Phase quadrants

Figure 6-14 The four types of 2D phase quadrants, corresponding to frequency modes that are real-real (RR), imaginary-real (IR), real-imaginary (RI), and imaginary-imaginary (II). (Reproduced from A. E. Derome, Modern NMR Techniques for Chemistry Research, Pergamon Press, Oxford, UK, 1987, p. 207.)...

Whether quadrant II or quadrant III should get more test attention depends on the test phase. Typically in lower level test phases quadrant III gets more attention, and in higher level test phases quadrant II is slightly more important. 

Historically, the first and most important capacitance method is the vibrating capacitor approach implemented by Lord Kelvin in 1897. In this technique (now called the Kelvin probe), the reference plate moves relative to the sample surface at some constant frequency and tlie capacitance changes as tlie interelectrode separation changes. An AC current thus flows in the external circuit. Upon reduction of the electric field to zero, the AC current is also reduced to zero. Originally, Kelvin detected the zero point manually using his quadrant electrometer. Nowadays, there are many elegant and sensitive versions of this technique. A piezoceramic foil can be used to vibrate the reference plate. To minimize noise and maximize sensitivity, a phase-locked... 

Fourier transformation in (Fti), spectra are obtained with real (R) and imaginary (/) data points. For detection in the quadrature mode with simultaneous sampling, a complex Fourier transformation is performed, with a phase correction being applied in F. (c) A normal phase-sensitive transform P— RR and I- RI. (d) Complex FT is applied to pairs of columns, which produces four quadrants, of which only the RR quadrant is plotted. 

Figure 3.6 The first set of Fourier transformations across <2 yields signals in V2, with absorption and dispersion compronents corresponding to real and imaginary parts. The second FT across /, yields signals in V, with absorption (i.e., real) and dispersion (i.e., imaginary) components quadrants (a), (b), (c), and (d) represent four different combinations of real and imaginary components and four different line shapes. These line shaptes normally are visible in phase-sensitive 2D plots.

In developing some of the relationships, it is helpful to use a four-quadrant diagram in which each quadrant represents a species in a lipid or water phase. The diagram below shows a typical distribution of an acid, AH, between two phases where ion partitioning is assumed to be negligible. The partition coefficent, P, is the ratio of the concentration of AH in the octanol to the concentration of AH in the aqueous phase. The distribution coefficient, D, is the ratio of the concentration in the octanol to that of all forms in the water. This is also called the apparent partition coefficient. 

Sometimes the log P from a two-phase titration using Equations 18 or 19 is low, compared with shake-flask values. We attribute this to ion-pair partitioning. The quadrant diagram. Figure 8, is helpful for developing the pertinent equations. The amount of each species in each phase is shown in the appropriate sector. 

If [X] is the concentration of AH in the aqueous phase (Figure 8), the concentration of AH in the octanol is P times this. The aqueous-phase concentration of the ion is [X] times the degree of dissociation. Multiplying this product by the ion-pair partition coefficient Pj (or P -) gives the concentration of the ion pair in the octanol. The actual amount of species in a phase is given by its concentration times the volume of the phase. At the pK, the amount of neutral species equals the amount of ionized species. Setting the sum of the two terms in the left quadrants equal to the sum of the two terms on the right, one can derive Equation 20. The equivalent e ression for bases is Equation 21. 

Figure 8. A quadrant diagram to aid derivation of equation 20. V is the volume of octanol or water phase [X] is the concentration of AH in the aqueous phase P is the partition coefficient of AH P is the partition coefficient of the ion pair A"Na+ pKa is the aqueous single-phase pKa pK is the two-phase pKa.

The singular point seen in the first quadrant of the phase plane (nA, nB)... 

The route for the less-dense phase leaving the rotor is fairly simple. When the separated less-dense phase flows inward and up over the LW, the liquid is thrown out into a channel (like a rain gutter) and moves to one of four sets of exit channels, one in each quadrant of the rotor. Each set of channels consists of a series of holes or a single rectangular channel that allows the less-dense phase to be flung by the rotor into the lower collector ring. 

However, the openloop unstable 85% conversion cases are very different. The curves start on the positive real axis but move counterclockwise, going from the first to the second to the third and eventually back to the second and then the first quadrants before finally ending with a phase angle of —360°. Figure 3.7 gives an enlarged view... 

The features discussed above have been observed in recent NSDI experiments, for argon irradiated by few-cycle pulses. The experimental findings exhibit very good agreement with the theory. In particular, the shift of the (Ti >T2 ) distribution from the first to the third quadrant takes place around the predicted critical phases [48]. 

Figure 4 Distribution of all six possible time orderings, labeled (ijk), onto the four quadrants spanned by time coordinates r and T (shown for the — k + k2 + k3 phase matching direction). 

On the other hand, when < > 5 l/(4a), the phase diagram is of the singular-node type, Fig. 7.12, similar to free vibration with viscous damping. Under these conditions it is possible that some particles starting in the second or third quadrants will have trajectories that terminate at the origin. This means that the x = 0 surface is reached after an infinite time. The practical implication of this observation is that these particles will never be collected. 

From item 2b, the characteristic J -sei (i.e., J c) passes very close to the critical point and items 3 and 4d signify that the corresponding jSf-set transverses the phase space from the transmissive quadrant to the reflective quadrant along the real unstable manifold. 

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