|
NMR
This site is running |
NMR /
SpinEcho< One pulse and the Free Induction Decay | Index | Spin echoes with matlab > NMR Spin Echo The spin echo was discovered in 1950 by E. L. Hahn, four years after the first detection of nuclear magnetic resonance signals, performed independently, nearly at the same time, by F. Bloch at Stanford and by E. M. Purcell at Harvard. The original paper, Phys Rev. 77, 297 (1950) and the subsequent more detailed description, Phys. Rev. 80, 580 (1950), are relatively easy to read. It is instructive to compare that description with the present one. The appearence of an echo after appropriate spin manipulation is evident if the FID decays rapidly, e.g. because of the inhomogeneity of the static field, {$ B_0 $}. The nuclei of a macroscopic sample are then subject to slightly different values of the field. We can imagine a Gaussian envelope of the field intensities, with a negligible variation in direction, resulting in a Gaussian distribution of precession frequencies (i.e. of isochromats). The echo sequence which is easiest to explain consists of a radio frequency pulse corresponding to a nutation of {$ \pi/2 $}, followed by a free spin evolution for a time interval {$ \tau $}, and by a second pulse along the same rf axis, corresponding to a nutation of {$ \pi $} (e.g. with radio frequency field twice as intense as that of the first pulse). The echo will develop at time {$ \tau $} after the second pulse (if we neglect pulse duration) and it is rather straightforward to understand its occurrence by identifying isochromats with a representative classical spin vector. Let us discuss the echo formation in the rotating frame: the first pulse turns all isochromats from the {$ \hat{z}^\prime = \hat z $} direction into, say, the {$ -\hat {y}^\prime $} rotating frame axis if the radio frequency field was along {$ \hat {x}^\prime $}. The subsequent free evolution will determine a decay of the signal because all isochormats precess at slightly diffenerent frequencies and eventually interfere destructively with one another. Pinpoint the attention on two isochromats, {$ -\omega $} in the rotating frame, lagging behind by the angle: {$ (1)\qquad\qquad -\phi=-\omega\tau $} and {$ \omega $}, running in front by the opposite angle {$ \phi $}, with respect to the average spin direction , which is always along {$ -\hat y $}. The second pulse determines a precession of all spin around {$ \hat {x}^\prime $} of the angle {$ \pi $}. This means that the slow isochromat, {$ -\omega $}, will now lie in the rotating precession plane at an angle {$ \pi+\phi $} from {$ - \hat {y}^\prime $}, i.e at an angle {$ \phi $} in front of {$ +\hat {y}^\prime $}. Likewise the fast isochromat, {$ \omega $}, will lie at an angle {$ -\phi $} behind {$ \hat {y}^\prime $}. The subsequent free evolution will bring all isochromats to coincide along {$ \hat {y}^\prime $}, exactly after a time {$ \tau $}. But the echo described by Hahn was different, and rather more complicated to treat with vectors (although such a description is given in the second Hahn reference above). It consisted of two equal length pulses. It is easy to realize that two {$ \pi $} pulses would yield no echo whatsoever: the first turns all isochromats along {$ -\hat z $}, and, disregarding the effect of relaxation towards thermodynamic equilibrium, the second brings them back to the start position. Hahn's own description of the echo is with two {$ \pi/2 $} pulses, separated by the interval {$ \tau $}. Considering the effect of the finite duration tw of each pulse the echo appears at time {$ \tau+t_w/2 $} after the end of the second pulse and, neglecting dynamic T2 processes, its amplitude is reduced by a factor x with respect to the ideal FID. < One pulse and the Free Induction Decay | Index | Spin echoes with matlab > |