Electrical/Magnetic Properties

When two atoms approach each other, their atomic orbitals overlap and split into two molecular orbitals — a bonding orbital (lower energy) and an antibonding orbital (higher energy). In a crystal containing ~10²³ atoms, this splitting produces ~10²³ closely spaced energy levels that effectively merge into a continuous 'band.' The band formed from valence orbitals is called the valence band; the next higher band is the conduction band. Electrons filling these bands follow the Pauli exclusion principle and Aufbau principle just as in isolated atoms.

The electrical behavior of a solid is determined by two factors: (1) whether the valence band is fully or partially filled, and (2) the size of the band gap (E_g) between the valence band and conduction band. CONDUCTORS (metals): The valence band is partially filled, or the valence and conduction bands overlap. Electrons at the Fermi level can move freely into unoccupied states — electrical conduction requires almost no energy. Example: copper (Cu), where the 4s band is half-filled. INSULATORS: The valence band is completely filled and is separated from the conduction band by a large band gap (E_g > 5 eV). At room temperature, thermal energy (~0.025 eV) is far too small to promote electrons across the gap. Example: diamond (C), E_g ≈ 5.5 eV. SEMICONDUCTORS: Similar band structure to insulators but with a small band gap (E_g ≈ 0.1–3 eV). At room temperature, some electrons are thermally promoted into the conduction band, leaving positive 'holes' in the valence band. Both electrons and holes act as charge carriers. Example: silicon (Si), E_g ≈ 1.1 eV; germanium (Ge), E_g ≈ 0.67 eV. Conductivity of semiconductors increases with temperature (opposite of metals), because more electrons are promoted across the gap.

Intrinsic semiconductors (pure Si or Ge) have limited conductivity. Doping — deliberate introduction of trace impurities — dramatically increases conductivity. n-type doping: adding a Group 15 element (e.g., phosphorus, P) to Si introduces an extra electron per dopant atom. These donor electrons occupy energy levels just below the conduction band and are easily promoted, increasing electron density. p-type doping: adding a Group 13 element (e.g., boron, B) creates a 'hole' — an empty orbital just above the valence band. Nearby electrons jump into this hole, making the hole migrate as a positive charge carrier. A p–n junction (the interface between p-type and n-type regions) is the basis of diodes, transistors, solar cells, and LEDs.

Diamagnetic materials have no unpaired electrons — all electrons are in pairs with opposite spins. When placed in an external magnetic field, the paired electrons generate a tiny opposing magnetic field (by Lenz's law applied at the quantum level), causing a very weak repulsion from the field. Diamagnetism is a universal property of all matter, but it is masked when stronger magnetic effects are present. Examples: NaCl, H₂O, Cu²⁺ (d¹⁰ configuration), Zn²⁺ (d¹⁰). Diamagnetic materials are slightly repelled by magnets.

Paramagnetic materials have one or more unpaired electrons. Each unpaired electron has a net magnetic moment due to its spin (and to a lesser extent, its orbital angular momentum). In the absence of an external field, these moments are randomly oriented and cancel. When an external field is applied, the moments partially align with the field, causing weak attraction toward the magnet. This alignment is overcome by thermal energy, so paramagnetism decreases with increasing temperature (Curie's Law: χ = C/T, where χ is magnetic susceptibility and T is absolute temperature). Examples: O₂ (two unpaired electrons), Fe³⁺ (five unpaired electrons, d⁵ high spin), Mn²⁺. The magnetic moment μ (in Bohr magnetons) is approximated by the spin-only formula: μ = √(n(n+2)) B.M., where n = number of unpaired electrons.

Ferromagnetic materials (Fe, Co, Ni, and certain alloys) exhibit an extreme form of paramagnetism. Within microscopic regions called magnetic domains, quantum mechanical exchange interactions force neighboring electron spins to align parallel to each other — even without an external field. Individual domains may be randomly oriented in an unmagnetized sample, but an external field aligns all domains, producing a strong net magnetization that persists after the field is removed (permanent magnets). Above the Curie temperature (T_C; e.g., 1043 K for Fe), thermal energy disrupts domain alignment and ferromagnetism collapses to paramagnetism. Antiferromagnetism (alternating antiparallel spins, e.g., MnO) and ferrimagnetism (unequal antiparallel spins, e.g., Fe₃O₄ — magnetite) are related phenomena.

• \mu — Magnetic moment (B.M. (Bohr magnetons)): The net magnetic dipole moment arising from unpaired electron spins; 1 B.M. = 9.274 × 10⁻²⁴ J T⁻¹
• n — Number of unpaired electrons (dimensionless): Count of electrons with no paired counterpart of opposite spin in the ion or atom

Problem. An unknown transition metal ion M³⁺ has a measured magnetic moment of approximately 5.92 B.M. (a) How many unpaired electrons does M³⁺ have? (b) Identify M³⁺ and determine whether it is paramagnetic or diamagnetic. (c) Silicon has a band gap of 1.1 eV. Would you classify it as a conductor, insulator, or semiconductor? Explain.

Answer. Fe³⁺ has 5 unpaired electrons, is paramagnetic with μ ≈ 5.92 B.M. Silicon (E_g = 1.1 eV) is a semiconductor.

• Metallic Bonding and the Electron Sea Model — Band theory is the quantum mechanical refinement of the electron-sea model; understanding metallic bonding provides the conceptual foundation before band theory.
• Electron Configuration and Spin — Magnetic properties (paramagnetic, diamagnetic) are determined directly by whether electrons are paired or unpaired, making electron configuration knowledge essential.
• Crystal Field Theory — Crystal field theory explains high-spin vs. low-spin d-electron configurations in transition metal complexes, directly affecting their paramagnetic moments.
• Types of Crystalline Solids — Electrical and magnetic properties depend on the type of solid (ionic, covalent, metallic, molecular); this article provides essential structural context.