Some important points

Close Packed Structures

In order to analyze close packing of constituent particles (ions, molecules or atoms) in a lattice, we get into the assumption that constituent particles (ions, molecules or atoms) are hard spheres of identical shape.



Image 1: Different Types of Layers form closed packed structures

What is Close Packing?

The packing of constituent particles inside lattice in such a way that they occupy maximum available space in the lattice is known as Close Packing.

Close Packing is done in three ways, namely:


One Dimensional Close Packing


Two Dimensional Close Packing


Three Dimensional Close Packing



One Dimensional Close Packing

In one dimension close packing, the spheres are arranged in a row touching each other. In one-dimensional close packing, each sphere is in direct contact with two of its neighbor spheres. The number of nearest spheres to a particle in a lattice is called Coordination Number. Since there are two spheres in direct contact with the one sphere, the coordination number of one-dimensional close packing is 2.

Two Dimensional Close Packing

Two-dimensional close packing is done by stacking rows of hard spheres one above the other. This can be done in two ways:


AAA type


ABA type



AAA Type



Image 2: AA type packing

The packing in which one sphere touches two spheres placed in two different rows one above and one below is called AAA type close packing. The coordination number of AAA type two-dimensional close packing is 4. The AAA type is formed by placing one-dimensional row directly one above the other in both horizontal and vertical directions. It is also called two-dimensional square close packing as the rows of spheres when arranged in vertical and horizontal alignments form a square.

AB Type



Image 3: AB Type Packing

The packing in which the spheres in the second row are located in the depressions of the first row. The ABA type close packing is formed by placing one-dimensional row let’s say B type over the A type close packing and this series continues to form a two dimensional hexagonal. The coordination number of ABA Type packing is 6 as each sphere is in direct contact with 6 other spheres.

In ABA type close-packing we find triangular empty spaces called voids. These are of two types:


Apex of triangle pointing upwards


Apex of triangle pointing downwards
 



Three Dimensional Close Packing

The formation of real lattices and structures take place through three-dimensional close packing. They are formed by stacking two-dimensional layers of spheres one above the other. This can be done by two ways:


Three-dimensional close packing from two dimensional square close packed layers


Three-dimensional close packing from two dimensional hexagonal close packed layers



Three-dimensional close packing from two-dimensional close packed layers

Formation of three-dimensional close packing can be done by placing the second square closed packing exactly above the first one. In this close packing, the spheres are aligned properly in horizontally and vertically. Similarly, by placing more layers one above the other, we can obtain a simple cubic lattice. The unit cell of the simple cubic lattice is called the primitive cubic unit cell.

Three-dimensional close packing from two dimensional hexagonal close packed layers

Three-dimensional close packing can be formed with the help of two-dimensional hexagonal packed layers in two ways:


Stacking the second layer over the first layer


Stacking the third layer over the second layer
 



Stacking the second layer over the first layer

Suppose we take two hexagonal close packed layer ‘A’ and place it over the second layer B ( as both layers have different alignment of spheres) such that spheres of the second layer are placed in the depressions of the first layer. We observe that a tetrahedral void is formed when a sphere of the second layer is right above the void (empty space) of the first layer. Adding further we notice octahedral voids at the points where the triangular voids of the second layer are placed right triangular voids of the first one in such a way that triangular space doesn't overlap. Octahedral voids are bordered by six spheres.



Image 4: Types of voids in three-dimensional closed packing

If there are ‘N’ closed spheres, then:


Number of Octahedral Voids equals to “N”


Number of Tetrahedral Voids equals to “ 2N”



Stacking the third layer over the second layer

There are two possible ways of placing the third layer over the second layer:


By Covering Tetrahedral Voids


By Covering Octahedral Voids



Covering Tetrahedral Voids

In this kind of three-dimensional packing, the spheres of the third layer are aligned right above the spheres of the first layer. If we name the first layer as A and second layer as B, then the pattern will be ABAB… so far and so forth. The structure formed is also called hexagonal close-packed structure also known as HCP.

Covering Octahedral Voids

In this kind of packing the third layer, spheres are not placed with either of the second layer or first layer. If we name the first layer as A, second as B and then the third layer will be C (as it is now a different layer) then the pattern will be ABCABC… The structure formed is also called cubic closed packed (ccp) or face-centred packed cubic structure (fcc). For Example metals like copper and iron crystallize in the structure.

The coordination number in both cases will be 12 as each sphere in the structure is in direct contact with 12 other spheres. The packing is highly efficient and around 74% of the crystal is completely occupied.



Image 5: ABC Type of Close Packing

The Fourteen Bravais Lattices

Although for simplicity we have so far chosen to discuss only a two dimensional space lattice, the extension of these concepts to three dimensions apply equally well. If the seven crystal systems discussed in  the table, are represented by their primitive unit cells, then we shall have seven possible lattice types. But sometimes the smallest primitive unit cell does not display the full symmetry of the lattice. In such a case, unit cell with the highes symmetry is chosen. For illustration, we may take the face centered cubic structure (FCC) of silver.

In the face-centred representation there are four silver atoms associated with this unit cell. (Eight corner atoms = 8 x 1/8 = 1 atom; Six face atoms = 6 x 1/2 = 3 atoms; Total = 4 atoms.)

We may now choose a primitive unit cell as shown by heavy lines. It contains only one silver atom per unit cell. Although this representation has the smallest number of atoms per unit cell, but them it does not display the full summetry of structure. The face-structure ( thin lines ) represents cubic symmetry, whereas, the primitive cell unit cell represents a rhombohedral. The unit cell for these reasons is chosen as face-centered cubic.

It was showm by A. Bravais in 1848 that all possible three dimensional space lattice are of fourteen distinct types. These fourtenn lattice types ( also known as Bravais lattices ) are derived from seven crystal systems. The unit cells for these fourteen Bravais lattices are show in the figure.

Triclinic

Monoclinic

Orthorhombic

Tetragonal

Rhombohedral

Hexagonal

Cubic

All crystalline solids can be represented by one of these lattice structures. Simple unit cells are labelled as P (primitive). Non primitive unit cells are of three type and described as:

• body centered (I);
• face centered (F);
• base centered (C).

Cubic system is represented by three type of untis cells. The primitive, also known as simple cubic has an atom or molecule at each point at corner. The number of atmos or molecules per unit cell is one. The other type of unit cell representing cubic symmetry is body centered cubic. It is made up of point at the corners of a cube with an additional point in the centre of the cube. In face centered cubic symmetry there are points at the corners of the cube with additional points at the centre of each face. The number of atoms or molecules per unit cell are four. The unit cell dimensions and angles of fourteen Bravais lattices are given in the table.

N°	Crystal System	Unit cell dimensions and angles	Bravais lattices
1	cubic	a=b=c ; α=β=γ=90°	Primitive Face centered Body centered
2	Orthorhombic	a≠b≠c; α=β=γ=90°	Primitive Face centered Body centered Base centered
3	Tetragonal	a=b≠c; α=β=γ=90°	Primitive Body centered
4	Monoclinic	a≠b≠c; α=γ=90° β≠90°	Primitive Base Centered
5	Rhombohedral	a=b=c; α=β=γ≠90°	Primitive
6	Triclinic	a≠b≠c; α≠β≠γ≠90°	Primitive
7	Hexagonal	a=b≠c; α=β=90° γ=120°	Primitive

Symmetry and Crystal System

Crystal may be distinguished from one another by their external morphology which generally appears to have some regularity of arangement. It is soon recognised that the regularity of external arragement implies a regularity of internal arrangement. This regularity is sometimes obscured by the presence of some over-grown or destroyed faces. Strictly speaking, classification of crystals is made on the basis of their characteristic internal structure which possesses some define symmetry. The symmetry of solid body may be defined in terms of the various elements of symmetry it possesses. Suppose we are looking at a cube and if for a moment we take our sight away from this cube and in the meantime someone rotates the cube through 90° about an imaginary axis passing through the centre of the top and bottom faces and if we look at the cube again, we will no find any change. This second orientation of crystal is exactly similar to the first orientation. In a similar way, the cube can be rotated twice more to get similar orientations. Such a crystal which has more than one orientation in space and which are indistinguishable from another is said to possess symmetry. The imaginary axis about which this rotation operation has been performed is a symmetry element, called axis of ratation. In this case the crystal has been rotated through 90° and so it possesses a 360/90 = 4-fold axis of rotation. Since a cube consists of six faces, there will be three 4-fold axis of rotation. (see the figure)

Similary, a cube may be brought into coincidence with itsef by rotation through one third of a revolution (120°) about a body diagonal. Such an axis of symmetry is called a 360/120 = 3-fold axis of symmetry. Since there are four body diagonals in a cube, it has four 3-fold axis of rotazion (see the figure). The figure depicts one of the six axes of 2-fold symmetry, emerging from opposite edges of a cube. The one centre of symmetry of a cube is represented simply at the mass centre (middle point) of a cube.

If we take a hexagon, we observe that it can be brought into coincidence with itself by rotation through 60°. So a hexagonal crystal posesses 360/60 = 6-fold axis of rotation.

a crystal may possess sevaral symmetry elements and these are:

1. axis of rotation,
2. axis of rotation-inversion,
3. plane of symmetry,
4. centre of inversion.

However, solids can be divided into sevendistinct types on the basis of only two of these symmetry elements. These are: axis of rotation, and axis of rotation-inverision.

An n-fold axis of rotation-inversion combines rotation through 360°/n with inversion through the center of symmetry of the crystal. Asoild of type shown in the figure possesses a four-fold axis of rotation-inversion. It is also represented as .

It can be shown that the onlyaxes of symmetry which are possible in crystals are 2-, 3-, 4- and 6-fold axis and it is found that the symmetry of all crystals can be described in terms of the following symmetry elements or combination thereof; 2-, 3-, 4- and 6- fold axes of simple rotation or axis of rotation-inversion,  and . For example, the 5-, 7-, 8- and higher fold symmetry does not occur in crystals. Why? The answer is that it is impossible to make a compact repeating pattern with these shapes. If we try to pack together some pentagons, we will soon find that some space is always left vacant. The only figures that can be packed together are the parallelogram (2-fold symmetry), the equilateral triangle (3-fold), the square (4-fold) and hexagon (6-fold). These are the only axial symmetries that are found in crystals. We may also add another axial symmetry, one-fold axis, which really means the absence of symmetry.

Classsification of solids

Solids are classified into categories:

• Amorphous solids
• Crystalline solids

The two types of solids have different characteristics.

• Amorphous Solids. An amporphous solidd is a substance whose constituents do not possess an orderly arangement. Important examples of amorphous solids are glass and plastics. Although amorphous solids consist of microcrystalline substance but the orderly arrangement is restricted to very short distances. These distances are of the same order of magnitude as the interatomic distances.
• Crystalline Solids. A crystalline solid is a substance whose constituents possess an orderly arrangement in a definite geometric pattern. Some very common examples of crystalline substances are sodium chloride, sugar and diamond. The main characteristica of crystalline substances are:
1. Orderly arrangement. The costituent units of crystalline solids are arranged in an orderly fashion which repeats itself over very long distances as compared to interatomic distances. The arrangement of bricks in a wall can be considered as an example.  The arrangement is so well defined that the entire pattern can be repeated provided the arrangement of a few atoms is known.
2. Crystals are always bounded by plane faces.
3. The faces of crystals always meet at some fixed angles. 
   For any particular substances the angle between corresponding pair of faces is alway the same in all crystals.
   Different crystals of the same substance may sometimes appear to be different from outside, ( either due to different rate of growth by different faces or due to some damage to the corners or edges) but the interfacial angle is always the same. For example, sodium chloride grows from water solution as cubes but from 15% aqueous urea as an octahedran. (see Paul E. Smith "The effect of urea on the morphology of NaCl crystals: A combined theoretical and simulation study

Imperfections in solids

Imperfections in Solids or Crystal defects

Irregularity in the arrangement of constituent particles in solids is called crystal defect or imperfection in solids. There are two types of crystal defects - Point Defects and Line Defects.

Point Defects: Irregularities or deviation from ideal arrangement of constituent particles around the point or atom in a crystalline solid is known as point defects.

Line Defects: Irregularities or deviation from ideal arrangement of constituent particles in entire row of lattice is known as line defects.

Point Defects: Point Defects are divided into three types:

(i) Stoichiometric Defects

(ii) Impurities Defects

(iii) Non-stoichiometric Defects

(i) Stoichiometric Defects: – It is a type of point defects which does not disturb the stoichiometry of solid. This is also known as Intrinsic or Thermodynamic Defects.

Types of stoichiometric defects: Vacancy Defects, Interstitial defects, Frenkel Defects, Schottky Defects.

Vacancy defects and Interstitial defects are found in non-ionic compounds while similar defects found in ionic compounds are known as Frenkel Defects and Schottky Defects.

(a) Vacancy Defects: When some lattice sites left vacant while the formation of crystal, the defect is called Vacancy Defects.

In vacancy defects, an atom is missing from its regular atomic site. Because of missing of atom the density of substance decreases, i.e. because of vacancy defects.

The vacancy defect develops on heating of substance.

(b) Interstitial Defects: - Sometime in the formation of lattice structure some of the atoms occupy interstitial site, the defect arising because of this is called Interstitial Defects.

In interstitial defect, some atoms occupy sites at which; generally there is no atom in the crystal structure. Because of the interstitial defects, the number of atoms becomes larger than the number of lattice sites.

Increase in number of atoms increases the density of substance, i.e. interstitial defects increase the density of substance.

The vacancy defects and interstitial defects are found only in non-ionic compounds. Such defects found in ionic compounds are known as Frenkel Defects and Schottky Defects.

(c) Frenkel Defects: It is a type of vacancy defect. In ionic compounds, some of the ions (usually smaller in size) get dislocated from their original site and create defect. This defect is known as Frenkel Defects. Since this defect arises because of dislocation of ions, thus it is also known as Dislocation Defects. As there are a number of cations and anions (which remain equal even because of defect); the density of the substance does not increase or decrease.

Ionic compounds; having large difference in the size between their cations and anions; show Frenkel Defects, such as ZnS, AgCl, AgBr, AgI, etc. These compounds have smaller size of cations compared to anions.

(d) Schottky Defects: Schottky Defect is type of simple vacancy defect and shown by ionic solids having cations and anions; almost similar in size, such as NaCl, KCl, CsCl, etc. AgBr shows both types of defects, i.e. Schottky and Frenkel Defects.

When cations and anions both are missing from regular sites, the defect is called Schottky Defect. In Schottky Defects, the number of missing cations is equal to the number of missing anions in order to maintain the electrical neutrality of the ionic compound.

Since, Schottky Defects arises because of mission of constituent particles, thus it decreases the density of ionic compound.

(ii) Impurities Defects: Defects in ionic compounds because of replacement of ions by the ions of other compound is called impurities defects.

In NaCl; during crystallization; a little amount of SrCl2 is also crystallized. In this process, Sr++ ions get the place of Na+ ions and create impurities defects in the crystal of NaCl. In this defect, each of the Sr++ ion replaces two Na+ ions. Sr++ ion occupies one site of Na+ ion; leaving other site vacant. Hence it creates cationic vacancies equal number of Sr++ ions. CaCl2, AgCl, etc. also shows impurities defects.

(iii) Non-stoichiometric Defects: There are large numbers of inorganic solids found which contain the constituent particles in non-stoichiometric ratio because of defects in their crystal structure. Thus, defects because of presence of constituent particles in non-stoichiometric ratio in the crystal structure are called Non-stoichiometric Defects.

Non-stoichiometric Defects is mainly of two types – Metal Excess Defects and Metal Deficiency Defects.

Metal Excess Defects: Metal excess defects are of two types:

(a) Metal excess defects due to anionic vacancies:

These type of defects seen because of missing of anions from regular site leaving a hole which is occupied by electron to maintain the neutrality of the compound. Hole occupied by electron is called F-centre and responsible for showing colour by the compound.

This defect is common in NaCl, KCl, LiCl, etc. Sodium atoms get deposited on the surface of crystal when sodium chloride is heated in an atmosphere of sodium vapour. In this process, the chloride ions get diffused with sodium ion to form sodium chloride. In this process, sodium atom releases electron to form sodium ion. This released electron gets diffused and occupies the anionic sites in the crystal of sodium chloride; creating anionic vacancies and resulting in the excess of sodium metal.

The anionic site occupied by unpaired electron is called F-centre. When visible light falls over the crystal of NaCl, the unpaired electron present gets excited because of absorption of energy and impart yellow colour.

Because of similar defect if present, crystal of LiCl imparts pink colour and KCl imparts violet.

(b) Metal excess defect due to presence of extra cations at interstitial sites:

Zinc oxide loses oxygen on heating resulting the number of cations (zinc ion) become more than anions present in zinc oxide.

The excess cations (Zn⁺⁺ions) move to interstitial site and electrons move to neighbouring interstitial sites. Because of this zinc oxide imparts yellow colour when heated. Such defects are called metal excess defects.

Metal Deficiency Defects:

Many solids show metal deficiency defects as they have less metals compare to ideal stoichiometric proportion. The less proportion of metal is compensated by same metals having higher valency. Such defects are shown generally by transition elements. Thus, when metal present less than ideal stoichiometric proportion in a solid, it is called metal deficiency defect.

Example – FeO is generally found in composion of Fe0.95O. In the crystal of FeO, missing Fe++ ions are compensated with Fe+++ ions in order to maintain neutrality.