Perturbation, weak

The influence of polymer architecture on intermolecular interactions in dilute solutions was investigated by membrane osmometry in toluene (good solvent for polystyrene), cyclohexane (theta or 0 solvent), and methylcyclohexane (poor solvent Striolo et al., 2001). The osmotic second virial coefficient (B22) measured for arborescent polystyrene in toluene was lower than for homologous linear polymers, as expected due to their smaller Rg. In a 0 solvent (cyclohexane), branching lowered the 0 temperature from 34.5 °C (linear homolog) to 32.2 °C (GO polymer). The 0 temperature for the GO polystyrene sample in methylcyclohexane was likewise lowered to 36 °C, as compared to values estimated between 60 and 70 °C for linear polystyrene samples. The experimental osmotic pressure data were successfully fitted with a molecular-thermodynamic equation suitable for colloids, indicating that the behavior of arborescent polystyrene molecules in dilute solution corresponds to a perturbed (weakly interacting or interpenetrable) hard sphere. 

If we suppose the medium is composed of a known part (the background) identified by the density po and the celerity co, and an unknown part (the perturbation) identified by p and c, the equation that describes acoustic propagation/diffiision phenomena in the medium (including the boundary and Sommerfeld conditions) result from the Pekeris equation and is given, for weak-scattering (pc poCo), by ... 

Perturbation theory is a natural tool for the description of intemioleciilar forces because they are relatively weak. If the interactmg molecules (A and B) are far enough apart, then the theory becomes relatively simple because tlie overlap between the wavefiinctions of the two molecules can be neglected. This is called the polarization approximation. Such a theory was first fomuilated by London [3, 4], and then refomuilated by several others [5, 6 and 7]. 

Assuming that the system has no pennanent dipole moment, tire existence ofP(t) depends on a non-stationary j induced by an external electric field. For weak fields, we may expand the polarization in orders of the perturbation. 

Adsorbates can physisorb onto a surface into a shallow potential well, typically 0.25 eV or less [25]. In physisorption, or physical adsorption, the electronic structure of the system is barely perturbed by the interaction, and the physisorbed species are held onto a surface by weak van der Waals forces. This attractive force is due to charge fiuctuations in the surface and adsorbed molecules, such as mutually induced dipole moments. Because of the weak nature of this interaction, the equilibrium distance at which physisorbed molecules reside above a surface is relatively large, of the order of 3 A or so. Physisorbed species can be induced to remain adsorbed for a long period of time if the sample temperature is held sufficiently low. Thus, most studies of physisorption are carried out with the sample cooled by liquid nitrogen or helium. 

Nomially the amplitude of the total incident field (or intensity of the incident light) is such that the light/matter coupling energies are sufficiently weak not to compete seriously with the dark matter Hamiltonian. As already noted, when this is tire case, tlie induced polarization, P is treated perturbatively in orders of the total electric field. Thus one writes... 

Since the electric field is a polar vector, it acts to break the inversion synnnetry and gives rise to dipole-allowed sources of nonlinear polarization in the bulk of a centrosymmetric medium. Assuming that tire DC field, is sufficiently weak to be treated in a leading-order perturbation expansion, the response may be written as... 

The influence of an applied magnetic field, as introduced in section Bl.5.2.2. is quite different from that of an applied electric field. A magnetic field may perturb the interfacial nonlinear response (and that of the weak bulk tenns), but it does not lead to any dipole-allowed bulk nonlmear response. Thus, in the presence of magnetic fields and magnetization, SHG remains a probe that is highly specific to surfaces and interfaces. It... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

However, by constructing a nested sequence of successively larger discrete spaces and approximations therein we hope to end up with some approximation of a unique invariant measure, which is then implicitly defined via the constructed sequence of subspaces. An expression of this mathematical consideration is the multilevel structure of the suggested algorithm - details see below (Section 3.2). In physical terms, we hope that the perturbations introduced by discretization induce a unique and smooth invariant measure but are so weak that they do not destroy the essential physical structure of the problem. 

The relative strengths and weaknesses of perturbation theory and the variational method, as applied to studies of the electronic structure of atoms and molecules, are discussed in Section 6. 

In the X-ray fluorescence spectmm of tin, as in those of other elements, transitions such as 3d- s and Ad- s, which are forbidden by the selection mles, may be observed very weakly due to perturbations by neighbouring atoms. 

AH distortions of the nematic phase may be decomposed into three basic curvatures of the director, as depicted in Figure 6. Liquid crystals are unusual fluids in that such elastic curvatures may be sustained. Molecules of a tme Hquid would immediately reorient to flow out of an imposed mechanical shear. The force constants characterizing these distortions are very weak, making the material exceedingly sensitive and easy to perturb. 

In HIC, the hydrophobic interactions are relatively weak, often driven by salts in moderate concentration (I to 2 M), and depend primarily on the exposed residues on or near the protein surface preservation of the native, biologically active state of the protein is an important feature of HIC. Elution can be achieved differentially by decreasing salt concentration or increasing the concentration of polarity perturbants (e.g., ethylene glycol) in the eluent. 

The early approaches to this model used perturbative expansion for weak coupling [Silbey and Harris 1983]. Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the a operators, 8c/8t = ih [H, d], have the form... 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

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