A Primer on Fourier Transform NMR Roger S. Macomber University of Cincinnati, Cincinnati, OH 45221 Although the pulsed-Fourier transform (PFT)methodology of NMR spectrhscopy has rendered the continuous wave (CW) technique nearly obsolete, many teachers still introduce the topic of NMR using the CW format because of its pedagogic simplicity. Thus, students must often await advanced courses to receive more than a "black box" introduction to the FT technique. This paper was written to give the beginning student some insight about how data are collected in the time domain, as well as how these data are transformed into the frequency domain. First, let us briefly review the CW technique as it is usually presented. As shown in Figure 1, a collection of identical magnetic nuclei (e.g., protons) immersed in a magnetic field collectively precesses around the axis of the field a t a frequency (u) governed by the magnetic field strength (Ho) and the magnetogyric ratio (7) of the nucleus: y=-
F i q r e 7. Recessh of nuchi amxd an external magnetic field. Boldfaceanow
I
1'0
j'",
Figure 2. Standard frequency domain NMR spectrum
YHO 2?r
The factor 2a converts angular frequency (w = THO,radlsec) into linear frequency (u, Hz). Once the system comes to equilibrium (which occurs a t a rate governed by TI, the spin-lattice or longitudinal relaxation time), there is a slight net magnetization in the +Z direction (Fig. 1). because the +Z orientation is slightly more stable than the -Z orientation. However, because the nuclear spins are randomly oriented around the Z-axis. there is no net maenetization in the x.v plane. Next, an alternating current of variable frequency in the radiufreauencv (11) region is passed through a coil whose axis lies in the x , i i l a n e , perpendicular to the applied magnetic field. This current gives rise to an oscillating magnetic field also perpendicular to the applied field. As the frequency (wid of the oscillating field is varied, there is a point a t which i t exactly matches the precessional frequencyof the nuclei. At this point, the system is said to be "in resonance."Energy is transferred from the radiation to the nuclei, causing a change in their spin orientation. This, in turn, induces a Faraday current in an appropriately situated receiver coil, and the signal is displayed on the recorder as a function of ui,. A typical "frequency domain" spectrum is thus generated, as
represents Me net magnetization.
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Figure 3. The effect of resonance between the precessing nuclei and an oscillaling lield of tho hesame frequency. M is the net nuclear magnetization.Mw is its component in the xy plane.
shown in Figure 2. The position of the peak, uo, gives the precessional frequency of the nuclei, as well a s the radiation frequency needed to cause "resonance." Now, let us modify the procedure. Instead of continuous rf current, a short but powerful pulse of rf current is sent through the transmitter coil. This "white" pulse is a mixture of all appropriate frequencies in the spectral region of interest, and it excites all the nuclei in the region simultaneously. Furthermore, the duration of this pulse is critically important for the following reasons. As the nuclei receive the pulse, there are several consequences. First, the nuclear spins "phase up," becoming a bundle of precessing nuclei which now has a component of net magnetization in the x,y plane (see Fig. 3). This x.v-magnetization induces an alternating current in the receiveicuil; whose uxis is in the s , y piane.l?he amnunt of x,v magnetization is controlled by rhesineuf the "tip - angle" &while the magnitude of or is determined by the duration of the pulse and its power. In most cases the optimum signal is obtained with a 90° tip angle, which requires a long enough pulse to tip the net magnetization vector perpendicular to the Z-axis. This is called a 90' pulse. At this point an alternating current has been induced in the
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In the PFT experiment, the same coil can be used for both transmission of the pulse and receiving the induced signal, for these two processes occur at separate times. Volume 62
Number 3
March 1985
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time
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Figure 4. Voltage induced in the receiver coil, collected digitally as a function of time.
receiver coil? The frequency of this current is equal to the (as yet unknown) precessional frequency of the nuclei, and its intensity is related, among other things (uide infra),to the pulse angle. Next, the intensity of this current is monitored as a function of time. However. intensitv data are collected digitally hy sampling the voltage at regular tntervals (called dwell time). A tvo~calset uf suchdata miaht look like that in is not obvious because, a t least for our Figure 4. The eves to recoenize the oattern, data ooints have not been coll&d often&ough. f i e experiment can he repeated with the dwell time reduced by a factor of five, thus affording five times as many points in the same length of time. Figure 5 shows the result-a more detailed and easily recognized cosine wave. So the first lesson is that points must be collected often enough to make the pattern recognizable. Actually, the pattern observed would more closely resemhle Fimre 6. Such a pattern is called the modulated f'ree induction decav ,(FID) signal because the current intensitv decreases with time. This decay is the result of spin-spin o i transverse relaxation. caused hv the deohasine" of the nuclei after the pulse, which reduces the net maynetizarion in the x , ) plane. The envelope of the damped cosine wave in Figure 6 describes an exponential decay whose decay time is T,',the spin-spin relaxation time.?'l'he halfwidth ( u , Y J of thesignal in the irequency domain spectrum (Fig. 2) Galso governed by Tz*. Still. the imoortant information is the precessional frequenc; of the nuclei. How is that information obtained from the data in Figure 6? First, note that the frequency of cosine wave is unaffected by the exponential decay (compare Figs. 5 and 6). Therefore, all that is needed is to measure the peak-to-peak or trough-to-trough time ( t o in Figure 6). Suppose this length of time is 2.0 X 10-8s. The frequency of the cosine wave would be uo = 1cyclel2.0 X s = 5.0 X lo7 Hz = 50 MHz. In a sense, we have just performed a Fourier transformation on the data3: we have taken digital data of a periodic function in the time domain and extracted the characteristic frequency. And most importantly, the entire spectrum (Fig. 2) observed from the CW experiment has been eenerated with a sinele ., rf oulse. . In cases that involve rither dilute samples or insensitive nuclei tsuch as IT,,II is often found in practice that one pulse does not give a sufficient signal-to-noise (SIN) ratio to allow accurate extraction of frequency information. The S I N ratio can be improved by repeating the pulseldata aquisition seauence, then adding the new data t o the original data. The numhe; ( n ) ~ f ' ~ u l s e ~ s e ~ t t erequired n c e s is &termind hy the desired SIN ratio, which increnses with \ n. There is an additional considerntion: one must allow rnough time between pulses for the nuclei to reach (a,r nearly reach) their origlnitl eouilibrium distribution. Tht,retnre. this "delnv time" w~llbe a function of T I of the nuclei. Most of the molecules examined by NMR have several sets of nuclei, all of the same type (e.g., protons), but each set has
Figure 5. The effectof dwell time on me signal. Compare with Figure 4
Figure 6. me complete modulated free induction decay signal.
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Journal of Chemical Education
a slight/) different prwessional frequency because of differences in their molecular environments. Furthermore, each set has different relaxation times. and there will he different numbers of nucle~within each set. These factors combine to give very complex FID signals. At this point it becomes necessary for a digital computer to mathematically recognize the oatterns and extract the freauencv and relative intensitv of each set of nuclei. I t is this extraction process that involveithe Fourier transformation" the FID signal is inteerated over time, and the values of u (or w ) and intensity are extracted to eenerate the conventional freauencv domain spectrum. The PFT-NMR experiment'can de summariied as involvine two maior stem: (1)the collection of data by a pulseldata aqkition/delay sequence, repeated enough times t o yield an FID signal possessing the desired SIN ratio, and (2) integration of the FID data, and extraction of the frequencylintensity information. For those interested in further reading on this topic, an excellent article on modern pulse nmr methods has recently been p ~ b l i s h e d . ~ I wish to thank Jerry Ackerman for a very helpful discussion.
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T,' is the "effective" transverse relaxation time, which is controlled both by the inherent rate of transverse nuclear relaxation (in a perfectly homogeneous field, T1) as well as contributions from field inhomogeneity. Usually the latter is the dominant contributor. Actually, the Fourier transform from the time domain into the frequency domain involves the integration below, where S(t)represents the FiD data:
im S(t)e-"'dt
Benn, R., and Gunther, H., Angew Chem. ht. Ed. Engl., 22, 350 (1983).
