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UNIT 4 SOLID STATE

Structure Introduction Objectives What are solids? Classification of Solids X-Ray Diffraction of Solids Principles of Dimaction Bragg Law and Bragg Equation Lattices Crystal systems and Bravais Lattices Cubic Unit Cells Calculation of Density of Solids Close-Packed Structures Voids Struch11.e~of Ionlc Solids Electrical Properties of Solids Metallic Conduction in Solids: Band Theory Ionic Conductioll in Solids Magnetic Properties of Solids Temperature Dependence of Magnetic Properties Summary Ternenal Questions

4.1 INTRODUCTION

Matter can exist in seven different states of varying extents of compactness. Of these, we nom~allycome across three states viz., solids, liquids and gases. The other four states are found in the cosmic bodies. In this unit we are going to learn about the solid state, a commonly encountered characterised by order and regularity. The last twenty years or so have seen a change in the perception of solid state , in particular the significance of understanding the relationship between chemical structure and physical properties. This has led to the development of newer materials with interesting properties ranging froin semiconductors to superconductors, which find divcrse applications. Study of solids is so very important that material science is one of the most sought after areas of research now a days. However, we are going to take up solid state at a very elementary level. You would learn about the basic structure of the solids, its detem~inationand some of the properties of solids. In the next unit you would learn about basic chemical thermodynamics.

Objectives

After studying this unit, you should be able to: classify solids on the basis ofthe nature of bonding between the species constituting them, define crystalline solids and differentiate them from amorphous substances, define crystal , and primitive and non prirnitivc unit cells describe the seven crystal systems and the fourteen Bravais lattices, state the principle of X-ray diffraction method for determining the structures of the solids, derive and use Bragg equation, describe close packed structures of metallic solids, discuss the structures of common ionic solids of MX,MX2 and M2X type and exnlain the electrical and mametic properties of solids. Structure 4.2 WHAT ARE SOLIDS?

A solid is an almost incompressible state of matter having a well defined rigid structure. It is composed of structural units - atoms, molecules or ions which are in close contact due to strong forces of attraction between one another. These forces of attraction are of different types and are responsible for the differences in the properties of the solids. The force of attraction may be of electrical nature as in case of chloride in which the sodium ions and chloride ions are attracted to each other or these may be of There are four different intermolecular type which hold different molecules together e.g., in , types of solids molecules are held together by intermolecular hydrogen bonding. Other possibilities are ionic, of strong chemical bonds between the constituent atoms as in case of or the molecular, . covalent and metallic bonding in which a large number of positive cores of the atoms are held metallic. together by a sea of electrons. In fact, the solids are sometimes classified on the basis of the forces of interaction or attraction between the structural units.

4.2.1 Classification of Solids There is a vast variety of solids and these can be classified on the basis of the nature of forces of attraction between the structural units constituting the solids as discussed above. Thus there are four different types of solids- ionic, molecular, covalent and metallic.

' Fig. 4.1: Ionic (NaCl), covalent (diamond), molecular (ice) and metallic () solids.

The characteristics and the properties of the different types of solids are compiled in Table 4.1.

Table 4.1: Characteristics and properties of different types of solids.

Constituent Forces of Melting Electrical Solid Species interaction Point conductivity between the constituent species Weak, dipolar or dispersion type - Metallic Hard and Aloms Electron Variable Conducting malleable-- delocalisation - Coulombic Ionic DoairHard and Ions 1 1 1 conducting I rhigh conducting Amorphous and Crystalline Solids Solid State There is another way to classify solids, let us perfom1 an activity to understand this. Take a few common solids like, common salt, sugar, benzoic acid and a piece of glass rod. Observe these solids closely -you may usc a hand lens to observe small particles of salt, sugar and benzoic acid. What do you observe? Thc salt, sugar and benzoic acid exhibit a number of definite smooth surfaces at definite anglcs to each other on the other hand the glass does not have these. Now try to melt these solids (othcr than NaCl, it has very high melting point) on a flame. Sugar and benzoic acid can be melted in a glass capillary while the glass rod can be put directly to the flame. You nlay observe that all thc solids other than glass undergo a sharp change from solid to liquid state at a fixed temperature while glass gradually softens over a range of temperature. The solids having smooth surfaces at definite angles to each other and showing a sharp change from solid to liquid state are called as crystalline solids while solids showing thc

' behaviour displayed by glass arc called as amorphous solids. Another point of differentiation betwecn the amorphous and crystalline solids is that while amorphous solids arc isotropic in naturc is., these exhibit same value of any property in all directions, the crystalline solids arc ai~isotropici.e., thc value ofphysical properties are different in different directions. We will, however, focus our attention on crystalline r solids only.

I The regularity and perfection of a crystalline solid suggests of an ordcred internal structure for them. Actually the form of crystal depends on the way it is grown. A slow cooling of a slightly supersaturated solution generally proceeds in such a way that it Thesc interfacial angles are 1 mcasured by an instrument allows all the naturally occurring faces of the to grow and develop highly called as contact symnletrical crystals. On the other hand a sudden cooling of a molten compound or goniometer. i highly supersaturated solution gives imperfect crystals. The quality of the crystal may vary but the anglcs between the perpendiculars to different faces called interfacial I angles are characteristic and distinct for a substance. This fact is known as Huay's law of constancy of interfacial angles. The highly ordercd internal structure of a crystalline solid is represented in terms of a three dimensional structure called lattice. You will lcam about the meaning of lattice and different types of crystal lattices and their representations in section 4.4.

4.3 X-RAY DIFFRACTION OF SOLIDS - In 1912 Laue suggested that the wavelength of X-rays may be of the order of the interatomic distances between atoms in a crystal and the crystal may serve as a diffraction grating for the X-rays. Tt was experimentally found to be so a little later. W.L. Bragg (1913) used monochromatic radiation for the scattcring experiment and in the process determined the structures of some common ionic compounds and provided a fundamental equation named after him. In the course of interaction with the crystal the X-rays are scattered by the electron cloud of an atom. Rragg suggested that the crystal contains a large number of stacks of planes from which the X-rays are reflected. Further, the angle of reflection from a given stack of planes depends on the wavelength of the X-rays and the interplaner distance. Let us learn about the meaning of diffraction and derive an cxprcssion for the diffraction of X-rays from a stack of parallel planes.

4.3.1 Principle of Diffraction The phenomenon of diffraction refers to the interference of waves due to the presence of an obstacle in their path. This can be exemplified by the bending of light round the edges of an object. Consider a beam of light passing through two slits (S, and S?),cut near to each other on a screen and falling on a second screen placed beyond the slits (Fig. 4.2). A series of dark and bright bands are observed on the screen, which are tn fhr rnnotnlrtive and destructive interference of the two beams passing through Structure

In phase combination

Source of light'

combination

Fig. 4.2: In phase and out of phase combination of the light waves originating from two point sources to bright ( for in-phase) and dark spots ( for out of phase) on the screen.

The amplitude is directly When the waves are in phase, the intensity is increased, this is known as constructive related to the intensity of interference; Fig. 4.3(a). When they are out of phase (known as destructive the beam. interference), the intensity is decreased, Fig. 4.3(b). If however, when the two waves are completely out of phase, these cancel each other, Fig. 4.3(c).

Fig. 4.3: Interference of two waves a) completely in phase b) partly out of phase and c) completely out of phase.

For diffraction to occur the dimensions of the diffracting object should be comparable to the wavelength of the light. Whether the beams are in-phase or out-of-phase will depend on the path difference between the two rays.

4.3.2 Bragg Law and Bragg Equation

If the path difference between the two rays is an integral multiple (n = 1,2,3 ...) of the wavelength of X-rays, then the two rays will be in-phase and the diffraction patterns will be bright (i.e. with enhanced intensity). This is called Bragg law. Stated mathematically, for a bright diffraction pattern,

Path difference = n h ... . . 4.1

Bragg derived an equation (Eq. 4.6) for X-ray diffraction of crystals. This equation is named after him. Some of the assumptions made by Bragg in deriving ths equation are given below: The incident waves are reflected by parallel planes of atoms in a crystal such that the angle of incidence, is equal to the angle of reflzction. This is called specular (mirror-like) reflection. Each plane reflects only a fraction of incideni radiation. When the reflections from parallel planes interfere constructively, the diffraction pattern arises. The wavelength of the X-rays is not changed on reflection; i.e., X-rays undergo elastic scattering on the lattice planes. Using geometric considerations, Bragg equation can be derived easily.

Let us take two parallel beams PA and QC incident at an angle 0 on the parallel planes EF and GH (Fig 4.4) of atoms in a crystal. The perpendicular distance (AC) between the two planes is d. The beam are reflected along AR and CS at an angle 0. The path difference between the two sets of incident and reflected beams (PAR and QCS) is the extra distance travelled by QCS as compared to PAR. To calculate the path difference, draw AB I QC and AD ICS.

Fig. 4.4: Reflection of X-rays from parallel crystal planes.

Path difference = (QC + CS) - (PA + AR) = (QB + BC) + (CD + DS) - (PA + AR) = BC + CD

[QB = PA and DS = AR, being opposite sides of the rectangles shown by the shaded portions in Fig.. 4.41

Since AC I GH, LACG = 90" = LACB + LBCG = LACB + 0

[LQCG and LBCG are same as 01

LACB = 90" - 0

In the right-angled AABC, LBAC + LACB + LCBA = 180" r Using Eq. 4.3 LBAC + (90" - 0) + 90" = 180"

LBAC = 180"- (180"- 0) = 0

Since AC = d, BC = d sin0

Similarly we can prove that CD = d sin0 Structure Using Eqs 4.3,4.4 and 4.5

Path difference = 2d sin 0

Again substituting in Eq. 4.1 we gct,

Bragg equation assumes nA = 2d sin 8 that incident X-rays are reflected specularly Eq. 4.6 is known as Bragg equation. It is useful in deternlination. In (mirror like) such that the this equation, 3L is the wavelength of X-rays used, d is the distance qr the spacing angle of incidence is equal to the angle of reflection. between the planes. The value of n gives the order of reflection. This assumption is convincing only because it If n = 1, it is first-order reflection. I explains the experimental If n = 2, it is second-order reflection and so on. ' results. SAQ 1 X-rays having a wavelength of 154 pm showed a first order diffraction from the surface of a crystal when the angle of incidence was 1 lo.Find out the interplaner distance of the diffracting lattice layers.

4.4 CRYSTAL LATTICES

In 1782 Abbe Huay proposed that the internal structure of the crystalline solids can be represented as an ordered arrangement of simple units like atoms, molecules or ions. Such a regular arrangement of basic units in three dimensions is called as a crystal lattice or simply a lattice. To understand the meaning of the term lattice let us first take a repetitive pattern in one dimension. A set of some unit repeated at a certain distance along a line e.g., a line of identical equally spaced telephone poles provide an example of a one dimensional pattern. This pattern can be represented in terms of a one dimensional lattice by indicating each pole by a point or a cross. Similarly, the repeating patterns on certain fabrics or wallpapers are examples of two dimensional lattices, Fig. 4.5(a). The repeat unit in the pattern is called as 'motif. We can obtain the corresponding two dimensional lattice by choosing an identical point in all the motifs, Fig. 4.5(b).

Fig. 4.5: An example of a) r two dimensional pattern and b) the corresponding lattice.

We can take any point to represent thc repeating motif but it should be same in all the motifs. Thus a lattice shows the arrangement of the basic units. We can generate the whole lattice by starting from any lattice point and moving fixed distances in the two directions as shown in Fig. 4.5. For this purpose we define the repeat distances called as unit vectors in the two directions indicated by arrows in the diagram. Solid State Since a vector has a magnitude as well as the direction, therefore the length as well as the directions of the unit vectors must be specified to define a lattice. For the pattem given in Fig. 4.6, the unit vectors will be different from the ones defined for the above pattem though the two have same repeating motif. In this case the two basis vectors a, and a2 are inclined at a certain angle cp while in case of Fig. 4.5, the basis vectors were at a right angle.

Fig. 4.6: A different two dimensional pattern with the same motif as in Figure 4.5.

Depending on the magnitude of the vectors and the angle cp between them we can obtain special cases of two dimensional (2D) lattices, some of these are given in Fig. 4.7.

Fig. 4.7: Some two dimensional lattices. . The geometric figure (a square, rectangle or a parallelogram etc.) created by 'these unit vectors is called as the repeat unit or the . Thus a repeat unit or a unit cell may be defined as the smallest repeating unit which on repeating in the two dimensions would create the complete 2-D lattice.

4.4.1 Crystal Systems and Bravais Lattices A three dimensional lattice will be defined in terms of three vectors and the corresponding angles between them. These boxes like moieties called as the unit cells can be used to generate the whde3-D lattice by stacking or aligning these unit cells in the three dimensions. Seven basic shapes are possible for these unit cells and correspondingly there are seven crystal systems. The seven crystal systems, the definitions of their unit cells and the examples are compiled in Table 4.2. The wide , range and variety of the available crystal morphologies can be grouped into just these seven different crystal systems. It means that any crystal, when classified on the basis of interfacial angles and the growth along different directions, will belong to one of these seven .

I A crystal system may have more than one possible crystal lattice. A unit cell that b I contains lattice points only at the comers is called as a primitive (P) unit cell and if a unit cell contains any additional lattice point(s) then it is called as centered unit cell. A &ber of centered unit cells are possible. A unit cell containing an additional lattice point each at the centers of its faces is called a face centered (F) unit cell. A body centered unit cell (I) has a lattice point at the center of the unit cell in additions tq the lattice points at the comers. In some cases, in addition to the lattice points at the corners there are two lattice points located at the centers of any two opposite faces. These are called as end centered (C) unit cells. All the seven crystal systems do not show all Structure are also indicated in Table 4.2. The seven crystal systems when combined with these possibilities gives rise to 14 Bravais lattices (Fig.4.8). We will discuss the Bravais lattices belonging to cubic crystal systcm in details. Table 4.2: The seven crystal systems and their possible lattice types. Systems Axes Angles Examples Possible lattice types Cubic a=b=c a = p = y = 90' NaCI, CsCl, Cu, P, F, I KC1, Zn blende Tetragonal a=b#c a=p=v=900 TiO, (rutilc), 1 p, 1 , 1 Sn (white tin) Orthorhombic 1 aic.b+c I a = 13 = y = 90" 1 CdS04,HgBr,, 1 P, F, I, C Rhombic S Rhombohedra1 a=b=c a = p = y # 90' CaC03(calcite), P

HgS () -- Hexagonal a=brc a = fi = 900. SiO,, ZnO, P 1 y = 120" C (graphite) Monoclinic a#b#c = = 900. KI03, NaHCO,, I P, I p # 90° PbCr04, S (Monoclinic) y + 90° NaHS04, CG2, CuS045H20,

Simple face centered Body centered

Monoclinic Hexagonal Triclinic

Simple Side Centered

Orthorhombic

Simple Side centered Body centered face centered

Tetragonal

Simple Body centered Solid State 4.4.2 Cubic Unit Cells Of the seven crystal systems, let us discuss cubic unit cells in somewhat details. The three possible unit cells viz., simple cubic, body centered cubic and the face centered cubic, belonging to cubic crystal system are shown in Figure 4.9.

j i .' a) b) - c) Fig. 4.9: a) primitive or simple b) body centered and c) face centered cubic unit cells.

Simple Cubic Unit Cell

The simple cubic system is a primitive unit cell with the lattice points at the comers of a Though simple cubic is . A lattice point at the comer of the unit cell is shared by eight unit cells therefore its quite-simpleit is not very contribution to the unit cell will be 118. Further since there are eight such lattice points common in the natural 8x1/8=1. crystals. Only in per cube the number of atoms per simple cubic unit cell comes out to be Thus a certain range of, there is one atom per simple cubic unit cell. Let us see how efficient is the packing in temperature exhibits this such a case. structure.

Fig. 4.10: A simple cubic unit cell, only two atoms are shown for the sake of clarity.

Fig. 4.10 shows a simple cubic unit cell in which only two atoms in the back face are shown for the sake of clarity. Since each edge of the cube has same length, let us take it to be, say a, then as per the figure, assuming the atoms to be spheres, the radius of the atom would be a12. The volume of such a unit cell will be, (edge)' or a3and the volume occupied by the atom ( there is only 1 atom per unit cell) will be

4 Volume occupied by the atom = - xr3 = ..... 3

The atomic packing fraction, the ratio of the volume occupied by the atoms in a unit cell to the volume of the unit cell, is

Thus in a simple cubic unit cell only about half of the volume is occupied by the atoms.

Body Centered Cubic Unit Cell

As discussed earlier, a body centered cubic (bcc) unit cell has a lattice point each at the Sodium, , , comers and one additional lattice point at the center of the cube. The atom in the center , and caesiurn of the cube belongs to the unit cell while the comer atoms, as in the case of simple cubic chloride etc. are typical unit cell, are shared by eight unit cells. Thus the number of atoms per unit cell will be [(8 examples of a hcc crystal. x 118) + 1 = 21. Let us see how efficient is the packing in case of a bcc unit cell. Structure

Fig. 4.11: A body centered cubic (kc) unit cell. (----)indicates the atoms of the front face, (.....) represents that of the back face while the body center atom is shown yith a bold outline.

From the above figure you can observe that in case of bcc lattice the atom at the body center is in touch with eight other atoms along the body diagonals. The body diagonal (shown in red) equals four times the radius of the atom. As you know that for a cube - with edge length, a, the body diagonal will be a, (it can be shown with the help of pythagorous theorem) therefore the radius of the atom would be & a14. The volume occupied by the atoms (2 in number) in the bcc unit cell will be

4 Volume occupied by the atoms = 2 x - nr3 3

which gives the atomic packing fraction to be,

' Thus a bcc structure is more densely packed (68%)as compared to the simple cubic structure (52%).

Face Centered Cubic Unit Cell As shown in Fig. 4.9, a face centered cubic unit cell has an atom at the center of each face besides the corner atoms. Thus it has eight lattice points at the comers and 6 at the face centers. As before since the comer atoms are shared by eight unit cells they contribute (8 x 118) = 1 atom to the unit cell and since each face is shared by two unit cells the six face centered atoms con&ibute 6 x 34 = 3 atoms to the unit cell. Thus a face centered unit cell has 1+3=4 atoms per unit cell.

Crystal lattice Number of lattice points i per unit cell Simple cubic

Body centered cubic

Face centered cubic solid State

Fig. 4.12: A face centered cubic (fcc) unit cell. Only the atoms of the back face are shown in true sizes; the rest of the atoms have been reduced in size for the sake of clarity.

SAQ 2 Calculate the atomic packing fraction for a face centered cubic (fcc) unit cell. You can use the Fig. 4.12, for this purpose.

Thus, in a solid crystal all the space in not occupied by the constituent species. A significant percentage of the space is vacant and it would affect the density of the solid - The nature of the vacant space, called voids, is an experimental property. Let us determine a relationship between the unit cell discussed in the next section. dimensions (determined from X-ray diffraction, discussed above) and the density of a solid.

4.4.3 Calculation of Density of Solids Mass You know that density =-- ..... 4.11 Volume In this unit, the cell-edge lengths and the distance X-ray measurement gives us the cell-edge length. If the cell-edge length is a m, (i.e., between the planes are a meter) then the volume of the unit cell = a3m3 ..... 4.12 given in m or pm units: but it is ysual to state such The mass of an atom of the substance is obtained by dividing the mass of one mole data in A unit also. 1 A = atoms [i.e., atomic mass (A) in kg mol-'1 by Avogadro constant (NA,which is equal to 10-'O m. 6.022~lo2' mol-').

A kprnol-' A Mass of an atom = -- =-kg ..... NA mol-I NA

A simple cubic structure has only one atom per unit cell; hence, mass of unit cell of a simple cubic crystal is given by Eq. 4.13. Substituting the proper values from Eq. 4.12 and 4.13 in Eq. 4.1 1 we get, A the density of a simple cubic cell = -kgm-j 4 LE?

Since, simple cubic, bcc and~fccunit cells have one, two and four atoms per unit cell, the densities of bcc and jkc are given by: 2A Density of a bcc cell =-kgm-3 ..... 4.15 4 a3 Structure ZA ' In general, the density of a cubic unit cell (p) =--i kgm-j ..... 4.17 N4 a where Z is thc number of net atoms per unit cell.

Rearranging Eq.4.17 we get,

ForZ= 1 or2or4, the The ccll-cdge length (a) and the density (p) of a crystal are experimentally unit cell is simple cubic determined. These valucs can be used in Eq 4.18 to calculate the value of Z. It thus or bcc orfic. provides a method of determining the nature of the lattice type of a crystalline solid. Let us work out an example.

Example packs in a cubic unit cell with a cell-edge length (a)of 3.524 x lo-'' m. The density (p) of nickel is 8.90 x lo3kg m-j. Let us find out the unit cell typc for nickel.

Since atomic mass of nickel is 58:7, A = 0.0587 kg mol-'

We calculate the value of Z using Eq. 4.18

= 4 (rounded to the nearest whole number)

since there are four atoms per unit cell, nickel has a fcc lattice.

Why don't you try the following SAQ?

SAQ 3 fom~sbcc crystals. Its cell-edge length is found to be 3.16 x lo-" m. Find the density of tungsten.

The powder diffractioil pattern for a crystalline solid car1 provide infornlation about the nature of the lattice type i.e., we can experimeiltally dctcrmine the value ofZ also. In such a situation, Equation 4.18 can be used to determine the value of Avogadro's constant. .Try the following SAQ. SAQ 4 Solid State

Silver crystallises in a fcc lattice and has a density of 1.05 x lo4 kg m". If the edge length of the cell is found to be 0.407 nm, determine the value of Avogadro's constant. The atomic mass of is 0.1078 kg mol-I.

4.5 CLOSE-PACKED STRUCTURES

The crystal structures of a large number of , alloys and inorganic compounds can be described geometrically in terms of a close-packing of identical spheres, held together by inter-atomic forces. Frequently, the positions of only one kind of atoms or ions in inorganic structures correspond approximately to those of identical spheres in a close- packing while the other atoms or ions are distributed among the voids. All such qtructures are referred to as close-packed structures though these nlay not truly be close- packed. Let us learn about the possible close packed structures and their significance.

The close-packed arrangement of identical spheres in a plane is shown in Fig.4. 13(a). You would observe that in such an arrangement each sphere is in contact with six other spheres Fig.4.13 (b). Such a layer is called a hexagonal close-packed layer.

Fig. 4.13: a) a close packed arrangement of identical spheres in two dimension b) hexagonal arrangement of spheres in (a) and C) two types of trigonal voids (marked as C and B) in the close packed arrangement.

Let this layer be called an A layer. It contains two types of triangular voids, one with the apex of the triangle pointing upwards and the other with the apex downwards. These are labelled as B and C respectively. When we place the second hexagonal close-packed layer on it to generate the 3-D packed structure, the spheres of the second layer can occupy either the B or C type trigonal holes but not both.

In thls process, the trigonal voids which are covered by the sphere get converted into tetrahedral hole. The trigonal holes which are not occupied have another trigonal hole on the opposite type (B type over C and C type over B type) and generate what are called as octahedral holes. We will discuss more about these a little later.

Fig. 4.14: Two layers of close packed spkres ,the second layer occupies only one type ( either B or C , given in Fig. 4.13 ) of triangular voids in the first layer. Structure Similarly when we place the third layer above a B layer this can be different from both A and B layers and called a C layer or it may just be a repetition of the A layer which means that the spheres in the third layer arc exactly on top of the first layer. Thus there are two possibilities ABC or ABA for the three layers. When we put the fourth layer on ABC it would just be a repetition of A layer and on further addition of layers the ABC pattern repeats. On the other hand when we put the fourth layer on ABA it would just be a repetition of B layer and we get the ABAB pattern.

b)

Fig. 4.15: a) Cubic closed packing (ccp) as a result of ABC pattern of close packed spheres (note ABC packing is equivalent to fcc structure); b) hexagonal closed packing (hcp) as a result of ABAB pattern of close packed spheres.

Any sequence of the letters, Thus, the third dimension in a three-dimensional close-packed structure is characterised A. B and C with no two by the number of layers, called identity periods, after which the stacking sequence successive letters alike repeats itself. The two most common close-packed sh-uctures which occur in nature are: represents a possihle manner of close-packing equal (i) the hexagonal close-packing (Izcp) with a layer stacking ABAB. sphereu. (ii) the cubic close-packing (ccp)with a layer stacking ABCARC.

These have identity periods of two and three layers respectively. In these three- dimensional close-packing patterns each sphere is surrounded by and touches 12 other spheres. This'is the maximum number of spheres that can be arranged to touch a givcn sphere and it provides the maximum packing density for an infinite lattice arrangement.

4.5.1 Voids When the atoms are arranged into these three dimensional close-packed structures of spheres some volume remains unoccupied in the form of gaps between the spheres which are called as the voids. There are two lunds of voids in these close packed structures. These are tetrahedral and octahedral voids.

Tetrahedral voids: When the triangular void in a close-packed layer has a sphere directly over it, there results a void with four spheres around it, as shown in Fig. .I..16(a). Such a void is called a tetrahedral void since the four spheres surrounding it are arranged on the comers c;f a regular . Fig. 4.16 (b). Solid State

Fig. 4.16: Tetrahedral void.

Octahedral voids: If a triangular void pointing up (of type B) in a close-packed layer is covered by a triangular void pointing down (of type C) or vice-versa in the adjacent layer, it forms a void which is surrounded by six spheres, Fig. 4.17(a). Such a void is called an octahedral void. The six spheres surrounding the void lie at the comers of a regular , Fig. 4.17(b).

Fig. 4.17: Octahedral void.

Number of tetrahedral and octahedral voids in close-packing of spheres The cubic close packed (ccp) structure gives rise to fcc lattice, Fig. 4.18(a). A fcc unit cell contains eight tetrahedral holes. To locate their positions divide the unit cell into eight small cubicals as shown in Fig. 4.18(b). The tetrahedral holes are located at the center of each of these cubicals Fig. 4.18(c). The tetrahedral arrangement of lattice points is shown for the two of the eight tetrahedral holes.

Fig. 4.18: 1,ocation of tetrahedral holes in a ccp (fcc) structure; the red dots indicate the tetrahedral voids while the black dots represent the fcc lattice points.

On the other hand, an octahedral hole is located at the center of the cube (main cube). Besides this, each edge center also locates an octahedral hole. Two such octahedral voids (in green colour) are shown in Figure 4.19(a); the blue dots represent the lattice The terms holes and points of the neighbouring unit cells. Since there are a total of 12 edges, there would voids are commonly used interchangeably. . be 12 such octahedral holes. Further since each edge is shared by four unit cells, the number of octahedral voids (located at the edge centers) per unit cell will be 12 x ?A = 3. Adding to it the octahedral void located at the cube center, there are a total of 3 + 1 = 4 octahedral voids per unit cell. All the octahedral voids present in a fcc unit cell are shown in Figure 4.19(b). Thus there are 8 tetrahedral'voids and 4 octahedral voids per fcc unit cell (or ccp packing). Structure

There are eight tetrahedral voids and four octahedral voids per fcc unit cell.

Fig. 4.19: Location of octahedral holes in a ccp (fcc) structure; the green dots indicate the octahedral voids while the black dots represent the fcc lattice points. The blue dots represent the lattice points of the neighbouring unit cells.

In an actual crystal structure a particular atomlion can best fit into one or the other kind of void depending on its size relative to that of the close-packed atomslions. The relative sizes of the ions or more appropriately, their radius ratios (r+/r-) can be used to predict their and the shape. The following table may be used for this purpose.

Radius ratio (r+/r-) Coordination Shape number

------Ah- p < 0.155 2 Linear ---- 0.155 - 0.225 3 Plane triangular

- - 0.225 - 0.414 4

0.414 - 0.732 Octahedral - 0.732 - 0.999 8 Body centered cubic ---- I_--

4.5.2 Structures of Ionic Solids As discussed above, it is easy to describe the structure of metallic solids as thcse consist of only one type of species (atoms) which can be put together in terms of close packed structures. On the other hand to describe the structures of ionic solids we need to specify the positions of both the cations as well as the anions in the crystal lattice. In an ionic compound composed of oppositely charged species, if the larger ions are closely packed then the smaller ions can be accommodated in the tetrahedral and 1 or octahedral holes.

The common ionic compounds have the general formulas as MX, MX2, and MXJ, where M represents the metal ion and X denotes the non metallic anion. We would discuss the structures of some ionic compounds of MX and MX2 types.

Structures of the Ionic Compounds of MX Type For the MX type of ionic compounds three structural forms are com~nonlyfound. These are , zinc sulphjde and chloride structures. Let us discuss these in some details. It is advised that you recapitulate the discussion on close packed structures and the number and location of the tetrahedral and octahedral holes in them. Solid State Structure In CsCl the cation and the anions are of comparable sizes (the radius ratio being 0.93) and adopt a bcc structure in which each ion is surrounded by 8 ions of opposite type. The Cs' ions (in the body center position) is surrounded by eight C1-ions located at the corners, Fig. 4.20. Strictly speaking, the CsCl structure is a bcc type structure and not apure bcc structure. As in case of pure bcc structure the species at the corners as well as in the body center are identical and not different as in case of CsC1. Alternatively, the CsCl unit cell may be represented with C1- ions at the body center and Cs' ions at the comers. In fact the CsCl structure can be visualised as consisting of two interpenetrating simple of Cs'and C1- ions respectively. NH4Cl and TlCl are examples of some other ionic compounds having similar structure.

Fig. 4.20: Cesium chloride structure.

Sodium Chloride Structure In case of NaCl the anion (Cl-) is much larger than the cation (Na') . The radius ratio of 0.52 suggests an octahedral arrangement i.e., each cation is surrounded by six anions. NaCl structure is a 6:6 structure which can be visualised as a ccp (or fcc) formed by the chloride ions and the smaller sodium ion occupying the octahedral holes. You would recall that the number of atoms1 ions in the fcc unit cell is 4 and also that there are 4 octahedral holes per unit cell. Thus a unit cell has the following composition; NadCl, or in other words there are four NaCl entities in a unit cell, Fig. 4.21.

Fig. 4.21: Sodium chloride structure.

Alternatively, the structure can be visualised as a ccp lattice of sodium ions and the chloride ions occupying the octahedral sites.

Zinc Sulphide Structure In case of zinc sulphide the radius ratio is s~illsmaller (0.40) that suggests a tetrahedral arrangemenl of the ions. ZnS crystallises in two different structures called as zinc blende and wul~zite.Zinc blende structure is based on a cr.p lattice while wurtzite is based on a hexagonal unit cell. We are going to discuss the zinc blende st~vctureonly. In this structure the sulphide ions form the ccp structure (i.e., these are locat.ed at the comers and the face centers of the cube) and the zinc ions are located in the alternate tetrahedral Structure holes (i.e., four of the eight tetrahedral holes are occupied), Fig. 4.22. This gives a cation to anion ratio of 4:4.

Fig. 4.22: Zinc blende structure of Zinc sulphide.

Structures of the Ionic Compounds of MX2 Type For the ionic compounds of MX2 type two types of structures are commonly found. These are or fluoride structure and rutile (adopted by Ti02).We shall discuss only the fluorite structure. In this structure the caZt ions form a,fcc arrangement and the fluoride ions are located in all the tetrahedral holes, Fig. 4.23. Since ~a"ion is in the,fcc arrangement, there are 4 calcium ions per unit cell. You would recall that in the,fcc structure there are 8 tetrahedral holes so there are eight fluoride ions per unit cell. Thus a unit cell contains 4ca2+ions and 8Fions i.e., the formula is c~TF; or CaF?. - R* rL

Fig. 4.23: or Fluorite structure; calcium ions (shown in yellow) occupy the ccp sites while the F- ions (shown in brown) are in the tetrahedral holes.

Structures of M2X Type Ionic Solids Some of the ionic compounds like NazO have antifluorite structure with the positions of cations and the anions in fluorite structures being interchanged. In Na,O the

, smaller sodium ions occupy the tetrahedral holes in the ccp structure formed by 02- ions. There are 40'-ions and 8Na' ions per unit cell, Fig. 4.24. A number of sulphides and oxides adopt antifluorite structure.

Fig. 4.24: Antifluorite structure adopted by Na20; oxide ions (shown in blue) occupy the ccp sites and the Na' ions ( shown i" orange ) are in the tetrahedral holes. Solid State 4.6 ELECTRICAL PROPERTIES OF SOLID MATERIALS

In the context of electrical properties our concern is whether the material conducts electricity or not and if it does what is the agent of conduction. That is whether the conduction takes place with the help of electrons or the ions are involved in the process? Depending on the answers to these questions the materials find wide range of applications. Let us understand these mechanisms of conduction and the applications of the solid materials on the basis of their electrical conductivity.

4.6.1 Metallic Conduction in Solids : Band Theory You are aware that metals conduct electricity due the presence of valence electrons. This may be explained in terms of electron sea model, which regards metals as a lattice of positive ions immersed in a "sea of electrons" which can freely migrate throughout the solid, or better in terms of molecular orbital theory. The metallic conductivity is not confined to metals and alloys only, many oxides and sulphides and even some conjugated organic molecules also follow this mechanism of electrical conduction. Let us learn about the molecular orbital explanation of the electrical conductivity.

Band Theory of Electrical Conduction You have learnt in Unit 3 that according to molecular orbital theory the atomic orbitals on the binding atoms combine to give molecular orbitals that are spread over the whole molecule. Let us take the example of bonding between lithium (or any other having a single s electron in its valence shell) atoms to understand the band theory of metallic bonding. When two lithium atoms combine we would get two molecular orbitals by the combination of one 2s orbital each on the two atoms. When three lithium atoms combine we would get three MOs and so on. The MO for Liz, Li3 and Li4 are shown in Fig. 4.25.

Fig. 4.25: The molecular orbitals of Liz, Li3, Lb and Li, molecules.

These new molecular orbitals extend over all the atoms of the metal. For large number of atoms, the molecular orbitals obtained are spread over a wide energy range. The ones on the lower end of the energy spread are bonding in nature while the ones at the higher energy end are antibonding. The orbitals of intermediate energy have both bonding and antibonding character in different regions. Every time we add Structure bonding and antibonding sharacter in different regions. Every time we add an atom, we get additional MOs, but since each atom contributes only a single valence electron a molecular orbital is filled for every two atoms. As a consequence the MOs are never more than half filled. If we extend this approach to a large "molecule", Li, containing a.very large number (n) of atoms, we get as many (n) ~~s'thatare very closely spaced in energy. This may be visualised as a band of allowed energies in which the lower half is occupied.

The upper half of the band remains empty and provides a 'green channel (!!)' for the electrons to move freely through the lattice and explain the highly conductive nature of metals. The mobility of the electron in metals is practically unaffected by temperature, but metals do suffer a slight conductivity decrease as the temperature rises. This is due to the disruption of the uniform lattice structure required for free motion of the electrons within the crystal by more vigorous thermal motions of the kernel ions.

You may raise a question that how does or any atom with an outer configuration of s2, conduct electricity, since in such a case all the molecular orbitals (or the band) will be completely filled. You are quite right, the MOs obtained from the s orbitals will be completely filled, however, in such.cases the empty p orbitals on the atoms also can combine and give a band. Though the energy of the 2p orbital of an isolated Be atom is quite high as compared to that of the 2s orbital , the bottom part of the 2p band overlaps the upper part of the 2s band. This provides a continuous conduction band, Fig. 4.26, with a number of unoccupied orbitals.

atoq~Pc Iticrlr~ulor urMt+lo orbitals

Fig. 4.26: The overlapping 2s and 2p bands for the Re, molecule. l'he empty levels in the p band help in conduction.

Only the outermost Thus in most metals there will be bands derived from the outernlost s-, p-. and d atomic orbitals form atomic levels. Some of these would have bands which will overlap as described bands; the inner orbitals remain localised on the above. Whereas in some cases such an overlap may not occur, and a continuous individual atoms and are stretch of energy levels are separated by a band gap or a forbidden 7one. Such a not involved in 'condine. substance is called as an insulator, Fig. 4.27(a). The band gap is sufficiently great to prevent any significant population of the upper band by thermal excitation of electrons The presence of a very from the lower one i.e., the difference in the energy of the filled and unfilled orbital is Intense electric field may be abls to supply the requilrd much more than that available by thermal means. Most molecular crystals and energy, in which case the covalent crystals such as diamond are insulators. insulaior undergoes drelectrlc breakdown. If however, the band gap is quite small so as to allow electrons in the filled band Solid State below it to jump into the upper empty band by thermal excitation, the solid is known as a semiconductor, Fig. 4.27(b). The conductivity of semiconductors increases with temperature. It is in contrast to metals, whose electrical conductivity decreases with temperature as discussed above. In many cases the excitation energy required to promote the electrons from the filled band to the empty band can be provided by absorption of light. In such cases the semiconductor acts as photoconductor. Se, Te, Bi, Ge, Si, and graphite are some of the examples of semiconductors.

I. Fig. 4.27: The band structure in a) an insulator and b) an intrinsic semiconductor and C) a semiconductor containing a dopant element.

The presence of an impurity in a nonconductor or insulator can introduce a new band into the system. If this new band happens to be in the region of band gap or the forbidden zone, it can reduce the band gap which in turn will increase the conductivity, Fig. 4.27(c). There is a lot of industrial interest in designing impurities (called dopes) that can fill the band gap to fit the desired application. The dopant elements are normally atoms whose valance shells contain one electron more or less than the atoms of the host crystal. For example, a atom introduced as an impurity into a silicon lattice possesses one more valence electron than Si. This electron is delocalised within the impurity band and serves as the carrier of charge. The resulting material is known as an n-type semiconductor. On the other hand if arsenic with only three valence electrons is used as a dopant then it creates an electron deficiency or hole. As this vacancy is filled by the electrons from silicon atoms the vacancy hops to another location, thus in such cases a positively charged hole acts charge carrier and the material is called asp-type semiconductor.

4.6.2 Ionic Conduction in Solids In most of the ionic and covalent solids the migration of ions does not take place to any appreciable extent. In such cases the conductivity is observed due to the presence of crystal defectb. Since the number of such defects are dependent on temperature- larger at higher temperature, the conductivity is more at high temperature. In some cases (e.g. P- alumina at 298 K) however, the materials have somewhat specialised structures containing open tunnels or layers that aid in the ionic conduction. This area of solid electrolytes is currently under active research. Structure The crystal defects may help in conduction through various mechanisms. Some of these are: a) Migration through cation vacancies In alkali halides the cations are highly mobile and can move to cation vacancies and in turn make their own site vacant, Fig. 4.28(a). Since in such cases the magnitude of conductance.depends on the number of cation vacancies, these may be increased either by heating or by adding some impurities.

b) Interstitial conductance Some crystalline solids contain a cation in the interstitial sites e.g., Ag' in AgCl crystals. These interstitial ions can move through interstitial spaces or may move by knocking off a lattice cation into an interstitial space, Fig. 4.28(b).

Fig. 4.28: Ionic conductance in solids a) migration through cation vacancies and b) interstitial conduction.

4.7 MAGNETIC PROPERTIES OF S-OLIDS Ferromagnetic materials will undergo a small Materials may be classified by their response to externally applied magnetic fields as mechanical change when diamagnetic, paramagnetic or ferromagnetic. These magnetic responses differ magnetic fields are considerably in their strength. Diamagnetism is a weak property shown by all applied, either expanding materials and opposes applied magnetic fields and result in repulsion. Paramagnetism, or contracting slightly. on the other hand, when present, is stronger than diamagnetism and produces This effect is called magnetostriction. magnetization in the direction of the applied field, and proportional to the applied field. It results in mild attraction. Ferromagnetic effects are much larger than either diamagnetic or paramagnetic effects and sometimes produce magnetisation effects that are many orders of magnitude greater than the applied field and result in strong attraction. Let us understand the meaning and origin of these properties in solid materials.

The orbital motion of electrons creates tiny atomic current loops, which produce magnetic fields. These magnetic fields are augmented by the magnetic moment originating from the spinning electron. The direction of the magnetic dipoles depends on the sense or the direction of the orbital motion and the electron spin. If the directions are reversed so are the effects. In case the atoms or ions constituting the solid contain all paired electtons then the magnetic moments of the paired electrons will nullify each other and there is no net magnetisation. Such solids will show diamagnetic behaviour i.e., when such a substance is kept in an external magnetic field, the number of lines of force passing through it are slightly less thlt that would pass through a vacuum. In most material's the magnetic moments of the electrons cancel and all materials are inherently diamagnetic. In some materials, containing unpaired electrons the cancellation of magnetic Solid State moments is incomplete and these exhibit a magnetisation which is proportional to the applied magnetic field. These materials are said to be paramagnetic i.e., when such a substance is kept in an external magnetic field, the number of lines of force passing through it.are slightly more than that would pass through a vacuum, Fig. 4.29. The paramagnetic substances, however, lose their magnetic property on removing the applied field.

Fig. 4.29: The magnetic lines of force inside a) diamagnetic material and b) paramagnetic material.

Certain materials when placed in applied magnetic field, develop magnetic fields that maybe upto a billion times more than those of the paramagnetic miterials placed in the same field. In such cases the applied field brings the dipole moments of the neighbouring atoms or ions into parallel orientations, Fig. 4.30. These parallel magnetic The term ferromagnetics dipoles get into a kind of 'loclung' and do not randomise even on removing the applied probably reflects the fact field. Such materials are called asferromagnetics and the phenomenon is called as that iron is its most . In antiferromagnetic substances the neighbouring spins adopt common example. antiparallel orientation and as a result the net macroscopic magnetic moment for the crystal is zero. On the other hand ferrimagnetism is somewhat in between ferromagnetism and antiferromagnetism. In this case there is pairing up of the neighbouring magnetic moments but the effect is not tgtal. There are more magnetic moments in one direction than the other i.e., there is a net magnetic moment.

Fig. 4.30: Alignment of individual magnetic moments in a) ferromagnetic b) anti ferromagnetic and c) ferrimagnetie materials. Structure 4.7.1 Temperature Dependence of Magnetic Properties Increasing the temperature increases the thermal energy possessed by the ions and electrons and as a consequence the order is lost. In case of paramagnetic substances an increase in temperature acts to disrupt the ordering and the magnetic susceptibility (X) decreases as shown in Fig. 4.3 l(a). It is said to follow Curie-Weiss law, however in case of ferromagnetic and antiferromagnetic substances the temperature dependence does not follow sinlilar curve. The ferromagnetic substances show a large value of magnetic susceptibility that decreases with increase in the temperature and above a temperature called as ferromagnetic , Tc, the material does not remain ferromagnetic instead it becomes paramagnetic, Fig. 4.3 I (b). On the other hand in case of aitiferromagnetic substances, the magnetic susceptibility increases with increase in the temperature and above a temperature called as Nee1 point, (TN)it acquires paramagnetic behaviour, Fig. 4.3 l(c).

Fig. 4.31: The temperature dependence of the magnetic properties of solids.

4.8 SUMMARY

A solid is an almost incompressible stdte of matter composed of structural units - atoms, molecules or ioris and having a well defined rigid structure. The forces of attraction between the structural units are of different types and are responsible for the differences in their proper-ties. These also form the basis for classifying solids into four different types -- ionic, molecular, covalent and metallic. Besides these the solids are also clnssified into crystalline and amorphous solids depending on their physical appearance and melting behaviour.

The regularity and perfection of a crystalline solid suggests of an ordered internal structure for them which can be determined with the help of X-ray diffraction. The internal structure of the crystalline solids can be represented as an ordered arrangement of its structural units in three dimensions, called as a crystal lattice or simply a lattice. The complete three di~~lensionalstructures or the lattice can be generated by repeating a small repcat unit or a unit cell in the threc dimensions. Corresponding to seven possible basic s11apt.s for these unit cells, there are seven crystal systems. Further, some crystal systems have Inore than one possible cqstal lattice with lattice points other than the ones at the corners. The seven crystal systems when combined with these possibilities gives rise to 14 :;,

For cubic crystal system three types of unit cells are possible with the lattice points at the comers only, comers and the body ccn:cr and the comer and the face centers. These are called as sir~plccubic, body centered cubic and the face centered cubic respectively. A relationship between the cell dimensions, the number of lattice points per unit cell Solid State and the density provides an important way of establishmg the type of unit cell if the density and the cell dimensions are known or in predicting the density of the solid when the cell data and the type of unit cell is known.

The crystal structures of metals, alloys and inorganic con~poundscan be described in terms of a close-packing of identical spheres, held together by inter-atomic forces. Thcsc form two dimensional layers which stack or pile up to generate the three dimensional structure. There are two conlrnon close-packed structures which occur in nature; the hexagonal close-paclung (hcp) with a layer stacking ABAB and the cubic close-paclung (ccp) with a layer staclung ABCABC. In such a packing of spheres some volume remains unoccupied in the form of gaps betwccn the spheres which are called as the voids. There are two kinds of voids, called tetrahedral and octahedral voids, in.these close packed structures. The structures of simple ionic solids can be described by specifying the positions of both the cations as well as the anions in the crystal lattice. In an ionic compound composed of oppositely charged species, if the larger ions are closely packed then the smaller ions can be accommodated in the tetrahedral and 1 or octahedral holes. The structures of some ionic compounds of MX and MX2 types, where M represents the metal ion and X denotes the nonmetallic anion have'been discussed.

Solids show a wide range of electrical and magnetic properties. The band theory , envisages the bonding betwccn a very large number (n) of atoms, to give an equal number of molecular orbitals that are very closely spaced in energy giving a kind of a band of allowed energics. Overlapping of different orbitals on thc bonding atoms give rise to a number of such bands. The relative disposition of these bands deternines whether the material is going to be a conductor or an insulator or a senliconductor. On the basis of their response to externally applied magnetic fields the solids may be classified as diamagnetic, paramagnetic or ferromagnetic. The magnetic nature of a solid is a consequence of the type of alignment of tiny magnets generated by the interaction of the orbital motion and the spin of the electron. The type of magnetic behaviour displayed by a material also depends on the temperature.

4.9 TERMINAL QUESTIONS

1. crystallises into a bcc lattice with a density of 8.93 x lo3kg m-3.If the edge length is found to be 0.33 1 nm, find the mass of a tantalum atom.

2. Coppcr crystallises into a fcc lattice with a density of 1.669 x lo4kg The atomic mass of copper is 63.5 u. Find the radius of a copper atom.

3. The first order reflection of X-rays (having a wavelength of 154 pm) from a set of parallel planes of crystals is observed at 29.30'. Compute the angles at which the second and third order reflections would be observed.

4. Rationalise the semiconducting property of solids in terms of band theory.

5. Discuss the magnetic properties of solids.