# Spin echo

Spin echo animation showing the response of spins (red arrows) in the blue Bloch sphere to the green pulse sequence

In magnetic resonance, a spin echo is the refocusing of spin magnetisation by a pulse of resonant electromagnetic radiation.[1] Modern nuclear magnetic resonance (NMR) and magnetic resonance imaging make use of this effect.

The NMR signal observed following an initial excitation pulse decays with time due to both spin relaxation and any inhomogeneous effects which cause different spins in the sample to precess at different rates. The first of these, relaxation, leads to an irreversible loss of magnetisation. However, the inhomogeneous dephasing can be removed by applying a 180° inversion pulse that inverts the magnetisation vectors.[2] Examples of inhomogeneous effects include a magnetic field gradient and a distribution of chemical shifts. If the inversion pulse is applied after a period t of dephasing, the inhomogeneous evolution will rephase to form an echo at time 2t. In simple cases, the intensity of the echo relative to the initial signal is given by e−2t/T2 where T2 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[3] and neutron scattering. Echoes were first detected in nuclear magnetic resonance by Erwin Hahn in 1950[4] , and spin echoes are sometimes referred to as Hahn echoes. In nuclear magnetic resonance and magnetic resonance imaging, radiofrequency radiation is most commonly used.[5][6]

In 1972 F. Mezei introduced spin echo neutron scattering, a technique that can be used to study magnons and phonons in single crystals. The technique is now applied in research facilities using triple axis spectrometers.

## Principle

The spin echo effect was discovered by Erwin Hahn when he applied two successive 90° pulses separated by short time period, but detected a signal, the echo, when no pulse was applied. This phenomenon of spin echo was explained by Erwin Hahn in his 1950 paper,[4] and further developed by Carr and Purcell who pointed out the advantages of using a 180° refocusing pulse for the second pulse.[7] The pulse sequence may be better understood by breaking it down into the following steps:

 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. A 90 degree pulse has been applied that flips the arrow into the horizontal (x-y) plane. 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. A 180 degree pulse is now applied so that the slower spins lead ahead of the main moment and the fast ones trail behind. Progressively, the fast moments catch up with the main moment and the slow moments drift back toward the main moment. Complete refocusing has occurred and at this time, an accurate T2 echo can be measured with all T2* effects removed. Quite separately, return of the red arrow towards the vertical (not shown) would reflect the T1 relaxation. 180 degrees is π radians so 180° pulses are often called π pulses.

Several simplifications are used in this sequence: no decoherence is included and each spin experiences perfect pulses during which the environment provides no spreading. Six spins are shown above and these are not given the chance to dephase significantly. The spin echo technique is more useful when the spins have dephased more significantly such as in the animation below:

## Spin echo decay

A Hahn echo decay experiment can be used to measure the spin–spin relaxation 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 π pulse. In simple cases, an exponential decay is measured which is described by the T2 time.

## Stimulated echo

Hahn's 1950 paper[4] 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 ${\displaystyle \tau }$, and then a second 90° pulse is applied such that the “pancake” is now in the x-z plane. After a further time ${\displaystyle T}$ a third pulse is applied and a stimulated echo is observed after waiting a time ${\displaystyle \tau }$ after the last pulse.

## Photon echo

Hahn echos have also been observed at optical frequencies.[3] 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.

## References

1. ^ J. E. Tanner & E. O. Stejskal (2003). "Restricted Self‐Diffusion of Protons in Colloidal Systems by the Pulsed‐Gradient, Spin‐Echo Method". The Journal of Chemical Physics 49: 1768. Bibcode:1968JChPh..49.1768T. doi:10.1063/1.1670306.
2. ^ Malcolm H. Levitt; Ray Freeman (1979). "NMR population inversion using a composite pulse". Journal of Magnetic Resonance 2: 473–476. Bibcode:1979JMagR..33..473L. doi:10.1016/0022-2364(79)90265-8.
3. ^ 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.
4. ^ a b c Hahn, E.L. (1950). "Spin echoes". Physical Review 80: 580–594. Bibcode:1950PhRv...80..580H. doi:10.1103/PhysRev.80.580.
5. ^ A Awad; R F Spetzler; J A Hodak; C A Awad; et al. (1986). "Incidental subcortical lesions identified on magnetic resonance imaging in the elderly. I. Correlation with age and cerebrovascular risk factors.". Am Heart Assoc 2.
6. ^ Bruce Gelerter. "PEMF Relieves Pain".
7. ^ 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.

## 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 978-0-471-48922-1.
• Arthur Schweiger; Gunnar Jeschke (2001). Principles of Pulse Electron Paramagnetic Resonance. Oxford University Press. ISBN 978-0-19-850634-8.