Simple Crystal Structures and X-ray Diffraction
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Crystal structures are most often determined by diffraction of X-rays from single crystals of sample, although structure determination from polycrystalline powders is also important, especially via neutron diffraction, as described below. As noted in the previous section, in order to generate a structure, (i.e. an image), it is necessary to recombine the beams that make up the diffraction pattern. Unfortunately this cannot be carried out by using lenses, and the process must be done mathematically. The technique is simple in principle. A small crystal of the material, of the order of a fraction of a millimeter in size, is mounted in a beam of X-rays. Each plane of atoms in the crystal gives rise to a narrow diffracted beam. The position of each beam is recorded, along with the intensity of each spot. Each member of the resulting data set comprises a position, intensity and hkl index.
Metal, salts, and most minerals-the materials of the Earth-are made up of crystals. People have known about such crystals as salt and quartz for centuries, but it wasn’t until the twentieth century that crystals were interpreted as regular arrays of atoms. X rays were used in 1912 to confirm that each crystal is a three dimensional orderly arrangement-a crystalline latticework of atoms. The atoms in a crystal were measured to be very close together about, the same distance apart as the wavelength of X rays. The German physicist Max von Laue discovered that acteristic pattern. X-ray diffraction pattern on photographic film show crystals to be neat mosaics of atoms sited on regular lattices, like three dimensional chessboards or children’s jungle gyms. Such metals as iron, copper, and gold have relatively simple crystal structure. Tin and cobalt are a little more complex. All metals contain a jumble of many crystals, each almost perfect, each with the same regular lattice but at some inclination to the crystal nearby. These metal crystals can be seen when a metal surface is etched, or cleaned, with acid. You can see the crystal structures on the surface of galvanized iron exposed to the weather, or on brass doorknobs etched by the perspiration of many hands.
Lattice Vibrations and Thermal Properties of Solids
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The basis of crystal structures is often described in terms of ions for the interpretation of the properties of solids. The valence electrons are considered to have been placed in the force field of the lattice of ions. The word ions when used in this general way stands for ions in ionic crystals, ion cores in metals and covalent crystals, and atoms in a rare gas solid. The roles of ionic and electronic motions are crucial to the determination of the properties of solids. While some properties depend heavily on the electronic motion, several others are closely linked to the ionic dynamics. Lattice heat capacity, thermal expansion and hardness are some examples of properties that belong to the latter class. In the present chapter, a classical theory will be developed to describe small vibration of atoms in crystalline solids in terms of normal modes (independent motions of characteristic frequency) of motion of constituent ions. In a normal mode has the same amplitude in each cell, but varies from one unit cell to the other across the crystal like a wave with a certain wave vector. Such a wave is called lattice wave and the vibration* with which it is associated is commonly known as lattice vibration.
The physical properties of a solid can be roughly divided in to those that are determined by the electrons and those that relate to the movement of the atoms about their equilibrium positions. In the latter category are, for example, the sound velocity and also the thermal properties: specific heat, thermal expansion, and - for semiconductors and insulators - the thermal conductivity. The hardness of a material is also determined, in principle, by the movement of the atoms about their equilibrium positions. Here, however, structural defects generally play a decisive role.
Specific heat: Specific heat is the energy required to increase the temperature of a solid by a degree. If energy dQ is supplied to 1 g of a solid to increase its temperature by dT, dQ/dT is the specific heat c per gram.
Thermal expansion: Thermal expansion is the dilatation of a solid due to change in temperature. It is a property with both technical and scientific significance.
Thermal conductivity: Like thermal expansion, thermal conductivity is also directly related to the anharmonicity of lattice vibrations. In the absence of anharmonicity, thermal expansion becomes zero whereas thermal conductivity becomes infinite.
There are many fundamental properties of solids, e.g. electrical and thermal conductivities, magnetic and optical properties, etc, depend up on their electronic structure. We can understand many physical properties of solids in terms of electron theory of solids. The development of the electron theory of solids started in the beginning of the 20th century. Today, it is the basis for the classification of all solids. When free electron theory applied to metals, it is the basis for the classification of all solids. When free electron theory applied to metals, it explains forces of cohesion and repulsion, binding the energy levels and the behavior of conductors and insulators and magnetic materials. According to this model, the valence electrons of the constituent atoms become conduction electrons and move about freely through the volume of the metal. The first version of the free electron model was introduced by P. Drude in the early 1900s, with improvements soon after by H. A. Lorentz. This is now known as Drude-Lorentz free electron theory.
One of the defining physical properties of a metal is its electrical conductivity. Electrical conductivity in a solid is attributable to electrons that are free to move (i.e. that gain energy) under the influence of an applied electric field. The metallic bonding model, allows conductivity to be understood most easily. In this model, the electrons on the atoms making up the solid are allocated to energy bands that run throughout the whole of the solid. A simple one-dimensional band-structure diagram, called a flat-band diagram allows the broad distinction between conductors, semiconductors and insulators to be understood.
There are several ways of defining a semiconductor. Historically, the term semiconductor has been used to denote materials with a much higher conductivity than insulators, but a much lower conductivity than metals measured at room temperature. Today there are two more types of conductors: superconductors and semiconductors and insulators. This definition is not complete. What really distinguishes metals from semiconductors is the temperature dependence of the conductivity. While metals (except for superconductors) and semimetals retain their metallic conductivity even at low temperatures, semiconductors are transformed into insulators at very low temperatures. In this sense semiconductors and insulators are actually one class of materials, which differs from metals and semimetals, which from another class. This classification is directly connected to the existence of a gap between occupied and empty states, i.e., an energy gap, in semiconductors and insulators.
The wide-ranging optical properties observed in solid state materials can be classified into a small number of general phenomena. The simplest group, namely reflection, propagation and transmission. This shows a light beam incident on an optical medium. Some of the light is reflected from the front surface, while the rest enters the medium and propagates through it. If any of this light reaches the back surface, it can be reflected again, or it can be transmitted through the other side. The amount of the light transmitted is therefore related to the reflectively at the front and back surfaces and also to the way the light propagates through the medium.
Refraction causes the light waves to propagate with a smaller velocity than in free space. This reduction of the velocity leads to the bending of light rays at interfaces described by Snell’s law of refractions. Refraction, in itself, does not affect the intensity of the light wave as it propagates.
Absorption occurs during the propagation if the frequency of the light is resonant with the transition frequencies of the atoms in the medium. In this case, the beam will be attenuated as it progresses. The transmission of the medium is clearly related to the absorption, because only unabsorbed light will be transmitted. Selective absorption is responsible for the coloration of many optical materials. Rubies, for example, are red because they absorb blue and green light, but not red.
Luminescence is the general name given to the process of spontaneous emission of light by excited atoms in a solid state material. One of the ways in which the atoms can be promoted into excited states prior to spontaneous emission is by the absorption of light. Luminescence can thus accompany the propagation of light in an absorbing medium. The light is emitted in all directions, and has a different frequency to the incoming beam.Luminescence does not always have to accompany absorption. It takes a characteristic amount of time for the excited atoms to re-emit by spontaneous emission. This means that it might be possible for the excited atoms to dissipate the excitation energy as heat before the radiative re-emission process occurs. The efficiency of the luminescence process is therefore closely tied up with the dynamics of the de-excitation mechanisms in the atoms.
Scattering is the phenomenon in which the light changes direction and possibly also its frequency after interacting with the medium. The total number of photons is unchanged, but the number going in the forward direction decreases because light is being re-directed in other directions. Scattering is therefore has the same attenuating effect as absorption. The scattering is said to be elastic if the frequency of the scattered light is unchanged or inelastic if the frequency changes in the process. The difference in the photon energy in an inelastic scattering process has to be taken from the medium if the frequency increases of given to the medium if the frequency decreases.
Phonons are not the only quanta of collective excitations that can be present in a solid. Other quasi particles that appear in solids with magnetic properties are the magnons. These are spin waves in magnetically ordered solids. From the point of view of magnetic properties, the magnetism of crystalline solids can be disordered (diamagnetism and paramagnetism) or ordered (Ferro-magnetism, antiferromagnetism, ferromagnetism. Disordered magnetism appears in those materials where the induced magnetization M = ΧHis proportional to the applied field H. In diamagnetic materials the susceptibility is small, negative, and positive and proportional to the inverse we of the temperature. In ordered magnetic solids, all individual spins are parallel in ferromagnetic materials, or can be divided in different sub lattices (usually two), In each sub lattice the spins being parallel. If the magnetic moments of different sub lattices compensate each other, the material is called antiferromagnetic, otherwise we have a ferromagnetic material. Ferromagnetic materials are composed of domain being magnetized at saturation, with all spins aligned in a certain direction that varies randomly from one domain to another. If an external magnetic field is applied, the walls between domains can be shifted and /or the spins in each domain can rotate. When a very small field is applies on a ferromagnetic material with an initially zero magnetization, the walls between domains are shifted (spatially) irreversibly from an energy minimum to another, so that when the field is removed the material attains a non-vanishing remnant magnetization. This magnetization value can be maintained indefinitely at low applied magnetic fields, and depends on the history of the probe. To bring again the magnetization at zero, a magnetic field-the coercitive magnetic field Hc-must be applied; this field is a measure of the necessary field to displace a wall over an energy barrier.
Progress in science in general, and in physics in particular, is characterized by many great discoveries both in theory and experiment. On superficial observation these sudden advances may look like a series of accidental occurrences. But in most cases such an impression would be wrong, in many cases even totally misleading. Very often discoveries occur when the time is ripe, i.e. when certain preconditions have been met, such as adequate technical capability, or attainment of a necessary intellectual and theoretical level. The discovery of superconductivity-the property of certain conductors to display zero DC electrical resistance.
It was a true discovery, and indeed a remarkable one, because there were no valid arguments around to predict such phenomenon. Yet, to draw the conclusion that it was accidental would be unjustified. The necessary technical basis and opportunity for the discovery had been solidly established in the same group by the liquefaction of the inert gas helium in 1908. And research, both experimental and theoretical, on the electrical conductivity of metals at temperatures approaching the absolute zero, was ongoing and considered an important issue by leading physicists. The graph of superconductors and normal conductors with the temperature is given below: