bluuebunny/crossref_metadata_2025 huggingface dataset:
Thermodynamic relationships
Authors: Newnham Robert E.
Published: November 2020

In the next few chapters we shall discuss tensors of rank zero to four which relate the intensive variables in the outer triangle of the Heckmann Diagram to the extensive variables in the inner triangle. Effects such as pyroelectricity, permittivity, pyroelectricity, and elasticity are the standard topics in crystal physics that allow us to discuss tensors of rank one through four. First, however, it is useful to introduce the thermodynamic relationships between physical properties and consider the importance of measurement conditions. Before discussing all the cross-coupled relationships, we first define the coupling within the three individual systems. In a thermal system, the basic relationship is between change in entropy δS [J/m3] and change in temperature δT [K]: . . . δS = CδT, . . . where C is the specific heat per unit volume [J/m3 K] and T is the absolute temperature. S, T, and C are all scalar quantities. In a dielectric system the electric displacement Di [C/m2] changes under the influence of the electric field Ei [V/m]. Both are vectors and therefore the electric permittivity, εij , requires two-directional subscripts. Occasionally the dielectric stiffness, βij , is required as well. . . . Di = εijEj Ei = βijDj. . . . Some authors use polarization P rather than electric displacement D. The three variables are interrelated through the constitutive relation . . . Di = Pi + ε0Ei = εijEj. . . . The third linear system in the Heckmann Diagram is mechanical, relating strain xij to stress Xkl [N/m2] through the fourth rank elastic compliance coefficients sijkl [m2/N]. . . . xij = sijklXkl. . . . Alternatively, Hooke’s Law can be expressed in terms of the elastic stiffness coefficients cijkl [N/m2]. . . Xij = cijklxkl. . . . When cross coupling occurs between thermal, electrical, and mechanical variables, the Gibbs free energy G(T, X, E) is used to derive relationships between the property coefficients. Temperature T, stress X, and electric field E are the independent variables in most experiments.
Specific heat and entropy
Authors: Newnham Robert E.
Published: November 2020

Before beginning the discussion of directional properties, we pause to consider specific heat, an important scalar property of solids which helps illustrate the important thermodynamic relationships between measured properties. Heat capacity, compressibility, and volume expansivity are interrelated through the laws of thermodynamics. Based on these ideas, similar relationships are established for other electrical, thermal, mechanical, and magnetic properties. Several atomistic concepts are introduced to help understand the structure–property relationships involved in specific heat measurements. The heat capacity or specific heat is the amount of heat required to raise the temperature of a solid by 1K. It is usually measured in units of J/kg K. Theorists prefer to work in J/mole K, and older scientists sometimes use calories rather than joules. One calorie is 4.186 J. For solids and liquids, the specific heat is normally measured at a constant pressure: where ΔQ is the heat added to increase the temperature by ΔT. Measurements on gases are usually carried out at constant volume: Electrical methods are generally employed in measuring specific heat. A heating coil is wrapped around the sample and the resulting change in temperature is measured with a thermocouple. If a current I flows through a wire of resistance R, the heat generated by the wire in a time Δt is given by . . . ΔQ = I2R
Piezoelectricity
Authors: Newnham Robert E.
Published: November 2020

The prefix “piezo” (pronounced pie-ease-o) comes from the Greek word for pressure or mechanical force. Piezoelectricity refers to the linear coupling between mechanical stress and electric polarization (the direct piezoelectric effect) or between mechanical strain and applied electric field (the converse piezoelectric effect). The equivalence between the direct and converse effects was established earlier using thermodynamic arguments (Section 6.2). The principal piezoelectric coefficient, d, relates polarization, P, to stress, X, in the direct effect (P = dX) and strain, x, to electric field E (x = dE). Thus the units of d are [C/N] or [m/V] which are equivalent to one another. Typical sizes for useful piezoelectric materials range from about 1 pC/N for quartz crystals to about 1000 pC/N for PZT (lead zirconate titanate) ceramics. To understand how the piezoelectric effect varies with direction and how it is affected by symmetry, it is necessary to determine how piezoelectric coefficients transform between coordinate systems. Since polarization is a vector and stress a second rank tensor, the physical property relating these two variables must involve three directions: . . . Pj = djklXkl . . . . In the new coordinate system . . . P'i = aijPj = aijdjklXkl . . . . Transforming the stress to the new coordinate system gives . . . P'i= aijdjklamkanlX'mn = d'imnX 'mn. . . . Thus piezoelectricity transforms as a polar third rank tensor. . . . d'imn = aijamkanldjkl . . . . In general there are 33 = 27 tensor components, but because the stress tensor is symmetric (Xij = Xji), only 18 of the components are independent. Therefore the piezoelectric effect can be described by a 6 × 3 matrix.
Ferroic crystals
Authors: Newnham Robert E.
Published: November 2020

Twinned crystals are normally classified according to twin-laws and morphology, or according to their mode of origin, or according to a structural basis, but there is another classification that deserves wider acceptance, one that is based on the tensor properties of the orientation states. An advantage of such a classification is the logical relationship between free energy and twin structures, for it becomes immediately apparent which forces and fields will be effective in moving twin walls. The domain patterns in ferroelectric and ferromagnetic materials are strongly affected by external fields, but there are many other types of twinned crystals with movable twin walls and hysteresis. These materials are classified as ferroelastic, ferrobielastic, and various other ferroic species. As explained in the next section, each type of switching arises from a particular term in the free energy function. Ferroic crystals possess two or more orientation states or domains, and under a suitably chosen driving force the domain walls move, switching the crystal from one domain state to another. Switching may be accomplished by mechanical stress (X), electric field (E), magnetic field (H), or some combination of the three. Ferroelectric, ferroelastic, and ferromagnetic materials are well known examples of primary ferroic crystals in which the orientation states differ in spontaneous polarization (P(s)), spontaneous strain (x(s)), and spontaneous magnetization (I(s)), respectively. It is not necessary, however, that the orientation states differ in the primary quantities (strain, polarization, or magnetization) for the appropriate field to develop a driving force for domain walls. If, for example, the twinning rules between domains lead to a different orientation of the elastic compliance tensor, a suitably chosen stress can then produce different strains in the two domains. This same stress may act upon the difference in induced strain to produce wall motion and domain reorientation. Aizu suggested the term ferrobielastic to distinguish this type of response from ferroelasticity, and illustrated the effect with Dauphine twinning in quartz. Other types of secondary ferroic crystals are listed in Table 16.1, along with the difference between domain states, and the driving fields required to switch between states.
Diffusion and ionic conductivity
Authors: Newnham Robert E.
Published: November 2020

The phenomenon of atomic and ionic migration in crystals is called solidstate diffusion, and its study has shed light on many problems of technological and scientific importance. Diffusion is intimately connected to the strength of metals at high temperature, to metallurgical processes used to control alloy properties, and to many of the effects of radiation on nuclear reactor materials. Diffusion studies are important in understanding the ionic conductivity of the materials used in fuel cells, the fabrication of semiconductor integrated circuits, the corrosion of metals, and the sintering of ceramics. When two miscible materials are in contact across an interface, the quantity of diffusing material which passes through the interface is proportional to the concentration gradient. The atomic flux J is given by where J is measured per unit time and per unit area, c is the concentration of the diffusing material per unit volume, and Z is the gradient direction. The proportionality factor D, the diffusion coefficient, is measured in units of m2/s. This equation is sometimes referred to as Fick’s First Law. It describes atomic transport in a form that is analogous to electrical resistivity (Ohm’s Law) or thermal conductivity. There are several objections to Fick’s Law, as discussed in Section 19.5. Strictly speaking, it is valid only for self-diffusion coefficients measured in small concentration gradients. Since J and Z are both vectors, the diffusion coefficient D is a second rank tensor. As with other symmetric second rank tensors, between one and six measurements are required to specify Dij , depending on symmetry. The relationship between structure and anisotropy is more apparent in PbI2. Lead iodide is isostructural with CdI2 in trigonal point group.m. The self-diffusion of Pb is much easier parallel to the layers where the Pb atoms are in close proximity to one another. Diffusion is more difficult along Z3 = [001] because Pb atoms have a very long jump distance in this direction. The mineral olivine, (Mg, Fe)2SiO4, is an important constituent of the deeper parts of the earth’s crust.
Electro-optic phenomena
Authors: Newnham Robert E.
Published: November 2020

Optical beams can be controlled by manipulating the refractive indices and absorption coefficients with applied electric fields. In communication systems electro-optic effects are used in phase and amplitude modulation, in beam deflectors, and in tunable filters. Three such effects are illustrated in Fig. 28.1. Lead lanthanum zirconate titanate (PLZT) is a transparent electroceramic that can be prepared in several different ferroelectric forms with large electro-optic coefficients. When prepared in a normal ferroelectric form it can be used in two different ways. A light-tunable shutter is constructed by coating a multidomain ceramic of PLZT with a photoconducting layer and transparent electrodes. A bias voltage on the electrodes is transferred to the ceramic when the photoconductor is illuminated. The electric field alters the domain structure and the degree of light scattering, controlling the intensity of light. Fully poled ferroelectric ceramics exhibit the linear electro-optic effect Using planar electrodes the PLZT is poled perpendicular to the optical beam. Polarizer and analyzer are positioned in the ±45◦ positions, and light intensity is controlled by altering the birefringence with an electric field. The third experiment utilizes a pseudo-cubic PLZT composition with a large quadratic electro-optic effect. No poling is required in this case. With polarizer and analyzer again in the ±45◦ positions, the transmitted light intensity is proportional to E2 rather than E. Linear and quadratic electro-optic coefficients are defined in terms of the fieldinduced changes in the optical indicatrix: . . . Bij(E) − Bij(0) = Δ
Optical activity and enantiomorphism
Authors: Newnham Robert E.
Published: November 2020

When plane-polarized light enters a crystal it divides into right- and lefthanded circularly polarized waves. If the crystal possesses handedness, the two waves travel with different speeds, and are soon out of phase. On leaving the crystal, the circularly polarized waves recombine to form a plane polarized wave, but with the plane of polarization rotated through an angle αt. The crystal thickness t is in mm, and α is the optical activity coefficient expressed in degrees/mm. The polarization vector of the combined wave can be visualized as a helix, turning α ◦/mm path length in the optically-active medium. Because of the low symmetry of a helix, optical activity is not observed in many high symmetry crystals. Point groups possessing a center of symmetry are inactive. In relating α to crystal chemistry it is convenient to divide optically-active materials into two categories: Those which retain optical activity in liquid form, and those which do not. It has long been known that optically-active solutions crystallize to give optically-active solids. This follows from the fact that molecules lacking mirror or inversion symmetry can never crystallize in a pattern containing such symmetry elements. Thus one way of obtaining optically-active materials is to begin with optically-active molecules, as in Rochelle salt, tartaric acid and cane sugar. Few of these crystals are very stable, however, and the optical activity coefficients are usually small, typically 2◦/mm. The same is true of many inorganic solids, though they are seldom optically active in the liquid state. For NaClO3 and MgSO4·7H2O, α is about 3◦/mm. Quartz and selenium, however, have coefficients an order of magnitude larger, showing the importance of helical structures to optical activity. Both compounds crystallize as right- and left-handed forms in space groups P312 and P322, with helices spiraling around the trigonal screw axes. Quartz contains nearly regular SiO4 tetrahedra with Si–O distances of 1.61 Å. Levorotatory quartz belongs to space group P312 and contains right-handed helices; enantiomorphic dextrorotatory quartz crystallizes in P322. Trigonal selenium also contains helical chains.
Magneto-optics
Authors: Newnham Robert E.
Published: November 2020

The magneto-optic properties of interest are the Faraday Effect, Kerr Rotation, and the Cotton–Mouton Effect. In 1846, Michael Faraday discovered that when linearly polarized light passes through glass in the presence of a magnetic field, the plane of polarization is rotated. The Faraday Effect is now used in a variety of microwave and optical devices. Normally the Faraday experiment is carried out in transmission, but rotation also occurs in reflection, the so-called Kerr Rotation that is used in magneto-optic disks with Mbit storage capability. Other magneto-optic phenomena of less practical interest include the Cotton– Mouton Effect, a quadratic relationship between birefringence and magnetic field, and magnetic circular dichroism that is closely related to the Faraday Effect. A number of nonlinear optical effects of magnetic or magnetoelectric origin are also under study. Almost all these magnetooptical effects are caused by the splitting of electronic energy levels by a magnetic field. This splitting was first discovered by the Dutch physicist Zeeman in 1896, and is referred to as the Zeeman Effect. When linearly polarized light travels parallel to a magnetic field, the plane of polarization is rotated through an angle ψ. It is found that the angle of rotation is given by . . . ψ(ω) = V(ω)Ht, . . . where H is the applied magnetic field, t is the sample thickness, ω is the angular frequency of the electromagnetic wave, and V(ω) is the Verdet coefficient. Faraday rotation is observed in nonmagnetic materials as well as in ferromagnets. The Verdet coefficient of a commercial one-way glass is plotted as a function of wavelength in Fig. 31.1(a). Corning 8363 is a rare earth borate glass developed to remove reflections from optical systems. A polarized laser beam is transmitted through the glass parallel to the applied magnetic field. The plane of polarization is rotated 45◦ by the Faraday Effect. The transmitted beam passes through the analyzer that is set at 45◦ to the polarizer. But the reflected waves coming from the surface of the glass and from the analyzer are rotated another 45◦ as they return toward the laser.
Sea-level changes
Authors: Hallam Tony
Published: November 2020

In earlier times many geologists clearly became cynical about what they had learned as students about Earth history from their stratigraphy courses. ‘The sea comes in, the sea goes out.’ This is indeed one of the most striking signals that emerges from study of the stratigraphic record in a given region: a succession of marine transgressions and regressions on the continents. Little scientific rigour was, however, applied to the subject, and students were left with no overarching explanation to provide any intellectual stimulation. Since the 1970s things have begun to change for the better, as less emphasis has been placed on learning the names of rock formations and fossil zones and more on the dynamic aspects of what to many ranks as a fascinating subject. This entails studying changing geographies and climates within a framework supplied by plate tectonics, the successions of strata being subjected to ever-more-rigorous sedimentological and geochemical analysis, and global correlation continually improved by further study of stratigraphically useful fossils. How do we infer sea-level changes from a given succession of sedimentary rocks? In essence we use facies analysis, which is based upon a careful study of the sediment types and structures, together with a study of the ecological aspects of the contained fossils, or palaeoecology. These features can be compared with those of similar sediments that are being deposited today, or similar organisms living today. Comparisons of this kind were practised by the likes of Cuvier as well as Lyell. Consider, for example, the Cretaceous succession in southern England. The oldest strata, well exposed on the coast from Sussex to Dorset, are known as the Wealden, and consist mainly of sandstones and siltstones that were deposited in a coastal plain (non-marine) setting. They are overlain by the Lower Greensand, a sandy unit of Aptian–Lower Albian age laid down in a very shallow marine environment. These conditions are revealed, not just by the types of fossils, which include the exclusively marine ammonites, but also by the distinctive green clay mineral glauconite, which gives its name to the rock formation and occurs today only in marine settings.
Case Study of Dynamic Simulation
Authors: Bulatov Vasily , Cai Wei
Published: November 2020

The preceding chapter focused on the dislocation core structure at zero temperature obtained by energy minimization. In this chapter we will discuss a case study of dislocation motion at finite temperature by molecular dynamics (MD) simulations. MD simulations offer unique insights into the mechanistic and quantitative aspects of dislocation mobility because accurate measurements of dislocation velocity are generally difficult, and direct observations of dislocation motion in full atomistic detail are still impossible. The discussion of this case study is complete in terms of relevant details, including boundary and initial conditions, temperature and stress control, and, finally, visualization and data analysis. In Section 3.1 we discussed a method for introducing a dislocation into a simulation cell. It relies on the linear elasticity solutions for dislocation displacement fields. To expand our repertoire, let us try another method here. The idea is to create a planar misfit interface between two crystals, such that subsequent energy minimization would automatically lead to dislocation formation.
Line Dislocation Dynamics
Authors: Bulatov Vasily , Cai Wei
Published: November 2020

In the preceding chapters we have discussed several computational approaches focused on the structure and motion of single dislocations. Here we turn our attention to collective motion of many dislocations, which is what the method of dislocation dynamics (DD) was designed for. Typical length and time scales of DD simulations are on the order of microns and seconds, similar to in situ transmission electron microscopy (TEM) experiments where dislocations are observed to move in real time. In a way, DD simulations can be regarded as a computational counterpart of in situ TEM experiments. One very valuable aspect of such a “computational experiment” is that one has full control of the simulation conditions and access to the positions of all dislocation lines at any instant of time. Provided the dislocation model is realistic, DD simulations can offer important insights that help answer the fundamental questions in crystal plasticity, such as the origin of the complex dislocation patterns that emerge during plastic deformation and the relationship between microstructure, loading conditions and the mechanical strength of the crystal. So far, two approaches to dislocation dynamics simulations have emerged. In the line DD method to be discussed in this chapter, dislocations are represented as mathematical lines in an otherwise featureless host medium. An alternative approach is to rely on a continuous field of eigenstrains, in which regions of high strain gradients reveal the locations of the dislocation lines. This representation leads to the phase field DD approach, which will be discussed in Chapter 11. Line DD has certain similarities with the models discussed in the previous chapters, but, at the same time, is rather different from all of them. For example, the representation of dislocations by line segments in line DD method is similar to the kinetic Monte Carlo (kMC) model of Chapter 9. However, having to deal with multiple dislocations on large length and times scales necessitates a more economical treatment of dislocations in the line DD method. Thus, line DD usually relies on less detailed discretization of dislocation lines and treats dislocation motion as deterministic.
Basic laws of thermodynamics
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

In this chapter, we first introduce the basic laws of thermodynamics and R17the most important thermodynamic functions. Even though many of the concepts introduced here will be familiar to many readers with a background in elementary physics, this chapter should not be ignored as it presents these concepts in the language of physical chemistry. Since these concepts form the basis of physical chemistry, this subject will make no sense without a firm footing in these fundamentals. Thermodynamics focuses on the thermal behavior of macroscopic systems (i.e. systems containing a very large number of particles). Thermal processes include both heat exchange between a system and its surroundings and work. The general scheme of a thermodynamic description of such processes can be described as in the picture: Thermodynamic descriptions are usually based upon experimental observations. Experiments can characterize the thermodynamic state of the system in terms of a small number of measurable parameters (e.g. temperature T and pressure p). The generalization of these measurements yields thermodynamics laws. Thermodynamic laws identify state functions that describe the system behavior solely in terms of the system parameters and not on how the system came to be in a particular state. Changes in the state functions during some process depend on only the intial and final states of the system but not on the path between them. Therefore, these changes can be determined from calculations based on a very small set of data. Thermodynamics can be used to answer such questions as (1) is a particular process possible? (2) can the system spontaneously evolve in a particular direction?, and (3) what is the final or equilibrium state? all under a given set of conditions. Equilibrium can be understood as the state in which the system parameters no longer evolve, there are no fluxes of matter or energy through the system, and for which all small disturbances decay. According to the zeroeth law of thermodynamics any isolated system will eventually evolve to an equilibrium state and will never spontaneously leave this state (without a substantial external disturbance).
Thermodynamics of stressed systems
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

As every school child knows, the difference between a solid and a liquid is that a liquid takes the shape of the container in which it is placed while the shape of a solid is independent of the shape of the container (providing the container is big enough). In other words, we must apply a force in order to change the shape of a solid. However, the thermodynamic functions described heretofore have no terms that depend on shape. In this chapter, we extend the thermodynamics discussed above to include such effects and therefore make it applicable to solids. However, since this is a thermodynamics, rather than a mechanics text, we focus more on the relationship between stress and thermodynamics rather than on a general description of the mechanical properties of solids. We start out discussion of mechanical deformation by describing the change of shape of a solid. We define the displacement vector at any point in the solid u(x, y, z) as the change in location of the material point (x, y, z) upon deformation: that is, ux(x, y, z) = x' - x, where the prime indicates the coordinates of the material that was at the unprimed position prior to the deformation. In linear elasticity, we explicitly assume that the displacement vector varies slowly from point to point within the solid where i and j denote the directions along the three axes, x, y, and z. Consider the small parallel-piped section of a solid with perpendicular edges shown in Fig. 7.1. We label the first corner as O, located at position (xO, yO, zO) and subsequent corners as A, B, . . . located at positions (xA, yA, zA), (xB, yB, zB), . . . The edge lengths are Δx, Δy, and Δz such that, for example, xA = xO + Δx. As a result of the deformation, the material originally at point O is displaced to point O' with coordinates (x'O, y'O, z'O).
Diffusion
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

Diffusion is associated with the random, thermal motion of atoms that produces a change in the macroscopic concentration profile. This process occurs in gases, liquids, amorphous and crystalline solids of metals, ceramics, polymers, semiconductors, etc. The investigation of diffusion provides valuable information about the atomic structure of materials and the defects within them. Perhaps, most importantly, diffusion controls the rates of a wide range of kinetic processes associated with the synthesis of materials, processes by which we modify materials, and processes by which materials fail. The most common driving force for diffusion in a single-phase systems is associated with the entropy of mixing of its constituents (recall that we showed that the entropy of mixing of gases and the components of an ideal solution are always positive—see Sections 1.2.6 and 3.3). Since diffusional processes occur through the thermal motion of atoms (see below), it will not be surprising to learn that the rate of diffusion increases with increasing temperature. However, note that while the mechanisms of thermal motion in gases (random collision of atoms with each other) and liquids (e.g. Brownian motion) necessarily lead to mixing, the mechanisms of mixing within a solid are not as obvious. In solids, thermal motion corresponds to the vibrations of atoms near their equilibrium positions. Since the amplitude of such vibrations is much smaller than the nearest-neighbor separation, it would seem that such thermal motions cannot lead to mixing. Thus, the question ‘‘how do atoms migrate in solids’’ is not so simple. The equations describing diffusion were suggested by the physiologist Fick in 1855 as a generalization of the equations for heat transfer suggested by Fourier in 1824. Fick’s equations for diffusion can be obtained by analogy with Fourier’s equations for heat transfer by replacing heat with the number of atoms, temperature with concentration, and thermal conductivity with diffusivity. Fick’s first law provides a relationship between atomic currents and concentration gradients. As discussed above, this relationship can be understood by analogy with thermal conductivity or electrical conductivity.
Introduction to statistical thermodynamics of gases
Authors: Bokstein Boris S. , Mendelev Mikhail I. , Srolovitz David J.
Published: November 2020

As we discussed earlier in this book, thermodynamics provides very general relations between the properties of a system. On the other hand, thermodynamics is unable to predict any of the individual properties without the addition of either empirical or microscopic information. For example, we used thermodynamics to obtain Raoult’s law from Henry’s law, but we cannot derive Henry’s law from thermodynamic principles. Statistical mechanics provides an approach to determine individual thermodynamic properties from microscopic considerations. When applied in the realm of physical chemistry, we refer to this approach as statistical thermodynamics. In this chapter, we provide a simplified derivation of the Gibbs distribution, which is the basis of much of statistical thermodynamics. We then use statistical mechanics to show how the properties of an ideal gas can be obtained from a small number of properties of the molecules in the gas. This will allow us to determine such quantities as the equilibrium and rate constants of gas phase chemical reactions. As a result, we will gain new insight into the phenomena which we have already considered on the basis of phenomenological thermodynamics or formal kinetics. This approach will also show how to determine some of the parameters we previously introduced as input data in our thermodynamic considerations. As we have already seen, a finite system will eventually come into equilibrium with its surroundings. We even showed that when thermodynamic equilibrium is established, the temperatures, pressures, and chemical potentials of the system and its surroundings are equal (see Section 1.5.2). However, we never discussed what equilibrium actually is. For example, does this mean that the energy of the system is truly constant or is it only constant on average? When the system has a particular energy, does this mean that it is in a unique physical state or can it be in any one of several states that have exactly the same energy? In the latter case, can we simply talk about the probability the system is in each of these states? If the energy can fluctuate, what is the probability that the system has a particular energy? A very general approach to these types of questions was suggested by Gibbs and is now known as Gibbs statistics.
Quantum Dynamics Using The Time-Dependent Schrödinger Equation
Authors: Nitzan Abraham
Published: November 2020

This chapter focuses on the time-dependent Schrödinger equation and its solutions for several prototype systems. It provides the basis for discussing and understanding quantum dynamics in condensed phases, however, a full picture can be obtained only by including also dynamical processes that destroy the quantum mechanical phase. Such a full description of quantum dynamics cannot be handled by the Schrödinger equation alone; a more general approach based on the quantum Liouville equation is needed. This important part of the theory of quantum dynamics is discussed in Chapter 10. Given a system characterized by a Hamiltonian Ĥ , the time-dependent Schrödinger equation is For a closed, isolated system Ĥ is time independent; time dependence in the Hamiltonian enters via effect of time-dependent external forces. Here we focus on the earlier case. Equation (1) is a first-order linear differential equation that can be solved as an initial value problem.
Spectroscopy
Authors: Nitzan Abraham
Published: November 2020

The interaction of light with matter provides some of the most important tools for studying structure and dynamics on the microscopic scale. Atomic and molecular spectroscopy in the low pressure gas phase probes this interaction essentially on the single particle level and yields information about energy levels, state symmetries, and intramolecular potential surfaces. Understanding environmental effects in spectroscopy is important both as a fundamental problem in quantum statistical mechanics and as a prerequisite to the intelligent use of spectroscopic tools to probe and analyze molecular interactions and processes in condensed phases. Spectroscopic observables can be categorized in several ways. We can follow a temporal profile or a frequency resolved spectrum; we may distinguish between observables that reflect linear or nonlinear response to the probe beam; we can study different energy domains and different timescales and we can look at resonant and nonresonant response. This chapter discusses some concepts, issues, and methodologies that pertain to the effect of a condensed phase environment on these observables. For an in-depth look at these issues the reader may consult many texts that focus on particular spectroscopies. With focus on the optical response of molecular systems, effects of condensed phase environments can be broadly discussed within four categories: 1. Several important effects are equilibrium in nature, for example spectral shifts associated with solvent induced changes in solute energy levels are equilibrium properties of the solvent–solute system. Obviously, such observables may themselves be associated with dynamical phenomena, in the example of solvent shifts it is the dynamics of solvation that affects their dynamical evolution. Another class of equilibrium effects on radiation– matter interaction includes properties derived from symmetry rules. A solvent can affect a change in the equilibrium configuration of a chromophore solute and consequently the associated selection rules for a given optical transition. Some optical phenomena are sensitive to the symmetry of the environment, for example, surface versus bulk geometry. 2. The environment affects the properties of the radiation field; the simplest example is the appearance of the dielectric coefficient ε in the theory of radiation–matter interaction.
Beyond first-order logic
Authors: Hedman Shawn
Published: November 2020

We consider various extensions of first-order logic. Informally, a logic 𝓛 is an extension of first-order logic if every sentence of first-order logic is also a sentence of 𝓛. We also require that 𝓛 is closed under conjunction and negation and has other basic properties of a logic. In Section 9.4, we list the properties that formally define the notion of an extension of first-order logic. Prior to Section 9.4, we provide various natural examples of such extensions. In Sections 9.1–9.3, we consider, respectively, second-order logic, infinitary logics, and logics with fixed-point operators. We do not provide a thorough treatment of any one of these logics. Indeed, we could easily devote an entire chapter to each. Rather, we define each logic and provide examples that demonstrate the expressive power of the logics. In particular, we show that none of these logics has compactness. In the final Section 9.4, we prove that if a proper extension of first-order logic has compactness, then the Downward Löwenhiem–Skolem theorem must fail for that logic. This is Lindstrom’s theorem. The Compactness theorem and Downward Löwenheim–Skolem theorem are two crucial results for model theory. Every property of first-order logic from Chapter 4 is a consequence of these two theorems. Lindström’s theorem implies that the only extension of first-order logic possessing these properties is first-order logic itself. Second-order logic is the extension of first-order logic that allows quantification of relations. The symbols of second-order logic are the same symbols used in first-order logic. The syntax of second-order logic is defined by adding one rule to the syntax of first-order logic. The additional rule makes second-order logic far more expressive than first-order logic. Specifically, the syntax of second-order logic is defined as follows. Any atomic first-order formula is a formula of second-order logic. Moreover, we have the following four rules: (R1) If φ is a formula then so is ￢φ. (R2) If φ and ψ are formulas then so is φ ∧ ψ. (R3) If φ is a formula, then so is ∃x φ for any variable x.
A uniform method for proving lower bounds on the computational complexity of logical theories
Authors: Compton Kevin J. , Ward Henson C.
Published: November 2020

In this chapter we present a method for obtaining lower bounds on the computational complexity of logical theories, and give several illustrations of its use. This method is an extension of widely used procedures for proving the recursive undecidability of logical theories. (See Rabin [1965] and Eršov et al. [1965].) One important aspect of this method is that it is based on a family of inseparability results for certain logical problems, closely related to the well-known inseparability result of Trakhtenbrot (as refined by Vaught), that no recursive set separates the logically valid sentences from those which are false in some finite model, as long as the underlying language has at least one non-unary relation symbol. By using these inseparability results as a foundation, we are able to obtain hereditary lower bounds, i.e., bounds which apply uniformly to all subtheories of the theory. The second important aspect of this method is that we use interpretations to transfer lower bounds from one theory to another. By doing this we eliminate the need to code machine computations into the models of the theory being studied. (The coding of computations is done once and for all in proving the inseparability results.) By using interpretations, attention is centred on much simpler definability considerations, viz., what kinds of binary relations on large finite sets can be defined using short formulas in models of the theory. This is conceptually much simpler than other approaches that have been proposed for obtaining lower bounds, such as the method of bounded concatenations of Fleischmann et al. [1977]. We will deal primarily with theories in first-order logic and monadic second-order logic.
Constraint Logic Programming: A Survey
Authors: Jaffar Joxan , Maher Michael J.
Published: November 2020

Constraint Logic Programming (CLP) began as a natural merger of two declarative paradigms: constraint solving and logic programming. This combination helps make CLP programs both expressive and flexible, and in some cases, more efficient than other kinds of programs. Though a relatively new field, CLP has progressed in several and quite different directions. In particular, the early fundamental concepts have been adapted to better serve in different areas of applications. In this survey of CLP, a primary goal is to give a systematic description of the major trends in terms of common fundamental concepts. Consider first an example program in order to identify some crucial CLP concepts. The program below defines the relation sumto(n, 1 + 2 + . . . . . + n) for natural numbers n. . . . sumto(0, 0). . . . . . sumto(N, S) :- N >= 1, N <= S, sumto(N - 1, S - N). . . . The query S <= 3, sumto(N, S) gives rise to three answers (N = 0, S = 0), (N = 1, S =1), and (N = 2, S = 3), and terminates.
Intermolecular Forces
Authors: Gray C. G. , Gubbins K. E.
Published: November 2020

Knowledge of the intermolecular potential for simple (i.e. monatomic) molecules has increased greatly in recent years. For polyatomic molecules, on the other hand, such knowledge is still rather meagre, and much more is needed. One needs to know (i) what is the pair potential? (ii) how important are the triplet and other multibody potentials in liquids? These multibody potentials have been studied very little for polyatomic liquids (see refs. 28-35 and §§ 1.2.3, 2.10, and 4.10), and are usually taken into account, if at all, by an effective pair potential. There have been, especially at short range, relatively few theoretical evaluations of the pair potential for diatomic or polyatomic molecules (see, e.g. refs. 21-7 and 40-55a). The most reliable existing knowledge has been obtained from binary collision experiments, or, for the longrange part of the potential, from measurements of properties of single molecules. Examples include molecular beam scattering, induced birefringence, pressure and dielectric virial coefficients, and collision-induced absorption (including gas dimer spectra, which can also be studied by beam resonance spectroscopy) which yield values for the parameters (e.g. Lennard-Jones constants, polarizabilities, dipole moments, quadrupole moments, octopole moments, etc. - see also Appendix D) occurring in the expressions for the intermolecular potentials. The shape of the repulsive core of the potential can be inferred approximately from the molecular structure and charge density as determined experimentally, for example by electron and X-ray diffraction or by quantum calculations. As an example of the last point we show in Fig. 2.1 a contour map for the theoretically calculated charge density of N2 , the prototype molecule for simple nonpolar molecular fluid studies. Over 95 per cent of the total electronic charge is contained within the outermost (0.002 au) contour, and the dimensions of this contour are sometimes used to define a theoretical size of the N2 molecule. The dimensions shown on Fig. 2.1 agree roughly with dimensions obtained experimentally from Lennard- Jones diameters in gases (virial coefficients and viscosity) and so-called van der Waals radii from X-ray diffraction studies of solids.
Textual Profiles of Genes
Authors: Raychaudhuri Soumya
Published: November 2020

Using algorithms to analyze natural language text is a challenging task. Recent advances in algorithms, and increased availability of computational power and online text has resulted in incremental progress in text analysis (Rosenfeld 2000). For certain specific applications natural language processing algorithms can rival human performance. Even the simplest algorithms and approaches can glean information from the text and do it at a rate much faster than humans. In the case of functional genomics, where an individual assay might include thousands of genes, and tens of thousands of documents pertinent to those genes, the speed of text mining approaches offers a great advantage to investigators trying to understand the data. In this chapter, we will focus on techniques to convert text into simple numerical vectors to facilitate computation. Then we will go on to discuss how these vectors can be combined into textual profiles for genes; these profiles offer additional biologically meaningful information that can complement available genomics data sets. The previous chapter introduced methods to analyze gene expression data and sequence data. The focus of many analytical methods was comparing and grouping genes by similarity. Some sequence analysis methods like dynamic programming and BLAST offer opportunities to compare two sequences, while multiple sequence alignment and weight matrices provide a means to compare families of sequences. Similarly, gene expression array analysis approaches are mostly contingent on distance metrics that compare gene expression profiles to each other; clustering and classification algorithms provide a means to group similar genes. The primary goal of applying these methods was to transfer knowledge between similar genes. We can think of the scientific literature as yet another data type and define document similarity metrics. Algorithms that tap the knowledge locked in the scientific literature require sophisticated natural language processing approaches. On the other hand, assessing document similarity is a comparatively easier task. A measure of document similarity that corresponds to semantic similarity between documents can also be powerful. For example, we might conclude that two genes are related if documents that refer to them are semantically similar.
Analyzing Groups of Genes
Authors: Raychaudhuri Soumya
Published: November 2020

The analysis of large-scale genomic data (such as sequences or expression patterns) frequently involves grouping genes based on common experimental features. The goal of manual or automated analysis of genomics data is to define groups of genes that have shared features within the data, and also have a common biological basis that can account for those commonalities. In utilizing algorithms that define groups of genes based on patterns in data it is critical to be able to assess whether the groups also share a common biological function. In practice, this goal is met by relying on biologists with an extensive understanding of diverse genes that decipher the biology accounting for genes with correlated patterns. They identify the relevant functions that account for experimental results. For example, experts routinely scan large numbers of gene expression clusters to see if any of the clusters are explained by a known biological function. Efficient definition and interpretation of these groups of genes is challenging because the number and diversity of genes exceed the ability of any single investigator to master. Here, we argue that computational methods can utilize the scientific literature to effectively assess groups of genes. Such methods can then be used to analyze groups of genes created by other bioinformatics algorithms, or actually assist in the definition of gene groups. In this chapter we explore statistical scoring methods that score the ‘‘coherence’’ of a gene group using only the scientific literature about the genes—that is whether or not a common function is shared between the genes in the group. We propose and evaluate such a method, and compare it to some other possible methods. In the subsequent chapter, we apply these concepts to gene expression analysis. The major concepts of this chapter are described in the frame box. We begin by introducing the concept of functional coherence. We describe four different strategies to assess the functional coherence of a group of genes. The final part of the chapter emphasizes the most effective of these methods, the neighbor divergence per gene. We present a discussion of its performance properties in general and on its robustness given imperfect groups. Finally we present an example of an application to gene expression array data.