Internal state

The theory coimecting transport coefficients with the intemiolecular potential is much more complicated for polyatomic molecules because the internal states of the molecules must be accounted for. Both quantum mechanical and semi-classical theories have been developed. McCourt and his coworkers [113. 114] have brought these theories to computational fruition and transport properties now constitute a valuable test of proposed potential energy surfaces that... 

One possibility is to use hyperspherical coordinates, as these enable the use of basis fiinctions which describe reagent and product internal states in the same expansion. Hyperspherical coordinates have been extensively discussed in the literature [M, 35 and 36] and in the present application they reduce to polar coordinates (p, p) defined as follows ... 

This expression is still oversimplified, as it ignores the fact that the molecules A and B have internal states and that the cross section a depends on these states a depends also on the internal states of the products C and D. Letting die indices and / denote the internal states of the reagents and products respectively, we find that G in equation (A3.11.174) must be replaced by y /and the Boltzmaim average must now include the... 

Hamiltonian H. Fonnally, at least, these expressions avoid the detennination of scattering wavefrmctions and any infonnation related to the internal states of the reagents or products. The frmdamental derivations in this area have been presented by Miller [44] and by Schwartz et al [45]. 

The frequency (number per second) of /transitions from all g. degenerate initial internal states and from the p. d E. initial external translational states is equal to tire reverse frequency from the g degenerate final internal states and the pyd final external translational states. The detailed balance relation between the forward and reverse frequencies is therefore... 

The reduced potential matrix elements which couple the internal states n and j are... 

When the initial and final internal states of the system are not well-separated in energy from other states then the closed-coupling calculation converges very slowly. An effective strategy is to add a series of correlation temis involving powers of the distance r. between internal particles of projectile and target to the tmncated close-coupling expansion which already includes the important states. 

For many reaction products and for the detection of molecules in their ground vibrational level, some laser-based spectroscopic method must be employed, rather than observation of spontaneous emission. The simplest spectroscopic method for detemiining concentrations of specified product internal states would involve the... 

Coherent anti-Stokes Raman spectroscopy (CARS) [59] has also found utility in the detemiination of the internal state distributions of products of chemical reactions. This is one of several coherent Raman spectroscopies based on the... 

Sauder D G and Dagdigian P J 1990 Determination of the internal state distribution of NC produced from the H + NG2 reaction J. Chem. Phys. 92 2389-96... 

Irvine AML, Smith I W M, Tuckett R P and Yang X-F 1990 A laser-induced fluorescence determination of the complete internal state distribution of CH produced in the reaction H + NG2 CH + NC J. Chem. Phys. 93 3177-86... 

Dagdigian P J, Varley D F, Liyanage R, Gordon R J and Field R W 1996 Detection of DCI by multiphoton ionization and determination of DCI and HCI internal state distributions J. Chem. Phys. 106 10 251-62... 

Sinha M P, Schulz A and Zare R N 1973 Internal state distribution of alkali dimers in supersonic nozzle beams J. Chem. Phys. 58 549-56... 

The occurrence of predissociation opens up a new family of observable quantities. It is possible to measure not only linewidths or lifetimes, but also the internal state distributions of the fragments. All these quantities are sensitive to the intennolecular potential and can be used to test or refine proposed potential surfaces. 

Accuracy of Pyrometers Most of the temperature estimation methods for pyrometers assume that the objec t is either a grey body or has known emissivity values. The emissivity of the nonblack body depends on the internal state or the surface geometry of the objects. Also, the medium through which the therm radiation passes is not always transparent. These inherent uncertainties of the emissivity values make the accurate estimation of the temperature of the target objects difficult. Proper selection of the pyrometer and accurate emissivity values can provide a high level of accuracy. 

K set of referential internal state variables I velocity gradient tensor... 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Specific applications of the theory are not considered in this chapter. Only one example, that of small deformation classical plasticity, is worked out in Section 5.3. The set of internal state variables k is taken to be comprised of... 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

Motivated by the qualitative observations made above, a set of internal state variables deseribing the internal strueture of the material will be intro-dueed ab initio, denoted eolleetively by k. Their physieal meaning or preeise properties need not be established at this point, and they may inelude sealar, veetor, or tensor quantities. The following eonstitutive assumptions are now made ... 

Inelastic Loading. The strain lies on the elastic limit surface = 0, and the tangent to the strain history points in a direction outward from the elastic limit surface > 0. The material is said to be undergoing inelastic loading, and k is assumed to be a function of the strain s, the internal state variables k, and the strain rate k... 

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