Secondary minimum state

The second type of quantum monodromy occurs in the computed bending-vibrational bands of LiCN/LiNC, in which the role of the isolated critical point is replaced by that of a finite folded region of the spectrum, where the vibrational states of the secondary isomer LiNC interpenetrate those of LiCN [9, 10]. The folded region is finite in this case, because the secondary minimum on the potential surface merges with the transition state as the angular momentum increases. However, the shape of the potential energy surface in HCN prevents any such angular momentum cut-off, so monodromy is forbidden [10]. 

Nuclei can be trapped in the secondary minimum of the fission barrier. Such trapped nuclei will experience a significant hindrance of their y-ray decay back to the ground state (because of the large shape change involved) and an enhancement of their decay by spontaneous fission (due to the thinner barrier they would have to penetrate.) Such nuclei are called spontaneously fissioning isomers, and they were first observed in 1962 and are discussed below. They are members of a general class of nuclei, called superdeformed nuclei, that have shapes with axes ratios of 2 1. These nuclei are all trapped in a pocket in the potential energy surface due to a shell effect at this deformation. 

For situations i) and ii) the coagulated state, i.e. with the particles in intimate contact, is desirable. For other purposes, the flocculated state is required, i.e. with the particles still essentially individual and separated by a thin layer of liquid, thus giving control of the rheological properties of the sytem. Frequently, the secondary minimum plays a significant role in flocculated systems. 

On a kinetic basis the presence of a pronounced secondary minimum should lead to a steady state condition in which the rate of particles entering the secondary minimum to form associated units should be balanced by their rate of egress back to single particles. Direct evidence for this situation has been obtained recently using optical microscope observations on particles of diameter 2 ym (3 ). In addition, by coupling the microscope with a high speed camera, observations were made on particles over a period of time. This gave values for the life-time of doublets in the associated state and also revealed the fact that the particles in an associated unit could be quite... 

Figure 5.6 shows an example of a total interaction energy curve for a thin liquid film stabilized by the presence of ionic surfactant. It can be seen that either the attractive van der Waals forces or the repulsive electric double-layer forces can predominate at different film thicknesses. In the example shown, attractive forces dominate at large film thicknesses. As the thickness decreases the attraction increases but eventually the repulsive forces become significant so that a minimum in the curve may occur, this is called the secondary minimum and may be thought of as a thickness in which a meta-stable state exists, that of the common black film. As the... 

Kinetic equations for reversible adsorption and reversible coagulation are established when the interaction potential has primary and secondary minima of comparable depths. The process is assumed to occur in two successive steps. First the particles move from the bulk of the fluid to the secondary minimum. A fraction of the particles which have arrived al the secondary minimum move further to the primary minimum. Quasi-steady state is assumed for each of the steps separately. Conditions are identified under which rates of reversible adsorption or coagulation at the primary minimum can be computed by neglecting the rate of accumulation at the secondary minimum. The interaction force boundary layer approach has been improved by introducing the tangential velocity of the particles near the surface of the collector into the kinetic equations. To account for reversibility a short-range repulsion term is included in the interaction potential. 

Equation [54] is generally derived on the basis of the quasi-steady-state flow-rate of particles between the bulk and the primary minimum and involves no accumulation at the secondary minimum. One can now... 

By splitting the quasi-steady-state assumption of diffusion of particles under the action of the interaction force field into two parts, kinetic equations which account for accumulation at both the primary and secondary minimum are formulated. Conditions are established under which, after a short transient, reversible adsorption or coagulation can be treated by neglecting accumulation at the secondary minimum. The effect of tangential velocity of particles on the rate of reversible adsorption is analyzed and a criterion established when the effect... 

Figure 3 Ground state energy of a chain (a) The dimerized, degenerate, ground state chain (equilibrium atomic displacement u0) (b) The nondegenerate ground state case, with a secondary minimum at an energy AE above the ground state. Such a minimum may be absent.

The introduction of the van der Waals potential in combination with a Yukawa potential produces a curve in which the primary minimum is always deeper than the secondary minimum. This must be so because the primary minimum state is that for which the particles have coalesced and the valency of the nth plate Zn has dropped to zero since Z —> 0 as Xmn —> 2a, —> 0 as —> 2a, and the van der Waals force... 

A definite prediction of DLVO theory is that charge-stabilized colloids can only be kinetically, as opposed to thermodynamically, stable. The theory does not mean anything at all if we cannot identify the crystalline clay state (d 20 A) with the primary minimum and the clay gel state (d 100 to 1000 A) with the secondary minimum in a well-defined model experimental system. We were therefore amazed to discover a reversible phase transition of clear thermodynamic character in the n-butylammonium vermiculite system, both with respect to temperature T and pressure P. These results rock the foundations of colloid science to their roots and... 

Vy has a maximum at h = ho). This curve has a secondary minimum at larger h. Its depth is too shallow to maintain the observed arrangement of membranes in the lamellar phase. Therefore, the curve for n = 1.3 mM corresponds to the disordered dispersion of membranes as was observed experimentally at = 1.3 mM, that is, the state II. The minimum of V is deepened and correspondingly the position of potential minimum shifts to smaller h with increasing n as seen in the curves for n = 50 mM and = 150 mM, which therefore correspond to state El. With a further increase of n, the curve again has a sharp minimum at h = ho as shown by the curve for n = 500 mM, which should correspond to state IV. The curve for n = 0.05 mM has a special character it has a sharp minimum at h = ho and a very shallow minimum at very large h and a maximum between them. Here, we assume that the... 

From these arguments it would be anticipated that once a steady-state condition was achieved the percentage of sin e particles would become constant. With a further increase in electrolyte concentration, however, a deepening of the secondary minimum would occur and therefore more particles would reside m secondary minima. The experimental observations shown in Fig. 16 appear to be in accord with this. 

State (g) represents the case of weak and reversible flocculation. This occurs when the secondary minimum in the energy distance curve is deep enough to cause flocculation. This situation can occur at moderate electrolyte concentrations, in particular with larger particles the same occurs with sterically and electrosterically stabihsed suspensions. It also occurs when the adsorbed layer thickness is not very large, particularly with large particles. The minimum depth required to cause... 

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