Pulsed/Fourier Transform NMR Spectroscopy Dallas L. Rabensteln Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G X 2 State-of-the-art NMR spectroscopy is generally done in the
transform )FP ('l./ . . mode (1-3). A hieh-nowered nulsed/Fourier .~ ~~. rf pulse is applied to the sample, giving'rise'to a t;aniient re~
~
~
~
~~
~
snonse. the free induction decav (FID). The FID is samded cfigitaIiyy,stored in a data acquiktion system, and then FOUrier-transformed to eive the familiar freauenw . - domain spectrum (4). A maior a d v a n b e of the P A T method over the contimuous wave ( d w ) methodof measuring NMR spectra is that signal averaging to enhance sensitivity can be done much more efficiently. In the CW mode, the spectral region of interest is slowly scanned by varying either the rf frequency or the magnetic field while keeping the other constant. Signal averaging is done by repetitive scanning of the spectrum with coherent addition of the individual scans. In the P/ET mode, the rf pulse excites nuclei over a wide spectral range, and thus each FID contains, simultaneously, signals from the entire spectral region of interest. Signal averaging is done by repetitive application of the rf pulse, with coherent addition of successive FID's. The increase in sensitivity in both the CW and PAW modes is theoretically equal to fl, where N is the number of scans.' However, signal averaging is more efficient with PlFT NMR because the time per FID is much shorter than the time per scan in CW NMR. To achieve maximum sensitivity enhancement by signal averaging with P/FT NMR requires the careful selection of experimental parameters (5.13,and generally involves compromises, e.g., a trade-off of maximum sensitivity enhancement for auantitative intensitv relationshins amone the resonances. fn this article, the dependence of iignal int;?nsity on exoerimental oarameters is considered. The behavior of the nuclear magn&zation during the repetitive application of the pulse-data acquisition sequence is considered from the classical point of view, from which some relations between signal intensity and experimental parameters can be obtained.
-
rotating around the z' axis a t the rf frequency (the carrier frequency), which is near the Larmor frequency of the nucleus. Because the two possible orientations of the individual nuclear magnetic moments for spin-one-half nuclei are of slightly different energies and because nuclei in each of the orientations are randomly distributed around the direction of B, the ensemble of nuclei in the sample gives rise to a net magnetization, Mo,which is colinear with and in the direction of Bo (Fig. 1). From this point on, the P/FT experiment will be discussed in terms of the behavior of this magnetization. Magnetization along the y'axis gives rise to the NMR signal. At equilibrium, this is zero. Application of the rf pulse with its magnetic component, B1, directed along the x' axis rotates M oabout the I' axis through an angle a given by where y is the magnetogyric ratio of the nucleus, B1 the power of the pulse, and t , the length of the pulse (generally in the microsecond range). With the appropriate combination of B1 and t,, Mo is rotated through 90' and becomes colinear with y' (Fig. lb), after which it gives rise to the FID (Fig. lc). The magnitude of the FID at a given time is proportional to they'
The PIFT NMR Experiment
It is convenient to discuss the P/FT NMR experiment in terms of the behavior of nuclear magnetization in a rotating .?-axis coordinate system (Fig. 1).The spectrometer magnetic field Bo is applied along the t'axis, and the x'and y' axes are
' The increase in sensitivity, as
iven by the signal-to-noise ratio Nscans is since signal strenath increases as N, while noise increases as ~ N ( Q . (SIN). from madding
A,
F!gm 1. The net macroscopicmagretizetionat equilibriunanda m a 90' wise (top)and me hee indunion decay obtained aner the 90° pulse (bottoml.
Volume 61 Number 10 October 1984
909
Figwe 2. Schematic representation of Ih-8 pulse-relaxatlonsequence. Mag netlzation along Ih-8 y' axls decays by spinapln relaxation and recovers along me i axis by spln-lanlce relaxation. In this example. T, = $re. component of the nuclear magnetization, Myr,a t that time. If the Larmor freauencv of the nuclei and the carrier frequency are different, the signal generated by the magnetization in the x'v' lane and the carrier are altematelv in and out of phase, g k n g rise to a frequency-modulated FID, the modulation frequency being equal to the difference between the Larmor and carrier frequencies. The FID in Figure 1would be obtained if there were a single resonance in the spectral region; the FID for a sample having more than one resonance is the sum of the individual FID's.
Time Between Pulses / T1 Figure 3. Relalve steady state signal Intensity as a functionof time between pulses (In "nib of T,) for flip angles of 30'. 60°, and 90'.
Spln-Lalca and Spln-Spin Relaxatlon After the 90° pulse, the system is in a nonequilibrium condition. It begins to return to equilibrium immediately by two relaxation h e s s e s : the component of the magnetization in the x'y' plane, M,v, decays to zero by spin-spin relaxation
where Mo is the magnetization along z'at equilibrium, t is the time followingthe rf pulse, and Tz the spin-spin or transverse relaxation time, while the magnetization returns to equilibrium along the z' axis by spin-lattice relaxation according to
M,,= Mo(l - e-'lTl)
(3)
where TI is the spin-lattice or longitudinal relaxation time. As M,rY, decays, the magnitude of the signal decreases. For small- to medium-sized molecules, T1 and Tz will have similar values. Generally, however, the effective Tzis somewhat less than the true Tz due to inhomogeneity in Bo, which causerr the individual magnetic vectors to fan out. Since TZ< TI, M , y and thus the signal will generally decay to zero before M,' recovers to Ma. The pulse-relaxation sequence is depicted schematically in Figure 2. For convenience, the carrier frequency and the Larmor frequency are set equal. Thus, the magnetization in the x'y' plane remains colinear with the y' axis, and the FID consists of a simple exponential decay. For the example shown in Figure 2, T I = 5Tz. Slgnal Averaglng with 90" Pulses Sensitivity can be increased hy thecoherent addition of the FlD's from re~etitiveoulses. If the time between oulses (the M,'~ , will return to 0.9933 of its repetition t&e, t,) i s & 5 ~ eauilibrium value between oulses. Thus. M,' will be 1.00 Mn immediately after the first 600 pulse and 0.6933 of this valu; immediatelv after each successive oulse. Generallv, one or more pulses are applied before dataacquisition is s-&d to allow the maenetization to reach a steadv state condition, in which case t& sensitivity obtained by co:adding N FID's All be 0.9933fl times that from a single 90' pulse. The time between pulses should be a5 T2 to avoid Um formation of spin echoes, which can cause phase and intensity distortions. This wndition is considered to be satisfied in all the samples discussed. 910
Joumal of Chemical Education
0.01 001
' """"
01
'
10
,
''L-,.
100
Time Between Pulses 1 TI Flgve 4. Dependence ofsignel-tc-nolseratlo m tlma behveen pulses forlllp angles of 30'. 60°, and 90'. Ttm slgnal-tpnoiseratio Is given relative to h t which wwld be obtained with ol = 90° and a repetition time of 5Tt.
It is clear from Figure 2 that the signal ie essentially zero during meater than 80% of the time in this experiment. If the time getween pulses is decreased to TI, M,. &ill have recov. ~ will ered to 0.632 Moat the t i i e the next pulse is a ~ p l i e dMy' he 1.00 Mo immediately after the first 90° pulse and 0.632 of this value immediately following each successive pulse. Thus, with these conditions the signal is partidy saturated after the first pulse and the theoretical increase in sensitivity is reduced to 0 . 6 3 2 n t i m e s that obtained from a single 90' pulse. If t , is decreased to 0.5 T1, the theoretical increase in sensitivity is reduced to 0.393fi. As the time between successive 90° pulses decreases, the steadv state sienal intensitv decreases. as shown in Fieure 3. t ow ever, as t i e time betieen pulses'decreases, more FIDs can be accumulated in the same amount of t i i e . For examnle. five times as many FIDs can be accumulated when t , = i ~ rather than 5T1, giving a total increase in signal strength of 0 . 6 3 2 m as compared to 0 . 9 9 3 3 n for the 90° pulsedT1 conditions. Thus. a more relevant comoarison when selectine " pulse conditions is the sensitivity enhancement in the same total time. When this comparison is made, the total increase in signalstrength using90° pulses with I , = IT, and0.5T1 are 1.423 and 1.244 times that obtained in the same total time with
;
the 90° pulse-5T1 conditions. The dependence of the signalto-noise ratio (SIN) on time between 90° pulses for the same total experimental time is plotted in Figure 4. The SIN is relative to that obtained with the 90' pulse-ST1 conditions. The maximum SIN is obtained with a time of 1.25T1between successive 90° pulses. Signal Averaglng with Pulses Less than 90' It is not necessary that a 90' flip angle be used. For a flip angle Q90°,My' and M,' immediately after the pulse are Mo sin a and Mn cos a. res~ectivelv.Thus. if a < 90'. the sienal strength is decreaseb, b i t less t i e is required for l k g i t u c h d relaxation since M,, is neater than zero immediatelv after the pulse. For example, if'a = 60°, the signal following a single pulse is 0.866 of that following a single 90° pulse and M,, is 0.5Mo. The recovery of M,, is described by Mo - M,, = -Ke-LIT1 (4) where t = time after the pulse and K depends on flip angle, time between pulses, and number of the pulse in a repetitive pulse experiment. For example, if a = 60° and t . = IT1, then My, = 0.866Mo and M,, = 0.500Mo at t = 0 after the first nulse. Substitution into eon. 0.50Mo for -K so that . (4) . . vields " reco\.ery of M,, after the first pulse is given by M,. = Mot1 0.5e-'/"1) At the time of the second oulse. M.. has recovered to 0.816~0.Immediately after the second pulse, My = 0.707Mo and M,, = 0.408Mo. From eqn. (4), recovery of M,, after the second pulse is given by M,, = Mo(1- 0.592 e-'lT~). At the time of the third pulse, M,, has recovered to 0.782Mo. As pulsing continues, a steady state is reached after the fifth pulse. Once the steady state is reached, M,. at the time of a pulse is given (2) by
where E = e-',IT1. The components of magnetization along the z' and y' axes immediately after the pulse reach steadystate values (2) of: (1 - E ) cos a M,. = Mo M
-
1-Eoosa (1 - E ) sin a
''- 1-Ecosa
Relative SIN values obtained in a constant total experimental time with 60° and 30° pulses are also plotted as a function of the repetition time in Figure 4. The SIN values are relative to that obtained with a = 90' and t . = 5T1. SIN reaches a maximum at 1.25T1, 0.63T1, and 0.14T1 for 90°,60°, and 30" pulses, respectively. At other repetition times, the flip angle which gives the maximup SIN can be calculated with the equation cos a = e-',/TI
The flip angle given by this equation is the so-called Emst angle (5). Resonance lntensltles Resonance intensities are often one of the main parametern extracted from an NMR spectrum, e.g., todetermine number of nuclei at a oanicular chemical shift in structure elucidation or to determke relative concentrations in analytical experiments. When signal-averaging under conditions where M,, does not recover to Mo between pulses, the signals are partially saturated and resonance intensities are distorted. If. as is almost always the case, the different renonmcen in the sbectrum have different T I values. thev will be saturated to different extents. ~eneraliythe pulseconditions are selected on the basis of the longest T I of the resonances of interest. Thus, other resonances will relax more completely between pulses and the resonance having the longest TI will be most saturated. For example, if the minimum repetition time2 is 0.14 TI, where TI is the longest T1 of the resonances of interest, the Ernst angle is 30". From eqn. (I),My, immediately after the 30' pulse for the resonance having this TI is 0.264Mo. If the TI of a second resonance is only one half this TI, My, for that resonance will be 0.353Mo. Thus, the resonance having the shorter TI is less saturated and, for an equal number of nuclei, will be 33.9% more intense. For resonances having TI values much less than the maximum TI, My, approaches 0.500Mo and their intensity can be up to 89.4% larger than that of the resonance having the longest TI. Thus, the larger the range of TI values, the larger the range of intensity differences for resonances from the same nqmber of nuclei. The maximum range of intensity differences (miniplum resonance intensity/maximum resonance intensity) is given by
Mo
The relative steady state signal intensity obtained from repetitive 30' and 60' pulses are also plotted as a function of the time between pulses in Figure 3. When the time between pulses is long, the largest relative signal strength is obtained s short, this with 90" oulses. When the time between ~ u l s e is isnot thecase."or example, thesteady statesignal intensity from 60° pulses is larger than that from 90' pulses when the time between pulses is less than l.:llT~.Likewise, thesteady state siend intensitv from 30' pulses is larger than that from 60° p&es when the time between p&es is less than 0.33T1. An important conclusion be reached from the results in Figure 3 is that, when optimizing pulse conditions for maximum sensitivity enhancementin a constant total experimental time, not only is it better to use the minimum repetition time allowed by Tz but also to use the pulse angle at which the steady state signal is largest for that repetition time. The relative SIN values of signals S1 and Sz measured with the conditions a l , t , and ~ an, t,s in the same total experimental time, is given by (SIN), ( I - E M - E2 cos a d sin a1 6 -= (8) (SIN), (1 - Ez)(l - E l cos a d sin a z 6 whereE, = e-t,.l/T~. E? = e - t d T ~ . Mi is larger the smaller lhe flip angle when successive pulses are applied since Mi at t = 0 is larger.
(9)
ID,,
=
l-E 1 - E coa a
where E is calculated using t , a i d the longest TI of the resonances of interest. For the Ernst angle, this reduces to IDm=-
1
(11)
l+E
Thus, one consequence of optimizing pulse conditions for
0 00
I0
20
30
40
50
60
Time Between Pulses I T1 Figure 5. The m a x i m flip angle whch can be asd at me Indicated maxlmm m m level$a$ a functionof the rspet;tlan tme. Volume 61
Number 10 October 1984
911
sensitivity enhancement is that relative intensities of resonances are generally not equal to relative numbers of nuclei. Quantitative PlFT NMR When a quantitative relationship between resonance intensities is important, the range of intensities obtained using the Ernst angle may be too large. If so, compromise pulse conditions can be used which give the optimum sensitivity within the maximum tolerable intensity error limits. The % error in intensity will be largest for the resonance having the longest T1 (7). The maximum % error, c, is given by f
=
100E(1- cw a) 1-Ecosa
(12)
Generally, a limit can he set on the maximum error tolerable for the p&poses of the experiment. The maximum pulse angle with which the maximum intensity error will be less than c is given by
An Example The effect of ~ u l s econditions on sensitivitv will be demonstrated with ' HNMR spectra for a DzO solkion of methanol and acetic acid (Fig. 6). T I values for the HOD, CHnOD. . and CH3C02D resonances were determined by the inversion-recoverymethod (8)t~ be 17.1,9.3,and 5.8 s, res~ectivelv. The three spectra shown in Figure 6 were each measured with a = 90° and the same number of scans but with t, = 5,1.09, and 0.16 times the T1 of the HOD resonance. As predicted by eqn. (7)and Figure 3, the signal intensity decreases as t , decreases. Also, the longer the T I ,the larger the relative decrease in intensity (e.g., HDO versus CH3C02D). In Figure 7 are shown spectra measured using t . = 3.4 s (0.2T1HOD) and a = 90°, 60°, 35O (the Ernst angle at t. = 0.2T1).20'. and 10".The same number of FIDs were co-added to each spectrum. The theoretically predicted and experimentally found dependence of the HOD resonance in-
The maximum pulse angle is plotted as a function of the repetition time for various error levels in Figure 5. The minimum t. combined with the maximum flin anele mav not eive maximum sensitivity. Rather, the repeti'tion'times and guise angles at which maximum sensitivity is obtained at various error levels are: t . = 4.5T1,a = 84" (1%error level); t . = 3.5T1, a = 81° (2.5%):t . = 2.5T1.a = 79' (5%):t . = 2.OT1. a = 73" As an example, suppose we want the intensity error due to T I differences to be 5% or less for a sample for which the minimum time between nulses (=5TJ is 1.5Tq. The maximum pulse angle for c = 5% a i d t. = 1.5Fl is calckated to be 35" with eqn. (13).However, if t , and a a r e increasedto 2.5T1and 7g0,6 is still 5%, but a signal 1.3 times as intense will be obtained in the same total experimental time.
10
60
1.0
2.0
PPU
Flgue 7. 'H NMR spectra ofthe methanol/acetic acid solution measured with a repetition time of 0.2Jt and the flip angles given in the figure. The number of FlDs was the same for each spectrum. For a repetition time of 0.2T,, lhe Emst angle is 35O.
F i w 6. 'H NMR spectra of melkml and acetic acid in D20solution. All thee s p e m were measured with a flip angle of 90' and the same number of scans; the time between pulses is given beside each scsbum. me 5 values are: 17.1 s (HOD). 9.3 s (CH~W)and 5.8 s (CH3C02D).'
912
Journal of Chemical Education
Figue 8. The HOD resonanceas a functlon of the repettion tlme. The flip angle was 90' and the same total experimentaltime was usad to &in each spectrum. The noise is ploned wilh the gain increased by 4.
-
tensity on a is: lo0 < 90° < 60' 260 < 35O. In a similar experiment using t, = 5T1 (HOD) and a = SO0, 60°, and 30°, the intensity of the HOD resonance in the a = 60° and 30° spectra was 0.86 and 0.47 that of the resonance in the a = 90° spectrum, as compared to theoretical values of 0.87 and 0.50. To demonstrate the dependence of SIN on t., spectra were measured as a function oft, while keeping a = SOo and the total experimental time constant. Results are shown for the HOD resonance in Figure 8. As T is decreased, the signal becomes more saturated, however the signal increases because the number of FID's which can be collected in the experimental time increases. Since more scans are being averaged, the noise also increases,' the net result being that SIN reaches a maximum and then decreases as t, is decreased. For the t, values used in Figure 8, the theoretical values for SIN relative to the SIN of the spectrum measured with t, = 5T1 are 1.25 (t, = 2.9T1), 1.44 (1.25T1), 1.25 (0.5T1), and0.91 (0.2T1).The relative values calculated from the spectra are 1.24,1.44,1.31, and 1.05, respectively. Finally, to test the use of eqn. (12) t o ensure quantitative results within a specified error limit, spectra were measured using the pulse conditions predicted t o give optimum sensitivity with maximum intensity errors of 2.5 and 20%. For comparison, spectra were measured using the same flip angle and number of scans but with t . = 7T1. Thus, the maximum intensity differences between resonances in corresponding spectra are predicted t o be 2.5% and 20%, with the intensity differences decreasing in the order HOD > CH30D > CH3C02D. The intensity differences found experimentally were: 2.4% (HOD), 0.3% (CH30D)and 0% (CH3C02D)(the-
oretical maximum error 2.5%);and 16%(HOD), 5% (CHnOD), and 0% (CHFOyD) (theoretical maximum error 20%). The differences in relative intensity within a spectrum will generally be considerably less than the maximum error used in eqn. (12) since the range of TI values generally will be limited, and thus, all resonances will be saturated to some extent. Summary
As in other forms of spectroscopy, sensitivity in P/lT NMR can be increased by signal averaging. However, the increase in sensitivity is somewhat more complicated than simply being proportional to f l due to the relatively slow rates a t which nuclear magnetization relaxes to equilibrium. Pulse conditions can he optimized for maximum sensitivitv enhancement. One consequ&ce, however, is that relative resonance intensities are generally not equal to relative numbers of nuclei. If quantitative information is of interest, compromise pulse conditions can he used which eive some sensitivitv enhancement while keeping error limb due to partial saturation a t a tolerable level. (1) F m m , T.C.. and B s k e r , E. D.,"Puk and Fwrier Tnrnaform NMR,"Acadrmie Prraa, New York, 1971. (2) Shsw, D.. "Fouriernmalorm NMRSpeetroampy,"Elsevia SeientificPubl. Co.. N n s
.".-,...".
" " . I ,
.*"C
(3) Smith, W. B., Bnd F'mh.T. W., J. CHEM. EDUC., 65, 70 (1976). (4) Marshsl1.A. G., and Comiaarow, M. B., J.CHeM.EDuc., 52,638 (1975) (5) Ernst, R. R., and Andemon, W. A.,Rm SC;. Imt..37.93 (1966). (6) Jonu, D. E., m d Sternlieht, H., J. Mog Res., 6,167 (1972). (7) Cwkson, D. J., and Smith, B. E,Anal. Chem, 54.2591 i19821. (8) Ref. Ill, p. 20.
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