Center of force

What displacements, as compared with the positions in the bulk, are permitted for each of the centers of force present in the surface layers ... 

It is supposed that this displacement can occur only in the direction normal to the rupture surface, not parallel to the latter. When all the above distances are systematically varied until the minimum of the total interaction energy is reached, then the most probable position of all 6 centers of force is found. It appears that the distance between the Li nuclei in the outermost and the H nuclei in the second layer is smaller (by 0.00032 angstrom) than in the bulk of the crystal, and the distance between the H nuclei in the external and the Li nuclei in the penultimate layer is... 

The results of theoretical potential calculations (18, 19) suggest considerable energetic heterogeneity within the zeolite cavities. In such calculations the probe molecule is, however, represented as a point center of force, and for polyatomic molecules the effect of molecular rotation will reduce any such variations in potential through the cavity. For such systems the idealized model, which assumes a uniform potential throughout the free volume, may not be too unrealistic. 

NUCLEAR POTENTIAL. The potential energy V of a nuclear particle as a function of its position in the field of a nucleus or of another nuclear particle. A central potential is one that is spherically symmetric, so that V is a function only of the distance r of the particle from the center of force. A noncentral potential, on the other hand, is one that is not spherically symmetrical, or one that depends upon the relative directions of the angular momenta associated with the particle and the center of force, as well as upon the distance r. A negative potential corresponds to an attractive force, while a positive potential corresponds to a repulsive force. 

Since the outcome of the collision only depends on the relative motion of the reactant molecules, we begin with an elimination of the center-of-mass motion of the system. From classical mechanics it is known that the relative translational motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass fj, = rri nif)/(m + mB), relative to a fixed center of force. This result can be generalized to molecules by introducing proper relative coordinates, to be described in detail in Section 4.1.4. 

The internal (vibrational and rotational) motion of molecule A is the same as that of the pseudo-molecule, while the center of mass of molecule B is at the fixed center of force. The force from the center of force on the pseudo-molecule is determined as the force between A and B, with A at the position of the pseudo-molecule and B at the position of the center of force. The scattering geometry is illustrated in Fig. 4.1.1. The pseudo-molecule moves with a velocity v = va — vB relative to the fixed center of force. We have drawn a line through the force center parallel to v that will be convenient to use as a reference in the specification of the scattering geometry. In addition to the internal quantum states of the pseudo-molecule and velocity v, the impact parameter b and angle are used to specify the motion of the molecule. 

Fig. 4.1.1 Trajectories corresponding to the relative motion of two molecules. The filled circle represents the pseudo-molecule and the cross the fixed center of force. Two trajectories are shown, for the same quantum states of the reactants, but with different phases of the internal degrees of freedom, as explained in the text.

With an eye on later applications we will assume that the potential energy depends not only on the coordinates q, but also on some parameters r , rs, r , which may for instance be outside centers of force that exert external forces on the molecules of the gas.178... 

The gas model exerts a certain reaction force at all times on the above-mentioned centers of force or on the piston. In a certain phase (q, p) of the motion it acts along the parameter r with the generalized force... 

The Spherically Symmetric Model is clearly unsatisfactory for predicting Cg values. Calculations are now in progress with the Na, Ca, O, Si, and A1 atoms treated as charged point centers of force. 

For mono-atomic gases the spacing between the particles is considered large enough so that we can approximate the particles as points, or point centers of force. For this reason, in kinetic theory a gas molecule is characterized by its position r and its velocity c. 

Fig. 2.2. Scattering of a molecule by a fixed center of force O. Molecule 1 is at rest...

Due to the symmetry properties of a binary collision, it is convenient to illustrate the basic ideas considering a one body scattering problem that is concerned with the scattering of particles by a fixed center of force. 

We first state that as a particle approaches the center of force, its orbit will deviate from the incident straight line trajectory. After passing the center of force, the force acting on the particle will eventually diminish so that the orbit once again approaches a straight line (as sketched in Fig. 2.3). In general the final direction of motion is not the same as the incident direction, and the particle is said to be scattered. 

In our discussions of ion-solid interactions, we restrict ourselves to central forces where the potential V is a function of r only (V = K(r)), so that the force is always along r. We need to consider only the problem of a single particle of mass, Mc, moving about a fixed center of force, which will be taken as the origin of the... 

Fig. 3.5. Plan view of trajectory of a particle moving with the impact parameter b and with a repulsive center of force (French, 1971)...

PVC has many points of attachment along the chain. Introduction of a plasticizer separates the macromolecules, breaks the attachments, and masks the many centers of force for intermolecular attraction, thus producing an effect similar to what exists in a polymer with fewer points of attachment. 

Relative Viscosity of Suspensions One of the most interesting derivations of the T vs. (() dependence (covering the full range of concentration) was published by Simha [1952]. He considered the effects of concentration on the hydrodynamic interactions between suspended particles of finite size. (Note that previously the particles were simply considered point centers of force that decayed with cube of the distance.) Simha adopted a cage model, placing each solid, spherical particle of radius a inside a spherical enclosure of radius b. At distances x < b, the presence of other particles does not influence flow around the central sphere and the Stokes relation is satisfied. This assumption leads to a modified Einstein [1906, 1911] relation ... 

In (3.12) the center of force is taken at the center of each bond. This approximation becomes exact as Aj 0. The presence of an external force implies that space is no longer isotropic. Thus, let r = R, and... 

Fig. 2.9. Trajectories of a particle which interacts with a center of force via a Lennard-Jones potential for various reduced impact parameters b = bla. As b increases from 0, the path changes from a head-on to a glancing collision. Plots are for different reduced collision energies, T = pclJle (a) 0.8, (b) 4, (c) 20, and (d) a hard-sphere potential. For the smaller b the trajectories are not greatly affected by changes in T although the deflection angle is energy dependent. For the larger b there are qualitative differences, most evident in (a), where T is the smallest. [Data from J. O. Hirschfelder, C. F. Curtiss, and R. B. Byrd, Molecular Theory of Gases and Liquids (New York John Wiley, 1954), pp. 1132-1146.]...

Modeling polyatomic molecular interactions require a more complicated function. One approach is through interaction-site models, which represent a molecule as an arrangement of sites or centers of force, which are usually the centers of larger atoms or groups such as methyl, each interacting with similar sites on the other molecule, as in Equation 5.5 ... 

The main aim in this paragraph is to close the inverse collision terms in the Boltzmann equation. Consider a scattering event, as sketched in Fig. 2.4. In the following analysis the particle collision is viewed in a frame where the origin of the relative vector r is fixed. This framework is also known as a fixed center of force. The line joining the... 

---