Spin echo

In magnetic resonance, a spin echo is the refocusing of precessing spin magnetisation by a pulse of resonant radiation. Modern nuclear magnetic resonance and magnetic resonance imaging rely heavily on this effect.

The NMR signal observed following an initial excitation pulse decays with time due to both spin-spin relaxation and any inhomogeneous effects which cause different spins to precess at different rates e.g. a distribution of chemical shifts or magnetic field gradients. Relaxation leads to an irreversible loss of magnetisation (decoherence), but the inhomogeneous dephasing can be reversed by applying a 180° or inversion pulse that inverts the magnetisation vectors. If the inversion pulse is applied after a period $t$ of dephasing, the inhomogeneous evolution will rephase to form an echo at time $2t$ . The intensity of the echo relative to the initial signal is given by $\exp(-2t/T_2)$ where $T_2$ is the time constant for spin-spin relaxation.

Echo phenomena are important features of coherent spectroscopy which have been used in fields other than magnetic resonance including laser spectroscopy^[1] and neutron scattering. Echoes were first detected in nuclear magnetic resonance by Erwin Hahn in 1950^[2] , and spin echoes are sometimes referred to as Hahn echoes. In nuclear magnetic resonance and magnetic resonance imaging, radiofrequency radiation is most commonly used.

1 Principle
2 Hahn echo decay
3 Stimulated echo
4 Photon echo
5 See also
6 Animations and Simulations
7 References
8 Textbooks

Principle

The spin echo concept of Erwin Hahn was explained in his 1950 paper,^[2] and was further developed by Carr and Purcell who pointed out the advantages of a 180 degree refocusing pulse.^[3]

The pulse sequence may be better understood by breaking it down into the following steps:

The spin echo sequence. A) - The vertical red arrow is the average magnetic moment of a group of spins, such as protons. All are vertical in the vertical magnetic field and spinning on their long axis, but this illustration is in a rotating reference frame where the spins are stationary on average. B) A 90 degree pulse has been applied that flips the arrow into the horizontal (x-y) plane. C) Due to local magnetic field inhomogeneities (variations in the magnetic field at different parts of the sample that are constant in time), as the net moment precesses, some spins slow down due to lower local field strength (and so begin to progressively trail behind) while some speed up due to higher field strength and start getting ahead of the others. This makes the signal decay. D) A 180 degree pulse is now applied so that the slower spins lead ahead of the main moment and the fast ones trail behind. E) Progressively, the fast moments catch up with the main moment and the slow moments drift back toward the main moment. F) Complete refocusing has occurred and at this time, an accurate $T_2$ echo can be measured with all $T_2*$ effects removed. Quite separately, return of the red arrow towards the vertical (not shown) would reflect the $T_1$ relaxation.

Several simplifications are used in this animation: no decoherence is included and each spin experiences perfect pulses during which the environment provides no spreading.

180 degrees is $\pi$ radians so 180° pulses are often called $\pi$ pulses.

Hahn echo decay

A Hahn echo decay experiment can be used to measure the dephasing time, as shown in the animation below. The size of the echo is recorded for different spacings of the two pulses. This reveals the decoherence which is not refocused by the $\pi$ pulse. In simple cases, an exponential decay is measured which is described by the $T_2$ time.

Stimulated echo

Hahn's 1950 paper^[2] showed that another method for generating spin echoes is to apply three successive 90° pulses. After the first 90° pulse, the magnetization vector spreads out as described above, forming what can be thought of as a “pancake” in the x-y plane. The spreading continues for a time $a$ , and then a second 90° pulse is applied such that the “pancake” is now in the x-z plane. After a further time $b$ a third pulse is applied and a stimulated echo is observed after waiting a time $a$ after the last pulse.

Photon echo

Hahn echos have also been observed at optical frequencies ^[1]. For this, resonant light is applied to a material with an inhomogeneously broadened absorption resonance. Instead of using two spin states in a magnetic field, photon echoes use two energy levels that are present in the material even in zero magnetic field ^[4].

Animations and Simulations

http://scratch.mit.edu/projects/nevit/872879 Spin Echo Simulation
The animation show pulse sequences like spin echo sequence

References

^ ^a ^b Kurnit, N. A.; Abella, I. D.; Hartmann, S. R. (1964). "Observation of a photon echo". Physical Review Letters 13: 567–568. Bibcode 1964PhRvL..13..567K. doi:10.1103/PhysRevLett.13.567.
^ ^a ^b ^c Hahn, E.L. (1950). "Spin echoes". Physical Review 80: 580–594. Bibcode 1950PhRv...80..580H. doi:10.1103/PhysRev.80.580.
^ Carr, H. Y.; Purcell, E. M. (1954). "Effects of Diffusion on Free Precession in Nuclear Magnetic Resonance Experiments". Physical Review 94: 630–638. Bibcode 1954PhRv...94..630C. doi:10.1103/PhysRev.94.630.
^ http://www.physics.montana.edu/oct/web/basics_of_octs.htm

Textbooks

Ray Freeman (1999). Spin Choreography: Basic Steps in High Resolution NMR. Oxford University Press. ISBN 978-0-19-850481-8.

Malcolm H. Levitt (2001). Spin Dynamics: Basics of Nuclear Magnetic Resonance. Wiley. ISBN 978-0471489221.

Arthur Schweiger, Gunnar Jeschke (2001). Principles of Pulse Electron Paramagnetic Resonance. Oxford University Press. ISBN 978-0198506348.