14.2 Physics of Solids: Elasticity, Band Theory, Superconductors & Magnetic Materials

Physics of Solids: Elasticity, Band Theory, Superconductors & Magnetic Materials

The properties of a solid, such as whether it stretches, conducts or can be magnetised, follow from how its atoms are arranged and how its electrons are bound. This section links those microscopic ideas to the numbers you are asked to calculate.

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1. Classification of Solids

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2. Stress, Strain and Elastic Moduli

Stress is force per unit area, $\sigma = \frac{F}{A}$ (unit $\text{N/m}^2 = \text{Pa}$). Strain is the fractional change in size and has no unit.

The minus sign in $K$ makes it positive, because an increase in pressure reduces volume. Liquids and gases have a bulk modulus but no shear modulus, since they cannot resist a steady shearing force.

Worked example: A steel wire 2.0 m long with cross-sectional area $1.0 \times 10^{-6}\text{ m}^2$ supports a 100 N load. With $Y = 2.0 \times 10^{11}\text{ Pa}$:

$\Delta L = \frac{FL}{AY} = \frac{100 \times 2.0}{(1.0 \times 10^{-6})(2.0 \times 10^{11})} = 1.0 \times 10^{-3}\text{ m} = 1.0\text{ mm}$

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3. The Stress–Strain Curve

For a ductile metal wire stretched to breaking:

1. Proportional limit: up to here, stress $\propto$ strain (Hooke's law), a straight line.
2. Elastic limit: up to here, the wire returns to its original length when unloaded.
3. Yield point: beyond it the wire undergoes plastic deformation; it stays permanently stretched.
4. Ultimate tensile stress (UTS): the maximum stress the material can bear.
5. Breaking point: the wire fractures.

• Ductile materials (copper, mild steel) show a large plastic region and can be drawn into wires.
• Brittle materials (glass, cast iron) break shortly after the elastic limit with little plastic flow.
• Rubber shows large elastic strain but does not obey Hooke's law well.

Strain Energy

The work done in stretching a wire elastically is stored as strain energy, equal to the area under the force–extension graph:

$U = \frac{1}{2}F\,\Delta L = \frac{1}{2}k(\Delta L)^2$

and the energy per unit volume is $\frac{1}{2}\times \text{stress} \times \text{strain}$. For the wire above, $U = \frac{1}{2}(100)(1.0 \times 10^{-3}) = 0.05\text{ J}$.

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4. Energy Bands: Conductors, Insulators and Semiconductors

In an isolated atom electrons occupy sharp energy levels. In a solid, the levels of neighbouring atoms overlap and spread into energy bands:

• Valence band: the highest band normally filled with electrons.
• Conduction band: the next higher band; electrons here are free to move and carry current.
• Forbidden gap $E_g$: the energy range between them that electrons cannot occupy.

Intrinsic and Extrinsic Semiconductors

• An intrinsic (pure) semiconductor conducts by thermally generated electron–hole pairs; a hole is a vacant bond that behaves like a positive charge carrier.
• Doping with a small amount of impurity greatly increases conductivity:
  • n-type: a pentavalent impurity (phosphorus, arsenic, antimony) donates an extra electron. Electrons are the majority carriers; the impurity atoms are donors.
  • p-type: a trivalent impurity (boron, aluminium, gallium, indium) creates a hole. Holes are the majority carriers; the impurity atoms are acceptors.
• Both doped types are still electrically neutral overall. The p-n junction built from them is the basis of the diode and transistor (Section 14.3).

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5. Superconductors

Certain materials lose all electrical resistance below a critical temperature $T_c$.

• Mercury, the first superconductor discovered (Onnes, 1911), has $T_c \approx 4.2\text{ K}$.
• Some ceramic copper oxides, such as yttrium barium copper oxide, superconduct at around 90 K or above, so they can be cooled with liquid nitrogen (77 K).
• A superconductor expels magnetic field from its interior (the Meissner effect), which allows magnetic levitation.
• Uses include powerful electromagnets for MRI scanners and particle accelerators, magnetically levitated trains, and loss-free power transmission research.

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6. Magnetic Properties of Solids

Every electron has a magnetic moment from its spin and orbital motion. How these moments combine decides the magnetic behaviour:

In ferromagnets, neighbouring atomic moments line up spontaneously within small regions called domains. In an unmagnetised sample the domains point in random directions; an external field makes favourably aligned domains grow and others rotate. Above the Curie temperature thermal agitation destroys the alignment and the material becomes paramagnetic (about $1043\text{ K}$, or roughly $770^\circ\text{C}$, for iron).

Hysteresis

When a ferromagnet is taken through a full cycle of magnetisation, the flux density $B$ lags behind the magnetising field $H$, tracing a hysteresis loop:

• Retentivity (remanence): the $B$ left when $H$ returns to zero.
• Coercivity: the reverse $H$ needed to reduce $B$ to zero.
• The area of the loop is the energy lost as heat per unit volume per cycle.