Wave first order

It has been postulated that jet breakup is the result of aerodynamic interaction between the Hquid and the ambient gas. Such theory considers a column of Hquid emerging from a circular orifice into a surrounding gas. The instabiHty on the Hquid surface is examined by using first-order linear theory. A small perturbation is imposed on the initially steady Hquid motion to simulate the growth of waves. The displacement of the surface waves can be obtained by the real component of a Fourier expression ... 

Figure 1.1 shows a typical stress-volume relation for a solid which remains in a single structural phase, along with a depiction of idealized wave profiles for the solid loaded with different peak pressures. The first-order picture is one in which the characteristic response of solids depends qualitatively upon the material properties relative to the level of loading. Inertial properties determine the sample response unlike static high pressure, the experimenter does not have independent control of stresses within the sample. 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

When the pressures to induce shock-induced transformations are compared to those of static high pressure, the values are sufficiently close to indicate that they are the same events. In spite of this first-order agreement, differences between the values observed between static and shock compression are usually significant and reveal effects controlled by the physical and chemical nature of the imposed deformation. Improved time resolution of wave profile measurements has not led to more accurate shock values rather. 

With nanosecond time resolution, sensitive, accurate detectors, studies of these release waves have proven to be particularly revealing. First-order descriptions of release properties were obtained with rudimentary instrumentation from the earliest studies [65A01] it has required the most sophisticated modern instrumentation to provide the necessary detail to obtain a clear picture of the events. Characteristically different profiles are encountered in the strong-shock, elastic, and elastic-plastic regimes. 

Release waves for the elastic-plastic regime are dominated by the strength effect and the viscoplastic deformations. Here again, quantitative study of the release waves requires the best of measurement capability. The work of Asay et al. on release of aluminum as well as reloading, shown in Fig. 2.11, demonstrates the power of the technique. Early work by Curran [63D03] shows that limited time-resolution detectors can give a first-order description of the existence of elastic-plastic behavior on release. 

Curran [61C01] has pointed out that under certain unusual conditions the second-order phase transition might cause a cusp in the stress-volume relation resulting in a multiple wave structure, as is the case for a first-order transition. His shock-wave compression measurements on Invar (36-wt% Ni-Fe) showed large compressibilities in the low stress region but no distinct transition. 

The indicated transition pressure of 15 GPa is in agreement with the published data with shock-wave structure measurements on a 3% silicon-iron alloy, the nominal composition of Silectron. A mixed phase region from 15 to 22.5 GPa appears quite reasonable based on shock pressure-volume data. Thus, the direct measure of magnetization appears to offer a sensitive measure of characteristics of shock-induced, first-order phase transitions involving a change in magnetization. 

Table 6.2 summarizes the low pressure intercept of observed shock-velocity versus particle-velocity relations for a number of powder samples as a function of initial relative density. The characteristic response of an unusually low wavespeed is universally observed, and is in agreement with considerations of Herrmann s P-a model [69H02] for compression of porous solids. Fits to data of porous iron are shown in Fig. 6.4. The first order features of wave-speed are controlled by density, not material. This material-independent, density-dependent behavior is an extremely important feature of highly porous materials. 

In the adiabatic approximation the form of the total wave function is restricted to one electronic surface, i.e. all coupling elements in eq. (3.12) are neglected (only the terms with i = i survive). Except for spatially degenerate wave functions, the diagonal first-order non-adiabatic coupling element is zero. 

Another aspect of wave function instability concerns symmetry breaking, i.e. the wave function has a lower symmetry than the nuclear framework. It occurs for example for the allyl radical with an ROHF type wave function. The nuclear geometry has C21, symmetry, but the Cay symmetric wave function corresponds to a (first-order) saddle point. The lowest energy ROHF solution has only Cj symmetry, and corresponds to a localized double bond and a localized electron (radical). Relaxing the double occupancy constraint, and allowing the wave function to become UHF, re-establish the correct Cay symmetry. Such symmetry breaking phenomena usually indicate that the type of wave function used is not flexible enough for even a qualitatively correct description. 

These are zero-, first-, second-, th-order perturbation equations. The zero-order equation is just the Schodinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy, Wi, and the first-order correction to the wave function, 4< i. The th-order energy correction can be calculated by multiplying from the left by 4>o and Integrating, and using the turnover rule ( o Ho, ) = (, Ho o)... 

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

The first-order correction to the wave function can be obtained by multiplying (4.32) from the left by a function other than 4>o(4 ) and integrating to give... 

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

The last equation shows that the second-order energy correction may be written in terms of the first order wave function (c,) and matrix elements over unperturbed states. The second-order wave function correction is... 

The zero-order wave function is the HF determinant, and the zero-order energy is just a sum of MO energies. The first-order energy correction is the average of the perturbation operator over the zero-order wave function (eq. (4.36)). 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

In order to achieve a high aceuraey, it would seem desirable to explicitly include terms in the wave functions which are linear in the intereleetronie distanee. This is the idea in the R12 methods developed by Kutzelnigg and co-workers. The first order correction to the HF wave funetion only involves doubly exeited determinants (eqs. (4.35) and (4.37)). In R12 methods additional terms are included which essentially are the HF determinant multiplied with faetors. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

Pt2,V and Pt y have been investigated at 1393 K and 1224 K respectively and we have explored the [100] and [110] planes of the reciprocal lattice. The measured Intensities have been Interpreted in a Sparks and Borie approach with first order displacements parameters and using a model Including 29 a(/ ) for PfsV and 21 for PtsV. In figure 1 is displayed the intensity distribution due to SRO a q) in the [100] plane. As for PdjV, the diffuse intensity of Pt V is spread along the (100) axes with maxima at the (100) positions, whereas the ground state is built on (1 j 0) concentration wave ( >022 phase). 

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