Instrumentation
Resolution Enhancement with Multiple-Pulsed Techniques in P u l s e d Fourier Transform Nuclear Magnetic R e s o n a n c e Spectroscopy Dallas L. Rabenstein Department of Chemistry University of Alberta Edmonton, Alta., Canada
High resolution nuclear magnetic resonance spectroscopy is now usually performed in the pulsed Fourier trans form mode (1, 2). A high-power radio frequency pulse of short duration is applied to the sample and causes a transient response, the free induction decay, in the time domain. T h e free induction decay (FID) is sampled and stored in a data acquisition system. Fourier transformation of the free in duction decay gives the familiar fre quency domain spectrum. T h e experiment as described uses a one-pulse sequence, t h a t is, a single radio frequency pulse is applied, fol lowed by the acquisition of the FID. If t h e signal-to-noise ratio is low, sig nal averaging is done by repetitive ap plication of the one-pulse sequence, with coaddition of the individual FID's. State-of-the-art N M R spec trometers also have the capability of doing experiments in which a series of rf pulses is applied to the sample prior to acquisition of the FID, gener ally for the purpose of measuring spinlattice (ΤΊ) and spin-spin (T 2 ) relaxa tion times. This article describes how these multiple-pulse techniques can also be used to enhance resolution. 0003-2700/78/A350-1265$01.00/0 © 1978 American Chemical Society
Two general approaches will be de scribed. One is based on the "sorting" of resonances according to differences in their TVs and/or T2's, and the other is based on spin-spin coupling modu lation (J modulation) of resonance in tensity in the spin-echo multiple-pulse experiment.
One-Pulse Sequence I t is convenient to begin with the one-pulse sequence. Figure 1A, a rep resentation of the one-pulse sequence, shows the pulse followed by the FID. Mt is the magnitude of the signal from the nuclear magnetization at time t. T h e origin and shape of the F I D can be accounted for in terms of the be havior of the nuclear magnetization in a rotating 3-axis coordinate system (frame 1 of Figure 2A). T h e spectrom eter magnetic field Hç, is applied along the 2 axis, and the x' and y' axes rotate around the ζ axis at the carrier frequency of the spectrometer (which usually is near the Larmor frequency of the nuclei). Because the two possi ble orientations of the individual nu clear magnetic moments for spin onehalf nuclei are of slightly different en ergy and because nuclei in each of these orientations are randomly dis tributed around the direction of Ho, the ensemble of nuclei in the sample gives rise to a residual magnetization, MQ, which is colinear with and in the direction of HQ- From this point the
N M R experiment will be discussed in terms of the behavior of this residual magnetization. Magnetization along the y' axis is detected by the spectrometer to give the N M R signal. At equilibrium, this is zero. Application of an rf pulse along x' rotates M0 about the x' axis through an angle a given by Equation 1: a = yHitw
(1)
where y is the magnetogyric ratio of the nucleus being investigated, H i the intensity of the rf pulse, and tw the length of the pulse (generally in the range of microseconds). With the ap propriate combination of Hi and tw, Mo is rotated through 90° and be comes colinear with the y' axis, giving rise to a signal. This is a nonequilibrium condition, and the system returns to equilibrium by two relaxation processes. T h e mag netization in the x'y' plane relaxes by spin-spin relaxation according to Mfy' = M o e - t / T *
(2)
where t is the time following the 90° pulse, while the magnetization returns to equilibrium along the ζ axis by spin-lattice relaxation according to M(2 = M 0 ( l - e - t / r i )
(3)
As the magnetization returns to equi librium, the magnitude of the signal
ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978 · 1265 A
decreases. At the same time the com ponent of the magnetization along y' is modulated at a frequency equal to the difference between the Larmor frequency and the spectrometer car rier frequency. This gives rise to fre quency modulation of the signal as shown in Figure 1. T h e FID's in Fig ure 1 are those for a single resonance frequency; the FID for a sample hav ing more than one resonance is the sum of the individual FID's. Fourier transformation of the overall F I D gives the frequency domain spectrum. Spin-Echo Sequence
180°
90°
180° I OU -». -4
χ2
M ·+«
Acquisition
Figure 1. Representations of (A) one-pulse sequence, (B) spin-echo sequence, (C) inversion-recovery sequence, and (D) inversion-recovery-spin-echo (IRSE) se quence
Generally, the major contribution to relaxation in the x'y' plane is loss of phase coherence due to inhomogeneity in H>; thus, the effective T 2 , T\, is a function of the natural T 2 and HQ inhomogeneity. T h e spin-echo ex periment was developed by H a h n for purposes of eliminating the effect of Ho inhomogeneity on the measure m e n t of T 2 's (3). T h e H a h n spin-echo sequence con sists of a 90° pulse, a delay interval of length T 2 , a 180° pulse, and a second delay of length τ 2 . T h e pulse sequence is represented by sequence Β in Figure 1, and the behavior of the nuclear magnetization in the rotating coordi nate system by sequence Β in Figure 2. Following the 90° pulse, the mag netization in the x'y' plane fans out due to loss of phase coherence from spin-spin relaxation and inhomogene ity in Ho. Inhomogeneity in Ho con tributes to the fanning out because nuclei in some parts of the sample precess more rapidly than the average while others precess more slowly. Fol lowing the 180° pulse, the individual vectors precess at the same rate as be fore so t h a t those precessing more rap idly now catch up with the average, those precessing more slowly fall back to the average, and an echo results
Figure 2. Behavior of nuclear magnetization during (A) one-pulse sequence, (B) spin-echo sequence, and (C) inversion-recov ery sequence 1266 A · ANALYTICAL CHEMISTRY, VOL. 50, NO. 13, NOVEMBER 1978
along the y' axis. T h e echo maximum is a t time 2τ 2 after the 90° pulse. W i t h this pulse sequence, effects due to Ho inhomogeneity on TJ are self-elimi nating so t h a t the net loss in magneti zation at the echo maximum is due only to the natural spin-spin relaxa tion. T h e second half of the echo is also a free induction decay and differs from t h a t following the 90° pulse only in amplitude, due to the spin-spin re laxation which has occurred during the interval 2r 2 (4). If the spectrum consists of a number of resonances t h a t have different T2's, the decrease in amplitude at the echo maximum will be different for the different reso nances with a relative enhancement of signals from those resonances with the longer T 2 's. For example, the mag netization a t the echo maximum is 0.819 Mo, 0.368 M 0 , 0.135 M 0 , and 0.018 Mo for T 2 values of 10τ 2 , 2r 2 , τ 2 , and τ 2 /2, respectively. If the T 2 's of the different resonances are sufficient ly different and τ 2 is properly chosen, the magnetization in the x'y' plane from the resonances with the shorter TVs will have relaxed to zero at time 2r 2 , so t h a t only those resonances hav ing the longer T 2 's will be observed in the spectrum obtained by Fourier transformation of the second half of the echo (5). Resolution enhancement by the H a h n spin-echo sequence is illustrated in Figure 3 by the proton N M R spec t r a for a solution containing bovine serum albumin (BSA) and iV-methylglycine. T h e top spectrum was ob tained with the one-pulse sequence. T h e broad, rather featureless peaks are due to the protons of the BSA with the iV-methylglycine giving the two singlets superimposed at 2.84 and 3.72 ppm. T h e bottom spectrum was ob tained from the second half of the echo in a H a h n spin-echo sequence with τ 2 = 0.060 s. The 7Ys of the pro tein resonances are so short t h a t by 0.120 s after the 90° pulse, their mag netization has decayed to zero in the x'y' plane, leaving the spectrum for the iV-methylglycine. T h e signal cen tered around 4.80 p p m is the residual signal from the HDO solvent, most of which has been eliminated by the presaturation technique (6). Our recent work on h u m a n red blood cells provides another example of an application of t h e H a h n spinecho sequence for resolution enhance ment (7). Spectrum A in Figure 4 is the normal one-pulse spectrum and is due mainly to resonances from he moglobin. Spectra Β and C are spinecho spectra obtained with r 2 's of 0.020 and 0.060 s, respectively. Be cause hemoglobin is a large molecule, the T 2 's of most of its resonances are short. As Figure 4 shows, the hemoglo bin signals disappear as r 2 is increased
iw
