Echoa

Echoa, a young spin-wrangler, calls a scattered swirl of spins back into one bright point.

The Physics

In a magnetic field, nuclear spins precess at slightly different rates because the field is not perfectly uniform, so after a pulse knocks them over they fan out and the signal fades. Erwin Hahn found in 1950 (“Spin Echoes”, Phys. Rev. 80, 580) that a second pulse makes them run back together, and a signal reappears: the spin echo. In the version by H. Y. Carr and E. M. Purcell (Phys. Rev. 94, 630, 1954), a π/2 pulse is followed after a time τ by a π pulse that flips the fan over. The fast spins are now behind the slow ones, catch up, and all line up again at 2τ.

t_echo = 2τ

• τ: time between the first and second pulse (s)
• t_echo: time of the echo after the first pulse (s)

Hahn's own experiment used two π/2 pulses; the π refocusing pulse and the train of repeated π pulses came from Carr and Purcell. The echo is the basis of much of modern NMR and MRI.

Honest caveat: The echo undoes only dephasing from a static spread in fields. Real, random spin–spin relaxation (T₂) and diffusion are not reversed, so each echo is smaller than the last.

On the sticker

Echoa holds a crook as a swirl of cyan spins gathers into a bright point in front of him. Printed text: ECHOA with the line ECHO AT 2τ. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.