Energy singularity

As the temperature of a surface is raised above the roughening temperature Tr, its stiffness is reduced and it no longer appears as a facet in the equilibrium shape. On the microscopic level, this corresponds to a shift from the low entropy, low energy singular surface toward the high entropy surface populated by islands and adatoms.18-20 At ambient temperature, low-index copper surfaces are below the roughening temperature in vacuum. However, in... 

There is clearly a possible singularity in (itj) if - p) vanishes. Let the energy scale be chosen such that the ground-state energy = 0. Then the ground-state occupancy is... 

If one assumes that tlie singular part A of the Helmholtz free energy is such a fimction... 

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

Elements of the matrix —are usually small in the vicinity of a conical intersection and can be added to to give a corrected diabatic energy matrix. As can be seen, whereas in Eq. (15) contains both the singular... 

Beutler, T. C., Mark, A. E., van Schaik, R. C., Gerber, P. R., van Gunsteren, W. F. Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations. Chem. Phys. Letters 222 (1994) 529-539... 

In the following we devise, following [14], an efficiently implementable scheme which leads to favorable error bounds independently of the highest frequencies under the mere assumption that the system has bounded energy. The scheme will be time-reversible, and robust in the singular limit of the mass ratio m/M tending to 0. 

Buetler T C, A E Mark, R C van Schaik, P R Gerber and W F van Gunsteren 1994. Avoiding Singularities and Numerical Instabilities in Free Energy Calculations Based on Molecular Simulations. Chemical Physics Letters 222 529-539. 

As indicated, an implicit assumption of the JKR theory is that there are no interactions outside the contact radius. More specifically, the energy arguments used in the development of the JKR theory do not allow specific locations of the adhesion forces to be determined except that they must be associated with the contact line where the two surfaces of the particle and substrate become joined. Adhesion-induced stresses act at the surface and not a result of action-at-a-distance interatomic forces. This results in a stress singularity at the circumference of the contact radius [41]. The validity of this assumption was first questioned by Derjaguin et al. [42], who proposed an alternative model of adhesion (commonly referred to as the DMT theory ). Needless to say, the predictions of the JKR and DMT models are vastly different, as discussed by Tabor [41]. 

Although still preliminary, the study that provides the most detailed test of the theory for the electronic properties of the ID carbon nanotubes, thus far, is the combined STM/STS study by Oik and Heremans[13]. In this STM/STS study, more than nine individual multilayer tubules with diameters ranging from 1.7 to 9.5 nm were examined. The 7-Fplots provide evidence for both metallic and semiconducting tubules[13,14]. Plots of dl/dV indicate maxima in the ID density of states, suggestive of predicted singularities in the ID density of states for carbon nanotubes. This STM/ STS study further shows that the energy gap for the semiconducting tubules is proportional to the inverse tubule diameter l/<7, and is independent of the tubule chirality. 

The existence of carbon nanotubes with diameters small compared to the de Broglie wavelength has been described by Iijima[l,2,3] and others[4,5]. The energy band structures for carbon nanotubes have been calculated by a number of authors and the results are summarized in this issue by M.S. Dresselhaus, G. Dres-selhaus, and R. Saito. In short, the tubules can be either metallic or semiconducting, depending on the tubule diameter and chirality[6,7,8]. The calculated density of states[8] shows singularities... 

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