topic/ in chemical injtrumentation
edited by FRANKA. SETTLE, JR. Virginia Lexington. Military VA 24450
The Fourier Transform in Chemistry Part 1. Nuclear Magnetic Resonance: Introduction Roy W. King and Kathryn R. Williams University of Florida, Gainesville, FL 3261 1 Previoua articles in this feature have described the fundamental concepts of the Fourier transform (FT) ( 1 3 ) and its application to infrared spectroscopy (IR) ( 4 4 . Although the FT method has been applied t o a number of instrumental techniques most chemists would prohahly list nuclear magnetie resonance spectroscopy along with IR spectroscopy as the principal heneficiaries of FT methodology. There have been same notable similarities in the historical development of FT-IR and FT-NMR. The interferometer, which is the central component of an FT-IR spectrometer, was invented almost a century ago. Its widespread application to infrared measurements is now possible because of the availability of microcomputers and fast Fourier transform (FFT) programs, which are necessary to make the interferogram intelligible to the user. In addition to the desirable energy-throughput characteristics of the interferometer, the principal advantage of FT-IR is the short time required to ohtain alarge number of interferograms. Averaging these interferograms improves the signdnoise ratio, thus achieving the multiples oduantage (5). The greatly increased sensitivity of the FT-interferometric methad has allowed the infrared spectroscopist to use a wide range of accessories, which, although available for many years, could not he used effectively with dispersive instruments. The pulse method of observing nuclear magnetic resonance has been known since 1950, but, like interferometric IR, pulse NMR was not a common tool of chemists until Fourier transform methods became readily available. As with FT-IR, one of the major henefits of the use of F T in NMR has been the increased sensitivity made possible by the multiplex advantage, along with the higher magnetic fields that have become available in the last 20 years. The NMR spectroscopist can now routinely study insensitive nuclei, such as '3C and 15N,whose natural-abundance signals are too weak to he easily observed with swept fields. This increased capability has hy itself led to the application of NMR to many interesting and complex chemical systems. However, the benefits of the F T method have been enhanced even more by the introduction of techniques using sequences of RF pulses of various amplitude and phase. Sequence methods hearing acronyms such as APT, DEPT, COSY, and NOESY now provide the chemist with the ability to determine the
number of hydrogens bonded to eachcarhon in a molecule, to unravel complex coupling patterns, and even to elucidate connections between bonded and nonbonded atoms. These pulse sequences represent an entirely new set of experiments that could not have been developed without the availability of the Fourier transform technique. It is the goal of this series of papers to provide the reader with an understanding and appreciation of these new tools of nuclear magnetic resonance spectroscopy. The f r s t two papers describe fundamental concepts of the pulse method and some of the special considerations for its use. The suceeedingpapers will present some of the simpler pulse sequence methods and will provide same practical examples of their appli-
Volume 66
cation t o problems determination.
in structure
Fundamental Concepts of Magnetization It is expected that the reader will have some familiarity with the interpretation of a simple proton NMR spectrum. Nevertheless, to be able to explain the pulse NMR process, the fundamental concepts of the behavior of nuclei in a magnetic field should be reviewed first. The reader who is interested in a more in-depth treatment is referred to books hv Farrar and Becker (7l and Slichter (8)o i t o a standard text (9, i0j on NMR. (Continued on page A214)
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A nucleus with nonzero spin, I,behaves as a small bar magnet, with a magnetic moment, p, given by: yh[I(I + I)]'" (1) p= 9, -" where h is Plamk's constant and y is the rnagnetogyric ratio, an inherent property of the nuclide. The observed value of the magnetic moment is the projection of p on the z axis, ,A,. According to the results of quantum mechanics, = rnnhl2z (2) where rnr is the magnetic quantum number. The allowed values of rnr are I , (I-I), . . .O, .. . (-I + 1). -I, giving rise to 21 + 1possible states for the nucleus. In the absence of a magnetic field the energy of the nucleus is independent of mr, hut the degeneracy is removed when an external field is applied. In a field of magnitude Be, whose direction defines then axis, the energies are given by
E
= -rSo = -rnnhBd2=
(3)
In this series of papers discussion will be limited to spin 112 nuclei, which give rise to only two states corresponding to rnr = +I12 and -112, also often called the a and @ spin states. The energy spacing between the two states is, therefore,
AE = yhBd2r
(4)
which corresponds in frequency to
fo = yBd2r (Hz)
(5) Ifthe magnetogyric ratio is positive (e.g., 'H
and W), then the +I12 state lies lower in energy, but the opposite is true for nuclei with negative y*s (e.g., mSi and I6N). The magnetogyricratios of some important spin ane-half nuclei are given in the table.
In an external magnetic field, the magnetic moment p lies at an angle 0 = cos-'(p,/lrlI with respect to the field, as shown in Figure 1.According to classical mechanics,which is still useful far the visualization of manv -~~~~ NMR concepts, the interaction of the spinning nucleus with the field causes the mag~
~~
Figwe 1. Acmding to clagtical mechanics, an individual maanetic mament orecarses abaut the axis 01 the applied magnetic field.M e the vecla is preCe6Slng BbOut the positive 1Wd axis, rrhich is the Iowa energ, aats it 7 is pos~ve.
-
netic moment toprecess ahout the magnetic field Bo.The frequency of this motion is called theLarmorfrequency, and isgiven by exactly the same expression as the quantum mechanical result (eq 5). The two energy states corresnond to orecession of u about the axis aligned with (Lower energy state rf y is positive) or opposed ta the appl~edfield The value of AE given by eq 4 IS much lesa than kT at room temperature. A aample containing a collection of spin 112 nuclei at equilibrium in the Bo field will have a small eicess population in the lower energy state,
Magnetogyrlc Ratlo8 of Some Common Spln-112 Nuclel and ~armiwFraquemles In a Flxed Magnetlc Fleld 017.0461 terla
as dictated by the Boltvnann distribution (Fig. 2a). As F i i e 2b shows, at equilibrium the individual mametic moment vectors are randomly distributedin two cones about the +and -2 axes. If 7 is positive, slightly more than half the nuclei will be aligned with the applied field. This excess spin population results in a net magnetization in the +n direction (Fig. 2c).
The RF Magnetlc Fleld In an NMR experiment an oscillating magnetic field. B,, is applied perpendicularIY to the static field Bo. This is achieved bv passage of a radiofrequency current through a coil whose axis is usually assigned as the x direction (Fig. 3a). The oscillating field along the x axis ia equivalent to the vector sum of two magnetic vectors of equal amplitude, counter-rotating in the xy plane (Fig. 3b). If BI is oscillating exactly at the Larmor frequencyof the nuclei being observed,then one of the component vectors of B, rotates precisely in step with the precessing nuclei; the vector that rotates in the opposite sense does not interact. Figure 4 again shows the magnetic moments of the Boltzmann excess population of spins. In order for a signal to he detected, there must be a component of the magnetization in the horizontal plane. At equilibrium there is no observed signal, because the vectors are uniformly distributed in a cone centered on the z axis. When the frequency of BI exactly matches the Larmor frequen-
Figure 4. Magnetlc moments of the Boikmann excesa spin population. (a)At equilibrium me vectws are unitormlv diskibuted about the dtrection oi the soiisld. (b) The nuclei bunch togsther and move in phaw h e n the appi~ed6, tmid metches ms uum frequency. cy, some of the nuclei are induced to flip their spins, causing anet absorption of energy from the BI field. At the same time the nuclear magnetic moments are forced to move toward one side of the cone and to precessas agroup following the in-step component of the B1 vector. The resulting r y
a
b
c
Figue 2. (a)Energy level diagsm for 2ntotal nuclei (positive y), showlng the excess population in the lower energy slate. (b) Schematic representalion showing a greater number of spins aligned with the magnetic field (lower energy state).(c)Excess spin population aligned with thefteld resuning in a net magnetization in the +z direction. The achrai excess is much smaller then shown (about 50 ppm in a 7-T field raUw man the 3 of 13 shown). A214
Journal of Chemical Education
+X
a
+X
b
Figure 3. (a) Fussage of RF cunant through a mil oriented along the x axis produces an oscillating B, field. which (b) can be expressed as the vectwsum of two counter-rotatingveaDrs of equal length.
component can bedetected by a receiver coil in the horizontal nlane. The transition bf nuclei between the lower and upper energy levels is analogous to absorption of UVIviaible photonaof the cor. rect energy by amolecular chromophore, except that the modes of emission of energy differ. In NMR the situation is complicated because of the very small Boltzmann excess of spins in the lower energy state. A B1 magnetic field aseillatinn a t the Larmor frequency induces both upward and downward spin transitions of a nucleus with equal probability. As long as there is a greater number of nuclei in the lower energy state, the rate of energy absorption exceeds the rate of emission, and a signal is observed. However, unless there is some means for the nuclei to return to the lower energy state, continued B, irradiation will cause the snin populations ;o become equal. In this s t a k of ~aturalionthere is nu net absorption of energy and no signal. In contrast to many other systems (e.g., electronically excited states), spontaneous emission occurs to a negligible extent. Energy can be effectively transferred from a nuclear spin to its environment only by interaction with a Lwal mametie field that is oscillatine" a t the Lar" mor frequency fa. To understand the origins of this field it is necessary to take a closer look at the microscopic environment of the nucleus,
Nuclear Relaxation A nucleus does not exist in isolation, but as part of a molecule, which, if it is in a nonviacous liquid, is perpetually tumbling a t random. This motion may be represented mathematically by a plot versus time of a geometric property of the molecule. The parameter of particular importance to NMR relaxation is the angle of an internudear veetar withthe Boaxis. Like any other pbysical process, the time dependence of molecular motion is related by the Fourier transform to a distribution of motional frequencies. The time domain behavior is often described in terms of the correlation time. r,, which may be considered as the average time over which a mctlecule maintains a certain geometric arrangement. For molecular tumbling 7, corresponds approximately to theaverage time for amolecule or functional group to turn through one radian and is of the order of 10-12-10-'0 s for a small molecule in a nonviscous liquid. Fourier analysis shows that motional frequencies span the range from zero to over 117, radls. For a 10-"-s correlation time this corresponds to a frequency range of zero to more than 1011 Hz, which means that there will be a frequency component of molecular motion a t the L m o r frequency of any nucleus in any foreseeable magnetic field. It is this malecular motion that, in combination with local magnetic fields, allows the nuclei to exchange energy with their surroundings (historically referred to as "the lattice") and return toward the equilibrium population. This process of spin-lattice (or longitudinal) relarotion is nominally fist-order. The rate constant is the reciprocal of TI, the spin-lattice relaxotion time, which is the parameter commonly used to specify the efficiencv of the nrocess. In diamagnetic systems the principal sourceof the required local magnetic field is the direct through-space interaction of two
nuclear magnetic dipoles. F i i e 5 shows such an interaction between a ISC nucleus and anroton that is near. but not necessarilv bondLd to, the carbon. The magnetic field strength that is induced at thecarbon nucleus by the proton magnetic moment depends on the angle, J., between the magnetic moment vector and the intemudear vector. The direction of the former is defied by the fixed magnetic field (cf. Fig. l ) , but, as the molecule tumbles, the internuclear vector reorients with remeet to Bo. and II.chanees. The valie of the "kduc;d fielz is - ~ avernee -~ ~zeru, au it baa nu effect un chemical shifla. However, the rapidly fluctuating magnetic fields do provide an important means of re-
-~--~
~~
~~~~
~~~~~~
Figure 5. Repesemation of lhe dipolar interaction showing WOalentations d the imernuclear vector wlth respect m lhe fixed magnetic field.m a n g l e 0 lo definad by me directionof60,but # changes as the molecule tumbles laxation called the dipolar mechanism. The magnitude of the associated TI depends on the sixth power of the internuclear distance
and on the inverse of the correlation time. In liquids dipolar relaxation produces typical proton or carbon TI values of about 10-2to 10's; a small molecule in a nonviscous medium will have a short correlation time and a long TI, but the TI values of a viscous,polymer sample will be very short. Another source of fluctuating magnetic fields, and consequent reduction in T,,is the presence of paramagnetic entities. An unpaired electronic spin, because of its large magnetic moment, is a much more effective relaxer than a nuclear spin. For this reason, oxygen must be carefully excluded in eaperiments sensitive to TI effects. On the other hand, the addition of smallamounts of paramagnetic compounds can reduce the inconveniently long TL's of isolated carbons. Chromium(II1) acetvlacetonate.. Cr(acach. . is commonly used for this purpose. There are also several other relaxation processes that may compete with the dipolar mechanism (7), but they are usually of less practical importance for spin-112 nuclei. The xy magnetization that gives rise to the NMR signal decays as TI effects cause the nuclei to lose their excess energy and return t o their equilibrium orientations about the z axis. There is also another class of relaxation processes that causes the loss of detectable magnetization (i.e., the grouping of the magnetic moment vectors that is depicted in Fig. 4b) without transfer of energy to the surroundings. I t is convenient to define an overall time constant far the loss of detectable magnetization, called T i , which is of great practical significance because it determines the spectral resolution. (Continued on page A216)
ments. In the CW operational mode the nuclei are successively brought into resonance by slowly varying either the radio frequency or the Bo magnetic field. The expression comes from the continuous application of the RF while the resonance is being obsewed. In pulse methods the nuclei in the entire frequency region are excited by a single burst of RF power (a pulse), and the resulting signal is measured after the RF is turned off. Except for extremely simple systems, Fourier transformation is required to convert the resulting signal into a usable form, and therefore pulse NMR is often called FT-NMR. Fourier analysis is also essential to understanding the effect of the pulse itself. A pulse of monochromatic RF with a rectangular envelope can be described in the freauencvdomain asa band of freouencies cenrered at the RF frpqneney. ~ i d u r e6 s h o w [he frequency-domain signal for two d~fferent pulse lengcha. The center valw, whreh is the monochromatic frequency produced by the pulse generator, is often called the speetrometer frequency, f,. The Heisenberg vrineide states that there is a minimum uncertaihy in the simultaneous specification
Spectral resolution is defined interma of the full width observed NMR line width (AfIfilz, a t half maximum intensity), which is equal to l/(rT;). Ti is called the effective spinspin, or effective transuerse, relaxation time. The word "effective" is included because usually the major contributor to the linewidth is lack of homogeneity of the Bo magnetic field. True spin-spin relaxation (Td arises from the same effects that determine Tx, except that TZrelaxation can be promoted by low-frequency motions as well as those in the range of fo. Other processes that do not affect TI, such as spin exchange by nearby nuclei and chemical exchange effects, can cause Tz to be less than TI, but the reverse can never occur. For liquids the two are usually equal. If TIis very short, then the resulting small value of Ti will produce line broadening.
.
The RF Pulse Until the mid-1970's most commercial NMR's were eontimow-wave (CW) instru-
1-fs
(H3
Figure 6. me Fourier transform of a rectangular RF pulse f, in amplitude A is lha sinc function.
(a) b = 25ps: (b) b =
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Journal of Chemical Education
low. The value of A is 2 X
10'
duration wi!h frequency f, and
for each plot
of bath the energy (and hence the frequency) of a system and the duration of the measurement. As Figure 6 shows, this means that, as the pulse length decreases, the width of the frequency hand must increase. For a typical pulse length of 10 ps, the flat central portion of the frequency hand, where the amplitude is within 1%of the peak value, is about 16000 Hz wide. For a 7.05-T field V, = 300 MHz for protons, 75 MHz for '3C) this region easily spans the range of chemical shifts of the comm#mly ohserved nurlei (15 ppm or 4500 Hz for prownu.. 2U0 .. oom or 1W00 Hz for "C). However, as Figure 6 also shows, because the irradiation is soread over a wider freauencv . . hand, a shorter pulse results in a lower power at the center frequenry. An RF EenerauBr for pulw NMH should be eapahle of delivrring a short, high-power puke with a precisely delined envelope, and the pwhe should he desipned for efficient conversion of RF power into the B, field. Before discussing how the nuclei respond in a pulse experiment, it is useful to reconsider the behavior of the spins in the fixed magnetic field, Bo. Figure 7a again shows schematically the excess population of spins aligned with Bo (+z direction) and precessing about it at the Larmor frequency, fo. Figure 7h shows an alternate representation of the situation, in which the axes are allowed to rotate a t the Larmor frequency. The soins now amear to he fixed in orientation with respect tu the r' and f axes. In either ease the resultant of the population excess e the magnetiration vector. M, which a t equilibrium points along the +z axis. Now consider the application of an RF field whose frequency, f,, is the same as the
..
Figwe 7. Alternate cwrdinate axes for a nuclear spin system in a b magnetic field oriented along the raxis. la) . . Whenthe xand "axes are fixed. the nuclear momsnll precess aba* the r axls a1 the Larmor frequency (0) Wnen thecoordmate system rotates about the z axis at the amw frequency. the momems appear to be stationary. In either case the rewltam is a bulk magnetization vector pointing in the r direction. rotation frequency of the frame. The applied RF is still composed of two counterFig. 3), but now the rotating vectors (6. component vector that rotates in the same direction as the frame appears to he stationary. As far as the nuclei are concerned, the other component may he ignored because its frequency (twice f.) is totally outside the range of chemical shifts. In the rotating frame the stationary vector behaves as a nonoscillating magnetic field, BI, perpendicular to Bo. Assuming for the time being that f, corresponds exactly to the Larmor frwuencv. the nuclei tend to orecess about the'^, fiild (Fig. 8). Becausehe RF is applied for a very short time period, the effect is only a partial revolution of the net magnetization vector, which is said ta he "tipped"
~'olume66
Figure 8. me efisct of an RF pulse. (a) Viewed in the rotating hame, the nuclei tendto preces about the 6, magnetic field. (b)Afierthe 6, field has been removed, the magnetization vector makes an angle a with the zaxis.
toward the -y' axis (assuming that the B, field points in the +x' direction when viewed from the origin).' The tip angle, u, is given (in radians) by a = yBltp
(6)
The RF pulse is usually designated by the value of a that it oroduces and the axis to . 90. which it is aonlied. The term (.~ 1 2..) or refers to a pulse directed along the +x' axis. (The prime is usually omi!wd from the juhscript.) At the end of a 90; pulse the magnetization vector points along the -y' axis. The length of a typical SO0 pulse is about s. For protons this requires a BI field of about 6 X lo-' T , compared to BI fields of the order of T for CW work. Bn fields are much larger and range from 1.4 tb 14 T , ~~~~~~
~~
~~
~
.~ .~ ~
~
~
(Continued on page ,4218)
Number 9
September 1989
A217
corresponding to proton frequencies of 60600 MHz.
Because the pulse produces a wide hand of freauenciea of almost eaual amolitude. the nuclei throughout the entire &mica1 rhift range are all effectively tipped by the same anglekAsfar an the excitation pulrie is concerned, each chemically shifted nucleus may he considered to belong to a frame rotating at the corresponding L m o r frequency. However, to understand what happens after excitation it is necessary to assign the frame to a single frequency, which is chosen to be the spectrometer frequency, fa, the value a t the center of the frequency hand.
The Free Induction Decay Figure9 shows the behavior of the m w e tization following excitation by a YO, pulse. The Bo field in still present, and the nuclei continue to precess-about it. Focusing for the moment an the nuclei whose Larmor frequency corresponds exactly to the frequency of the rotating frame, this means that, in the absence of relaxation effects, the magnetization vedor would remain directed 9a). However, TI and along the -)/axis (Fi. T.;prorersea hothaet toredure the-)'cornponent of the magnetization. Bo inhomopeneities, the principal source of T i effects, cause groups i f nudei in differentpmts of the sample to experience slightly different Local Bo fields. Figure 9b shows only two of the many such microscopic sections of the sample (called isochromats because the field is the same within the section). One group is shown precessing slightly ahead of the frame, and the other group is lagging hehind. The net effect is a decrease in the -y'eomponent of the magnetization. At the same time TI relaxation processes cause a gradual return of the magnetization towards the z axis (Fig. 9c), further decreasing the component along -y'. Because field inhomogeneities usually cause T; to be leas than TI, an intermediate situation often o c m as in Figure 9d, where the -y' component of the magnetization decays to zero before the spin population can achieve the Boltzmann distribution. At some later time (after at .~~~~ least 5 times T I ) the magnetization has again returned r u i t s equrlihrrum value (Fig. 9e). Returning now to the laboratory frame, the -y'component corresponds to a magnetization vector rotating in the xy plane. Just as in the CW mode of oneration, a receiver coil (usually the same coa used t i excite the nuclei) is situated in the xy plane, where it senses the rotating magnetization. As the -y' component decreases, the oscillating voltage from the coil decays, and it reaches Lero when the condition of the spins rorresponds to the situation depicted in Figure 9d. The record of the receiver voltme in the time domain in called the free induction decay (FID), because the nuclei are allowed to precess "freely" in the absence of the B1 field. The FID is the Fourier partner of the NMR frequency spectrum. The FID's and transformed spectra for some simple proton systems are shown in Figure 10. Figure 10a corresponds to a system containing only one type of proton (HzO) observed with the spectrometer fre~~
Journal of Chemical Education
Figure 9. Bahaviw of nuclei aner a 90, pulse. The frame is rotating at the spemmmter frequency. 1,. which is assumed to be equal to the L m r h e quency. (a1 Immediately aner the pulse the tipped mag~tizationvector is fixed In the rotating frame. (b) T, effects (mainly 6 inhomogeneity) cause locaiiv. divement Larmor freouencles. Onlv two divfngent lsochromats are shown hers. Ic) At me ssme tune T,relaxation istaking me magna zatm back toward me zaxis. (d) Ti effeclshave resuned in complete loss of magnetization in the x'y' plane, but the nuclei have not returned to equliibrium. (el Camplete restoration of equilibrium r axis magnetl?allon. e
ing exponential with time constant Ti.The corresponding frequency spectrum shows thatthe Larmor frequency is slightly shifted from the spectrometer frequency. In a sample (ethyl benzoate) that contains several protons with different chemical shifts and splitting effects, the various rotation frequencies produce a romplicated interference pattern (Fie.10c) that would be totally unintelligible without the aid of cbe Fourier
transform.
The oscillatory character of the FID, especially the one shown in Figure lob, may remind the CW NMR user of the "ringing" seen on the tails of rapidly swept peaks. The behavior of the nuclei is indeed similar in both situations. In the CW mode the mag netic vectors are caused to rotate as a group when f, or Bo, whichever is swept, meeta the resonance condition. After resonance is past the nuclei continue to precess "freely", aa in the puke experiment. The difference between the nuclear precession frequency and the frequency of the applied RF is observed as the ringing pattern. The amplitude of the oscillations decays with the same Ti time constant discussed above. The ringing tail and the FID of a aingle resonance do not appear exactly the same, however. The oacillations of the FID have a constant frequency, whereas the frequency of the ringing tail oscillations is proportional to the steadily increasing difference between the ~ a r m o iand sne&meter freauencies. A rineine tail as &solaved on a chart recorder shows a decreasing period with increasing time, and it usually appears shorter than a corresponding FLD because of the limited frequency response of the record& and its associated circuitry. Thin paper has covered the basics of the interaction of spin-%nuclei with static and RF maemtic fields. Practical aoolications of p u l s ; F T - ~ will ~ ~ be dbcu&d in subsequent articles in this series.
-~~
~~
~
. .
Literature Cited 1. Glssser, L.J. Ckm. Rduc. 1987,M,A128-*Z53. 2. Glaawr.L.J. Chm. Rdue. 1987.M.A26&A288.
1. Parmr,T. C.; Bseksr, E. D.R'bo and F m e r TmwformNMR; Aeadeadic:New York, 1971.
8.
SLichter, C. P. Rimipk8 ofMmetir R e r o ~ n r e 2od . ed.: Cardcna. M.: Fuide, P.: Queiuer, H.J..Edr.; springer series in Solid-sta* sciences: springer: N w York 1978:Vol1, Chaptvs 1and 2.
9. W i , R. K. Nuclear Magnetic R e s o ~ n c oSprfm$copy; Longmao: l k i o u , Eassr. U.K.,1988.
Fiwe 10.FID's ard transfomwrd hequency spectra.(a) Singb~mtonsystsm (HD conmining approx. 50 ppm Mn2+ to bmadsn the line fa better prepemation) with Lsrmw frequency equal lo f,. (b) Single-pmton system with~hequencysligMly&then fs.(c)Compound(mibenroateat 300 MHr) with prmns in several envlronmentll.
lo. Bovsy, F. A Nuclear Mngogtic Rcao-ce Smtmscopy, 2nd ed.;Aead.mie: snn Diego, 1888 A. E. MalernNdU1 Techniques for Chemis-
11. -me,
tg.RearamhPergamon:N w York, 1987.
12. FuLuahirnP E.: Raeds., B. W. EIWn'rn"td film NMR;Ad&rm-Welley: Reading. MA. 1981. G.: Wokaun. A. Rincioler
13. Emst. R R.:Bodenhsmn.
'
quency, f,, exactly equal to the Larmor frequency,fo. In the data acquisition process f. is subtracted electronically from the obsewed signal prior to digitization. The subtraction has the effect of placing the detector in the r o t a t i i frame, so that in this situation the output is nonoacillatory. The FID shows a simple exponential decay with time constant equal to Ti.In F i i 10h the
same single proton sample is observed, hut this time the spectrometer frequency is slightly greater than fo. In the frame rotating at f,, the magnetization vectar is not stationary. The vector's rotation frequency corresponds to the difference between fo and the spectrometer frequency, so the FID consists of sinuaoidal oscillations of constant frequency whose envelope is a deereas-
Volume 66
Rltdirection of the precession of nuclei about the R,- manetic field ard of the tioolno of the net ma~nairationM a by the RF field (precession aba* 6,) is mmlstent with nnern convernim (13). The final resuns are unanected if boa directions are reversad, 86 in m e texb. The actusl trajeomry of a nuclear spin vectu when ~xcnedby a pulse that is not exactly at the Lemw heqvency is q u k cmpiicated, but f o r b namly the perturbations c d by oftresonance excltatlon cancel out almosl entirely in practice (11, pp 6+72; 12. pp 5W6Q 13, pp 99-1 11).
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September 1989
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