Volume 172, number 3,4 CHEMICAL PHYSICS LETTERS 7 September 1990 Experimental verification of the exact response of nuclear spins to an exponentially shaped pulse Max A. Keniry ’ Research School of Chemistry, Australian National University,GPO Box 4. Canberra, ACT 2601, Australia and B.C. Sanctuary * Department of Chemistry McGill University, 801 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 Received 17 May 1990; in final form I3 June 1990 This paper presents an experimental determination of the offset dependence of the response of a nuclear spin to an exponen- tially shaped pulse There are two limiting cases: If the decay constant, a, is greater than w, Ix, 100% inversion is not possible. If the decay constant, a, is less than q/a, 100% inversion is possible.In the latter case, the agreementwith the theoretically derived response curve is excellent.In the former case, agreementbetween theory and experimentis very good but not exact. In addition, if the pulse length of a squarex pulseof maximum radiofrequency field strength w, is known,then the pulse length of an exponen- tial K pulse with the same maximum field strength may be calculated. 1. Introduction With the advent of inexpensive and versatilewave- form generators, selectiveexcitation in nuclear mag- netic resonancehas attracted renewedinterest [ l-31. The output from the transmitter is modulated in am- plitude and/or phase by the waveform generator re- sultingin a pulse shape that resembles the waveform stored in the waveform generator. Historically, rec- tangular pulse shapes were the first to be employed in Fourier transform NMR [ 41, and although ade- quate for many NMR experiments they have severe limitations [ 5] when applied to magneticresonance imaging or NMR experiments requiring selectiveex- citation. These limitations may be alleviated by us- ing alternative pulse shapes. In the limit of a weak radiofrequency field, the re- sponse of a weakly coupled spin system to a single arbitrarily shaped pulse is easy to calculate. The so- ’ To whom correspondence should be addressed. 2 On leave at the Australian National University, Research Schoolof Chemistry, 1989-1990. called linear responsetheory predicts the responseof a spin system to a pulse is simply the Fourier trans- form of the pulse shape. The assumptions of linear response theory are, however, rarely satisfied in the typical experiments carried out by NMR spectro- scopists.We must resort to non-linear techniques to calculate the excitation response for the most fre- quently used flip angles, x/2 and 71. Despite enor- mous advances in instrumentation in the area of se- lective excitation, the response of a weakly coupled spin systemto commonlyused pulseshapeshas been calculated analytically for only two waveforms, the rectangular waveform and the phase-modulatedhy- perbolic secant waveform [ $61. One must resort to perturbative techniques to obtain an approximate solution for pulse shapes such as a Gaussian wave- form or a sine waveform [ 5,7,8]. In this Letter we experimentally verify the theo- retically derived response of a weakly coupled spin system to two different casesof exponentiallyshaped pulses [ 81. We demonstrate in both cases an excel- lent correlationbetween the experimentalresultsand the predicted response. 0009-2614/90/% 03.50 0 1990- Elsevier Science Publishers B.V. (North-Holland) 295