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### Lattice defects and their dimensionality

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So far, we have used the fact that ideal crystals
show perfect translational order to describe their structure by identifying a repeat unit and a pattern to describe how exactly it is repeated to fill space. All this is based on perfect symmetry. In reality, crystals are neither ideal nor infinitely large. These deviations from the ideal structure as represented by the space group are called lattice defects.

Lattice defects can be classified according to their dimensionality
as zero-, one- or two-dimensional defects. Point defects
are zero-dimensional: an atom isn't where it is supposed to be according to the ideal description that we have, or it is the wrong type of atom. An isolated atom on the surface of a crystal can also be regarded as a point defect.

Dislocations
are one-dimensional or linear defects: An additional lattice plane is inserted part-way into the crystal, causing a linear discontinuity at its end while the structure on either side of the line remains perfect.

Grain boundaries
separate small crystals which each can be described as perfect. A two-dimensional network of distortions is needed to adapt the misaligned lattices of the individual crystalline grains to each other. The surface
of a finite crystal can also be seen as a two-dimensional defect, as can be stacking faults
where the periodic repetition of layers, e.g. in a close-packed
structure is disrupted.

### Basic point defects

Perhaps the most obvious type of point defect is a vacancy:
an atom is missing from its lattice point. This is illustrated in the figure by a purple square.

If the orange atoms are anions (negatively charged) and the green ones are positive cations, then a negative charge is missing in the lattice at the position of the vacancy. In other words, the vacancy has a positive charge relative to the lattice.
As a result, nearby anions are drawn into the gap, and nearby cations are pushed away from it - they all adapt to the local distortion of the lattice potential.

The lattice as a whole doesn't collapse around the vacancy because the small adjustments to atom positions balance each other out.

An interstitial
atom is one that is located in between regular lattice positions. Of course, there is little space available, and the extra atom distorts the local potential quite dramatically. Nearby atoms are repulsed, particularly where like charges are involved.

Vacancies and interstitials are both highly mobile (see diffusion
below). If one of each meet, they will usually annihilate,
i.e. the interstitial atom will simply fill the vacancy and become a regular lattice atom. But, conversely, a vacancy-interstitial pair can be formed spontaneously if a phonon, a lattice vibration, supplies sufficient energy for an atom to be ejected from its lattice point.

This defect creation and annihilation mechanism means that at above-zero temperature, where there are always phonons active, a certain number of interstitials and vacancies will always be present: it is the crystal's way of absorbing thermal energy. The defect creation energy $E_{def}$
needed to form a pair depends on the particular material. If we know it and the thermal energy, $k_BT$, we can calculate the concentration of these intrinsic defects
using the Boltzmann distribution:
$$\frac{n}{N-n}\approx\frac{n}{N}=\exp{\left(-\frac{E_{def}}{k_BT}\right)}\qquad,$$ considering that the number $n$ of defects has to be small compared to the number $N$ of atoms in the crystal.

Impurities
are the third type of point defect. A 'wrong' atom is placed on a regular lattice point. If the charge on the impurity atom is (even fractionally) different from that of a regular lattice atom, the lattice will be distorted. The picture shows a doubly charged cation (purple) on a site which is regularly occupied by a cation with a single charge (orange). The net result is a single extra charge relative to the lattice,
which will attract the neighbouring (green) anions and repel the cationic neighbours.

 NaNa× Na+ ion on a regular site vO•• oxygen vacancy, i.e. a missing O2- ion Lii• interstitial Li+ ion CuNi' Cu+ impurity on a Ni2+ site

A common way of describing point defects is the Kröger-Vink notation.
For each atom, its chemical symbol is given along with the chemical symbol of the atom that would normally occupy its lattice point and the charge relative to the lattice. Positive relative charges
are indicated by dots, negative ones by dashes, and atoms that are neutral relative to the lattice are marked by a '×' symbol. Some examples are given in the table. The symbol for a vacancy or an interstitial is 'v' or 'i', respectively - always in lower case to avoid confusion with vanadium and iodine atoms or ions.

### Compound point defects

Compound defects
are several point defects associated with each other. These can be transient or persistent,
depending on whether their proximity stabilises or destabilises the crystal in terms of its free enthalpy.

A Schottky pair
is a vacancy near the surface
of a crystal, where the missing atom is placed on a regular lattice point on the surface. This is one mechanism by which vacancies are created. Since the atom, once it has found its surface site, is indistinguishable from other atoms on surface lattice points, there is no reason for the vacancy to remain at its position, so a Schottky pair is always a transient defect.

The Schottky pair formation mechanism can be described as $$\mathrm{A_A^{\times}+v_A'=v_A'+A_A^{\times}\qquad\qquad B_B^{\times}+v_B^{\bullet}=v_B^{\bullet}+B_B^{\times}}$$ where $\mathrm{A^+}$ is a cation, $\mathrm{B^-}$ an anion, and the second term on each side of the equation refers to an entity on the surface.

A Frenkel pair
is an ion removed from its lattice position and located in an interstitial site, i.e. a combination of a vacancy and an interstitial:
$$\mathrm{A_A^{\times}=A_i^{\bullet}+v_A'\qquad\qquad B_B^{\times}=B_i'+v_B^{\bullet}}$$ Frenkel pairs are created and annihilated continuously due to lattice vibrations - the higher the temperature, the more this happens. A Frenkel pair as such is transient; however, as both defects are highly mobile, it is likely that they will separate before they can annihilate.

The Schottky and Frenkel mechanisms constitute thermodynamic equilibria.
The higher the temperature, the more the equilibrium will be biased towards the disordered side of the equation, favouring defects over regular lattice configurations. The Schottky and Frenkel equilibria ensure that there is always a minimum number of vacancies present in any crystal - the intrinsic vacancy
concentration. Additional vacancies can be introduced by doping,
i.e. by introducing impurities deliberately in order to create vacancies to balance the charge of the impurity relative to its location in the lattice.

via Wikimedia
under this CC licence.

Another common type of compound defect is an electron trapped in an anion vacancy.
In this case the electron balances the "missing" charge of the anion that would normally occupy the site in the regular lattice. This type of defect is known as an F-centre
(from German Farbzentrum - colour centre) because of its tendency to produce vibrant colours in otherwise colourless or transparent materials. F-centres often occur when a material is exposed to ionising radiation: the radiation excites an electron from one of the atoms into the continuum, and the free electron is subsequently trapped by an intrinsic vacancy.

### Point defects and diffusion

rotary interchange

vacancy diffusion

interstitial diffusion

random walk

### Self diffusion and directed diffusion

Fick's law:
$$j_N=\frac{{\rm d}N}{A{\rm d}t}=-D\frac{{\rm d}c}{{\rm d}z}$$ diffusion coefficient, $D$, in m2/s,
$$D=D_0\exp{\left(-\frac{E_a}{k_BT}\right)}$$ activation energy, $E_a$,

lithium-ion battery

cathode: LiNiO2 = NiO2 + Li+ + e-
anode: Li+ + e- + 6C = LiC6