Linear and Planar Densities

Coordinate Type Index Symbols Representative Equation Equation Symbols 

Non-hexagonal [uvw, [UVW] fX2- = ) Xj = tail coordinate —x axis X2 = head coordinate —x axis 

Hexagonal uutw a = head coordinate—axis a = tail coordinate—Ui axis 

The two previous sections discussed the equivalency of nonparallel crystallographic directions and planes. Directional equivalency is related to linear density in the sense that, for a particular material, equivalent directions have identical linear densities. The corresponding parameter for crystallographic planes is planar density, and planes having the same planar density values are also equivalent. 

Linear density (LD) is defined as the number of atoms per unit length whose centers lie on the direction vector for a specific crystallographic direction that is. 

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Linear and planar densities are important considerations relative to the process of slip— that is, the mechanism by which metals plastically deform (Section 7.4). Slip occurs on the most densely packed crystallographic planes and, in those planes, along directions having the greatest atomic packing. 

Crystallographic directional and planar eqnivalencies are related to atomic linear and planar densities, respectively. 

A density functional study of the molecular structures and vibrational frequencies of the linear and planar monocyclic isomers of the C series (2 s n < 18) was performed recently by Hutter et al. [236]. Thqr found that for n s 9 the linear chains are generally more stable than the ring structures. Their results confirmed that for n s 10 the cyclic structures dominate. 

M. H. Wagner, P. Ehrecke, P. Hachmann, and J. Meissner, A Constitutive Analysis of Uniaxial, Equibiaxial and Planar Extension of a Commercial Linear High-density Polyethylene Melt, J. Rheol., 42, 621-638 (1998). 

Consider two parallel planar ion-penetrable membranes 1 and 2, which may not be identical, at separation h in a symmetrical electrolyte solution of valence z and bulk concentration n (Fig. 16.1). We take an x-axis perpendicular to the membranes with its origin at the surface of membrane 1. The electric potential i/ (x) at position X between the membranes (relative to the bulk solution phase, where is set equal to zero) is assumed to be small so that the linearized Poisson-Boltzmann equation can be employed. Membranes 1 and 2, respectively, consist of N and M layers. All the layers are perpendicular to the x-axis. Let the thickness and the density of membrane-fixed charges of the ith layer of membrane j (7=1, 2) be and The linearized Poisson-Boltzmann equation for the /th layer... 

This paper reports experiments and numerical simulations related to a linear low-density polye ylene (LLDPE) and a low-density polyethylene (LDPE), in a significant number of axisymmetric and planar mixed flows. Converging and abrupt contraction geometries involving short and long dies were considered as well as extrudate swell flows occurring at the exit of the ducts under investigation. 

These linkages allow delocalization of electron density between the terminal aromatic rings, and have the effect if retaining the rigidity and planarity of the central core. The delocalized electron density can enhance the molecules anisotropic polarizability. Interestingly, the analogue of the above molecule in which the two aromatic rings are joined by a double bonded rather than a triple bond does not necessarily form a liquid crystal phase. The tram isomer retains the overall linear profile and can form a liquid crystalline phase whereas the cis does not. ... 

Dislocations do not move with the same degree of ease on all erystallographie planes of atoms and in all crystallographic directions. Typically, there is a preferred plane, and in that plane there are specific directions along which dislocation motion occurs. This plane is called the slip plane it follows that the direction of movement is called the slip direction. This combination of the slip plane and the slip direction is termed the slip system. The slip system depends on the crystal structure of the metal and is such that the atomic distortion that accompanies the motion of a dislocation is a minimum. For a particular crystal structure, the slip plane is the plane that has the densest atomic packing—that is, has the greatest planar density. The slip direction corresponds to the direction in this plane that is most closely packed with atoms—that is, has the highest linear density. Planar and linear atomic densities were discussed in Section 3.11. 

The problem of the sign of AR/R for the divalent tin compounds was investigated by Lees and Flinn (16). In the relationship between the quadrupole splitting and chemical shift for the stannous compounds, two distinct correlations became apparent—compounds with a linear covalent bond, and compounds with a predominantly planar bond. Furthermore, there exists a linear relationship between the number of 5 p electrons and the chemical shift and hence the total 5 electron density. Using free tin ion wave functions in a self-consistent field calculation, they showed that the direct eflEect of adding 5 electrons is considerably... 

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