attributable to a type of interatomic bonding found in graphite that does not exist in Diamond.

Thus by studying the crystal structure and bonding nature of different materials, we can investigate the reasons for the similar or dissimilar nature of the selected materials in terms of different properties or different parameters.

The crystal structure and symmetry of a material play a vital role in determining many of its physical properties, such as , electronic band structure, and optical transparency, etc.

Based on the atomic arrangement in a substance, solids can be broadly classified as either crystalline or non-crystalline. In a crystalline solid, all the atoms are arranged in a periodic manner in all three dimensions whereas in a non-crystalline solid the atomic arrangement is random or non periodic in nature. A crystalline solid can either be a single crystalline or a polycrystalline. In the case of single crystal the entire solid consists of only one crystal and hence, periodic arrangement of atoms continue throughout the entire material. A polycrystalline material is an aggregate of many small separated by well- defined grain boundaries and hence periodic arrangement of atoms is limited to small regions of the material called as grain boundaries as shown in Fig. The noncrystalline substances are also called amorphous substances materials. Single crystalline materials exhibit long range as well as short range periodicities while long range periodicity is absent in case of poly-crystalline materials and non-crystalline materials.

Crystallinity : Repeating or periodic array over large atomic distances. 3-D pattern in which each atom is bonded to its nearest neighbors.

Crystal structure: the manner in which atoms, ions, or molecules are spatially arranged.

Above , in the first and third pictures, we can see how a crystal actually looks , while in the second picture ,we can see the structure of a crystal and a which is known as a .

Unit cell: small repeating entity of the atomic structure. The basic building block of the crystal structure. It defines the entire crystal structure with the atom positions within.

• By repeating the pattern of the unit cell over and over in all directions, the entire crystal can be constructed.

To define the unit cell parameters or lattice parameters, first we define crystallographic axes. These axes are obtained by the intersection of the three non-coplanar faces of the unit cell. The angle between these faces or crystallographic axes are known as interfacial or interaxial angles. The angles between the axes Y and Z is α, between Z and X is β and between X and Y is γ. The translational vectors or primitives a, b, c of a unit cell along X, Y, Z axes and interaxial angles α, β, γ are called cell parameters. These cell parameters are shown in Fig. The cell parameters determine the actual size and shape of the unit cell.

Lattice: 3-D array of points coinciding with atom positions (center of spheres):

CRYSTALLINE MATERIALS

A crystalline material consists of primarily organized crystal structure. A crystal is: a solid composed of atoms, ions, or molecules arranged in a pattern that is repetitive in three-dimensions. Each crystal structure within a specific is defined by a unit cell. A unit cell is the smallest repeatable subsection of the crystal. In thinking about crystals, it is often convenient to ignore the actual atoms, ions, or molecules and to focus on the geometry of periodic arrays. The crystal is then represented as a lattice, that is, a three-dimensional array of points (lattice points), each which has identical surroundings.

fig:- Lattice points.

Each crystal lattice is defined by a crystal system. In three-dimensions, there are seven crystal systems: triclinic, monoclinic, orthorhombic, hexagonal, rhombohedral, tetragonal, and cubic. These collections of systems are called the Bravais lattices. One example of a crystalline material is . Iron has a Body Centered Cubic (BCC) unit cell: Fig:- BCC Unit Cell

It is grouped in the .

Crystal Systems: Possible Unit Cell Shapes

• Goal is to Quantitatively Describe : (a) Shape and Size of the Unit Cell (point symmetry)

(b) Location of the Lattice Points (translational symmetry)

For (a) to specify the Crystal System and the Lattice Parameters For (b) to define the ―Bravais‖ Lattice

Crystal Systems • Unit cells need to be able to ―Stack‖ them to fill all space. • This puts restrictions on Unit Cell Shapes • Work, Pentagons Don’t!

Different types of unit cell are possible, and they are classified based on their level of symmetry.

Symmetry

Symmetry is a set of mathematical rules that describe the shape of an object.

Do you know that there is only ONE object in the geometrical universe with perfect symmetry? It’s a SPHERE. Infinite planes of symmetry pass through its center, infinite rotational axes are present, and no matter how little or much you rotate it on any of its infinite number of axes, it appears the same!

Crystal:

By definition crystal is a periodic arrangement of repeating ―motifs‖( e.g. atoms, ions). The symmetry of a periodic pattern of repeated motifs is the total set of symmetry operations allowed by that pattern

• Let us apply a rotation of 90 degrees about the center (point) of the pattern which is thought to be indefinitely extended. This pattern will be unchanged as a result of this operation, i.e. the operation maps the pattern onto itself. • Also a linear shift (translation) of this pattern by a certain amount (e.g. by the length of a small square) results in that same pattern again. •The total set of such symmetry operations, applicable to the pattern is the pattern's symmetry, and is mathematically described by a so-called Space Group.

The Space Group of a Crystal describes the symmetry of that crystal, and as such it describes an important aspect of that crystal's internal structure.

Translational Symmetry

A Space Group includes two main types of symmetries (i.e. symmetry operations) (I) The Translational Symmetries, and

(II) The Point Symmetries Translations, i.e. executable shifting movements, proceeding along a straight line and on a certain specified distance, such that the operation does not result in any change of the shifted pattern.

Typically the translational symmetries are macroscopically not visible because the translation lengths are in the order of Å.

Point Symmetries

It is a macroscopically visible symmetry operation: after it has been applied to the crystal at least one point remains where it was !! These operations are :

•Reflection in a point (inversion) – Center of Symmetry! •Reflection in a plane or Mirror Symmetry ! •Rotation about an imaginary axis –Rotational Symmetry! •Rotation-and- after-it-inversion or Roto-inversion!

Center of Symmetry

Example: The item consists of two asymmetric faces: then every part of the item can also be found on the opposite side of some point (center of symmetry.) at the same distance. Reflection in a point or inversion.

Mirror Symmetry

Rotational Symmetry

A point around we rotate – symmetry axis

Figure looks the same n times in a 360° rotation. n-fold symmetry!! 6 6 / / / % / / / 99 two-fold symmetry Z S N three-fold symmetry four-fold symmetry six-fold symmetry One-fold symmetry = No symmetry!!

N-fold Roto-Inversion Symmetry

The object will be transformed into itself after the following two step operation (e.g. the 4-fold roto-inversion): • a rotation of 90 degrees along the axis; • followed by the inversion with respect to a point on the axis

Objects may have more than one kind of symmetry

The Sets of Basic Symmetry Elements for Crystals

• 1 –fold rotation (rotation through 360 degrees); symbol: none • 2 - fold rotation (rotation through 180 degrees); symbol: 2 • 3 - fold rotation (rotation through 120 degrees); symbol: 3 • 4- fold rotation (rotation through 90 degrees); symbol: 4 • 6- fold rotation (rotation through 60 degrees); symbol: 6

In the case of crystals the only above rotation axes can occur

Crystal Classes

With all these point symmetries (i.e. Rotation, Reflection, and Roto- inversion) combinations can be made, which themselves are again cover operations, and this results in a total of 32 unique possibilities.

Thus all crystals can be classified in 32 CRYSTAL SYMMETRY CLASSES according to their symmetry content, i.e. specific set of symmetry elements

For example: The highest symmetrical Cubic (Hexakis hedric) Class possess the following symmetry elements: 1. Three 4-fold rotation axes. 2. Four 3-fold rotation axes. 3. Six 2-fold rotation axes. 4. Three primary mirror planes. 5. Six secondary mirror planes. 6. Center of symmetry. The lowest symmetrical class Triclinic (Hemihedral) involves 1-fold rotation axis, thus no symmetry at all.

The Crystal Systems

In turn these symmetry classes, because some of them show similarities among each other, are divided among the different Crystal Systems. There are six Crystal System :

1. The CUBIC (also called Isometric system) 2. The TETRAGONAL system 3. The HEXAGONAL system 4. The ORTHORHOMBIC system 5. The MONOCLINIC system 6. The TRICLINIC system

Every Crystal System involves a number of Crystal Classes.

CRYSTALLOGRAPHIC AXES

For representing the type of distribution of lattice points in space, seven different coordinates. systems are required. These coordinate systems are called crystal systems. The crystal systems are named on the basis of geometrical shape and symmetry. The seven crystal systems are: (1) Cubic (2)Tetragonal (3) Orthorhombic (4) Monoclinic (5) Triclinic (6) Trigonal (or Rhombohedral) and (7)Hexagonal. Space lattices are classified according to their symmetry. In 1948, Bravais showed that 14 lattices are sufficient to describe all crystals. These 14 lattices are known as Bravais lattices and are classified into 7 crystal systems based on cell parameters. The Bravais lattices are categorized as primitive lattice (P); body-centered lattice (I); face-centered lattice (F) and base-centered lattice (C). These seven crystal systems and Bravais lattices are described below.

The point of intersection of the three axes is called the AXIAL CROSS. By using these crystallographic axes we can define six large groups or crystal systems that all crystal forms may be placed in:-

1. CUBIC (or ISOMETRIC) System -I

The three crystallographic axes are all equal in length and intersect at right angles to each other. a = b = c alpha=beta=gamma=90 degree.

In the cubic system, there are three Bravais lattices; they are simple (primitive); body-centered and face-centered. Examples for cubic systems are Au, Cu, Ag, NaCl, diamond, etc. In simple cubic lattice, lattice points or atoms are present at the corners of the cube. In a body- centered cube, atoms are present at the corners and one atom is completely present at the center of the cube. In the case of face-centered cube, atoms are present at corners and at the centers of all faces of the cube.

2. Tetragonal crystal system:

In this crystal system, two lengths of the unit cell edges are equal whereas the third length is different. The three edges are perpendicular to one another i.e., a = b ≠ c and α = β = γ = 90°. In tetragonal system, there are two Bravais lattices; they are simple and body-centered. These are shown in Fig. Examples for tetragonal crystal systems are TiO2, SnO2, etc.

3. Orthorhombic crystal system:

In this crystal system, unit cell edge lengths are different and they are perpendicular to one another i.e., a ≠ b ≠ c and α = β = γ = 90°. There are four Bravais lattices in this system. They are simple, face centered, body centered and base centered. These are shown in Fig.

Examples for orthorhombic crystal system are BaSO4, K2SO4, SnSO4, etc.

4. Monoclinic crystal system:

In this crystal system, the unit cell edge lengths are different. Two unit cell edges are not perpendicular, but they are perpendicular to the third edge i.e., a ≠ b ≠ c; α = γ = 90° ≠ β. This crystal system has two Bravais lattices; they are simple and base centered. These are shown in Fig. Examples for Monoclinic crystal systems are CaSO4.2H2O (gypsum), Na3AlF6 (cryolite), etc.

5. :

In this crystal system, the unit cell edge lengths are different and are not perpendicular i.e., a ≠ b ≠ c and α ≠ β ≠ γ ≠ 90° and all the angles are different. This crystal exists in primitive cells only. This is shown in Fig. Examples for triclinic crystal systems are K2Cr2O7, CuSO4. 5H2O, etc.

6. Trigonal or Rhombohedral crystal system:

In this crystal system, all the lengths of unit cell edges are equal. The angles between the axes are equal but other than 90° i.e., a = b = c and α = β = γ ≠ 90°. The is simple only as shown in Fig. Examples for Rhombohedral crystal systems are As, Bi, Sb, etc.

7. Hexagonal crystal system:

In this crystal system, two sides of the unit cell edge lengths are equal and the angle between these edges is 120°. These two edges are perpendicular to the third edge, and not equal in length i.e., a = b ≠ c and α = β = 90°; γ = 120°. The Bravais lattice is primitive only. This is shown in Fig. The atoms in these crystal systems are arranged in the form of a hexagonal close pack.

The fourteen Bravais lattices of seven crystal systems are summarized in Table:

SELECTED CRYSTAL STRUCTURES :

AX-Type Crystal Structures/AB type

Compounds Some of the common ceramic materials are those in which there are equal numbers of cations and anions. These are often referred to as AX compounds, where A denotes the cation and X the anion. There are several different crystal structures for AX compounds; each is normally named after a common material that assumes the particular structure

1. Rock Salt Structure :

Perhaps the most common AX crystal structure is the chloride (NaCl), or rock salt, type. The for both cations and anions is 6, and therefore the cation–anion radius ratio is between approximately 0.414 and 0.732.

Fig: NaCl Structure

A unit cell for this crystal structure is generated from an FCC arrangement of anions with one cation situated at the cube center and one at the center of each of the 12 cube edges. An equivalent crystal structure results from a face-centered arrangement of cations. Thus, the rock salt crystal structure may be thought of as two interpenetrating FCC lattices, one composed of the cations, the other of anions. Some of the common ceramic materials that form with this crystal structure are NaCl, MgO, MnS, LiF, and FeO.

Figure shows a unit cell for the cesium chloride (CsCl) crystal structure;

The coordination number is 8 for both ion types. The anions are located at each of the corners of a cube, whereas the cube center is a single cation. Interchange of anions with cations, and vice versa, produces the same crystal structure. This is not a BCC crystal structure because ions of two different kinds are involved.

Fig : ZnS Structure

A third AX structure is one in which the coordination number is 4; that is, all ions are tetrahedrally coordinated. This is called the zinc blende, or , structure, after the mineralogical term for (ZnS). A unit cell is presented in All corner and face positions of the cubic cell are occupied by S atoms, while the Zn atoms fill interior tetrahedral positions. An equivalent structure results if Zn and S atom positions are reversed. Thus, each Zn atom is bonded to four S atoms, and vice versa. Most often the atomic bonding is highly covalent in compounds exhibiting this crystal structure which include ZnS, ZnTe, and SiC.

2. GRAPHITE:

Graphite is a crystalline allotrope of carbon, a semimetal, a native element , and a form of coal. Graphite is the most stable form of carbon under standard conditions. Therefore, it is used in thermochemistry as the standard state for defining the heat of formation of carbon compounds. Graphite occurs in metamorphic rocks as a result of the reduction of sedimentary carbon compounds during metamorphism. It also occurs in igneous rocks and in meteorites.] associated with graphite include quartz, calcite, micas and tourmaline. In meteorites it occurs with troilite and silicate minerals. Small graphitic crystals in meteoritic iron are called cliftonite. Structure Graphite has a layered, planar structure. The individual layers are called graphene. In each layer, the carbon atoms are arranged in a honeycomb lattice with separation of 0.142 nm, and the distance between planes is 0.335 nm.] Atoms in the plane are bonded covalently, with only three of the four potential bonding sites satisfied. The fourth electron is free to migrate in the plane, making graphite electrically conductive. However, it does not conduct in a direction at right angles to the plane. Bonding between layers is via weak van der Waals bonds, which allows layers of graphite to be easily separated, or to slide past each other.The two known forms of graphite, alpha (hexagonal) and beta (rhombohedral), have very similar physical properties, except for that the graphene layers stack slightly differently.

The alpha graphite may be either flat or buckled.] The alpha form can be converted to the beta form through mechanical treatment and the beta form reverts to the alpha form when it is heated above 1300 °C.

PROPERTIES 1.High strength 2.Good chemical stability at elevated temperature and in non oxidising atmosphere 3.High 4.Good machinability 5.High temp refractories and insulation

Fig: (a) 3-D Graphite structure (b) Graphite Solid (c) Structure (d) Van der Waals bonds in Graphite structure 3. DIAMOND

The crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group14 also adopt this structure, including α-, the and , and silicon/germanium alloys in any proportion

Crystallographic structure-Diamond cubic follows the face-centered cubic Bravais lattice. The lattice describes the repeat pattern; for diamond cubic crystals this lattice is "decorated" with a motif of two tetrahedrally bonded atoms in each primitive cell, separated by 1/4 of the width of the unit cell in each dimension. The diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by 1/4 of the width of the unit cell in each dimension.

The of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the vertices of the structure and are as large as possible without overlapping) is π√3/16 ≈ 0.34 significantly smaller (indicating a less dense structure) than the packing factors for the face-centered and body- centered cubic lattices.

Properties: 1.EXTREMELY HARD 2. LOW ELECTRICAL CONDUCTIVITY 3.HIGH REFRACTIVE INDEX

APPLICATION: Industrial are industrially utilised to grind and cut other softer materials. The surface of the drills, dies ,bearings,knives and other tools have been coated with diamond film to increase the surface hardness.

Fig : (a) Diamond 3-D structure (b) Diamond.

NON-CRYSTALLINE MATERIALS

Amorphous solids have two characteristic properties. When cleaved or broken, they produce fragments with irregular, often curved surfaces; and they have poorly defined patterns when exposed to x-rays because their components are not arranged in a regular array. An amorphous, translucent solid is called a glass. Almost any substance can solidify in amorphous form if the liquid phase is cooled rapidly enough. Some solids, however, are intrinsically amorphous, because either their components cannot fit together well enough to form a stable crystalline lattice or they contain impurities that disrupt the lattice. For example, although the chemical composition and the basic structural units of a quartz crystal and quartz glass are the same—both are SiO2 and both consist of linked SiO4 tetrahedra—the arrangements of the atoms in space are not. Crystalline quartz contains a highly ordered arrangement of silicon and atoms, but in quartz glass the atoms are arranged almost randomly. When molten SiO2 is cooled rapidly (4 K/min), it forms quartz glass, whereas the large, perfect quartz crystals sold in mineral shops have had cooling times of thousands of years. In contrast, aluminum crystallizes much more rapidly. Amorphous aluminum forms only when the liquid is cooled at the extraordinary rate of 4 × 1013 K/s, which prevents the atoms from arranging themselves into a regular array.

The lattice of crystalline quartz (SiO2). The atoms form a regular arrangement in a structure that consists of linked tetrahedra.

In an amorphous solid, the local environment, including both the distances to neighboring units and the numbers of neighbors, varies throughout the material. Different amounts of thermal energy are needed to overcome these different interactions. Consequently, amorphous solids tend to soften slowly over a wide temperature range rather than having a well-defined melting point like a crystalline solid. If an amorphous solid is maintained at a temperature just below its melting point for long periods of time, the component molecules, atoms, or ions can gradually rearrange into a more highly ordered crystalline form.

Some more differences between crystalline solids and amorphous solids is listed in Table:

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REFERENCES:

1. https://www.pmec.ac.in/images/1_Science%20and %20Engineering%20of%20Materials.pdf

2. https://web.iit.edu/sites/web/files/departments/ac ademic-affairs/academic-resource- center/pdfs/Crystal_Structures.pdf

3. https://depts.washington.edu/matseed/ces_guide/ crystalline.htm

4. https://www3.nd.edu/~amoukasi/CBE30361/Lectu re__crystallography_A.pdf

5. https://chem.libretexts.org/Courses/Howard_Uni versity/General_Chemistry%3A_An_Atoms_First_ Approach/Unit_5%3A_States_of_Matter/Chapter _12%3A_Solids/Chapter_12.01%3A_Crystalline_a nd_Amorphous_Solids

6. Book- Material Science and Engineering - V. Raghavan

7. Material Science and Engineering- William Callister, David Rethwisch