Solid State Notes

Learning Objectives

• Describe general characteristics of solids

• Distinguish amorphous and crystalline solids

• Define unit cell

• Describe different types of voids and close-packed structures

• Calculate the packing efficiency of different types of cubic unit cell

• Solve numerical problems involving unit cell dimensions

• Explain point defects in solids

Introduction

• Matter exists in three states: solid, liquid, and gas.

• Solids have definite volume and shape.

• Atoms/molecules/ions are tightly held in an ordered arrangement.

• Different solids have different properties, useful for various applications.

• Understanding the structure-properties relationship helps in synthesizing new solid materials.

• The chapter covers characteristics, classification, structure, properties, crystal defects, and their significance.

6.1 General Characteristics of Solids

• Unlike gases, atoms, ions, or molecules in solids are held together by strong forces of attraction.

• Characteristics:

  • Definite volume and shape.

  • Rigid and incompressible.

  • Strong cohesive forces.

  • Short interatomic, ionic, or molecular distances.

  • Constituents have fixed positions and can only oscillate about their mean positions.

6.2 Classification of Solids

• Based on the arrangement of constituents, solids are classified into:

  • Crystalline solids

  • Amorphous solids

• Crystal: From the Greek word “krystallos,” meaning clear ice.

Classification of Solids

• Crystalline Solids:

  • Ionic crystals (e.g., NaCl, KCl)

  • Covalent crystals (e.g., diamond, $SiO_2$)

  • Molecular crystals (e.g., naphthalene, anthracene, glucose)

  • Metallic crystals (e.g., Na, Mg, Cu, Au, Ag)

  • Atomic solids (e.g., frozen Group 18 elements)

• Amorphous Solids: e.g., Glass, rubber.

Crystalline vs. Amorphous Solids

• Crystalline Solids:

  • Long-range orderly arrangement of constituents.

  • Definite shape.

  • Anisotropic.

  • True solids.

  • Definite heat of fusion.

  • Sharp melting points.

  • Examples: NaCl, diamond, etc.

• Amorphous Solids:

  • Short-range, random arrangement of constituents.

  • Irregular shape.

  • Isotropic (like liquids).

  • Pseudo solids (or supercooled liquids).

  • Heat of fusion is not definite.

  • Gradually soften over a range of temperature and can be molded.

  • Examples: Rubber, plastics, glass, etc.

Isotropy vs. Anisotropy

• Isotropy: Uniformity in all directions; identical values of physical properties (refractive index, electrical conductance, etc.) in all directions.

• Anisotropy: Property depends on the direction of measurement. Crystalline solids are anisotropic.

6.3 Classification of Crystalline Solids

6.3.1 Ionic Solids

• Structural units are cations and anions.

• Bound by strong electrostatic attractive forces.

• Cations are surrounded by as many anions as possible, and vice versa.

• Possess definite crystal structure; many are cubic close packed.

• Example: Arrangement of $Na^+$ and $Cl^-$ ions in NaCl crystal.

Characteristics:

• High melting points.

• Do not conduct electricity in the solid state (ions are fixed).

• Conduct electricity in molten state or when dissolved in water (ions are free to move).

• Hard (strong external force needed to change the relative positions of ions).

6.3.2 Covalent Solids

• Constituents (atoms) are bound together in a three-dimensional network by covalent bonds.

• Examples: Diamond, silicon carbide.

• Very hard and have high melting points.

• Usually poor thermal and electrical conductors.

6.3.3 Molecular Solids

• Constituents are neutral molecules.

• Held together by weak van der Waals forces.

• Generally soft and do not conduct electricity.

Types of Molecular Solids:

• Non-polar Molecular Solids:

  • Constituent molecules are held together by weak dispersion forces or London forces.

  • Low melting points; usually liquids or gases at room temperature.

  • Examples: Naphthalene, anthracene, etc.

• Polar Molecular Solids:

  • Constituents are molecules formed by polar covalent bonds.

  • Held together by relatively strong dipole-dipole interactions.

  • Higher melting points than non-polar molecular solids.

  • Examples: Solid $CO<em>2$, solid $NH</em>3$, etc.

• Hydrogen-Bonded Molecular Solids:

  • Constituents are held together by hydrogen bonds.

  • Generally soft solids at room temperature.

  • Examples: Solid ice ($H_2O$), glucose, urea, etc.

6.3.4 Metallic Solids

• Lattice points are occupied by positive metal ions and a cloud of electrons pervades the space.

• Hard and have high melting points.

• Excellent electrical and thermal conductivity.

• Possess bright luster.

• Examples: Metals and metal alloys (Cu, Fe, Zn, Ag, Au, Cu-Zn, etc.).

6.4 Crystal Lattice and Unit Cell

• Crystalline solid: Characterized by a definite orientation of atoms, ions, or molecules in a three-dimensional pattern.

• Crystal lattice: Regular arrangement of these species throughout the crystal.

• Unit cell: A basic repeating structural unit of a crystalline solid.

• A crystal is considered to consist of a large number of unit cells, each in direct contact with its neighbors and similarly oriented in space.

• Coordination number: The number of nearest neighbors surrounding a particle in a crystal.

• Unit Cell Characteristics:

  • Edge lengths or lattice constants: a, b, and c

  • Angles between the edges: $\alpha$, $\beta$, and $\gamma$

6.5 Primitive and Non-Primitive Unit Cell

• Primitive Unit Cell: Contains only one type of lattice point, made up from the lattice points at each of the corners.

• Non-Primitive Unit Cells: Additional lattice points on a face or within the unit cell.

• Seven Primitive Crystal Systems: Cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic, and rhombohedral. Differ in the arrangement of their crystallographic axes and angles.

• Bravais Lattices: 14 possible crystal systems.

Crystal Systems and Lattice Parameters

• Cubic: a = b = c, $\alpha = \beta = \gamma = 90^\circ$

• Tetragonal: a = b ≠ c, $\alpha = \beta = \gamma = 90^\circ$

• Orthorhombic: a ≠ b ≠ c, $\alpha = \beta = \gamma = 90^\circ$

• Rhombohedral: a = b = c, $\alpha = \beta = \gamma ≠ 90^\circ$

• Hexagonal: a = b ≠ c, $\alpha = \beta = 90^\circ, \gamma = 120^\circ$

• Monoclinic: a ≠ b ≠ c, $\alpha = \gamma = 90^\circ ≠ \beta$

• Triclinic: a ≠ b ≠ c, $\alpha ≠ \beta ≠ \gamma$

14 Bravais Lattices

• Cubic:

  • Primitive

  • Body-centered

  • Face-centered

• Tetragonal:

  • Primitive

  • Body-centered

• Orthorhombic:

  • Primitive

  • Body-centered

  • Face-centered

  • End-centered

• Monoclinic:

  • Primitive

  • End-centered

• Triclinic

• Trigonal (Rhombohedral)

• Hexagonal

Number of Atoms in a Cubic Unit Cell

6.5.1 Primitive (or) Simple Cubic Unit Cell (SC)

• Each corner is occupied by an identical atom/ion/molecule.

• Atoms touch along the edges, not diagonally.

• Coordination number: 6.

• Nc = Number of atoms at the corners.

• Number of atoms in a SC unit cell: $\frac{N_c}{8} = \frac{8}{8} = 1$

6.5.2 Body-Centered Cubic Unit Cell (BCC)

• Each corner is occupied, and one atom occupies the body center.

• Corner atoms do not touch each other but touch the body-center atom.

• Coordination number: 8.

• The atom at the body center belongs only to that unit cell.

• Number of atoms in a BCC unit cell: $(\frac{N_c}{8}) + 1 = (\frac{8}{8}) + 1 = 2$

6.5.3 Face-Centered Cubic Unit Cell (FCC)

• Identical atoms lie at each corner and in the center of each face.

• Corner atoms touch those in the faces but not each other.

• Face-center atoms are shared by two unit cells, each contributing $\frac{1}{2}$.

• Number of atoms in a FCC unit cell: $(\frac{N<em>c}{8}) + (\frac{N</em>f}{2}) = (\frac{8}{8}) + (\frac{6}{2}) = 1 + 3 = 4$

6.5.4 Calculations Involving Unit Cell Dimensions

• X-Ray Diffraction Analysis: Most powerful tool for determining crystal structure.

• Interplanar distance (d) between two successive planes of atoms can be calculated using Bragg’s equation: $2d \sin{\theta} = n \lambda$

  • $\lambda$: Wavelength of X-ray used for diffraction.

  • $\theta$: Angle of diffraction.

  • n: Order of diffraction.

• $d = \frac{n \lambda}{2 \sin{\theta}}$

• Edge length of the unit cell can be calculated using these values.

6.5.5 Calculation of Density

• Density ($\rho$) of the crystal can be calculated using the edge length of a cubic unit cell.

• $\rho = \frac{\text{mass of the unit cell}}{\text{volume of the unit cell}}$

• Mass of the unit cell = (total number of atoms belonging to that unit cell) x (mass of one atom)

• Mass of one atom $\text{m} = \frac{M}{N<em>A}$, where M is molar mass and $N</em>A$ is Avogadro's number.

• Mass of the unit cell = $\frac{n M}{N_A}$

• Volume of the unit cell = $a^3$ for a cubic unit cell where a is the edge length.

• Density of the unit cell: $\rho = \frac{n M}{a^3 N_A}$

Example

• Barium has a body-centered cubic unit cell with an edge length of 508 pm. What is the density of barium in g/cm³?

• Given: n = 2, M = 137.3 g/mol, a = 508 pm = 5.08 x $10^{-8}$ cm

• $\rho = \frac{2 \times 137.3}{(5.08 \times 10^{-8})^3 \times 6.023 \times 10^{23}} = 3.5 g/cm^3$

6.6 Packing in Crystals

• Atoms/ions/molecules are treated as hard spheres.

• Tend to pack as close as possible to maximize attractive forces.

• Discusses packing of identical spheres to create cubic and hexagonal unit cells.

6.6.1 Linear Arrangement of Spheres in One Direction

• Only one possibility – spheres in contact with two neighboring spheres.

6.6.2 Two-Dimensional Close Packing

• Two different ways:

  • AAA… type: Linear arrangement repeated in two dimensions. All spheres align vertically and horizontally. Each sphere contacts four neighbors.

  • ABAB… type: Spheres of the second row fit in the depression of the first row. Each sphere contacts six neighbors. ABAB… type is the closest arrangement.

6.6.3 Simple Cubic Arrangement

• Repeating the AAAA type two-dimensional arrangements in three dimensions.

• Spheres in one layer sit directly on top of those in the previous layer. All layers are identical.

• Each sphere is in contact with 6 neighboring spheres.

• Coordination number: 6

Packing Efficiency

• Percentage of total volume occupied by constituent spheres.

• Packing fraction = $\frac{\text{Total volume occupied by spheres in a unit cell}}{\text{Volume of the unit cell}} \times 100$

• Consider a cube with edge length ‘a’.

• Volume of the cube = $a^3$

• 'r' is the radius of the sphere. From the figure, a = 2r => r = $\frac{a}{2}$

• Volume of the sphere = $\frac{4}{3} \pi r^3 = \frac{\pi a^3}{6}$

• In a simple cubic arrangement, the number of spheres belonging to a unit cell is equal to one.

• Total volume occupied by the spheres in a simple cubic unit cell = $\frac{\pi a^3}{6}$

• Packing fraction = $\frac{\pi}{6} \times 100 = 52.38\%$

• Only 52.38% of the available volume is occupied.

6.6.4 Body-centered Cubic Arrangement

• The spheres in the first layer (A type) are slightly separated, and the second layer is formed by arranging spheres in the depressions between the spheres in layer A.

• The third layer is a repeat of the first. ABABAB pattern.

• Coordination number: 8.

• Here, the spheres are touching along the leading diagonal of the cube.

• Using the Pythagorean theorem,

• $\sqrt{3}a = 4r$

• $r=\frac{\sqrt{3}}{4}a$

• Volume of sphere $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi\frac{3\sqrt{3}}{64}a^3 = \frac{\pi\sqrt{3}}{16}a^3$

• Number of spheres belong to a unit cell in BCC = 2

• Total volume of sphere$=2 \times \frac{\pi\sqrt{3}}{16}a^3=\frac{\pi\sqrt{3}}{8}a^3$

• Packing Efficiency = \frac{\frac{\sqrt{3}}{8}\pia^3}{a^3}=\frac{\pi \sqrt{3}}{8} \times 100 = 68\%

• 68% of the available volume is occupied.

6.6.5 The Hexagonal and Face-Centered Cubic Arrangement

Formation of First Layer:

• In this arrangement, the first layer is formed by arranging the spheres as in the case of two-dimensional ABAB arrangements.

• The spheres of the second row fit into the depression of the first row.

• This first layer is designated as ‘a’.

Formation of Second Layer:

• The next layer is formed by placing the spheres in the depressions of the first layer. Let the second layer be ‘b’.

• In the first layer (a), there are two types of voids (or holes), and they are designated as x and y.

• The second layer (b) can be formed by placing the spheres either on the depression (voids/holes) x (or) on y.

• Consider the formation of the second layer by placing the spheres on the depression (x).

• Wherever a sphere of the second layer (b) is above the void (x) of the first layer (a), a tetrahedral void is formed.

• This constitutes four spheres – three in the lower (a) and one in the upper layer (b).

• When the centers of these four spheres are joined, a tetrahedron is formed.

• At the same time, the voids (y) in the first layer (a) are partially covered by the spheres of layer (b);

• Such a void in (a) is called an octahedral void. This constitutes six spheres – three in the lower layer (a) and three in the upper layer (b).

• When the centers of these six spheres are joined, an octahedron is formed.

• Simultaneously, new tetrahedral voids (or holes) are also created by three spheres in the second layer (b) and one sphere of the first layer (a).

• The number of voids depends on the number of close-packed spheres.

• If the number of close-packed spheres is ‘n,’ then the number of octahedral voids generated is equal to n, and the number of tetrahedral voids generated is equal to 2n.

Formation of Third Layer:

• The third layer of spheres can be formed in two ways to achieve the closest packing.

  • aba arrangement - hcp structure

  • abc arrangement – ccp structure

• The spheres can be arranged so as to fit into the depression in such a way that the third layer is directly over a first layer.

• This “aba” arrangement is known as the hexagonal close-packed (hcp) arrangement.

• In this arrangement, the tetrahedral voids of the second layer are covered by the spheres of the third layer.

• Alternatively, the third layer may be placed over the second layer in such a way that all the spheres of the third layer fit in octahedral voids.

• This arrangement of the third layer is different from the other two layers (a) and (b); hence, the third layer is designated (c).

• If the stacking of layers is continued in the abcabcabc… pattern, then the arrangement is called a cubic close-packed (ccp) structure.

• In both hcp and ccp arrangements, the coordination number of each sphere is 12 – six neighboring spheres in its own layer, three spheres in the layer above, and three spheres in the layer below.

• This is the most efficient packing.

• The cubic close packing is based on the face-centered cubic unit cell.

Calculation of Packing Efficiency

• From the relation $AC = 4r$

• $4r = \sqrt{2a}$

• $r=\frac{\sqrt{2}}{4}a$

• Volume of Sphere $\frac{4}{3} \pi r^3= \frac{4}{3}\pi(\frac{\sqrt{2}}{4}a)^3$

• $\frac{2\pi \sqrt{2}}{24}a^3$

• Total number of spheres belonging to a single FCC unit cell is 4

• Volume of spheres in one FCC unit cell = $4 \times \frac{2\pi \sqrt{2}}{24}a^3$
  $=\frac{\pi \sqrt{2}}{3}a^3$

• Packing Efficiency$\frac{\frac{\pi \sqrt{2}}{3}a^3}{a^3}=\frac{ \pi \sqrt{2}}{3} \times 100$
  $=74\%$

Radius Ratio:

• The structure of an ionic compound depends upon the stoichiometry and the size of the ions.

• Generally, in ionic crystals, the bigger anions are present in the close-packed arrangements, and the cations occupy the voids.

• The ratio of the radius of the cation and anion ($\frac{r<em>c}{r</em>a}$) plays an important role in determining the structure.

6.7 Imperfection in Solids:

• According to the law of nature, nothing is perfect, and so crystals need not be perfect.

• They are always found to have some defects in the arrangement of their constituent particles.

• These defects affect the physical and chemical properties of the solid and also play an important role in various processes.

• For example, a process called doping leads to a crystal imperfection, and it increases the electrical conductivity of a semiconductor material such as silicon.

• The ability of ferromagnetic material such as iron, nickel, etc., to be magnetized and demagnetized depends on the presence of imperfections.

Classification:

• Point Defects:

  • Stoichiometric Defects

  • Non-Stoichiometric Defects

  • Impurity Defects

• Line Defects

• Interstitial Defects

• Volume Defects

Stoichiometric Defects in Ionic Solid:

• This defect is also called intrinsic (or) thermodynamic defect.

• In stoichiometric ionic crystals, a vacancy of one ion must always be associated with either by the absence of another oppositely charged ion (or) the presence of same changed ion at the interstitial position, so as to maintain electrical neutrality.

6.7.1 Schottky Defect:

• Schottky defect arises due to the missing of an equal number of cations and anions from the crystal lattice.

• This effect does not change the stoichiometry of the crystal.

• Ionic solids in which the cation and anion are of almost similar size show Schottky defect.

• Example: NaCl.

• The presence of a large number of Schottky defects in a crystal lowers its density.

• For example, the theoretical density of vanadium monoxide (VO) calculated using the edge length of the unit cell is 6.5 $\frac{g}{cm^{-3}}$, but the actual experimental density is 5.6 $\frac{g}{cm^{-3}}$.

• It indicates that there is approximately 14% Schottky defect in VO crystal.

• The presence of Schottky defects in the crystal provides a simple way by which atoms or ions can move within the crystal lattice.

6.7.2 Frenkel Defect:

• Frenkel defect arises due to the dislocation of ions from its crystal lattice.

• The ion which is missing from the lattice point occupies an interstitial position.

• This defect is shown by ionic solids in which cations and anions differ in size.

• Unlike the Schottky defect, this defect does not affect the density of the crystal.

• For example, AgBr, in this case, the small $Ag^+$ ion leaves its normal site and occupies an interstitial position.

6.7.3 Metal Excess Defect:

• Metal excess defect arises due to the presence of a more significant number of metal ions compared to anions.

• Alkali metal halides NaCl, KCl show this type of defect.

• The electrical neutrality of the crystal can be maintained by the presence of anionic vacancies equal to the excess metal ions (or) by the presence of extra cations and electrons present in the interstitial position.

• For example, when NaCl crystals are heated in the presence of sodium vapor, $Na^+$ ions are formed and are deposited on the surface of the crystal.

• Chloride ions ($Cl^-$) diffuse to the surface from the lattice point and combine with the $Na^+$ ion.

• The electron lost by the sodium vapor diffuses into the crystal lattice and occupies the vacancy created by the $Cl^-$ ions.

• Such anionic vacancies occupied by unpaired electrons are called F centers.

• Hence, the formula of NaCl which contains excess $Na^+$ ions can be written as $Na_{1+x}Cl$.

• ZnO is colorless at room temperature. When it is heated, it becomes yellow in color. On heating, it loses oxygen, thereby forming free $Zn^{2+}$ ions.

• The excess $Zn^{2+}$ ions move to interstitial sites, and the electrons also occupy the interstitial positions.

6.7.4 Metal Deficiency Defect:

• Metal deficiency defect arises due to the presence of fewer cations than anions.

• This defect is observed in a crystal in which the cations have variable oxidation states.

• For example, in the FeO crystal, some of the $Fe^{2+}$ ions are missing from the crystal lattice.

• To maintain electrical neutrality, twice the number of other $Fe^{2+}$ ions in the crystal is oxidized to $Fe^{3+}$ ions.

• In such cases, the overall number of $Fe^{2+}$ and $Fe^{3+}$ ions is less than the $O^{2-}$ ions.

• It was experimentally found that the general formula of ferrous oxide is $Fe_xO$, where x ranges from 0.93 to 0.98.

6.7.5 Impurity Defect:

• A general method of introducing defects in ionic solids is by adding impurity ions.

• If the impurity ions are in different valance states from that of the host, vacancies are created in the crystal lattice of the host.

• For example, the addition of $CdCl_2$ to AgCl yields solid solutions where the divalent cation $Cd^{2+}$ occupies the position of $Ag^+$.

• This will disturb the electrical neutrality of the crystal.

• To maintain the same, a proportional number of $Ag^+$ ions leave the lattice.

• This produces a cation vacancy in the lattice; such kinds of crystal defects are called impurity defects.