doi:10.1016/S0009-2614(02)01033-3 
Copyright © 2002 Elsevier Science B.V. All rights reserved.
Design of Bloch–Siegert phase-shift self-compensated pulses for HCN triple-resonance experiments
Shanmin Zhang
, 
and David G. Gorenstein
Department of Human Biological Chemistry and Genetics, Sealy Center for Structural Biology, University of Texas Medical Branch, Galveston, TX 77555-1157, USA
Received 12 April 2002;
revised 17 June 2002.
Available online 1 August 2002.
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Abstract
A scheme to construct a Bloch–Siegert phase-shift (BSPS) self-compensated phase-incremented pulse (PIP) with a compensating PIP (similar to the idea by McCoy and Mueller) is presented, using the coherent averaging theory up to the second order. Under the condition of 2πΔfτ=kπ, where Δf is the difference of the chemical-shifts between the centers of the 13Cα and 13CO and τ is the pulse-width, the zero-order interaction between the PIPs and the 13Cα spins is minimized. Consequently, the BSPS of the 13Cα spins, caused by the two simultaneous PIPs, one applied at the center of the 13CO and the other at the other side of the 13Cα, becomes linear in a quite broad range as a function of the offset. Therefore, the BSPS can be removed by taking into account a scaling factor, λ=[1−(f12/Δf2)], of the pulse-width or by a first-order phase correction. In addition, this scheme introduces a much smaller disturbance to the transverse and longitudinal magnetizations of the 13Cα than that by a single PIP.
Fig. 1. The BSPS-compensated PIPs, one applied at the center of the 13CO and the other at the other side of the 13Cα for compensating the BSPS. The two PIPs can be 90° pulses, 180° inverting or refocusing pulses, or adiabatic inverting pulses.
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Fig. 2. Influences of the longitudinal magnetization, transverse magnetization, and the BSPS of the 13Cα caused by a single 180 PIP (a–c) and by the BSPS-compensated PIPs (A–C). The solid-triangles and open-circles (b, B and c, C) represent the results with an initial transverse magnetization along the x- and y-axis, respectively. The BSPS (solid-triangles) is corrected (solid-diamonds in C) when the scaling factor of the pulse-width, λ=[1−(f12/Δf2)], is taken into account. The single 180° PIP applied at the center of the 13CO is denoted by PIP22.22(2°, 4°, 0.5 μs, 5.75 kHz, 174), where the numerical values following PIP represent the frequency-shift Δf (in kHz), initial phase 
0, phase-increment Δ, time-increment Δτ, pulse-strength f1, and the number of steps of the PIP, respectively [10]. An initial phase 0=2° is used to compensate the universal phase-shift of the PIP (UPS=−(1/2)Δ), otherwise the PIP would have a phase of −2° rather than 0° (or x phase) [9 and 10]. The BSPS-compensated PIPs are then represented by PIP22.22(0°, 0°, 0.5 μs, 2f1cos(2πΔft+0), 180), where 2f1=11.12 kHz, 0=2°, and t=0, Δτ, 2Δτ,…
Fig. 3. Influences of the longitudinal magnetization, transverse magnetization, and the BSPS of the 13Cα caused by the BSPS-compensated PIPs (a–c) and by the same PIPs except with a shorter pulse-width (A–C). The BSPS-compensated PIPs are represented by PIP22.22(0°, 0°, 0.5 μs, 2f1cos(2πΔft+0), 135) where 2f1=14.82 kHz, 0=2°, and t=0, Δτ, 2Δτ,… The PIPs used in Fig. 3A–C has a total of 113 steps (instead of 135 steps) that corresponds to a pulse-width of 56.5 μs and 2πΔfτ=2.51π. The solid-triangles and open-circles (b, B and c, C) represent the results with an initial transverse magnetization along the x- and y-axis, respectively. The BSPS (solid-triangles) is corrected (solid-diamonds in c) when the scaling factor of the pulse-width, λ=[1−(f12/Δf2)], is taken into account.
Fig. 4. Inversion profiles (a) by a single PIP22.22(2°, 4°, 0.5 μs, 12.83 kHz,78), satisfying the condition of , and (b) by a simultaneous PIP22.22(0°, 0°, 0.5 μs, 2f1cos(2πΔft+0), 90) satisfying the condition of 0=2°, ωτ=2πΔfτ=2π and f1=11.12 kHz.
Abstract
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