2.4.16 · D5States of Matter (Quantitative)
Question bank — Defects — Schottky, Frenkel; non-stoichiometric defects
Before you start, make sure these plain-word anchors are solid — everything below leans on them:
- A vacancy is an empty lattice site (an atom that should be there is gone or moved).
- An interstitial site is a small gap between the normal sites — see Interstitial Sites (Tetrahedral & Octahedral Voids).
- Density — see Density of a Unit Cell. If atoms leave but (edge length) stays, drops.
- Electrical neutrality means total positive charge total negative charge, always.
True or false — justify
State true/false, then give the reason — a bare T/F scores zero.
Frenkel defects lower the density of a crystal.
False. The ion only relocates from its site into an interstitial gap; nothing leaves the crystal, so both mass and volume are unchanged, hence is unchanged.
Schottky defects lower the density of a crystal.
True. A cation and an anion are actually removed from the crystal while the edge length (and so the volume) stays the same, so in the mass falls while is fixed.
In a Schottky defect the number of missing cations can differ from the number of missing anions.
False. Removing one leaves net ; only removing one too restores neutrality, so cation and anion vacancies must come in equal numbers.
An F-centre is a stray metal atom sitting in the crystal.
False. The extra metal is ionised; its released electron is trapped in an anion vacancy, and that trapped electron — not any atom — is the F-centre that absorbs light.
Frenkel defects usually involve the anion jumping into an interstitial site.
False. The cation jumps, because cations are smaller and can squeeze into the tiny interstitial holes; large anions cannot fit.
Non-stoichiometric compounds still obey a fixed whole-number formula like .
False. By definition the cation:anion ratio drifts (e.g. , ), which is exactly what "non-stoichiometric" means.
Defects would still exist in a crystal cooled to absolute zero.
False. At the entropy reward in vanishes, so the energy cost dominates and the perfect (defect-free) crystal minimises — see Entropy and Gibbs Free Energy.
Heating a crystal increases the number of Schottky defects.
True. In the exponent's magnitude shrinks as rises, so grows exponentially with temperature.
A missing in is compensated by converting one to one .
False. The vacancy removes charge ; each restores only , so two conversions are needed per vacancy.
Both Schottky and Frenkel defects slightly increase electrical conductivity.
True. In both, ions (or the vacancy they leave) can hop to a neighbouring empty site, giving mobile charge and slightly raised ionic conductivity.
Spot the error
Each statement contains one flaw. Name it and correct it.
"AgBr can only show Schottky defects."
Error: it is limited to one type. AgBr actually shows both Schottky and Frenkel defects, because is small enough for interstitial jumps yet the ions are close enough in size for vacancy pairs.
"ZnO turns yellow on heating because ions jump into interstitial sites."
Error: wrong dominant mechanism. Hot ZnO loses , leaving oxygen (anion) vacancies that trap the freed electrons; the -interstitial route is much higher in energy and not the realistic cause.
"NaCl heated in Na vapour becomes yellow because extra Na atoms colour the crystal."
Error: the metal atom is not the chromophore. A leaves to bond with surface Na, and the electron trapped in the resulting anion vacancy (the F-centre) absorbs visible light, giving the colour.
"For any crystal the Schottky count is ."
Error: that form assumes . In general ; it only collapses to the form when the cation and anion site counts are equal.
"Frenkel defects need a large cation and a small anion."
Error: sizes are reversed. Frenkel needs a small cation (to fit interstitials) and a large anion, i.e. a large size difference — that is why and show it.
"In the factor of two appears because two atoms are removed."
Error: the is combinatorial, not a headcount. It falls out of independently vacating a cation site and an anion site when you minimise with Stirling's approximation — not from "dividing the atoms by two".
" shows only cation Frenkel defects because cations always move."
Error: it is the classic exception. In the anion is unusually small, so shows anion Frenkel defects.
Why questions
Answer with the mechanism, not just a restatement.
Why must Schottky defects always appear in pairs?
Because removing a single ion leaves the crystal charged; a cation vacancy needs an anion vacancy (and vice versa) to keep the whole crystal electrically neutral.
Why does Schottky lower density but Frenkel does not?
Schottky physically removes atoms (mass down, volume fixed → down); Frenkel merely moves an atom inside the same crystal (mass fixed → fixed).
Why do defects exist at all above ?
Making a defect raises entropy ; since and the term rewards disorder more as rises, some defects lower and are thermodynamically favoured — see Entropy and Gibbs Free Energy.
Why is a high coordination number linked to Schottky defects?
High Coordination Number goes with similarly sized ions packed tightly, which favours removing matched vacancy pairs rather than forcing a cation into a cramped interstitial.
Why are non-stoichiometric metal-deficient compounds limited to transition metals?
Restoring charge without adding atoms needs an ion that can take a higher oxidation state (); only elements with more than one oxidation state — transition metals — can do this.
Why does trapping an electron in a vacancy produce colour?
The trapped electron (F-centre) has quantised energy levels whose spacing matches visible-light photons, so it absorbs part of the spectrum and the crystal shows the leftover colour.
Why does metal-excess ZnO become an n-type semiconductor?
The electrons trapped in oxygen vacancies are loosely bound and easily promoted into the conduction band, donating negative carriers — see Semiconductors and Band Theory.
Why is a large cation–anion size difference needed for Frenkel defects?
A big size gap means the cation is small relative to the lattice, so it can physically fit into the tiny interstitial voids without unbearable strain — see Interstitial Sites (Tetrahedral & Octahedral Voids).
Edge cases
The boundary scenarios the topic quietly invites.
If in a compound, which Schottky formula applies?
The general one ; you may not substitute a single because that shortcut assumes equal site counts.
In the limit , what does predict?
The exponent , so and — recovering the perfect defect-free crystal at absolute zero.
In the limit of very high , does the exponential formula predict unlimited defects?
As the exponential , so (or ) — an upper bound set by the number of sites, not infinity; the crystal would melt long before then anyway.
What happens to composition as more cation vacancies form?
grows, more must oxidise to (two per vacancy), so the crystal becomes progressively more iron-deficient and more strongly p-type.
Can a compound show both Schottky and Frenkel defects at once?
Yes — AgBr is the standard case, because its ions are close in size (allowing vacancy pairs) yet is small enough to also jump into interstitials.
If an anion is very small, is anion Frenkel possible?
Yes — small anions can fit interstitial holes, so (small ) shows anion Frenkel defects, the exception to the "cations move" rule.
Does introducing F-centres change the compound's stoichiometry?
Yes — the F-centre in metal-excess NaCl comes with a missing , shifting the formula to , so it is a non-stoichiometric defect, not a Schottky one.
Recall One-line summaries to seal it
Schottky ::: equal cation+anion vacancies → density down, neutrality preserved. Frenkel ::: cation relocates to interstitial → density unchanged. F-centre ::: electron trapped in anion vacancy → colour + semiconduction. Metal deficiency ::: one vacancy needs two higher-oxidation-state cations.