Crystal Structure, Defects and Diffusion

Bragg diffraction in FCC

Bragg's law relates the wavelength \lambda, the interplanar spacing d and the diffraction angle \theta through n\lambda = 2d\sin\theta. For a cubic lattice of parameter a, the interplanar spacing is d_{hkl} = a/\sqrt{h^2+k^2+l^2}, so for a fixed wavelength \sin^2\theta is proportional to h^2+k^2+l^2. This relation lets you rank the reflections of any cubic lattice by their diffraction angle. In a face-centred cubic (FCC) lattice the structure factor vanishes unless h, k and l are either all even or all odd. The first allowed reflections are therefore 111, 200, 220 and 311, in that order of increasing 2\theta, while mixed-parity indices such as 100, 110 and 211 are systematically absent.

Because \sin^2\theta scales with the sum of squared indices, the angle of a higher reflection can be predicted from the first peak. For FCC the first peak is 111 and the second is 200, whose index sums of squares are 3 and 4. Hence \sin\theta_2 = \sin\theta_1\sqrt{4/3}, which gives a second angle only slightly larger than the first.

A powder pattern of two cubic metals with different structures can also be compared at the same wavelength and angle. Equal first-peak angles mean equal d-spacings, and the lattice parameters then follow from the different index sums of the first reflections of each structure. For an FCC metal (111) has \sqrt{3} in the denominator, while for a BCC metal (110) has \sqrt{2}, so the ratio of the two lattice parameters is \sqrt{3}/\sqrt{2}\approx 1.22.

Fick's second law carburizing

Fick's first law states that the diffusion flux is proportional to the concentration gradient, J = -D\,\dfrac{\partial C}{\partial x}, where D is the diffusion coefficient. Fick's second law combines this with mass conservation to give \dfrac{\partial C}{\partial t} = D\,\dfrac{\partial^2 C}{\partial x^2}. For flux, the magnitude is simply D multiplied by the local slope of the concentration profile, so the units follow from the units of C and x. For example, if C is given in atoms per mm^3 and x in mm, then the slope is in atoms per mm^4 and the flux is in atoms per mm^2 per second.

Carburizing is the process of enriching the surface of a low-carbon steel with carbon by holding it in a carbon-rich atmosphere. When the surface concentration C_s is held constant and the initial concentration is C_0, a semi-infinite solid obeys \dfrac{C_s - C_x}{C_s - C_0} = \mathrm{erf}\!\left(\dfrac{x}{2\sqrt{Dt}}\right). The error function is read from tables, and when the measured ratio is matched to a tabulated value of z the depth is x = 2z\sqrt{Dt}.

For a fixed concentration at a fixed temperature, D is constant and the argument x/(2\sqrt{Dt}) must stay constant. Therefore the depth scales with the square root of time, so t_2 = t_1 (x_2/x_1)^2 for the same concentration. Doubling the depth needs four times the time.

If the measured ratio corresponds to a tabulated \mathrm{erf}(z), then z = x/(2\sqrt{Dt}) can be solved for t = x^2/(4z^2D). Units must be consistent, so x in metres and D in square metres per second give time in seconds.

Diffusion coefficients: lattice, grain boundary, surface

Diffusion occurs along several paths in a polycrystal, and each has its own coefficient. Lattice (volume) diffusion takes place through the perfect crystal and is usually the slowest path. Grain boundary diffusion is faster because grain boundaries are more open, with a disordered structure that offers easier paths for atoms. Surface diffusion is fastest because atoms on a free surface have fewer neighbours to break bonds with. The usual ordering is therefore D_S > D_{GB} > D_L.

Each mechanism follows an Arrhenius form D = D_0\exp(-Q/RT), and the faster path has the smaller activation energy. Interstitial diffusion of small solutes such as carbon in BCC iron has a low activation energy because the atoms move through the interstices with little distortion. Self-diffusion of iron through the lattice requires vacancy formation and migration, so it has a much higher activation energy. Grain boundary diffusion lies between these two extremes.

Some items in this group test extraction metallurgy rather than diffusion. In coal analysis, proximate analysis reports moisture, volatile matter, fixed carbon and ash, while ultimate analysis reports carbon, hydrogen, nitrogen, sulphur and oxygen. In the blast furnace, wüstite is reduced by carbon monoxide in the middle and lower stack, and during converting of copper matte iron sulphide is oxidised and removed as slag, leaving white metal.

Vacancy formation enthalpy

A vacancy is a missing atom on a lattice site. The equilibrium mole fraction of vacancies is x_v = \exp(S_f/k_B)\exp(-H_f/k_BT), where H_f is the enthalpy of vacancy formation and S_f is the formation entropy. The vacancy concentration rises steeply with temperature, and the enthalpy can be found from two temperatures if the concentration ratio is known. Writing the ratio of concentrations at two temperatures gives \dfrac{H_f}{R}\left(\dfrac{1}{T_1}-\dfrac{1}{T_2}\right) = \ln\dfrac{x_2}{x_1}. Temperatures must be in kelvin, and the bracket is small, so a small error in temperature strongly affects the result.

If the formation entropy is given as a multiple of k_B, its contribution is simply \exp(S_f/k_B). The enthalpy term H_f/(k_BT) must be computed with k_B in eV per atom per kelvin when H_f is in eV, because the exponent must be dimensionless. The final vacancy fraction is then the product of the entropy factor and the enthalpy factor, often reported as a number times 10^{-4} or 10^{-5} at moderate temperatures.

Types of chemical bonding

Primary bonds are ionic, covalent and metallic, while secondary bonds are weaker van der Waals attractions. Ionic bonding arises from electrostatic attraction between oppositely charged ions, as in sodium chloride. Covalent bonding involves shared electron pairs in a directional network, as in diamond, where each carbon atom is sp3 bonded to four neighbours. Metallic bonding arises from a delocalised electron sea that binds positive ion cores, as in silver. Secondary bonding is the weak attraction between closed-shell atoms or molecules, as in solid argon.

Diamond is an electrical insulator because all its valence electrons are locked in covalent bonds and no free electrons are available for conduction. Its strong, rigid C–C bonds also give a very high elastic modulus and extreme hardness, and heat is carried efficiently by lattice vibrations, so its thermal conductivity is high.

Functional materials and applications

Functional materials are chosen for a specific physical response rather than for mechanical strength. Gallium arsenide is a III–V compound semiconductor with a direct band gap, which makes it useful for light-emitting and high-speed electronic devices. Barium titanate is a ferroelectric ceramic that shows the piezoelectric effect, converting mechanical strain into electric charge. Iron alloyed with about 4 wt.% silicon has high resistivity and low coercivity, making it a soft magnetic material for transformer cores. Yttrium–barium–copper oxide is a high-temperature superconductor that loses electrical resistance above the boiling point of liquid nitrogen.

Several physical laws and quantities tie these functional properties to measurable behaviour. The Wiedemann–Franz law states that the ratio of thermal to electrical conductivity of a metal is proportional to temperature, with the Lorenz number as the constant. The Hall voltage is inversely proportional to the carrier concentration, through R_H = 1/(ne) for a single carrier type. Above the Néel temperature an antiferromagnet becomes paramagnetic, and the Curie law makes paramagnetic susceptibility inversely proportional to absolute temperature.

Bragg diffraction in BCC

In a body-centred cubic (BCC) lattice, the structure factor is non-zero only when h+k+l is even. The first allowed reflection is therefore 110, followed by 200, 211 and 220. For the first peak, Bragg's law with n=1 gives d_{110} = \lambda/(2\sin\theta), and since d_{110} = a/\sqrt{2} the lattice parameter is a = \sqrt{2}\,d_{110}. The reflection 200 gives d_{200}=a/2, so the diffraction angle of a second-order peak follows directly from the same lattice parameter.