Properties in One Dimension

The magnetization M is defined as the magnetic moment per unit volume. The magnetic susceptibility, f, relates Mto the applied magnetic field, H  

The susceptibility is usually expressed as per unit volume, per mole, or per 

Here ps is the Bohr magneton, p is the effective number of Bohr magnetons, and ks is Boltzman s constant. Thus a plot of 1 // versus temperature is a straight line with an intercept at zero. Other contributions to the susceptibility of an isolated molecule include core diamagnetism and Van Vleck paramagnetism, both of which are small and temperature independent (231). 

---

We will start from a single, ideal, regular, and infinite chain, the perfect one-dimensionality system of the theorists. For several reasons one-dimensionality is unfavorable to conduction. (As a textbook for basic principles of transport properties in one dimension, see Ref. 29.) First, we know from Chapter 11 that the electronic structure of such a chain cannot be like that of a metal. Even the best chemist will not be able to prevent the Peierls transition. He will end up with a compound with a gap in the band structure that is, an insulating, or at best a semiconducting material. Second, since we are dealing with a pure one-dimensionality system, any defect in the... 

Just as any periodic structure has dimensionality, so do photonic crystals. Thus, photonic crystals can be 1, 2, or 3-dimensional in terms of their mode of operation. This dimensionality reflects the periodicity which is relevant to the desired optical properties, thus a simple layered structure will show properties in one dimension only (i.e. if light is perpendicular to the surface) whereas a regular 3-dimensional lattice will be dependent on the lattice parameters. 

The equilibrium properties of a fluid are related to the correlation fimctions which can also be detemrined experimentally from x-ray and neutron scattering experiments. Exact solutions or approximations to these correlation fiinctions would complete the theory. Exact solutions, however, are usually confined to simple systems in one dimension. We discuss a few of the approximations currently used for 3D fluids. 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

P. A. Monson. The properties of inhomogeneous square-well mixtures in one dimension. Mol Phys 70 401—423, 1990. 

This paper is organized as follows. Section 2 presents non-trivial properties of the velocity distribution functions for RIG for quasi and ordinary particles in one dimensions. In section 3 we find the state equation for relativistic ideal gas of both types. Section 4 presents the distribution function for the observed frequency radiation generated for quasi and ordinary particles of the relativistic ideal gas, for fluxons under transfer radiation and radiative atoms of the relativistic ideal gas. Section 5 presents a generalization of the theory of the relativistic ideal gas in three dimensions and the distribution function for particles... 

We have to be careful. The symmetry between columns and rows of the matrix Y is not complete. Closure is a property of the concentration profiles only and thus applies only in one dimension. The command mean (Y, 1) computes the mean of each column of Y and the resulting mean spectrum is subtracted from each individual spectrum. 

DFT simulations of DNA-like structures constituted of more than two bases, with an accent on their electronic properties, have become available only recently [58, 90-95, 114]. In our review, we mainly focus on periodic systems obtained by replicating a given elementary unit. For such periodic assemblies, prototypes of DNA-based wires, it is possible to define a crystal lattice in one dimension. This allows the extension of the concept of band dis-... 

An antiferromagnetically coupled spin dimer of two s=l/2 has a spin singlet as its lowest energy state. The first excited state is a spin triplet and the energy between these two states is denoted as the spin gap A. If such dimers are coupled in one dimension a dimerized spin chain is formed. Such spin chains exist in several spin systems, e.g. in CuGeC>3 [5], Properties of this and related compounds are further discussed in Ref. [3],... 

Mesomorphic phases have properties between those of solids and liquids. For example, rod-shaped or disk-shaped molecules often melt to form liquid crystals, which are fluid, but contain some order. One form of liquid crystals formed from rod-shaped molecules, called a nematic liquid crystal, is shown in Fig. 11. The molecules in the nematic phase are partially ordered in one dimension, with an orientation angle that can vary from one domain to another in the fluid. The forces... 

Nanomaterials are composed of structural entities - isotropic grains or particles, rods, wires, platelets, layers - of size, at least in one dimension, between 1 and 100 nm [1-3]. Larger particles are called submicron particles, smaller ones are known as clusters. Some physical properties of nanomaterials [4-6] differ from those of coarse-grained materials of the same chemical composition due to two essential features ... 

On the other hand, if we consider the so-called Fourier-Plancherel functions in one dimension defined by Eq. (3.2), all of them are situated in the complement C(u 1) as soon as the rotation angle is different from zero (a 0). It is evident that, in polar coordinates, the exponential factor exp(ikr) is going to have similar properties, since one has... 

---