The role of time symmetry in composite-pulse design is examzation. Whether it be rigid-body kinematics or spin kinematined by considering a phase-alternating composite pulse pair {p(I ics, the ER parameters, in conjunction with a computation-Å 1 2 ), p/2(I Å 1)}, where the spin-1 excitation pulse has been ally efficient compounding formula (9) to handle consecuderived from its spin-1 2 progenitor by halving the pulse durations. tive rotations, have become the method of choice for The quaternion calculus is used to define the quaternion elements parametrizing rotations. This composition rule is the basis (Euler-Rodrigues parameters) of each composite pulse. In this of the quaternion calculus used to calculate the response to manner, it is shown how an Euler-Rodrigues (ER) parametrizamultipulse sequences employed in spin-1 2 (9-15) and spintion of the consecutive rotations implicit in each composite pulse 1 (16) NMR. The enormous advantages of the ER paramecan be used to derive simple phase and amplitude relationships ters in comparison to the Euler angles, for example (17), between the members of such a {p(I Å 1 2 ), p/2(I Å 1)} pulse pair. have led to their use not only in NMR (9, 16) but also in
The simplicity and compactness of the ER parametrization is then describing Berry's phase in neutron spin resonance (18), used to identify optimal time-symmetric sequences for spin-1 exciand in spacecraft attitude kinematics (19).
tation by using the Lagrange multiplier method. ᭧ 1996 Academic
In NMR applications, as we will demonstrate in this paper, Press, Inc.
the ER parameters certainly offer much more than just computational efficiency, or ease of visualization in the spin-1 2 case. In this regard, Emsley and Bodenhausen ( 14) have Symmetry principles are now well recognized as an recently used an ER parametric representation of the spinimportant consideration in the design of NMR pulse se-1 2 pulse propagator to carry out numerically optimized design quences. In the case of isolated spin-1 2 nuclei, time symmeof shaped selective pulses. In our examination of time-symtry is one of the simplest and most fruitful symmetry prinmetric composite pulses, we begin by considering a phaseciples to be employed in the design of broadband excitaalternating composite pulse pair {p(I Å 1 2 ), p/2(I Å 1)}, tion and inversion pulses ( 1, 2 ) . On the other hand, for where the spin-1 p/2 excitation pulse has been derived from spin-1 nuclei, symmetry considerations of any kind have its spin-1 2 progenitor by halving the pulse duration(s), as yet to play a significant role in the design of efficient, first described by Levitt et al. (3). Since any pulse of any broadband excitation pulses. Most composite spin-1 excinet rotation angle (single or composite) is parametrized by tation pulse sequences either have been derived directly just four ER parameters, we will exploit the simplicity and from composite spin-1 2 inversion sequences by halving compactness of the ER parametrization to identify (and pulse durations ( 3 -5 ) , or have been found by numerical prove for the first time) simple phase and amplitude relationoptimization ( 6, 7 ) , irrespective of any explicit symmetry ships between the members of a {p(I Å 1 2 ), p/2(I Å 1)} considerations. Recently, however, Ramamoorthy and pulse pair, as well as symmetry properties. Such relation-Narasimhan ( 5 ) have pointed out that time-symmetric ships and properties often remain obscure, and certainly very phase-alternating composite pulses in a quadrupole-echo difficult to prove, whenever any of the fictitious spin-1 2 forsequence can be used to refocus quadrupole spin packets malisms (20) are used. We will further exploit the simplicity without any phase distortion. and compactness of the ER parametrization to identify opti-In this paper, such time-symmetric composite pulses will mal excitation sequences using the Lagrange multiplier be examined in some detail. By using an Euler-Rodrigues method. (ER) parametrization (8) of the consecutive rotations im-For both spin systems, we begin by establishing the ER plicit in each composite pulse, our examination will highlight parameters for single and phase-alternating composite some of the unique advantages and features of this parametripulses. For a two-level spin-1 2 system, whose Hamiltonian in the presence of a resonance offset D Å v 0 v 0 and RF irradiation of phase f can be written as * NSERC Research Reorientation Associate.