Spatial extent of forces

The convergence of several quantities depending on planar forces was examined in detail in Ref. 39, from which the main results are summarized in this Subsection. The notion convergence expresses mathematically what we intuitively feel as range of forces because, strictly speaking, the spatial extent of forces is infinite. Thus rather than asking how far do the forces extend , a 

This agrees with values of the force constants found ab initio and given in Tab. 5.1 was found to be of the same order as the 

Much more interesting is the spatial extent of the [100] transverse forces, and interactions up to the fifth neighbors were included for calculation of the phonon dispersion in Fig. 5.3.3. It is argued in Ref. 39 that the range of these forces is m = 5 2 -the error margin partly resulting from the present use of less precise local potentials. The most sensitive part of the transverse dispersion is the TA-branch a fairly steep slope at the origin (elastic constant becomes a flat dispersion near the 

Brillouin-zone edge this behavior is characteristic for covalent bonding. In Fig. 5.5.2a the frequency v(TA(X)) calculated by the ab initio force constant method is plotted versus the range of forces The frequency of the mode is given in terms of the transverse force constants as 

Inspection of Fig. 5.5.2a suggests that the limiting value is already attained at n=3 - although as many as 6 to 7 neighbor forces may be needed to stabilize its evolution , i.e. in order to achieve convergence. (Note that 4th- and 8th-nelghbor forces do not contribute at all to vibrations at the Brillouin-zone edge.) 

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The self-consistent calculations were performed on superc lls with m = 4 to 8 (quadrupled to octupled) and displacements lu ranging from 0.01 to 0.02 The Results from different supercells are consistent within <0.004 x 10 dyn/cm and the force constants given in Tab. 5.1 are those obtained on the smallest of the supercells tried (which we expect to limit the roundoff errors and thus to be more reliable) they will be discussed later. Writing down the equations of motion for the linear chains shown schematically in Fig. 5.3.1 leads to 2 x 2 secular equations, which have as solutions the phonon dispersion shown in Fig. 5.3.3 before reaching this point, however, two problems require an adequate treatment anharmonlclty and spatial extent of forces. They are the topics of the next two Subsections. 

The orientation used in the vibrational calculations on the linear chain is the one showed in Fig. 7.0.2a the cation plane is placed at the origin, and anion planes are at the sites 1 plane K =0 contains the atom at (000), the plane < =-M is the one containing the site a/4(lll). The dynamical matrix is given by eq. (5.7.1) and treatment of enharmonic effects - which are small in the [100] direction - is as in Ge, Chapter 5.4. Even the conclusions about the spatial extent of forces turn out to be similar to Ge (Chapter 5.5), including the even/odd ii zig-zag convergence. The essential difference is in presence of a macroscopic electric field. 

The correlation length corresponds to the spatial extent of the restoring force originating from an ordered region. When the temperature approaches the critical temperature Ttr, the restoring force vanishes. This can be formalized by letting diverge as... 

The above effect is explained by the spatial extent of the initial wavepacket on the potential surface. A larger wavepacket (smaller force constant) covers more of the potential, experiences a larger slope in the Qx dimension, and therefore increases the width of the low frequency progressions because the initial decrease in the overlap is fast. Conversely, a smaller wavepacket (larger force constant) covers less of the potential, experiences a smaller slope in the Qx... 

In the discussion so far we have considered only linear elastic fracture mechanics (LEFM) the term linear elastic means that the cracked specimen obeys Hooke s law to a good approximation. In the context of fracture mechanics, the requirement is that the extent of yielding in the neighbourhood of the crack tip is negligible, so that the force-deflection curves for test specimens are linear. In addition, the value of B must be sufficiently high that the deformation at the crack tip occurs under plane-strain conditions. It is found experimentally that these conditions are met if B, (IV — a), and a are all greater than 2.5(K,c/ffy). Provided these conditions are met, the spatial extent of the plastically deformed zone at the crack tip is less than 2 % of the above dimensions, the specimen fractures in plane-strain and the measured is a true material property. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

In the absence of pending groups or antiplasticisers, the motions of the hydroxypropyl ether units in the high-temperature part of the transition force motions of the crosslink points that are spatial neighbours of the moving HPE sequence. A crude estimate leads to an extent of cooperativity at high temperatures reaching more than six crosslink points in densely crosslinked networks. 

The typical case we consider is the most natural initial treatment of molecules of non-trivial spatial extent, and with some conformational flexibility treated by molecular-mechanics force fields. We will here denote that mechanical potential energy surface by the molecular type is denoted by the subscript a here. 

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