### The Spin Echo Effect

06Nov07

I actually performed this experiment as an undergraduate in physics. At that time I had little understanding of its theoretical underpinning, and hence was unable to fully appreciate its beauty. Two years on, I’m reading the paper describing the original experiment by Hahn, and the lab tech was right: it is a beautiful experiment. With intriguing implications for the foundations of statistical mechanics, to boot.

In brief, the experiment involves letting an initial ‘ordered’ state of spins evolve into a ‘disordered’ state, and subsequently recovering the initial ordered state by a judicious application of magnetic pulses. The experiment is done on nuclear spins in a liquid. The initially randomly aligned spins (left in diagram below) are aligned into an equilibrium ensemble along the z-axis by applying a constant magnetic field $H_0$ in the z-direction (right in diagram below).

After the spins are aligned along the z-axis, a short 900 rf pulse, parallel to the x-y plane, is applied. This causes the spins to begin precessing around the direction of $H_0$, that is, around the z-axis. In order to understand what happens next, it is convenient to switch into the frame of reference that is rotating around the z-axis with the average frequency $\omega$ with which the spins are rotating around the z-axis. We consider the precessional behaviour of two different nuclei in the liquid. When the rf pulse is first applied, they are induced to precess around the z-axis at the same rate, so in the rotating frame their x-y vectors are parallel (first panel of the diagram below). However, due to inhomogeneities in the constant applied field $H_0$, the nuclei do not all precess at the same rate (this is known as free induction decay). In the diagram below, the second panel shows the ‘faster’ nuclei (blue vector) as ‘ahead’ of the ‘slower’ nuclei (red vector) in the rotating frame. After a sufficiently long time, there is a wide enough spread of fast and slow nuclei that the collection of nuclei no longer looks ordered. Instead, it looks like just another ‘equilibrium’ set of spins which, though aligned with the z-axis, are not precessing at any central frequency. In short, it looks like the collection of nuclei that persisted before the 900 rf pulse was applied.

We will, however, recover order from this apparent disorder. After waiting for a long enough time $\tau$ for the spins to descend into apparent disorder, we apply a 1800 rf pulse to flip the spins about the y-axis, as indicated by the transition between the second and third panels in the diagram above. Now, instead of the red vector lagging behind the blue vector, the blue vector is ‘chasing’ the red vector. The originally slower spins become ‘ahead’ of the originally faster spins after the flip, and after a further time $\tau$, the faster spins ‘catch up’ with the slower spins. So at a time $2\tau$ after the first rf pulse, namely at a time $\tau$ after the second rf pulse, all the spins are aligned again (fourth panel in the diagram above). This results in an ‘echo’ of the magnetic moment of the liquid sample (which is measured by an inductive coil placed around the sample). The oscillographic traces of the initial magnetization, subsequent free induction decay, and echo can be seen in the following picture from Hahn:

This realignment of spins after achieving apparent disorder has been hailed by many as an example of reversing the increase of entropy that supposedly ensued after the first rf pulse, an apparent violation of the second law of thermodynamics. Rhim et al, for example, called this a “Loschmidt Demon” after the physicist Loschmidt, who objected to Boltzmann’s statistical mechanical derivation of a law of increasing entropy on the basis that any process in classical mechanics that can proceed forwards in time can just as well proceed backwards in time, thus violating the second law of thermodynamics. The “Demon” part comes from Maxwell’s Demon, a hypothetical intelligent being who violates the second law of thermodynamics by carefully shunting gas molecules into either side of a partitioned box according to their individual velocities. The spin echo experiment seems to justify Loschmidt’s claim (except in a quantum context): the road to disorder is not one-way, for we can regain the order lost in a system of precessing spins by an appropriate application of an rf pulse.

Whether the spin echo experiment constitutes a genuine violation of the second law of thermodynamics has been of some dispute, and also depends on which definition of entropy you choose to use. For further analysis of this experiment, see Lavis, Ridderbos, and Ridderbos and Redhead.