Defects: why every real crystal has some, and how many

Real crystals are never perfect, and the flaws are named by shape and by cause

Intuition

Picture a tray of eggs where every slot holds one egg. That is a perfect crystal. Now take one egg out, or squeeze an extra one into a gap, or swap one egg for a golf ball. The tray still looks like a tray, but it is no longer perfect. Those flaws are defects, and almost every property you care about (strength, colour, conductivity) depends on them.

At absolute zero every atom would sit still on its ideal lattice position. Real crystals are not like that. Lecture 18 says the imperfections are called defects, and they cause or change strength, conductivity and colour.

Defects are sorted two ways.

• By geometry: point (0-dimensional), linear (1-dimensional, also called dislocations), planar (2-dimensional) and bulk (3-dimensional).
• By cause: intrinsic defects are caused by thermodynamics. Extrinsic defects are not required by thermodynamics. Impurities are the usual example.

The most common extrinsic defects are impurities. They are point defects because they affect one lattice site, or a few. A substitutional impurity takes the place of a normal atom. The lecture’s examples are ruby (corundum with chromium, Al$_2$O$_3$:Cr), sapphire (corundum with other impurities such as Fe and Ti), emerald (beryl with chromium) and blue diamond (carbon with boron, C:B). An interstitial impurity sits in a hole between the regular atoms. fcc crystals have both octahedral and tetrahedral holes, and the lecture’s example is H in the octahedral sites of Pd.

Intrinsic point defects come as Schottky pairs or Frenkel pairs

Intrinsic point defects keep the crystal’s formula (they are stoichiometric) and keep it electrically neutral. There are two kinds.

Compare

Schottky defect

Think of a missing tooth and a missing tooth opposite it: a cation vacancy and an anion vacancy, drawn together as a pair. Example: NaCl, where Schottky defects explain the electrical and optical properties. Typical of materials that are highly coordinated and where the ions are close in size.

Frenkel defect

Think of a tooth that has moved out of its socket into a gap. A vacancy plus an interstitial, pulled together as a pair. Example: AgCl. Typical where the ions are of different size, so there are big interstitial holes. The cation is usually smaller, so it usually goes into the hole (an exception is CaF$_2$).

Exam detail

Schottky: NaCl, highly coordinated, ions of similar size. Frenkel: AgCl, ions of different size, large interstitial holes, small cation moves. Both are intrinsic and both keep stoichiometry and electroneutrality.

A defect costs energy, but it buys entropy, so some always form

Intuition

Imagine a car park with a thousand spaces and one missing car. There are a thousand places the gap could be. A completely full car park has exactly one arrangement. Having a gap lets the system pick from many arrangements, and nature likes having many choices. That “number of choices” is entropy. A gap costs a bit of energy, but the choice it unlocks pays for it.

Lecture 18 states it in two sentences. Creating a defect costs energy. Intrinsic defects form spontaneously because they increase entropy. The best number of defects is where the free energy $G = H - TS$ is lowest. The enthalpy term $H$ rises in a straight line with the number of defects. The $TS$ term curves. Their difference has a minimum at a small, non-zero defect count.

The sketch is schematic. It follows the shape of the slide 11 plot: $H$ rising linearly, $TS$ curving, and $G$ with a minimum at the equilibrium defect concentration.

The defect fraction is a Boltzmann factor, derived in four moves

Move 1, count the choices. How many ways can $n_d$ defects be spread over $N$ possible sites? That is “choose $n_d$ from $N$”.

$W = \frac{N!}{n_d!\,(N-n_d)!}$

Move 2, turn the count into entropy. Entropy is $k\ln W$. The lecture uses Stirling’s approximation to remove the factorials.

$\Delta S = k\ln W \approx k\left(N\ln N - (N-n_d)\ln(N-n_d) - n_d\ln n_d\right)$

The lecture names Stirling’s approximation but does not print it. The usual form is $\ln x! \approx x\ln x - x$. The extra $-x$ terms cancel between the three factorials, which is why they do not appear.

Move 3, write the free energy. Each defect costs $\Delta H_d$, so the enthalpy is $n_d\Delta H_d$.

$\Delta G = \Delta H - T\Delta S = n_d\Delta H_d - kT\left(N\ln N - (N-n_d)\ln(N-n_d) - n_d\ln n_d\right)$

Move 4, find the bottom of the curve. Set the slope to zero.

$\frac{d\Delta G}{dn_d} = 0 \quad\Longrightarrow\quad \Delta H_d - kT\left(\ln(N-n_d) - \ln n_d\right) = 0$

Rearranging gives the equilibrium ratio of defects to non-defects.

$\frac{n_d}{N-n_d} = \exp\left(-\frac{\Delta H_d}{kT}\right)$

Defects are far fewer than sites ($N \gg n_d$), so $N - n_d$ is almost $N$:

$\frac{n_d}{N} = \exp\left(-\frac{\Delta H_d}{kT}\right)$

Read it in plain words. A big cost $\Delta H_d$ makes defects rare. A higher temperature makes them more common. This is the same Boltzmann factor you met in module 4.

Try it: worked example

Differentiate the Gibbs energy of defect formation and show where Delta H_d - kT(ln(N - n_d) - ln n_d) comes from.

Show answer and steps

AnswerThe enthalpy term gives Delta H_d. The entropy bracket gives ln(N - n_d) - ln n_d, because the extra +1 and -1 cancel.

1. Start with $\Delta G = n_d\Delta H_d - kT\left(N\ln N - (N-n_d)\ln(N-n_d) - n_d\ln n_d\right)$.
2. The first term is a straight line in $n_d$. Its slope is $\Delta H_d$.
3. $N\ln N$ does not contain $n_d$. Its slope is zero.
4. For $-(N-n_d)\ln(N-n_d)$, the slope is $+\ln(N-n_d) + 1$. The inner $(N-n_d)$ shrinks as $n_d$ grows, which flips the sign.
5. For $-n_d\ln n_d$, the slope is $-\ln n_d - 1$.
6. Add the two entropy slopes: $\ln(N-n_d) + 1 - \ln n_d - 1 = \ln(N-n_d) - \ln n_d$. The $+1$ and $-1$ cancel.
7. Put it together: $\dfrac{d\Delta G}{dn_d} = \Delta H_d - kT\left(\ln(N-n_d) - \ln n_d\right)$. Set it to zero for equilibrium.

Now put numbers in.

Try it: worked example

Lecture 18 takes dH_d = 1 eV. What fraction of sites are vacant at 300 K, and at 1000 K?

Show answer and steps

AnswerAbout 10^-17 at 300 K and about 10^-7 at 1000 K.

1. Use $n_d/N = \exp(-\Delta H_d/kT)$ with $\Delta H_d = 1$ eV.
2. At 300 K the lecture gives $10^{-17}$. At 1000 K it gives $10^{-7}$.
3. Raising $T$ makes $\Delta H_d/kT$ smaller, so the exponential gets bigger. That is why 1000 K has more defects.
4. The lecture’s reading: at room temperature there are very few intrinsic defects, and even at 1000 K it is only a few sites in tens of millions. They still matter for some material properties.
5. Not from the lecture, a check: with $k = 8.617\times10^{-5}$ eV K$^{-1}$, $kT = 0.0259$ eV at 300 K, so $\Delta H_d/kT \approx 38.7$ and $e^{-38.7} \approx 10^{-17}$. This agrees.

Optional aside

Check the 1000 K number yourself. $kT$ at 1000 K is $0.0862$ eV, so $\exp(-1/0.0862) = \exp(-11.6) \approx 9\times10^{-6}$, about $10^{-5}$, not the slide’s $10^{-7}$. The 300 K value checks out: $\exp(-1/0.0259) \approx 2\times10^{-17}$. The slide’s message is the same either way (very few defects, many more when hot). If an exam asks you to calculate, show the working with the formula; if it asks what the lecture says, the slide states $10^{-7}$.

Exam detail

The defect fraction $n_d/N = \exp(-\Delta H_d/kT)$ and its two numbers ($10^{-17}$ at 300 K, $10^{-7}$ at 1000 K for 1 eV). Lecture 18 also tabulates formation enthalpies. Schottky (eV): MgO 6.60, CaO 6.10, LiF 2.34, LiCl 2.12, LiBr 1.80, LiI 1.30, NaCl 2.30, KCl 2.26. Frenkel (eV): UO$_2$ 3.40, ZrO$_2$ 4.10, CaF$_2$ 2.80, SrF$_2$ 0.70, AgCl 1.60, AgBr 1.20, $\beta$-AgI 0.70.

Lines and planes of misfit change how a solid deforms

A linear defect is a line of misfit through the crystal. Lecture 18 gives two.

• Edge dislocation. An extra half-row of atoms ends inside the crystal, like an extra half-sheet pushed into a stack of paper. Under shear it can move (slip), which allows plastic or ductile deformation. The slide compares it to a corn cob with a row that stops partway.
• Screw dislocation. The atoms are arranged in a helix around the line, like a spiral staircase. The slide shows a Burgers circuit and the Burgers vector $\mathbf{b}$.

A planar defect is a whole surface of misfit.

• Stacking fault. The layer sequence breaks. fcc stacks close-packed layers as ABC and hcp as ABA. A fault is a place where the expected sequence is broken.
• Grain boundary. A plane that separates regions (grains) of different crystalline orientation in a polycrystalline solid. Grain boundaries often come from uneven crystallisation conditions, so they are usually extrinsic.

Pitfall

Do not call every defect “intrinsic”. Intrinsic means thermodynamics requires it (Schottky and Frenkel pairs). Impurities and grain boundaries are extrinsic. The summary slide also says that all solids contain intrinsic defects at non-zero temperature, and that defects blur X-ray diffraction.

Recall

Why does a crystal at 300 K always hold a few vacancies, even though each one costs energy?

Each vacancy costs $\Delta H_d$ but raises the entropy because it can sit in many places. The free energy $G = H - TS$ is lowest at a small non-zero number of vacancies, given by $n_d/N = \exp(-\Delta H_d/kT)$.