The Fermi Level

The Fermi level is the maximum energy of the electrons in a material. Effectively there is a Fermi equation: EF = E max. This chapter examines the discrete electron energy levels in individual atoms as a consequence of the Pauli exclusion principle, the corresponding energy bands in a material composed of many atoms or molecules, and the way in which conductor, insulator and semiconductor materials depend on the position of the Fermi level relative to the energy bands. It explains: the concepts of electron mobility, mean free path and conductivity; the dielectric effect and capacitance; p-type, n-type, intrinsic and extrinsic semiconductors; and the behaviour of some simple microelectronic devices. Enrico Fermi was the son of a minor railway official in Rome. He had a meteoric scientific career in Italy, developing Fermi-Dirac statistics for the energies of fundamental fermion particles (such as electrons and protons), discovering the neutrino, and explaining the behaviour of different materials under bombardment from fast and slow neutrons. After initially joining Mussolini’s Fascist Party, he became unhappy at the level of anti-Semitism (his wife was Jewish) and left suddenly for America, immediately after receiving the Nobel Prize in Sweden. At Columbia and Chicago Universities and at Los Alamos National Labs, he played a key scientific role in developing controlled fission in an atomic pile, leading to the development of the atomic bomb towards the end of the Second World War, and the nuclear energy industry after the war.

Griffith’s Equation

Most materials fracture suddenly because they contain small internal and surface cracks, which propagate under an applied stress. Griffith’s equation shows how fracture strength depends inversely on the square root of the size of the largest crack. It was developed by Alan Griffith, while he was working as an engineer at Royal Aircraft Establishment Farnborough just after the First World War. This chapter examines brittle and ductile fracture, the concepts of fracture toughness, stress intensity factor and stBiographical Memoirs of Fellows ofrain energy release rate, the different fracture modes, and the use of fractography to understand the causes of fracture in broken components. The importance of fracture mechanics was recognised after the Second World War, following the disastrous failures of the Liberty ships from weld cracks, and the Comet airplanes from sharp window corner cracks. Griffith’s father was a larger-than-life buccaneering explorer, poet, journalist and science fiction writer, and Griffith lived an unconventional, peripatetic and impoverished early life. He became a senior engineer working for the UK Ministry of Defence and then Rolls-Royce Aeroengines, famously turning down Whittle’s first proposed jet engine just before the Second World War as unworkable because the engine material would melt, then playing a major role in jet engine development after the war, including engines for the first vertical take-off planes.

The Gibbs-Thomson Equation

The external surface of a material has an atomic or molecular structure that is different from the bulk material. So does any internal interface within a material. Because of this, the energy of a material or any grain or particle within it increases with the curvature of its bounding surface, as described by the Gibbs-Thomson equation. This chapter explains how surfaces control the nucleation of new phases during reactions such as solidification and precipitation, the coarsening and growth of particles during heat treatment, the equilibrium shape of crystals, and the surface adsorption and segregation of solutes and impurities. The Gibbs-Thomson was predated by a number of related equations; it is not clear whether it is named after J. J. Thomson or William Thomson (Lord Kelvin); and it was not put into its current usual form until after Gibbs’, Thomson’s and Kelvin’s time. J. J. Thomson was the third Cavendish Professor of Physics at Cambridge University. He discovered the electron, which had a profound impact on the world, notably via Thomas Edison’s invention of the light bulb, and subsequent building of the world’s first electricity distribution network. William Thomson was Professor of Natural Philosophy at Glasgow University. He made major scientific developments, notably in thermodynamics, and he helped build the first trans-Atlantic undersea telegraph. Because of his scientific pre-eminence, the absolute unit of temperature, the degree Kelvin, is named after him.

Fick’s Laws

Atoms and molecules are not completely immobile within a solid material. They move by jumping into vacancies or interstitial sites in the crystal lattice. The laws describing their motion were discovered by Adolf Fick in the mid-19th century, modelled on analogous laws for the flow of heat (Fourier’s law) and electricity (Ohm’s law). According to Fick’s first law, the rate at which atoms move is proportional to the concentration gradient, with the diffusion coefficient defined as the constant of proportionality. Fick’s second law generalises the first law to a wide range of situations and is called the diffusion equation. This chapter examines a number of characteristic diffusion profiles; the difference between self, intrinsic, inter- and tracer diffusion coefficients; the Kirkendall effect and porosity formation when different components move at different speeds; and the Arrhenius temperature dependence of diffusion. Fick was a physiologist and derived his laws initially to describe the flow of blood through the heart. He made advances in anatomy, physiology and medicine, developing methods of monitoring blood pressure, muscular power, corneal pressure and glaucoma. He lived at the time of Bismarck’s post-Napoléonic unification of Germany and the associated flowering of German science, engineering, medicine and culture.

Boltzmann’s Equation

Thermodynamics describes the relationship between heat, work, energy and motion. The key concepts are the conservation of energy and the maximisation of entropy (or disorder) as given by the first and second laws of thermodynamics. Boltzmann’s equation explains how the entropy of a material is related to the disorder of its atoms or molecules, as measured by the probability or the number of equivalent atomic or molecular structures. This chapter examines thermodynamic properties such as internal energy, enthalpy and Gibbs and Helmholtz free energy; physical properties such as specific heat and thermal expansion coefficient; and the application of thermodynamics to chemical reactions, solid and liquid solutions, and phase separation. Ludwig Boltzmann’s early life as the son of a minor tax official in Austria is described, as are: his scientific career in a series of Austrian and German universities; his philosophical arguments with Ernst Mach and the phenomenalists about whether atoms do or do not exist; his increasing moodiness, paranoia and bipolar disorder; and his ultimate suicide while trying to recuperate from depression in Trieste.

The Gibbs Phase Rule

Materials are made up of regions of space that are homogeneous in structure and properties, called phases. The number of different phases in a material depends on its temperature, pressure and composition, as given when the material is at equilibrium by the Gibbs phase rule. This was discovered by the American scientist J. Willard Gibbs during his ground-breaking investigations in the late 19th century into the thermodynamics of heterogeneous materials. This chapter explains the differences between solutions, mixtures and compounds; the use of phase diagrams to determine the structure of a material; and the way in which phase transformations can be used to change the structure of a material. Gibbs grew up in an academic family at Yale University in New Haven at the time of the American Civil War. He was the first person to receive an engineering doctorate in the United States, and he later became a fundamental theoretician of thermodynamics, statistical mechanics and vector fields.

Bravais Lattices

Most solid materials are crystalline, with their component atoms and molecules arranged in regular arrays throughout space. The French scientist Auguste Bravais showed that there are only 14 different ways of doing this, called the Bravais lattices, each with different symmetry. In other words, there is a Bravais equation for the number of different lattices: N L = 14. This chapter examines the relationship between Bravais lattices, crystal systems and symmetry groups, the use of Miller indices to describe crystal planes and directions, and the use of stereograms to describe crystal orientations. Bravais’ early life in the Ardèche in France is described, along with his exciting career during and after the French Revolution: as an officer in the French navy during the Barbary wars; as an explorer in North Africa, the Arctic and the Alps, notably leading the second scientific ascent of Mont Blanc; and as an environmental, geophysical and crystallographic scientist.

Hooke’s Law

When a material is stretched, the extension is proportional to the stretching force, with the elastic modulus defined as the constant of proportionality. This is called Hooke’s law and was discovered by Robert Hooke, just after the end of the English civil wars in the mid-17th century. This chapter examines the underlying atomic forces responsible for Hooke’s law, the use of tensors to describe three-dimensional stresses and strains in a material, and the relationships between the different elastic moduli under different loading conditions. Hooke was the son of a clergyman, born and brought up on the Isle of Wight, a royalist stronghold, where King Charles I fled after his imprisonment by Parliament, only to be recaptured and executed. Hooke was smuggled to London and then Oxford under the protection of Royalist academics, where he became a member of the group of intellectuals who, after the restoration of the monarchy, led the Enlightenment and set up the Royal Society. He took on many jobs: Lab Assistant to Robert Boyle, Curator at the Royal Society, Professor of Geometry at Gresham’s College, City Surveyor for the rebuilding of London after the Great Fire, and First Officer in Christopher Wren’s architectural firm. He was paranoid about his need for money and about people stealing his scientific ideas. He feuded with many of the great scientists of his age, claiming that he invented their ideas first, notably with Newton about his theories of gravity.

Bragg’s Law

The diffraction of X-rays is used as the main method for determining the atomic and molecular structures of inorganic and biological materials. The basic law of diffraction was discovered by Lawrence Bragg when he was a student at Cambridge University and he was just 22 years old. Bragg’s law explains how the angle of a diffracted X-ray beam varies with the wavelength of the X-rays and the spacing of the atoms and molecules in the material. This chapter examines the way X-rays are generated and scattered by electrons, atoms and crystals; the use of structure factors and Fourier transforms to calculate the intensity of the scattered X-rays; and the effect of using electrons or neutrons instead of X-rays. Bragg was born and brought up in Adelaide in Australia. He discovered Bragg’s law with the help of his father, William, after they had moved to England. Lawrence was a Professor at Manchester University, Cambridge University, and the Royal Institution; contributed to the development of range-finding, asdic, and sonar during the First and Second World Wars; and supervised Crick and Watson when they discovered the structure of DNA.

The Scheil Equation

Many materials are manufactured by solidification, either as a final product by casting, or as an intermediate ingot or bar. The Scheil equation describes the partitioning that takes place during solidification and the resulting spatial redistribution of solute, which makes it difficult to maintain a homogeneous material composition, and which leads to unwanted concentrations of harmful impurities. This chapter explains nucleation and growth processes during solidification, the resulting dendritic, faceted, equiaxed and columnar structures depending on thermal conditions and material type, coupled solidification of two-phase eutectic materials, and typical casting methods and associated structures and defects. Very little is known about Erich Scheil, who worked at the Max Planck Institute in Stuttgart in the mid-20th century.