Crystallography and Defects

Stress, strain and elastic moduli

Hooke’s Law states that the force required to stretch (or compress) a bar is proportional to the displacement from its equilibrium position.

where is the spring constant.

By applying a force to the rod, we do work on it (to stretch or compress the bonds). This work is stored as elastic potential energy. If we do the work adiabatically, then

where , alternatively, where . Therefore, from Hooke’s law to give the elastic potential energy:

The spring constant in Hooke’s law depends on the rod’s geometry: a larger cross-sectional area makes it stiffer (higher ), while a longer rod decreases stiffness (lower ). We introduce Young’s modulus, , which describes a material’s stiffness independently of its shape or size. To eliminate dependence on geometry, we divide Hooke’s law by and multiply by :

Note, we have changed terminology in the last step. is called Young’s modulus: it is an example of an elastic modulus, a material property that quantifies elasticity or stiffness.

Stress – Tensile and Compressive

We define the linear stress, , on a material as

The units can be given by , which is the same as pressure. The stress is produced within the material in response to applied external forces. Stress can be tensile when the body is pulled, or compressive when the body is pushed.

Strain (linear)

Applied forces cause the atoms of the body to be displaced from their equilibrium positions, resulting in a net change in its length. The strain is the fractional change in length

which is dimensionless.

Young’s modulus

Young’s modulus, , is the proportionality constant relating the linear (uniaxial) stress and strain:

with units . It is a measure of a material’s ability to withstand changes in length under tension (or compression). The stiffer a material, the higher its Young’s modulus. It is related to the spring constant in Hooke’s Law by .

Bulk modulus

Another elastic modulus of a material is the bulk modulus, which relates the fractional change in volume to the change in applied pressure (compressive stress):

which means an increase in pressure causes a decrease in volume.

Units are again . It is a measure of a material’s ability to withstand changes in volume under uniform pressure.

Shear modulus:

Shear stress is the force parallel to the section area divided by area:

Note that for fluids, which is why fluids cannot resist shear forces.

The relationship between the three moduli is:

where is called Poisson’s ratio, the ratio of transverse strain to longitudinal strain:

Mechanical properties of materials

When real materials are subjected to tensile stress, they deform linearly according to Hooke’s Law, with Young’s modulus () defining the proportional relationship between stress and strain. This linear behavior holds up to a certain point, called the proportionality limit. Beyond this, stress and strain are no longer directly proportional. If the load is removed before reaching the elastic limit, the material returns to its original shape (elastic deformation). Beyond the elastic limit (or yield point), the material undergoes plastic deformation, meaning it will not return fully to its original shape when unloaded. If the stress continues to increase, the material eventually fractures, which marks its breaking point. Ductile materials (e.g., metals like steel) exhibit significant plastic deformation before fracture. Brittle materials (e.g., glass, ceramics) fracture with little or no plastic deformation.

Elastic energy

Applying a force to a material causes it to deform. Consider the case of uniaxial tensile stress: if the external force stretches the material by , it does work on it. In the infinitesimal limit,

Thus the total work done in extending the material by is

Using Young’s modulus:

Thus:

We can use :

From the first law of thermodynamics, assuming no heat flow (), the work done on the material is stored as elastic strain energy:

Elastic energy is the potential energy stored in a material as work is performed to distort its shape.

The specific elastic energy (energy per unit volume) is:

The elastic energy corresponds to the area under the stress-strain curve (valid only in the linear elastic region).

Relationship of Young’s modulus to the interatomic potential

We can relate Young’s modulus to the interatomic pair potential. Young’s modulus applies to the elastic region where there is a linear relationship between applied force and displacement. We have already seen above that

where is the restoring force in the material. Comparing to the interatomic potential energy function, we see that such a quadratic approximation is valid close to the equilibrium separation between two atoms.

Thus, we can estimate by expanding the interatomic potential as a Taylor series about the equilibrium point, , up to the quadratic term,

is simply an offset (i.e., the minimum of the potential well) and at , thus

By comparison to we see that

This equation connects the macroscopic property to the interatomic potential.

In order to relate this to Young’s modulus, we make some simplifying assumptions:

  1. The material has a simple cubic lattice.
  2. Only nearest neighbour atoms contribute to the cohesive energy.

These assumptions allow us to use the simple pair potential and ensure that the equilibrium separation of the pair potential is equal to the cubic lattice constant, . We can then get the Young’s modulus considering just a single unit cell of the material where and :

Here, we have used just a single pair potential since the cross-section of one unit cell contains exactly one bond in the -direction.

Alternatively, we could imagine a macroscopic piece of material where,

where , and are integers and is the lattice constant. In this case, one atomic plane in the – plane contains bonds. The total spring constant for one atomic plane is then

Along the macroscopic length there are such planes and the total spring constant for the macroscopic sample is

(Adding more springs in parallel increases the stiffness, but adding more springs in series decreases the stiffness.) Young’s modulus is then:

With this expression we now have a way to calculate the Young’s modulus of a simple cubic material from knowledge of the interatomic pair potential energy function.

Morse potential example:

Thus

The Young’s modulus:

This result shows that the stiffness of the material depends directly on the interaction strength and the range parameter of the Morse potential, scaled by the lattice spacing .

Defects in crystals and their effects

Real crystals have imperfections, known as defects. These defects impact material properties (optical, thermal, electrical, magnetic, mechanical) and can be beneficial or detrimental.

Some examples of defects in crystals and their effects:

Classification of crystal defects by dimensionality

Defects can arise during crystal growth or due to external factors such as mechanical stress, temperature changes, or radiation exposure. Defects can be broadly categorised by their dimensionality:

  1. Point Defects (0D) – Localized at single atomic sites (vacancies, interstitials, substitutions).
  2. Dislocations (1D) – Extend along a line but distort the lattice in 2D (edge and screw dislocations).
  3. Planar Defects (2D) – Interfaces affecting the crystal structure (grain boundaries, stacking faults, surfaces).
  4. Volume Defects (3D) – Regions of distinct composition or phase (precipitates, voids, inclusions).

Types of crystal defects

Zero-dimensional:

One-dimensional:

Two-dimensional:

Three-dimensional:

Point defects

Vacancy defects: missing atoms in a crystal lattice

A vacancy is a point defect where an atom is missing from its expected lattice site, disrupting the periodicity of the crystal. It is an intrinsic defect (i.e., forms even in the absence of impurities).

Unreconstructed vs. reconstructed vacancy: A vacancy leaves a void in the lattice. In a reconstructed vacancy, nearby atoms relax and shift to minimize energy, often forming new local bonding configurations.

Impact on material properties: Vacancy defects influence electronic, optical, and mechanical properties by altering charge distribution, introducing localised states in the band structure, and modifying diffusion behaviour.

Temperature-dependent density of vacancies

The number of intrinsic defects increases with temperature. Suppose we start with atoms and have created vacancies; this means there must be atomic sites in total.

Let the Gibbs free energy change in forming a vacancy at one particular site be ; this will contain contributions from the changes in internal energy , volume and entropy of the material in forming this particular defect:

(In practice the term dominates, so depends only weakly on temperature.) The number of ways of arranging the vacancies on the atomic sites is

Using Stirling’s approximation we get a contribution to the entropy, the so-called configurational entropy, from the number of ways of arranging the vacancies (Boltzmann equation):

However,

So

since . Thus,

Hence the overall Gibbs free energy change (relative to the perfect crystal) when we form vacancies is

In equilibrium the number of vacancies will be the number that minimises :

assuming we can treat as a continuous variable. Therefore the equilibrium number of vacancies is

This grows rapidly as the temperature rises, causing the exponent to become less negative. This is the Boltzmann distribution.

Substitutional impurities: foreign atoms in the lattice

Interstitial defects: extra atoms in the crystal lattice

Interstitial site: A small gap between atoms in a crystal where an extra atom can fit.

Self-interstitial: A host atom forced into an interstitial site, causing lattice strain as surrounding atoms push outward. This is another example of an intrinsic defect.

Interstitial impurity: An interstitial impurity occurs when a foreign atom, usually smaller than the host atoms, occupies an interstitial site. This can affect diffusion and mechanical properties of the material.

Impact on material properties: Interstitial defects can increase hardness, enhance diffusion rates, and modify electronic behavior. Self-interstitials are often highly mobile and can cluster to form defect complexes.

Line defects

Edge dislocation

An example of a one-dimensional (1D) defect is the edge dislocation. We apply a shear force to the top half of the crystal, causing an increase in the atom density in the top half of the structure. This increase in density is accommodated by an extra “half-plane” of atoms that terminates at the edge dislocation, indicated by . The edge dislocation runs perpendicular to the page, i.e., into the crystal from the observed perspective.

The defect can be characterised by a Burgers vector, . This is defined with respect to the un-distorted crystal. Here, we go two atoms left, three atoms down, three atoms right, three atoms up and one atom left. To make a closed path, we must add a Burgers vector where in this example .

Screw dislocation

Another type of dislocation is the screw dislocation. In this case, the dislocation does not introduce a half-plane. Instead, the crystal becomes a type of spiral staircase around the dislocation line. In contrast to the edge dislocation, the Burgers vector is parallel to the line of the dislocation.

Plastic deformation and work hardening

Materials are found to be much easier to deform than they should be: many pure metals deform at less than 1% of the strength theoretically predicted by their bond strength.

The reason is the ease of motion of dislocations: the movement of the dislocation line facilitates the motion of an entire plane of atoms through the breaking and reforming of only a single row of atomic bonds.

During plastic deformation, e.g., bending or stretching a piece of metal in the plastic deformation portion of the stress-strain curve, the dislocation density increases significantly. Initially, the presence of dislocations makes deformation easier. However, as deformation continues, these dislocations interact and entangle, forming barriers to further dislocation motion. As a result, additional stress is required to move dislocations, thus the material hardens and becomes stronger, at the expense of reduced ductility. Annealing (heat treatment) can reverse work hardening by promoting recovery and recrystallization, which reduce the dislocation density.

Bragg diffraction

Crystalline solids form natural three-dimensional diffraction gratings for x-rays. The maxima are in directions as if the x-rays were reflected by parallel reflecting planes that contain regular arrays of atoms called crystal planes. The incident rays are said to reflect from these planes, whose separation is . The angle of incidence and the angle of reflection are represented with , defined relative to the surface of the reflecting plane rather than a normal to that surface.

Bragg’s law

By drawing the dashed perpendiculars, we find that the path length difference is

Thus, we have, as the criterion for intensity maxima for x-ray diffraction,

where is the order number of an intensity maximum.

Bragg’s law was derived by the British physicist W. L. Bragg and is named after him. Bragg and his father were awarded the Nobel Prize in Physics in 1915 for their work using x-rays to investigate crystal structures. The angle of incidence and reflection is referred to as a Bragg angle.

Bragg planes

There are many possible Bragg planes for diffraction. Since the separation between planes (the spacing) is different for each set of planes, diffraction will occur from each plane at different angles. The spacings are labelled with corresponding Miller indices as subscripts, i.e., , e.g., , , , etc.

Experimental crystallography

A monochromatic beam of x-rays with wavelength angstrom is incident on a sample. The sample acts as a reflection diffraction grating and x-rays are scattered in a variety of directions. A detector is then scanned by slowly increasing the angle , which is called the scattering angle. When the Bragg condition is met, a high intensity of scattered x-rays will be measured by the detector. When single crystal samples are used, the sample itself must also rotate by an angle . However, another method involves first grinding the sample into a powder so that all incident angles are present simultaneously and in this case only the detector rotates during the measurement.

Diffraction from cubic crystals

From geometric considerations, it can be shown that the spacings for cubic crystals with lattice spacing can be calculated using the following equation:

We can combine this with Bragg’s equation,

to calculate the scattering angle for a given experiment.

Identification of the structure of DNA: a famous example of x-ray diffraction

Rosalind Franklin (King’s College London) used hydrated DNA fibers, aligning many DNA molecules in parallel, to obtain data to identify the structure of DNA.

Although not a true single crystal, these fibers produced a diffraction pattern similar to a single crystal. X-rays were directed perpendicular to the fibers (i.e., side-on to the DNA strands), yielding an X-shaped pattern characteristic of a helix. The spot spacing along the “X” arms revealed a pitch of nm per turn. Two observations confirmed the double helix: the first diffraction spot gives a real-space width of nm per strand, too wide for a single helix. The missing 4th layer line (due to destructive interference) indicated a double-stranded structure.