Condensed-matter phase

Mn is the mass of the nucleon, jis Planck s constant divided by 2ti, m. is the mass of the electron. This expression omits some temis such as those involving relativistic interactions, but captures the essential features for most condensed matter phases. 

Another realistic approach is to constnict pseiidopotentials using density fiinctional tlieory. The implementation of the Kolm-Sham equations to condensed matter phases without the pseiidopotential approximation is not easy owing to the dramatic span in length scales of the wavefimction and the energy range of the eigenvalues. The pseiidopotential eliminates this problem by removing tlie core electrons from the problem and results in a much sunpler problem [27]. 

Since and depend only on die valence charge densities, they can be detennined once the valence pseudo- wavefiinctions are known. Because the pseudo-wavefiinctions are nodeless, the resulting pseudopotential is well defined despite the last temi in equation Al.3.78. Once the pseudopotential has been constructed from the atom, it can be transferred to the condensed matter system of interest. For example, the ionic pseudopotential defined by equation Al.3.78 from an atomistic calculation can be transferred to condensed matter phases without any significant loss of accuracy. 

Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

Stillinger, F. H., and Weber, T. A., Computer simulation of local order in condensed matter phases of silicon. Phys. Rev. B 31,5262-5271 (1985). 

Condensed matter phases and structures are commonly reached via symmetry breaking transitions. In such systems, when the continuous symmetry is broken, temporary domain-t5q)e patterns are formed. The domain structures eventually coarsen, and disappear in the long-time limit, leaving a uniform broken-symmetry state. This state possesses so-called long-range order (LRO), in which the spatially dependent order parameter correlation function does not decay to zero in the limit of large distances. 

As a method, TRXRD is still in its infancy. While it has already proven to be a powerful, information-dense structural and kinetic probe there is much room for improvement. In what follows I draw upon my own experience as to the limitations of the method and will address these at four levels facility, sample, instrumentation, and experimental design. The view held here is that deficits in the technique will be corrected more expeditiously if attention is drawn to them at an early stage. Alas, in some cases, a solution must await the development of new technologies. This exercise also serves to emphasize that TRXRD is not the panacea but rather another tool in an arsenal of physicochemical techniques with which to tackle critical issues in condensed matter phase science. 

Another frequent mistake among students is to try to apply the ideal gas law to calculate the concentrations of species in condensed-matter phases (e.g., liquid or solid phases). Do not make this mistake] The ideal gas law only applies to gases. To calculate concentrations for liquid or solid species, information about the density (pj) of the liquid or solid phase is required. Both mass densities and molar densities (concentrations) as well as molar and atomic volumes may be of interest. The complexity of calculating these quantities tends to increase with the complexity of the material under consideration. In this section, we will consider three levels of increasing complexity pure materials, simple compounds or dilute solutions, and more complex materials involving mixtures of multiple phases/compounds. 

Liquid-solid and solid-solid phase transformations are also known as condensed-matter phase transformations. Condensed-matter phase transformations, like other kinetic processes, are driven by thermodynamics. When a region of matter can lower its total free energy by changing its composition, structure, symmetry, density, or any other phase-defining aspect, a phase transformation can occur. 

In most condensed-matter phase transformations, pressure is typically not a chief controlling variable (although it is important in certain circumstances). This is in distinct contrast to the gas-solid processes discussed in the previous chapter, where gas-phase partial pressures typically played a cendal role. 

In order for a phase transformation to occur, a driving force must be present. For most of the condensed-matter phase transformations discussed in this chapter, the driving force is supplied by a change in temperature or composition. Temperature and composition are two of the primary processing knobs that materials engineers have at their disposal to manipulate the structure and property of materials for various applications. 

An important truth of nature has significant implications on the kinetics of condensed-matter phase transformations ... 

The energy cost to create surfaces/interfaces leads to a nucleation barrier in condensed-matter phase transformations. Therefore, nucleation-based phase transformations can only occur if the energy released by creating the new volume of the second phase sufficiently offsets the energy expended in creating the new interfacial area. This leads to a minimum viable nucleation size and thus helps determine the speed at which nucleation can proceed. These issues will be discussed in the next section ... 

For the case of nucleation in a condensed-matter phase, diffusion controls the rate at which fresh atoms can arrive at the surface of the nucleus. In this case, the nucleation rate may be expressed as... 

Condensed-matter phase transformations can be broadly divided into two main categories diffusional transformations and diffusionless (or fluxless ) transformations. 

A driving force must be present for a phase transformation to occur. The most common driving forces for condensed-matter phase transformation include temperature and composition, although pressure-induced phase transformations are... 

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