One of the early methods designed for the observation of nonideal refocusing pulses under this condition will be especially investigated. spin echoes was the Carr-Purcell (CP) sequence (1), which was originally proposed for reducing attenuation effects due
For the following analysis, it is assumed that spin-coupling effects can be ignored and during the pulse intervals the Bloch to molecular diffusion. Multiecho sequences based on the CP sequence are still widely used. In the ideal case, echoes equations are valid (12). Following the partition method (13-15), the magnetization after a sequence of n RF pulses can be are formed which decay exponentially with the transverse relaxation time T 2 , allowing T 2 measurements or T 2 -decomposed into contributors which are characterized by their individual coherence pathways. These coherence pathways can weighted NMR imaging.
But multiple refocusing of the signal introduces cumulative be labeled by vectors {p n , p n01 , . . . , p 1 } where the p i describes the magnetization state after the ith RF pulse and can take the errors if the 180Њ pulses are imperfect. Meiboom and Gill minimized this effect by introducing a 90Њ phase shift between the values /1 to indicate the transverse magnetization M xy Å M x / initial 90Њ pulse and the subsequent 180Њ pulses (CPMG se-iM y , 01 for its complex conjugate M* xy , and 0 for the longitudiquence) (2). Certain phase-alternating schemes were introduced nal magnetization M z . Since there is interest only in the transfor the compensation of cumulative pulse errors for all three verse components of the magnetization, p n can be constrained components of magnetization (3-5). Individual refocusing erto be /1. As the observed signal is the spatial integral over rors can also be reduced using composite pulses (6).
isochromats, the condition for getting maximal amplitudes is the Imperfect rephasing pulses introduce a growing number nulling of the spatially varying frequency offsets due to static of extra spin echoes and stimulated echoes which contribute field inhomogeneities (16). This occurs when the coherence increasingly as the number of rephasing pulses increases (7condition is satisfied. 9 ). The resulting deviation of the observed echo magnitudes
In the presence of B 0 and B 1 inhomogeneities, i.e., spatially from the desired exponential T 2 -dependent decay can lead varying B 0 and B 1 fields, a CP sequence with identical refocusing to significant errors in estimates of T 2 . Recent studies invespulses can be written as q 0 -[tq-t] n , where q 0 is the tigated this consequence of nonideal refocusing pulses in an excitation pulse and q is the rephasing pulse, which are ideally original CP and a CPMG sequence both numerically and 90Њ and 180Њ, respectively, and 2t is the echo time. experimentally (10,11).
After the nth q pulse, the coherence pathways of the In this Note, an analytical method for studying the effect of resulting contributors to the transverse magnetization are small pulse imperfections in Carr-Purcell sequences is used. It labeled by the vectors {1, p n , p n 01 , . . . , p 1 }, where p 1 refers applies to both single and composite pulses and allows insight to the interval between the excitation pulse and the first into the different compensational mechanisms and their efficienrephasing pulse, and p i (1 õ i £ n) is related with the cies in dependence upon relaxation and diffusion processes, reinterval between the (i 0 1)th and the ith refocusing pulse. flecting the variable compensational performance for odd-and At the time t after the nth rephasing, pulse echoes are even-numbered echoes. The beneficial effects of certain phaseformed if the following condition is fulfilled: alternating schemes in the case of phase dispersion due to the 1 / 2 p n / 2 p n 01 / 2 p n 02 / rrr / 2p 2 / p 1 Å 0.
[1] RF pulses or when phase is encoded for T 2 measurements in NMR imaging will be emphasized. In biological tissues, T 1 tends Equation [1] demands that p 1 Å {1. If relaxation processes to be significantly greater than T 2 . Therefore, the effects of during the pulse are negligible, the pulse action is assumed to be a pure rotation of the magnetization vector. Then a RF pulse can be described by the general rotation matrix † To whom correspondence should be addressed.