Antiphase vectors

This sequence of events can be represented schematically in a circle (Fig. 7.8) with the rotation rate ( 2i) in the center. This is simply the rotation of the two opposed (antiphase) vectors in the x -y plane at the frequency determined by the chemical shift of nucleus I. The Sz part just goes along for the ride because the antiphase relationship is retained throughout. Stand up with your arms outstretched at your sides your right arm represents the 1H net magnetization vector (13C = a) on the +x axis, and your left arm represents the... 

Figure 5.29. A simplified picture of multiple-quantum coherence in an AX system views this as being composed of groups of evolving antiphase vectors which have zero net magnetisation, and hence can never be directly observed.

Figure 5.44. The ID double-quantum filter. The sequence is derived from the 2D experiment by replacing the variable ti period with a fixed spin-echo optimised to produce antiphase vectors (A = 1 /2J) as required for the generation of double-quantum coherence. Signal selection is then as for the 2D experiment, and gradient selection may be implemented as in Fig. 5.41.

Antiphase boundaries (APBs) are displacement boundaries within a crystal. The crystallographic operator that generates an antiphase boundary in a crystal is a vector R parallel to the boundary, specifying the displacement of one part with respect to the other (Fig. 3.27), whereas the crystallographic operator that generates a twin is reflection (in the examples considered above). 

Figure 3.27 Antiphase boundaries (a, b) antiphase boundaries are formed when one part of a crystal is displaced with respect to the other part by a vector parallel to the boundary.

Because the vector describing an antiphase boundary always lies parallel to the boundary, there is never any composition change involved. 

As shown in Fig. 2.48 (e), the vectors of C13 — C13 doublet magnetizations are aligned in opposite directions when irradiated by the 90 pulse. Thus, in the INADEQUATE spectrum the C13 —C13 doublet signals will appear with the corresponding antiphase relationship, as shown in Fig. 2.49, which also demonstrates the effective suppression of the strong 13C — 12C signals of piperidine. Analysis of carbon-carbon coupling constants can be performed easily in this simple case. 

Fig. 8.19. Vector representation of a H-13C HMQC experiment. The first 90° pulse along y rotates the equilibrium magnetization of the proton spin, /H, from the z axis to the x axis. After a time /d = 1/2/Hx, the antiphase coherence 2/J1/ t (see Appendix IX) is at its maximum. A 90° pulse on carbon along y then transforms the antiphase coherence into a MQ (multiple quantum) coherence (the 2/J1/ component is shown). During t the MQ evolves (with a 180 refocusing pulse on proton in the middle), until a further 90 pulse on carbon along x transforms the —2/ / component (shown at its maximum for clarity) into a 2/ /f antiphase coherence. After the time fd, in-phase coherence of the proton spin develops. The latter is detected during h. Its initial intensity is modulated by the carbon Larmor frequency during t (if proton refocusing has been used), thus originating a proton-carbon cross peak.

Consequently, the resistivity decrease (increase) within the temperature range, Ms - MS(MS - As), is simply interpreted as the decrease (increase) in the concentration of antiphase boundaries as the cooperative shear movements of atoms take place by way of shear vector, S2 (S3). This suggests that the total resistivity of TiNi(lI) is given by the resistivity curve above 125 °C. Similarly, the resistivity curve below the Ms temperature is the total resistivity of TiNi(III). By extrapolating the total resistivity of TiNi(II) and TiNi(III) as shown in Fig. 13 (dotted lines), the total resistivity of TiNi(II) is shown to be lower than that of TiNi(III) within the transition temperature range, Ms - As. In order to confirm this interpretation, it was logical to obtain other data corresponding to the two extreme resistivity curves, Fig. 12 (a) and (c). 

The antiphase doublet (Fig. 6.14(c)) is dispersive because /-coupling evolution to the antiphase state moves the vectors by 90°, from the +x axis to the +/ and —/ axes. This dispersive antiphase doublet can be phase corrected by moving the reference axis from the +x axis to the +/ axis (90° zero-order phase correction). Now the C = a peak is positive absorptive and the C = ft peak is negative absorptive (Fig. 6.15) and the central 12CH3l peak is pure dispersive because the vector is on the -hx/ axis and the reference axis is now +y (90° phase error). 

Note that the 180 -y pulse on the 13C channel has no effect on Sv. The cosine term is just the product operator we started with, unaffected by the1H pulse, and the sine term is the operator we would get with a full 90° 1H pulse. Note that rotation of the lx magnetization vector by a XH B field on the / axis goes from x to — z to — x to +z as is incremented from 0° to 90° to 180° to 270° in the trigonometric expression. The first term is DQC/ZQC, which will not be observable in the FID—there are no more pulses in the sequence to convert it to observable magnetization. Only the second term represents full coherence transfer to antiphase 13C coherence, which will refocus during the final 1/(27) delay into in-phase 13C coherence ... 

At t = 21 Ja, you are back to the starting magnetization. With product operators we do not need to draw vector diagrams, because we treat the vectors as pure in-phase (Ia, Ib) and antiphase (2IaIb, 2IaIb) components. Now plug these results in wherever you see Ia or Ia as a result of chemical-shift evolution ... 

We spent a lot of time in Chapter 6 (Section 6.10) using the vector diagrams to understand the effect of 180° pulses in the center of a spin echo. This is easy to understand now that we have the product operator tools. In general, consider the effect of a 180° pulse on the in-phase and antiphase XH and 13C operators ... 

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