{"product_id":"echoa-spin-echo","title":"Echoa – The Spin Echo \/ Echo at 2τ Physics Sticker","description":"\u003cp\u003eEchoa, a young spin-wrangler, calls a scattered swirl of spins back into one bright point.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eIn 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.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003et\u003csub\u003eecho\u003c\/sub\u003e = 2τ\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eτ: time between the first and second pulse (s)\u003c\/li\u003e\n\u003cli\u003et\u003csub\u003eecho\u003c\/sub\u003e: time of the echo after the first pulse (s)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eHahn'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.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The echo undoes only dephasing from a static spread in fields. Real, random spin–spin relaxation (T\u003csub\u003e2\u003c\/sub\u003e) and diffusion are not reversed, so each echo is smaller than the last.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eEchoa holds a crook as a swirl of cyan spins gathers into a bright point in front of him. Printed text: \u003cstrong\u003eECHOA\u003c\/strong\u003e with the line \u003cstrong\u003eECHO AT 2τ\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50963980681319,"sku":"60324752946189765074","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50963980714087,"sku":"63494050579731557356","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50963980746855,"sku":"12166351811166722884","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50963980779623,"sku":"20103184324606376936","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/8195704070495666824_1200.jpg?v=1793657614","url":"https:\/\/pixelatedphysics.com\/de\/products\/echoa-spin-echo","provider":"Pixelated Physics","version":"1.0","type":"link"}