Material Science

Primary and secondary bonding

Primary (chemical) bonds are formed by the actual transfer or sharing of electrons between atoms, so that each atom attains a stable (noble‑gas) electronic configuration. They are strong and act between nearest neighbours. The three types are:

• Ionic bond — complete transfer of one or more electrons from an electropositive to an electronegative atom, giving $\text{Na}^+\text{Cl}^-$ type ions held by Coulomb attraction.
• Covalent bond — sharing of a pair of electrons between two atoms (e.g. $\text{H}_2$, diamond).
• Metallic bond — valence electrons form a shared "electron cloud" among the positive ion cores (e.g. Cu, Fe).

Bond energy is of the order of 100–1000 kJ/mol (≈ 1–10 eV per bond) (1 eV ≈ 96.5 kJ/mol), so these materials have high melting points, high elastic moduli and are hard and brittle.

Secondary (physical) bonds arise purely from electrostatic interaction between dipoles or fluctuating charge distributions; there is no transfer or sharing of electrons and the bonds are non-directional and non-saturable. The types are:

• Van der Waals bond — fluctuating dipole (London dispersion), dipole–dipole (Keesom) and dipole–induced dipole (Debye) attractions, e.g. solid argon, inert gases.
• Hydrogen bond — an intermediate case, e.g. ice, water.

Bond energy is only about 0.1–40 kJ/mol (≈ 0.001–0.4 eV per bond), the lower values for van der Waals and the upper for hydrogen bonds, so such materials are soft, have low melting/boiling points and low elastic moduli.

Fundamental differences

Category of ionic bonding and reason

Ionic bonding belongs to the primary (chemical) bonding category. The reason is that in an ionic bond there is an actual transfer of electrons from the cation to the anion, producing ions with stable electronic configurations; the resulting bond is due to strong Coulomb attraction between the oppositely charged ions. Its bond energy is of the order of 100–1000 kJ/mol (e.g. NaCl ≈ 640 kJ/mol), which is characteristic of primary bonds. Hence ionic bonding is a primary bond.

Attractive and repulsive terms for an ionic crystal

For a pair of ions in an ionic crystal, the potential energy as a function of separation $r$ is written as the sum of an attractive and a repulsive term:

$U(r) = -\underbrace{\frac{\alpha e^{2}}{4\pi\varepsilon_{0} r}}_{\text{attractive (Coulomb)}} + \underbrace{\frac{B}{r^{n}}}_{\text{repulsive}}$

where $\alpha$ is the Madelung constant (depends on crystal geometry), $e$ is the electronic charge, $\varepsilon_0$ the permittivity of free space, $B$ the repulsive constant and $n$ the Born exponent ($n \approx 8$–$12$, due to overlap of electron clouds and Pauli exclusion).

Equilibrium binding energy $E_b$

At equilibrium separation $r_0$ the net force is zero, i.e. $\left(\dfrac{dU}{dr}\right)_{r_0}=0$:

$\frac{\alpha e^{2}}{4\pi\varepsilon_{0} r_0^{2}} = \frac{nB}{r_0^{\,n+1}}$

$\therefore \quad B = \frac{\alpha e^{2} r_0^{\,n-1}}{4\pi\varepsilon_{0}\, n}$

Substituting $B$ in $U(r_0)$:

$U(r_0) = -\frac{\alpha e^{2}}{4\pi\varepsilon_{0} r_0} + \frac{\alpha e^{2} r_0^{\,n-1}}{4\pi\varepsilon_{0}\, n\, r_0^{\,n}} = -\frac{\alpha e^{2}}{4\pi\varepsilon_{0} r_0}\left(1-\frac{1}{n}\right)$

The binding energy is the energy required to separate the ions to infinity, i.e. $E_b = -U(r_0)$:

$\boxed{\,E_b = \frac{\alpha e^{2}}{4\pi\varepsilon_{0} r_0}\left(1-\frac{1}{n}\right)\,}$

This is the Born–Landé equation for the equilibrium binding energy of an ionic crystal.

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OR

Soft and hard magnetic materials

Applications (on the basis of magnetic hardness)

• Soft materials need to be magnetised and demagnetised easily and with little loss, so they are used in transformer cores, electric motor and generator armatures, relay and electromagnet cores, chokes, magnetic shielding, and (as ferrites) in high-frequency cores and recording heads.
• Hard materials need to keep their magnetisation against demagnetising fields, so they are used as permanent magnets in loudspeakers, microphones, compasses, magnetos, DC motors, energy meters and magnetic separators.

Option 1: Relevancy of Spectroscopic Techniques in Material Science and Working Principle of Raman Spectroscopy

1. Relevancy of Spectroscopic Techniques in Material Science (3 marks)

Spectroscopic techniques study the interaction of electromagnetic radiation with matter. They are indispensable in material science for the following reasons:

1. Compositional analysis: Techniques such as X-ray photoelectron spectroscopy (XPS), energy-dispersive X-ray spectroscopy (EDS) and Auger electron spectroscopy (AES) identify the elements present in a material, even in trace amounts.

2. Structural and phase identification: X-ray diffraction (XRD) and Raman spectroscopy reveal the crystal structure, phase, polymorph and crystallinity of a material.

3. Chemical state and bonding information: XPS and infrared (IR) spectroscopy give information about oxidation states, functional groups and the nature of chemical bonds.

4. Electronic structure: UV-visible and photoluminescence spectroscopy determine band gap, defect levels and optical transitions in semiconductors and nanomaterials.

5. Surface and interface study: Since most material properties (catalysis, corrosion, adhesion) are surface-controlled, surface-sensitive spectroscopies are essential.

6. Non-destructive and in-situ capability: Most spectroscopic methods are non-destructive and can be used during synthesis, heating or operation, making them valuable for real-time monitoring.

Thus, spectroscopic techniques provide information about composition, structure, bonding and electronic properties, which are essential for designing, characterizing and quality-controlling modern materials.

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2. Working Principle of Raman Spectroscopy (7 marks)

Basic principle: When monochromatic light (usually a laser) is incident on a sample, most of it is scattered elastically without any change in frequency. This is called Rayleigh scattering. A very small fraction (about 1 in $10^6$ to $10^8$ photons) is scattered inelastically, with a change in frequency. This inelastic scattering is called Raman scattering, discovered by C. V. Raman in 1928.

The change in frequency (Raman shift) corresponds to the energy of a vibrational or rotational mode of the molecule:

$$\Delta \bar{\nu} = \bar{\nu}_0 - \bar{\nu}_s$$

where $\bar{\nu}_0$ is the wavenumber of the incident light and $\bar{\nu}_s$ is the wavenumber of the scattered light.

Types of Raman scattering:

• Stokes lines: The scattered photon has lower energy than the incident photon. The molecule gains energy and moves to a higher vibrational level. These lines are more intense.
• Anti-Stokes lines: The scattered photon has higher energy than the incident photon. The molecule loses energy from an already excited vibrational state. These lines are weaker.

Quantum explanation: The incident photon excites the molecule to a short-lived virtual state (not a real energy level). The molecule then returns to a different vibrational level, emitting a photon of changed frequency. The energy difference between the initial and final vibrational levels equals the Raman shift.

Condition for Raman activity: A vibrational mode is Raman active only if it produces a change in the polarizability of the molecule during the vibration. This is different from IR spectroscopy, which requires a change in dipole moment. Hence, molecules such as $N_2$, $O_2$ and $CO_2$ (symmetric) that are IR inactive can be studied by Raman spectroscopy.

Schematic diagram:

   Laser (monochromatic source)
            |
            v
   +-------------------+
   |     Sample        |  <-- scattered light in all directions
   +-------------------+
            |
            v
   +-------------------+
   |  Lens / Collection|  (collects scattered light at 90°)
   +-------------------+
            |
            v
   +-------------------+
   |  Rayleigh filter  |  (removes strong Rayleigh line)
   +-------------------+
            |
            v
   +-------------------+
   |  Monochromator /  |
   |  Spectrometer     |
   +-------------------+
            |
            v
   +-------------------+
   |  Detector (CCD /  |
   |  photomultiplier) |
   +-------------------+
            |
            v
      Raman Spectrum
   (Intensity vs Raman shift)

Working steps:

1. A monochromatic laser beam is focused on the sample.
2. The sample scatters light in all directions; scattered light is collected by a lens at $90^\circ$ (or $180^\circ$).
3. A notch or edge filter removes the intense Rayleigh line.
4. The remaining light is dispersed by a monochromator/grating.
5. A sensitive detector (CCD or photomultiplier) records the intensity at each wavenumber.
6. The plot of intensity versus Raman shift gives the Raman spectrum, whose peak positions identify vibrational modes and hence the material.

Applications: identification of materials and polymorphs, study of carbon materials (diamond, graphite, graphene), strain and stress measurement, and non-destructive analysis of artworks and biological samples.

Phase equilibrium is the state of a system in which two or more phases (solid, liquid, gas) of the same substance coexist at definite temperature and pressure without any net transfer of matter from one phase to another. At such a state the chemical potential (or free energy per mole) of the substance is the same in all the coexisting phases, so the system is thermodynamically stable and its intensive properties remain constant with time. For example, ice and water kept together at 0 °C and 1 atm remain in equilibrium — ice melts and water freezes at equal rates.

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Nature of a one-component (unary) phase diagram

A one-component phase diagram (P–T diagram) is a graphical representation of the conditions of temperature and pressure under which the different phases of a single pure substance are stable. Since the system has one component, by the phase rule

$F = C - P + 2 = 1 - P + 2 = 3 - P$

where F = degrees of freedom, C = number of components, P = number of phases.

Features of the diagram:

1. Areas (regions): Each single-phase region (solid, liquid, vapour) has P = 1, so F = 2. Both temperature and pressure can be varied independently without changing the phase.

2. Curves (lines): Along each curve two phases coexist, so P = 2 and F = 1. Only one variable (either T or P) can be chosen freely; the other is fixed.

   • Fusion curve (solid–liquid): separates solid and liquid. For most substances it slopes to the right (e.g. CO₂); for water it slopes to the left because ice is less dense than water.
   • Vaporisation curve (liquid–vapour): separates liquid and vapour; it starts at the triple point and ends at the critical point.
   • Sublimation curve (solid–vapour): separates solid and vapour.

3. Triple point: The point where all three curves meet. Here P = 3, so F = 0 — the three phases coexist only at one definite temperature and pressure. For water it is at 0.01 °C and 4.58 mm Hg.

4. Critical point: The upper end of the vaporisation curve. Above the critical temperature and pressure the liquid and vapour phases become indistinguishable and only a single supercritical fluid phase exists. For water: 374 °C and 218 atm.

5. Metastable curves: The vaporisation curve can be extended beyond the triple point as the dashed curves — the supercooled liquid (below triple point) and superheated solid — which represent metastable states.

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Example: Phase diagram of water

• Solid region lies at low temperature and high pressure; liquid region in the middle; vapour region at high temperature and low pressure.
• The fusion curve (OB) slopes backwards (to the left) — the melting point of ice decreases as pressure increases, because ice has a larger volume than water (Clapeyron equation, dP/dT = ΔH/TΔV, gives a negative slope when ΔV is negative).
• The vaporisation curve (OC) starts at the triple point O (0.01 °C, 4.58 mm Hg) and rises to the critical point C (374 °C, 218 atm).
• The sublimation curve (OA) lies below the triple point; below 0.01 °C ice sublimes directly to vapour without melting.
• At 1 atm the melting point is 0 °C and the boiling point is 100 °C, both lying on the respective curves.

Thus the unary phase diagram completely describes the stability ranges of the solid, liquid and vapour phases of a pure substance and the exact conditions of their coexistence.

Electronic Polarisation and Electron–Photon (Absorption) Interaction

When electromagnetic radiation enters a solid, it interacts with the atoms, ions and electrons of the material. Of the several possible interactions, the two most important are:

1. Electronic polarisation (interaction of the electric field of the radiation with the electron cloud of atoms/ions), and
2. Electron–photon interaction / electronic absorption (interaction of the photon energy with electrons, leading to electronic transitions).

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1. Electronic Polarisation [4]

Mechanism: When an alternating electric field (E) of the incident electromagnetic wave acts on an atom, the positively charged nucleus and the negatively charged electron cloud are displaced in opposite directions. The centre of negative charge no longer coincides with the centre of positive charge, so an induced electric dipole moment (p) is created.

For small fields the induced moment is proportional to the field:

$$p = \alpha_e E$$

where (\alpha_e) is the electronic polarisability of the atom. The polarisation of the whole solid is

$$P = N\,\alpha_e E$$

where (N) is the number of atoms per unit volume. The relative permittivity (dielectric constant) is then given by the Clausius–Mossotti relation:

$$\frac{\varepsilon_r - 1}{\varepsilon_r + 2} = \frac{N\alpha_e}{3\varepsilon_0}$$

Because the electron cloud is very light, it can follow the field even at optical frequencies ((10^{14}–10^{15}\ \text{Hz})). Hence electronic polarisation is the only polarisation mechanism that operates in the visible and ultraviolet range. The frequency dependence is described by

$$\alpha_e(\omega) = \frac{e^2}{m_e(\omega_0^2 - \omega^2)}$$

where (\omega_0) is the natural frequency of the electron cloud. This gives rise to the refractive index (n = \sqrt{\varepsilon_r}) and to dispersion (variation of (n) with wavelength).

Examples:

• Glass (SiO₂) — the refractive index (n \approx 1.5) in the visible region arises almost entirely from electronic polarisation of Si⁴⁺ and O²⁻ ions. This is why a glass lens bends light and why a prism disperses white light into a spectrum.
• Diamond — strong electronic polarisation of tightly bound carbon electrons gives (n \approx 2.42) and its characteristic brilliance.
• Alkali halides (NaCl, KCl) — transparent in the visible region because their electronic polarisation is purely elastic (no energy absorption) at those frequencies.

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2. Electron–Photon Interaction (Electronic Absorption) [4]

Mechanism: A photon of energy (h\nu) can be absorbed by an electron in the solid only if the energy matches the difference between an occupied and an empty electronic state:

$$h\nu = E_f - E_i$$

The electron is then excited from a lower to a higher energy level. The types of transition are:

• Band-gap (fundamental) absorption: an electron is raised from the valence band to the conduction band. The threshold is the band-gap energy (E_g), so absorption starts at

$$\nu_0 = \frac{E_g}{h}, \qquad \lambda_0 = \frac{hc}{E_g}$$

• Impurity / defect absorption: transitions involving donor or acceptor levels inside the gap.
• Free-carrier (intraband) absorption: electrons in the conduction band absorb photons and move to higher states within the same band; important in metals and heavily doped semiconductors.

The absorption is quantified by the absorption coefficient (\alpha), defined through the Beer–Lambert law:

$$I = I_0 e^{-\alpha x}$$

where (x) is the depth travelled. The material is opaque when (h\nu > E_g) (strong absorption) and transparent when (h\nu < E_g) (photon passes through).

Examples:

• Silicon ((E_g \approx 1.12\ \text{eV})) — absorbs strongly for (\lambda < 1.1\ \mu\text{m}) (near-infrared and visible) but is transparent beyond that. This is the basis of silicon solar cells and photodetectors, where each absorbed photon creates an electron–hole pair.
• Glass — (E_g \approx 8–9\ \text{eV}), so it does not absorb visible photons and is transparent; it absorbs only in the ultraviolet.
• Ruby (Cr³⁺ in Al₂O₃) — absorption of green and violet light by Cr³⁺ ions leaves red light transmitted, giving the ruby its colour; the same absorption–emission cycle produces laser action in the ruby laser.
• Metals (Cu, Ag, Au) — free electrons absorb photons of almost all visible wavelengths and re-emit them, which is why metals are opaque and highly reflective.

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Conclusion: Electronic polarisation controls how a solid refracts and disperses light without absorbing it, while electron–photon interaction controls whether the solid absorbs light and how that absorbed energy is converted into electronic excitation. Together they explain the optical properties — refractive index, colour, transparency and opacity — of solid materials.

Synthesis of Carbon Nanotubes from Graphene — In Principle

Yes, carbon nanotubes (CNTs) can, in principle, be synthesized from graphene. A single-walled carbon nanotube (SWCNT) is geometrically nothing but a rolled-up sheet of graphene — a single layer of sp²-hybridized carbon atoms arranged in a hexagonal honeycomb lattice. If a rectangular strip of graphene is rolled into a seamless cylinder, the result is a SWCNT, with the tube's chirality (armchair, zigzag, or chiral) determined by the rolling direction. Multi-walled carbon nanotubes (MWCNTs) correspond to several concentric graphene cylinders nested inside one another. Therefore, the structural relationship is direct: graphene → rolled graphene → CNT. In practice, this "rolling" is achieved not by mechanically curling a graphene sheet but by bottom-up growth (arc discharge, laser ablation, or CVD), where carbon atoms assemble directly into tubular form on a catalyst; nevertheless, the topological transformation from a planar graphene sheet to a closed tube is geometrically and thermodynamically feasible, so CNTs are conceptually graphene-derived structures.

Applications of Carbon Nanotubes as Advanced Materials

1. Structural and composite materials. CNTs possess exceptional mechanical strength — tensile strength ~100 GPa (many times that of steel) and Young's modulus ~1 TPa — with very low density. They are used as reinforcing fillers in polymer, metal, and ceramic composites for lightweight, high-strength components in aerospace, automobiles, sports goods (bikes, rackets), and construction.

2. Electronics and nanoelectronics. Depending on chirality, SWCNTs can be metallic or semiconducting. They are used to fabricate field-effect transistors (CNTFETs), interconnects, logic circuits, and flexible/transparent conductive films, offering miniaturization beyond silicon limits.

3. Energy storage and conversion. High surface area and electrical conductivity make CNTs excellent electrode materials for supercapacitors, lithium-ion batteries, and fuel cells, improving charge storage, cycling stability, and power density. They also serve as catalyst supports in fuel cells.

4. Field emission devices. Sharp tips and high aspect ratio enable low-threshold field emission — used in field emission displays, electron guns, X-ray sources, and vacuum microelectronics.

5. Sensors and biosensors. CNTs' huge surface area and sensitivity to adsorbed molecules allow detection of gases (NO₂, NH₃, H₂), biomolecules (glucose, DNA), and environmental pollutants at very low concentrations.

6. Biomedical applications. Functionalized CNTs are explored for drug delivery, gene therapy, cancer targeting, bio-imaging, and as scaffolds in tissue engineering, owing to their ability to penetrate cell membranes.

7. Thermal management. CNTs have very high thermal conductivity (~3500 W m⁻¹ K⁻¹), so they are used as thermal interface materials, heat sinks, and in nanofluids for cooling electronics.

8. Future advanced materials. Ongoing research targets CNT-based nanoelectromechanical systems (NEMS), ultra-strong fibers and yarns, lightweight armor, space elevator concepts, quantum wires, and next-generation flexible electronics — positioning CNTs as key materials for nanotechnology-driven industries.

Solid solution (with reference to impurity atoms in solids)

When impurity atoms (called solute atoms) are added to a host solid (called the solvent), they may enter the host crystal lattice without forming a separate phase. The resulting homogeneous crystalline phase is called a solid solution. The impurity atoms may occupy normal lattice sites of the solvent atoms, giving a substitutional solid solution, or they may occupy interstitial voids, giving an interstitial solid solution.

In a substitutional solid solution, a solute atom replaces a solvent atom at a regular lattice point. This creates a substitutional impurity defect. The extent to which such substitution occurs is called the degree of substitutional defect or the solubility limit. It depends mainly on the relative features of the solute and solvent atoms.

Important features of solute and solvent atoms determining substitutional solid solubility:

1. Atomic size factor
   For extensive substitutional solid solution, the difference in atomic radii of solute and solvent should be small.
   If $r_{\text{solute}}$ and $r_{\text{solvent}}$ are the atomic radii, then
   $\frac{|r_{\text{solute}} - r_{\text{solvent}}|}{r_{\text{solvent}}} < 0.15$
   i.e. the size difference should be less than about 15%. A larger difference produces severe lattice strain and limits substitution.
   For interstitial solid solution, the solute atom must be much smaller; its radius should be less than about 0.59 times the solvent radius.

2. Crystal structure
   The solute and solvent should have the same crystal structure (e.g. both FCC, both BCC, or both HCP). If the structures differ, only limited substitutional solubility occurs.

3. Electronegativity
   The electronegativity difference between solute and solvent should be small. A large difference favours compound formation rather than solid solution. For good substitutional solubility, the difference is usually less than about 0.4.

4. Valency
   For metals, a metal of higher valency is generally more soluble in a metal of lower valency than the reverse. In ionic solids, charge neutrality must be maintained. If an impurity ion of different valency substitutes a host ion, charge-compensating defects such as vacancies or interstitials are created.
   Example: In NaCl, substitution of a Na$^+$ ion by a Ca$^{2+}$ ion creates a Na$^+$ vacancy to maintain electrical neutrality.

5. Temperature
   Solubility generally increases with temperature because higher thermal energy and entropy favour mixing of solute atoms into the solvent lattice.

6. Concentration and chemical affinity
   The degree of substitution increases with solute concentration up to the solubility limit. Strong chemical affinity between solute and solvent may lead to precipitation or compound formation, reducing substitutional solubility.

Thus, solid solutions are formed when impurity atoms are accommodated in a host lattice, and the extent of substitutional solid solution is governed mainly by atomic size, crystal structure, electronegativity, valency, temperature, and concentration.

(a) I–V characteristics of a p–n junction diode

The characteristic curve plots diode current $I$ against applied voltage $V$. Forward bias is taken positive, reverse bias negative.

        I (mA)
          |            /
          |           /
          |          /   ← forward bias
          |         /
          |        /
  ────────┼───────/──────── V (volts)
   reverse|      /  Vk (knee, ≈0.7 V Si)
   region |     /
   (µA)   |    /
          |   /
          |  /
          | /
          |/

Key features:

• Forward bias: For small $V$ the current is negligible. At the knee (cut-in) voltage $V_k$ (≈0.7 V for Si, ≈0.3 V for Ge) the current rises sharply and almost exponentially. The relation is $I = I_0\left(e^{V/\eta V_T}-1\right)$ where $V_T = kT/e \approx 26$ mV at room temperature and $\eta \approx 1$–2.
• Reverse bias: A small reverse saturation (leakage) current $I_0$ (of order µA for Ge, nA for Si) flows, nearly independent of voltage. At a large reverse voltage the breakdown occurs (Zener or avalanche) and current increases abruptly.

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OR

Antiferromagnetism vs ferrimagnetism

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(b) Phonons and their role in thermal conductivity

Phonons: In a solid the atoms vibrate about their mean positions. These lattice vibrations are quantised, and a quantum of lattice vibrational energy is called a phonon. Its energy is $E = \hbar\omega$ where $\omega$ is the angular frequency of vibration. Phonons behave like particles (quasi-particles) with momentum $\hbar\vec{k}$ and obey Bose–Einstein statistics. They are responsible for the vibrational (lattice) contribution to specific heat and to thermal conduction in non-metals.

Role in thermal conductivity: Heat in a solid is carried by two agents — free electrons (metals) and lattice waves, i.e. phonons (insulators and semiconductors).

• Phonons travel through the lattice and transport energy from the hot end to the cold end, so they directly produce lattice thermal conductivity $K_{ph}$.
• The phonon thermal conductivity is $K_{ph} = \tfrac{1}{3}\,C_v\,v_s\,\lambda$ where $C_v$ is the lattice specific heat, $v_s$ the speed of sound (phonon velocity) and $\lambda$ the mean free path of phonons.
• Phonon–phonon collisions (Umklapp processes) and scattering by impurities, defects and grain boundaries reduce $\lambda$, hence reduce $K_{ph}$. This is why thermal conductivity falls as temperature rises (more phonons, more collisions) and why disordered or alloyed materials conduct heat poorly.
• In insulators (diamond, glass, quartz) essentially all heat is carried by phonons; in metals phonons add to the much larger electronic contribution.

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OR

Importance of recycling and disposability in materials science

Materials are selected not only for performance but also for what happens to them at the end of their service life. These two issues matter because:

1. Environmental protection: Non-biodegradable and toxic materials (plastics, heavy metals, e-waste) pollute soil, water and air if dumped. Recycling and safe disposal reduce this damage.
2. Conservation of resources: Mineral ores and fossil fuels are finite. Recycled metals (Al, Cu, steel) and glass save raw material and energy — recycling aluminium saves about 95% of the energy needed to extract it.
3. Energy and cost saving: Re-melting scrap is cheaper and less energy-intensive than primary extraction, lowering production cost and carbon emission.
4. Waste reduction and landfill pressure: Growing municipal and industrial waste needs reduction at source; design for recyclability (easy separation, fewer material types, labelling) is now a design requirement.
5. Sustainability and legislation: Governments enforce disposal norms (e.g. e-waste and plastic rules), so materials must be chosen so that products can be reused, remanufactured or recycled.
6. Life-cycle thinking: Materials scientists now evaluate life-cycle assessment (LCA) — energy, emissions and cost from extraction to disposal — so selection balances performance with recyclability and disposability.

Thus recycling and disposability guide the choice of materials, their processing and product design, making technology sustainable.

(a) How lasers differ from ordinary light

Laser light differs from ordinary light in the following ways:

1. Coherence: Laser light is highly coherent — all the emitted photons are in phase in space and time. Ordinary light is incoherent, with random phases.
2. Monochromaticity: A laser emits light of essentially a single wavelength (very narrow linewidth). Ordinary light is a mixture of many wavelengths.
3. Directionality: Laser light is highly directional, forming a narrow, almost parallel beam with very small divergence. Ordinary light spreads out in all directions.
4. Intensity/Brightness: Because of its directionality and coherence, laser light has very high intensity (power per unit area) compared to ordinary light.
5. Polarization: Laser light is usually plane polarized, whereas ordinary light is generally unpolarized.

These properties arise because laser light is produced by stimulated emission (amplified by population inversion in a resonant cavity), while ordinary light is produced by spontaneous emission from independent atoms.

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(b) What is meant by Fermi gas? Brief introduction of Fermi gas on the basis of free electron model of electrons

Fermi gas: A Fermi gas is a collection of a large number of non-interacting particles obeying the Pauli exclusion principle, i.e. particles with half-integer spin (fermions) such as electrons. At absolute zero, the particles fill all available energy states up to a maximum energy called the Fermi energy ($E_F$); above $0\ \text{K}$ the occupation is governed by the Fermi–Dirac distribution:

$f(E)=\frac{1}{e^{(E-E_F)/k_BT}+1}$

In the free electron model of metals:

• Valence electrons are treated as a gas of free fermions moving freely inside the metal, with no interaction between them and a uniform potential inside (the potential well).
• The electrons occupy quantized energy levels in k-space; at $T=0\ \text{K}$ they fill states up to the Fermi wave vector $k_F$, giving the Fermi energy $E_F=\frac{\hbar^2k_F^2}{2m}=\frac{\hbar^2}{2m}\left(3\pi^2 n\right)^{2/3}$ where $n$ is the electron density.
• The density of states is $g(E)\propto E^{1/2}$, and the total number of electrons is obtained by integrating $g(E)f(E)\,dE$.
• This model successfully explains the heat capacity of metals (only electrons near $E_F$ contribute, giving a linear $T$ dependence), electrical and thermal conductivity (Drude–Sommerfeld), and the Wiedemann–Franz law.

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(c) How conductivity of materials is characterized in terms of energy band structures of solids

Solids are classified by the arrangement of energy bands and the size of the band gap $E_g$:

1. Metals (Conductors): The valence band and conduction band overlap, or the conduction band is partially filled. Electrons can move freely with no energy gap, so resistivity is low and conductivity is high. Conductivity decreases with rising temperature (due to increased lattice scattering).
2. Semiconductors: A small forbidden gap ($E_g \approx 0.1\text{–}3\ \text{eV}$, e.g. Si $\approx 1.1\ \text{eV}$, Ge $\approx 0.7\ \text{eV}$) separates a filled valence band from an empty conduction band. At $0\ \text{K}$ they behave as insulators; at room temperature thermal energy excites electrons across the gap, so conductivity is moderate and increases with temperature. Doping (n-type, p-type) greatly enhances conductivity.
3. Insulators: A large forbidden gap ($E_g > 3\ \text{eV}$, e.g. diamond $\approx 5.5\ \text{eV}$). The valence band is completely filled and the conduction band completely empty; thermal energy at room temperature cannot excite electrons across the gap, so conductivity is negligibly small.

Thus conductivity depends on the band gap width and the degree of band overlap/filling.

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(d) A list of a few historical examples of key synthetic discoveries

1. Urea (1828) — Friedrich Wöhler synthesized urea from ammonium cyanate, the first organic compound made from inorganic starting materials, disproving the vital force theory.
2. Aspirin / Acetylsalicylic acid (1897) — Felix Hoffmann at Bayer synthesized a stable, less irritating form of salicylic acid, the first widely used synthetic drug.
3. Bakelite (1907) — Leo Baekeland produced the first fully synthetic plastic (thermosetting phenol-formaldehyde resin).
4. Ammonia by the Haber–Bosch process (1909–1913) — Fritz Haber and Carl Bosch synthesized ammonia from nitrogen and hydrogen, enabling synthetic fertilizers.
5. Nylon (1935) — Wallace Carothers at DuPont synthesized the first fully synthetic fibre (polyamide).
6. Teflon / PTFE (1938) — Roy Plunkett accidentally synthesized polytetrafluoroethylene, a non-stick polymer.
7. Graphene (2004) — Novoselov and Geim isolated a single atomic layer of carbon, a landmark synthetic/2D material discovery.

Linear density is the number of atoms whose centres lie on a direction vector, per unit length of that vector:

$LD=\frac{\text{number of atoms centred on the vector}}{\text{length of the vector}}$

Step 1: lattice parameter. In BCC the atoms touch along the body diagonal, so $\sqrt{3}\,a = 4R$:

$a=\frac{4R}{\sqrt{3}}=\frac{4(0.124\ \text{nm})}{1.732}=0.286\ \text{nm}$

Step 2: the [110] direction. It runs along a face diagonal of the unit cell, from one corner atom to the opposite corner atom of the same face. Its length is

$L=a\sqrt{2}=0.286\times1.414=0.405\ \text{nm}$

Step 3: atoms on the vector. Only the two corner atoms at the ends lie on it (the body-centre atom is not on a face diagonal). Each is shared with the next cell along the line, so each counts as $\tfrac12$:

$N=\tfrac12+\tfrac12=1\ \text{atom}$

Step 4: linear density.

$LD_{[110]}=\frac{1}{a\sqrt{2}}=\frac{\sqrt{3}}{4\sqrt{2}\,R}=\frac{1}{0.405\ \text{nm}}\approx 2.47\ \text{atoms/nm}\ (2.47\times10^{9}\ \text{atoms/m})$

So about 2.47 atoms lie along each nanometre of a [110] line in BCC iron. This is lower than along [111], where atoms touch and $LD = 1/(2R) \approx 4.03$ atoms/nm, because [111] is the close-packed direction of BCC.

Note: 20.35 wt% C lies outside the Fe–Fe₃C diagram (cementite is only 6.67 wt% C). The intended alloy is 99.65 wt% Fe – 0.35 wt% C, a hypoeutectoid steel, and the solution uses that.

Data just below the eutectoid temperature (727 °C): $C_0 = 0.35$, $C_\alpha = 0.022$, $C_{\text{Fe}_3\text{C}} = 6.67$, eutectoid composition $C_e = 0.76$ (all wt% C).

(a) Total ferrite and cementite (lever rule across the $\alpha$ + Fe₃C field):

$W_\alpha=\frac{6.67-0.35}{6.67-0.022}=\frac{6.32}{6.648}=0.951$

$W_{\text{Fe}_3\text{C}}=\frac{0.35-0.022}{6.67-0.022}=\frac{0.328}{6.648}=0.049$

So the alloy has about 95.1% total ferrite and 4.9% cementite.

(b) Proeutectoid ferrite and pearlite. Ferrite that forms above 727 °C is proeutectoid. The remaining austenite, at the eutectoid composition, becomes pearlite. Apply the lever rule just above the eutectoid temperature:

$W_{\alpha'}=\frac{0.76-0.35}{0.76-0.022}=\frac{0.41}{0.738}=0.556$

$W_{p}=\frac{0.35-0.022}{0.76-0.022}=\frac{0.328}{0.738}=0.444$

So the alloy has about 55.6% proeutectoid ferrite and 44.4% pearlite.

Arsenic is a pentavalent impurity in silicon, so it acts as a donor. The majority carriers are electrons. Hence the material is n-type.

(b) Since the donor concentration is
[ N_D=10^{23}\ \text{m}^{-3} ] and at 300 K and 400 K the donors are essentially fully ionized, the electron concentration due to impurity is
[ n \approx N_D = 10^{23}\ \text{m}^{-3}. ]

The conductivity due to impurity is
[ \sigma = n e \mu_e ] where
[ e=1.6\times 10^{-19}\ \text{C}. ]

At 300 K, taking electron mobility in silicon as
[ \mu_e(300)=0.135\ \text{m}^2\text{V}^{-1}\text{s}^{-1}, ] [ \sigma_{300}=10^{23}\times 1.6\times 10^{-19}\times 0.135 ] [ \sigma_{300}=2.16\times 10^{3}\ \text{S m}^{-1}. ]

At 400 K, using lattice-scattering dependence
[ \mu_e \propto T^{-3/2}, ] [ \mu_e(400)=\mu_e(300)\left(\frac{300}{400}\right)^{3/2} ] [ =0.135\times 0.6495=0.0877\ \text{m}^2\text{V}^{-1}\text{s}^{-1}. ] Thus, [ \sigma_{400}=10^{23}\times 1.6\times 10^{-19}\times 0.0877 ] [ \sigma_{400}=1.40\times 10^{3}\ \text{S m}^{-1}. ]

[ \boxed{\sigma_{300}\approx 2.16\times 10^{3}\ \text{S m}^{-1},\qquad \sigma_{400}\approx 1.40\times 10^{3}\ \text{S m}^{-1}} ]

So the conductivity due to the added arsenic impurity is about 2.16×10³ S/m at 300 K and 1.40×10³ S/m at 400 K.

(a) Brass rod of length 0.35 m

Given:

• Initial temperature, (T_1 = 15^\circ\text{C})
• Final temperature, (T_2 = 85^\circ\text{C})
• Temperature rise, (\Delta T = 85 - 15 = 70^\circ\text{C})
• Linear expansion coefficient of brass,
  [ \alpha = 20.0 \times 10^{-6}\ ^\circ\text{C}^{-1} ]
• Young’s modulus of brass,
  [ E \approx 97\ \text{GPa} = 97 \times 10^9\ \text{Pa} ]

Since the ends are rigid, the rod cannot expand. Therefore, the thermal expansion is fully prevented, and a compressive stress develops.

Thermal strain: [ \varepsilon_{\text{thermal}} = \alpha \Delta T ] [ \varepsilon_{\text{thermal}} = 20.0 \times 10^{-6} \times 70 = 1.4 \times 10^{-3} ]

Because expansion is prevented, the mechanical strain is equal and opposite: [ \varepsilon_{\text{mechanical}} = -1.4 \times 10^{-3} ]

Stress: [ \sigma = E \varepsilon_{\text{mechanical}} ] [ \sigma = 97 \times 10^9 \times (-1.4 \times 10^{-3}) ] [ \sigma = -1.358 \times 10^8\ \text{Pa} ] [ \sigma = -135.8\ \text{MPa} ]

The negative sign indicates compressive stress.

[ \boxed{\text{Type: compressive; magnitude } = 135.8\ \text{MPa}} ]

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(b) If the brass rod is 1 m long

The thermal strain and the resulting stress do not depend on the length of the rod. The constraint condition is the same, so the stress remains the same.

[ \boxed{\text{Magnitude of stress } = 135.8\ \text{MPa (compressive)}} ]

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13. Maximum stress at the tip of an internal crack

Given:

• Radius of curvature at crack tip,
  [ \rho = 1.9 \times 10^{-4}\ \text{mm} ]
• Total internal crack length,
  [ 2a = 3.8 \times 10^{-2}\ \text{mm} ] Therefore, [ a = \frac{3.8 \times 10^{-2}}{2} = 1.9 \times 10^{-2}\ \text{mm} ]
• Applied tensile stress,
  [ \sigma_0 = 140\ \text{MPa} ]

Using the standard crack-tip stress formula: [ \sigma_m = 2\sigma_0 \sqrt{\frac{a}{\rho}} ]

Substitute the values: [ \frac{a}{\rho} = \frac{1.9 \times 10^{-2}}{1.9 \times 10^{-4}} = 100 ] [ \sigma_m = 2 \times 140 \times \sqrt{100} ] [ \sigma_m = 2 \times 140 \times 10 ] [ \sigma_m = 2800\ \text{MPa} ]

[ \boxed{\sigma_m = 2800\ \text{MPa} = 2.8\ \text{GPa}} ]

So, the maximum stress at the crack tip is 2800 MPa.

Model the particle as a full-shell (cuboctahedral) fcc cluster with $n$ complete atomic layers (shells) around a central atom:

$N_T=\frac{10n^3+15n^2+11n+3}{3},\qquad N_S=10n^2+2,\qquad N_I=N_T-N_S$

(The question lists "5" twice; the values 2, 5, 20, 50 and 100 are used.)

Discussion

• $N_T$ grows as $n^3$ (volume) while $N_S$ grows only as $n^2$ (area), so the ratio $N_S/N_T \approx 3/n$ for large $n$. Check: at $n=100$, $3/100 = 3\%$.
• For very small particles ($n\approx 2$) most atoms, over three-quarters, are on the surface. Around $n\approx 4$–5 the surface and interior counts are comparable, and by $n=100$ (a particle only about 20–40 nm across) the surface still holds about 3% of the atoms.
• Surface atoms have fewer neighbours (lower coordination number), so they carry unsatisfied bonds and excess surface energy. This is why nanoparticles are far more chemically reactive, are better catalysts, melt at lower temperatures, and sinter more easily than bulk material.
• The large surface-to-volume ratio therefore explains the size-dependent properties of nanomaterials, and these properties change quickly below about 10 nm.