含时射频脉冲作用下耦合自旋系统Magnus解存在的显式判据
An explicit criterion for existence of the Magnus solution for a coupled spin system under a time-dependent radiofrequency pulse
摘要：
详细推导了相互作用图象中Magnus展开式收敛和Magnus解存在的显式判据，即，弱耦合自旋（I=1/2）系统SnAMX中的指数传播子.（n= 1，2，3，.）其中只有自旋基团S受到时间相关的成形的选择性射频脉冲。推导是建立在与Maricq（J.Chem.Phys.86（1987）5647）相同的方案上。结果表明，该判据仅依赖于作用于系统的含时场的振幅，当非负振幅整形脉冲或弱振幅整形脉冲的翻转角小于2 π时，Magnus展开收敛，且存在Magnus解.然后，指数传播子可以被分解成一系列基本传播子的乘积，并且可以用于确定成形脉冲下自旋系统的时间演化。明确地得到了确定Magnus解中未知参数的线性微分方程组。另一种传播子的扩展形式的马格努斯解决方案也提出了描述的时间演化时，不符合标准。
英文摘要：
The explicit criterion is derived in detail for the convergence of the Magnus expansion and the existence of the Magnus solution in the interaction picture, i.e., the exponential propagator in the weakly coupled spin (I=1/2) system SnAMX... (n=1,2,3,...) in which only spin group S is subjected to a time-dependent shaped selective radiofrequency pulse. The derivation is built on the same scheme as the Maricq's (J. Chem. Phys. 86 (1987) 5647). It is shown that the criterion depends only upon amplitude of the time-dependent field applied to the system, and the Magnus expansion converges and the Magnus solution exists when the flip angle of a non-negative-amplitude shaped RF pulse or a weak-amplitude shaped pulse is smaller than 2pi. The exponential propagator then can be decomposed into a product of a series of elementary propagators and can be used to determine time evolution of the spin system under the shaped pulse. The linear differential equations to determine the unknown parameters in the Magnus solution are obtained explicitly. An alternative propagator in an expansion form for the Magnus solution is also proposed to describe the time evolution when the criterion is not met.
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