Solid State · TOPIC LESSON
Connect crystal defects and charge compensation to density, semiconductor carriers and collective magnetism.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. Check the current official syllabus separately; historical question frequency does not establish eligibility or predict future questions.
CONNECT THE CONCEPT TO THE EXAM
These selected examples are separate from the original practice below. Each identifies its exam, year and wording. They do not represent every paper or every selected year.
No source-verified exam questions have been published for this lesson yet. The worked examples and practice below are original teaching material, not past-exam questions.
01 / UNDERSTAND & APPLY
An ideal crystal repeats a structural pattern, but real solids contain vacancies, displaced particles, impurities, dislocations and grain boundaries. A defect is a departure from an ideal reference lattice, not necessarily a useless or damaged material. Controlled defects can improve mechanical, electrical or optical behaviour.
A vacancy is an unoccupied normal lattice site. An interstitial particle occupies a position between normal sites. A substitutional impurity replaces a host particle at a normal site. Point defects involve local sites, while dislocations are line defects and grain boundaries separate differently oriented crystalline regions.
Creating a defect usually costs energy, but the number of ways to distribute defects increases entropy. A finite equilibrium population can therefore be favourable at nonzero temperature. Rapid preparation or cooling can also trap nonequilibrium arrangements, so measured defects depend on processing history as well as composition.
For ionic crystals, charge neutrality is a central bookkeeping constraint. A missing positive ion has an effective negative charge relative to the perfect lattice, while a missing negative ion has an effective positive charge. Effective defect charge describes the difference from the reference site, not a new free particle with an arbitrary charge.
ORIGINAL PRACTICE · 1
At a normal lattice site.
An interstitial impurity occupies an additional space between normal sites; the distinction concerns location.
02 / UNDERSTAND & APPLY
A Schottky defect set removes ions in the stoichiometric ratio needed to preserve charge neutrality. In a simple MX crystal with M⁺ and X⁻, one cation vacancy accompanies one anion vacancy. In MX₂ with M²⁺ and X⁻, the neutral set instead contains one cation vacancy and two anion vacancies; “always one of each” is incorrect.
Because particles are absent from the crystal interior, Schottky vacancies reduce its mass per ideal lattice-site volume. In the simple constant-volume comparison, density decreases. The bulk cation:anion ratio remains stoichiometric when neutral formula-unit sets are missing.
A Frenkel defect pair displaces an ion from its normal site into an interstitial site, leaving a vacancy behind. The displaced ion remains in the solid, so the pair preserves total composition and approximately preserves density in the school-level fixed-volume model. Small cations in suitable lattices commonly provide examples; AgCl and AgBr can exhibit such disorder.
Neither name is a statement that only one defect exists in a whole sample or that a material can have only one possible defect mechanism. Temperature, pressure and composition determine populations. The essential distinction is missing stoichiometric sets versus vacancy–interstitial displacement pairs.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
An ideal MX crystal has 1.00×10⁶ cation sites and the same number of anion sites. It develops 2000 Schottky pairs, with no volume change. Find vacancy fraction on each sublattice and relative density.
Answer 0.200% vacancies on each sublattice; density decreases by 0.200%.
ORIGINAL PRACTICE · 1
Two.
Removing one +2 ion and two −1 ions removes net zero charge and one complete formula unit.
ORIGINAL PRACTICE · 2
The displaced ion remains inside the solid.
Mass is retained; with approximately unchanged volume the density is approximately unchanged.
03 / UNDERSTAND & APPLY
An anion vacancy can trap an electron, forming an F centre in a common alkali-halide description. The electron compensates the vacancy’s effective positive charge and can absorb visible light through transitions between defect-related states, producing colour. A missing anion is therefore not always compensated by a missing cation.
Metal excess can also occur through interstitial cations with compensating electrons in suitable solids. Nonstoichiometric formulas summarize the average composition over many lattice sites; a formula such as Fe₀.₉₅O does not imply a molecule containing a fractional individual iron atom.
Metal deficiency is often balanced by oxidation of some remaining cations. In the simplified Fe₁₋ₓO model containing O²⁻, Fe²⁺ and Fe³⁺, let y be the number of Fe³⁺ ions per oxygen. Charge balance gives 2(1−x−y)+3y=2, hence y=2x. The remaining Fe²⁺ amount is 1−3x per oxygen.
The fraction of occupied iron sites that are Fe³⁺ is y/(1−x), not simply y. This difference between a per-oxygen amount, a fraction of ideal sites and a fraction of occupied metal sites is a frequent source of errors. Real defect clustering and phase stability can be more complex than the simple isolated-vacancy model.
ORIGINAL WORKED EXAMPLE
For Fe₀.₉₅O in the stated Fe²⁺/Fe³⁺ vacancy model, find Fe²⁺ and Fe³⁺ amounts per oxygen and the fraction of iron that is Fe³⁺.
Answer 0.85 Fe²⁺ and 0.10 Fe³⁺ per O; about 10.5% of the iron is Fe³⁺.
ORIGINAL PRACTICE · 1
No; it is an electron associated with an anion vacancy.
An anion vacancy has effective positive charge relative to the ideal lattice and can trap an electron.
04 / UNDERSTAND & APPLY
An impurity with the same charge as the host ion can replace it without requiring charge compensation solely from that substitution, though size and bonding still affect solubility. An aliovalent impurity has a different charge and requires compensating defects or electronic carriers.
For example, Sr²⁺ replacing Na⁺ in NaCl introduces one extra positive charge relative to the original site. One additional Na⁺ vacancy can compensate each substituted Sr²⁺ in a simple vacancy-compensated model. The chloride sublattice can remain full.
Using one chloride site as the composition basis, a model formula is Na₁₋₂ₓSrₓ□ₓCl, where □ denotes a cation vacancy. The cation-site occupancies sum to (1−2x)+x+x=1 and charge sums to (1−2x)+2x=1, balancing Cl⁻. Both tests are necessary.
Doping does not always create vacancies: compensation may involve electrons, holes or changes in oxidation state depending on the host and conditions. A charge-balance equation identifies possibilities but does not by itself establish which microscopic mechanism is thermodynamically preferred.
ORIGINAL WORKED EXAMPLE
A model NaCl region has 1000 chloride sites, all occupied, and 1000 cation sites. Twenty cation sites contain Sr²⁺. If compensation occurs only by sodium vacancies, how many Na⁺ and vacancies are present?
Answer 960 Na⁺, 20 Sr²⁺ and 20 cation vacancies.
ORIGINAL PRACTICE · 1
Each substitution adds one positive charge relative to the host site.
A vacancy, electronic carrier or other compensating change is needed to preserve overall neutrality.
05 / UNDERSTAND & APPLY
For a perfect cubic crystal, ρ=ZM/(N_Aa³), where Z is formula units per cell, M is molar mass, a is cell edge and N_A is Avogadro’s constant. Defects alter actual mass or occupancy, and sometimes lattice dimensions. The perfect-cell formula must therefore be adjusted rather than treating every ideal site as occupied.
If neutral formula units are missing with fraction f and cell volume is unchanged, a simple model gives ρ_defective≈(1−f)ρ_ideal. For substitutional doping, changing atom masses also matters; adding a heavy ion and a vacancy cannot be analyzed by vacancy count alone without a mass balance.
Vacancies and interstitial sites can allow ionic hopping. A crystal with ions fixed at lattice sites may conduct poorly at low temperature, yet become an ionic conductor as defects and mobility increase. Ionic conduction transports ions; it is distinct from electronic conduction through electrons and holes.
In a dilute noninteracting vacancy model, the equilibrium fraction often has an activated form proportional to exp(−ΔG_f/k_BT), with the appropriate formation free energy and site counting. Higher temperature generally increases equilibrium defect populations, but measured conductivity also includes mobility and multiple possible carriers.
06 / UNDERSTAND & APPLY
In a solid, interaction among many atomic orbitals produces closely spaced allowed energy states organized into bands. A valence band and conduction band separated by an energy gap provide a useful model for many semiconductors and insulators. A metal has available states near the occupied states, through a partly filled band or overlapping bands, so carriers can respond readily to an electric field.
An ideal intrinsic semiconductor at finite temperature can generate electron–hole pairs by exciting electrons across the gap. The conduction-band electron and the corresponding valence-band hole both contribute to current. An insulator has a gap and carrier availability that make electronic conduction much less accessible under ordinary conditions; the difference is quantitative and context-dependent rather than a universal magic gap value.
In a common semiconductor model, σ=e(nμ_e+pμ_h), where n and p are electron and hole number densities in m⁻³, mobilities μ in m² V⁻¹ s⁻¹ and e is the positive elementary-charge magnitude. Conductivity has units S m⁻¹. Resistivity is 1/σ in Ω m, not the reciprocal of resistance without a geometry factor.
Metal resistivity commonly increases as temperature rises because increased lattice vibration scatters carriers. Intrinsic semiconductor conductivity often increases strongly with temperature because carrier generation dominates. Doped semiconductors have different temperature regimes, so neither trend should be asserted for every material at every temperature.
ORIGINAL WORKED EXAMPLE
A model semiconductor has n=1.0×10²¹ m⁻³, electron mobility 0.10 m² V⁻¹ s⁻¹ and negligible hole contribution. Use e=1.60×10⁻¹⁹ C to estimate σ and resistivity.
Answer σ≈16.0 S m⁻¹; resistivity≈0.0625 Ω m.
ORIGINAL PRACTICE · 1
No.
A hole is a missing electron in the band description and behaves as a positive mobile carrier; the nuclei need not move with it.
07 / UNDERSTAND & APPLY
Silicon has four valence electrons and forms a covalent network. A substitutional group 15 dopant such as phosphorus contributes an extra valence electron relative to silicon. In the simple donor model, a shallow level near the conduction band permits that electron to become mobile, producing n-type behaviour with electrons as majority carriers.
A group 13 dopant such as boron supplies one fewer valence electron. An acceptor level near the valence band can accept an electron, leaving a mobile hole, producing p-type behaviour. Majority carrier type does not mean the specimen acquires a large permanent net negative or positive charge. Ionized dopant centres compensate carrier charge overall.
At suitable temperatures with nearly complete dopant ionization and negligible compensation, n≈N_D for a donor-dominated sample or p≈N_A for an acceptor-dominated sample. If both kinds are present, compensation changes the net majority concentration. At high enough temperature intrinsic electron–hole generation can dominate.
In the nondegenerate thermal-equilibrium model, np=n_i². Increasing one carrier population through doping therefore decreases the minority population at the same temperature. This relation has assumptions; it should not be applied unchanged to strongly nonequilibrium illumination or very heavy degenerate doping.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
At a stated temperature, n_i=1.0×10¹⁶ m⁻³ and an equilibrium nondegenerate n-type sample has n=1.0×10²¹ m⁻³. Find p.
Answer p=1.0×10¹¹ m⁻³.
ORIGINAL PRACTICE · 1
No.
Mobile electrons are balanced by positively ionized donor centres in the usual electrically neutral bulk model.
08 / UNDERSTAND & APPLY
Diamagnetic response opposes an applied field and is usually weak; it is a background property of matter and dominates in many systems with no permanent local moment. Paramagnetic materials have moments that respond preferentially along a field but do not show the same spontaneous long-range ordering in the simple regime. Unpaired electrons are important, but their presence alone does not prove ferromagnetism.
Ferromagnetic order aligns neighbouring moments parallel within an ordered region, producing spontaneous magnetization below an ordering temperature. Iron, cobalt and nickel are familiar examples under ordinary conditions. Exchange interactions are quantum-mechanical in origin and cannot be reduced to a picture of tiny classical bar magnets merely touching.
In a simple antiferromagnet, opposing sublattice moments cancel. In a ferrimagnet, opposing sublattice contributions are unequal, leaving a net moment. Magnetite Fe₃O₄ is ferrimagnetic; MnO is a standard antiferromagnetic example below its Néel temperature. The ordering temperature must be respected when assigning a material’s behaviour.
A compensated antiferromagnet can have zero spontaneous net moment without being diamagnetic. Its local moments and collective arrangement still exist. Conversely, a ferromagnetic specimen can have a small overall moment when differently oriented domains cancel, even though each domain is internally ordered.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
Two ideal sublattices contribute +5 and −3 magnetic-moment units per repeat. What is the net, and which ordering description fits this antiparallel unequal case?
Answer Net +2 units; ferrimagnetic ordering in the stated two-sublattice model.
ORIGINAL PRACTICE · 1
Different microscopic arrangements can cancel at the specimen scale.
Antiferromagnetic sublattices cancel locally in the simple model, while ferromagnetic domains may cancel across larger regions.
09 / UNDERSTAND & APPLY
A domain is an ordered region with a particular magnetization direction. An applied field can favour growth or rotation of domains aligned with it. When the field is removed, some alignment may persist, giving remanence; a reverse field may be needed to reduce net magnetization to zero, giving coercivity.
Hysteresis means the response depends on magnetic history. Domain-wall motion and pinning by defects can influence coercivity and energy loss. A structural defect therefore can affect magnetic performance without changing every ion’s unpaired-electron count. Soft magnetic materials reverse relatively readily, whereas hard magnetic materials resist reversal; these terms do not simply mean mechanical softness or hardness.
Above the Curie temperature, ordinary ferromagnetic or ferrimagnetic long-range order is lost. Antiferromagnets have a Néel ordering temperature. Thermal disorder competes with collective alignment; heating does not require removal of every local electronic moment to remove the ordered state.
Solid-state properties connect composition, structure and interactions: vacancies affect transport, dopants affect carriers, and lattice defects can affect magnetic domain motion. No single defect label determines all these outcomes. Temperature, phase and the relevant measurement remain part of a complete answer.
ORIGINAL PRACTICE · 1
Yes.
Defect pinning can impede domain-wall motion, altering reversal behaviour even when overall composition is nearly unchanged.
10 / UNDERSTAND & APPLY
For a defect calculation, write an explicit composition basis, count occupied sites and vacancies, and impose charge neutrality. Distinguish fractions of ideal sites from fractions of occupied ions. Then calculate mass or density only with the volume and masses specified.
For electrical behaviour, identify ionic versus electronic carriers, band availability, dopant type and the temperature regime. For magnetic behaviour, identify local moments, their ordering and whether domain cancellation affects the measured specimen.
The worked examples and practice are original idealized cases. Their simplifying assumptions allow precise calculations without pretending that every real crystal has isolated defects, one carrier species or a single magnetic domain.
KEEP THE BIG IDEAS
READ WITH CONTEXT
The teaching notes, worked explanations and original practice are written for this lesson. Separately labeled selected exam questions identify their exam, year, source and whether wording is paraphrased or an exact excerpt. Educational references do not certify current exam coverage.
Educational reference inspected in HTML. Teaching narrative, numerical choices and solved problems are original.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Primary university or author/publisher educational HTML inspected; original models, calculations and diagrams.
Nonlinked scope reference for nonstoichiometry, impurity defects, electrical and magnetic classifications.
Analytics use a separately reviewed, incomplete historical corpus. Opening a topic does not change its counts or denominator.