Stationary phase, method

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

Kiisters, E., Preparative Chromatographic Separation of Enantiomers on Chiral Stationary Phases - Method Development, Scale-Up and Techniques CHIMICA OGGI July/August 1996,39. 

Noticeably, the phase / plays the role of a potential function, the requirement V/= 0 defines the catastrophe surface M, c corresponds to control parameters. The radiation intensity at the point c is represented by the oscillation integral (3.39a), see equation (3.34). The method of estimation of integrals of this type presented above is called the stationary phase method. 

As follows from our previous considerations, the integral (3.67) cannot be computed by the stationary phase method in the vicinity of the caustic due to appearance of the divergence associated with having by the potential function F(x u), i.e. the phase, a degenerate critical point on the caustic (at the return point x0). However, the Airy function can be computed by another method, see Section 3.4.5. The form of the Airy function is shown in Fig. 47. Let us recall that P2( k) is interpreted as the intensity of scattered light or the probability of finding the particle at the point u. 

Let us recall that in the case of computing the oscillatory integrals (3.25) in which the function F has a degenerate critical point, the stationary phase method, described in Section 3.4.2, fails. There are two basic methods of computation of such integrals which will be exemplified by the Airy function. The information provided below is brief and intended to facilitate the reader an access to suitable references (see bibliographical remarks at the end of this chapter). 

Ahn, H.-Y Shiu, G.K. Trafton, W.F. Doyle, T.D. Resolution of the enantiomers of ibuprofen comparison study of diastereomeric method and chiral stationary phase method. J.Chromatogr.B, 1994, 653, 163-169... 

Without going into details (Hassani, 1991), if one has to solve an integral of the (Al) type with a > 0, the saddle point approximation or the stationary phase method or the method of the steepest descendent requires its expansion around the point the solution of the extreme equation ... 

With a complex potential = eg i72t for the excited state (to allow for the decay) and a pure Coulombic potential (e) = e + X/R for the final state, Eq. (18) can be evaluated by means of the stationary phase method, which yields... 

A practical semiclassical approximation follows from performing the integrations in eq. (4.2.3b) via stationary-phase methods for each value of X while neglecting the phase of (A"fX, t/"[/f] A"iXi) in comparison with 5([i ], X lh. As a consequence, the stationary-phase quantities depend parametrically on the initial values of the slow variables. The stationary condition with respect to t is now approximately satisfied by K,A-V R,X, x))= E-E ), and the condition 85 = 0 leads to trajectories Q t A"j) for relative motion determined from the Newton-like equation... 

The quasiclassical amplitude in the momentum representation does not suffer from this feature, because boundary conditions Pz h) = Pu> Pzitz) = P2z determine the unique classical trajectory for the typical scattering potentials. Such amplitude cannot be obtained as a Fourier transform of the quasiclassical propagator in coordinate representation. So it is necessary to modify the stationary phase method for the evaluation of the path integral in momentmn representation. 

In some ceises the influence functional F may be teiken out of the integrals over the imit cell in (2.2.2). It is possible either when Pi(Rc) and Gqq(Ro) depend on Rq weakly compared to GUq is, when the crystal corrugation is much smaller than the phonon relief), or when S Ro) oscillates strongly enough for the evaluation of the integrals by the stationary phase method (the classical limit). In the latter case F is treated as a pre-exponential factor and may be replaced with its value at the stationary phase point. Then the general structural factor Hq can be factorized into the product of the form factor (diffi action intensities) and the dynamic structural factor ... 

Gas phase chromatography is a separation method in which the molecules are split between a stationary phase, a heavy solvent, and a mobile gas phase called the carrier gas. The separation takes place in a column containing the heavy solvent which can have the following forms ... 

Chromatography (Section 13 22) A method for separation and analysis of mixtures based on the different rates at which different compounds are removed from a stationary phase by a moving phase... 

Analytical separations may be classified in three ways by the physical state of the mobile phase and stationary phase by the method of contact between the mobile phase and stationary phase or by the chemical or physical mechanism responsible for separating the sample s constituents. The mobile phase is usually a liquid or a gas, and the stationary phase, when present, is a solid or a liquid film coated on a solid surface. Chromatographic techniques are often named by listing the type of mobile phase, followed by the type of stationary phase. Thus, in gas-liquid chromatography the mobile phase is a gas and the stationary phase is a liquid. If only one phase is indicated, as in gas chromatography, it is assumed to be the mobile phase. 

Thus far all the separations we have considered involve a mobile phase and a stationary phase. Separation of a complex mixture of analytes occurs because each analyte has a different ability to partition between the two phases. An analyte whose distribution ratio favors the stationary phase is retained on the column for a longer time, thereby eluting with a longer retention time. Although the methods described in the preceding sections involve different types of stationary and mobile phases, all are forms of chromatography. 

Method f2.i describes the analysis of the trihalomethanes CHCI3, CHBr3, CHChBr, and CHClBr2 in drinking water using a packed column with a nonpolar stationary phase. Predict the order in which these four trihalomethanes will elute. 

Three general methods exist for the resolution of enantiomers by Hquid chromatography (qv) (47,48). Conversion of the enantiomers to diastereomers and subsequent column chromatography on an achiral stationary phase with an achiral eluant represents a classical method of resolution (49). Diastereomeric derivatization is problematic in that conversion back to the desired enantiomers can result in partial racemization. For example, (lR,23, 5R)-menthol (R)-mandelate (31) is readily separated from its diastereomer but ester hydrolysis under numerous reaction conditions produces (R)-(-)-mandehc acid (32) which is contaminated with (3)-(+)-mandehc acid (33). 

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