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Chapter 17 Continuous wave and rapid scan correlation NMR
362
Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.
Chapter 17 C O N T I N U O U S W A V E A N D RAPID SCAN CORRELATION NMR P.S. Belton
Institute of Food Research, Norwich Laboratory, Norwich Research Park, Colney, Norwich, NR4 7UA, UK.
1 INTRODUCTION Historically, time domain and frequency domain NMR diverged very early in the history of NMR, they were only reunited in the 1970's by the development of Fourier transform methods. Continuous wave (CW) methods were conceptually analogous to other spectroscopic methods in use, such as dispersive infrared or ultraviolet spectroscopy, and in the early stages of development of NMR were the only practical means of obtaining spectra. With the development of Fourier transform instruments they have largely fallen out of favour except for some low cost applications in low field magnets. However for many applications outside the research laboratory, there is a need to trade off cost against performance. In addition high fields may not be beneficial in an industrial setting where stray field problems can lead to difficulties. Under some circumstances, therefore, there can be applications for continuous wave NMR. In order to see what thesc may be it is nccessar3.' to undcrstand the constraints on the technique, but also to examine where it might have advantages. In the remainder of the chapter this is done, first by outlining the principles of CW NMR and then considering the routes to optimum signal to noise ratio and spcctral acquisition time and finally by outlining some areas where application of the techniques may be useful. 2 THE PRINCIPLES OF CW NMR A good account of the principles and practice of CW NMR is given in the books by Martin. Delpuech and Martin (198(I) and Abragam (1978). This section draws heavily on these. Conceptually a non-multiplexing approach to spectroscopy will involve the sequential measurements of intensities associated with frequency elements. In NMR the resonance
363 frequency is linearly related to the magnetic field strength so that it is practically easier to observe at one frequency and linearly vau, the field strength. In this case resonance lines with different chemical shifts come sequentially in to resonance. If the lineshape is to be undistoned by saturation or scan rate effects, two criteria must be met. These are : 1) The scanning process must be adiabatic, that is the nuclear magnetisation maintains a constant angle with respect to the effective magnetic field (Abragam, 1980). 2) The radio frequency strength must be sufficiently small so that saturation does not occur. The definition of criterion 1 is given by Abragam (1980) as dBo dt
<
< yB~ 2
(l)
Where Bo is the main magnetic field, t is time, g is the magnetogyric ratio and B 1 is the radiofrequency field. The criterion for condition 2 may be deduced (Martin, Delpeuch and Martin, 1980) from the Bloch equations for adiabalic passage. For the absorption mode signal these give U=
"Y B~ A (o T2 2 Mo 1 + T22 A ( 0 2 +
7"2 T~ T2
(2)
and 7" B~ T2 Mo
v= 1 +
T2 2 A
(_z)+ 7" 2 B, T, T2
(3)
A ~..~= (o.~-o~o) e~o is the resonance fiequcncy, that is the frequency of the resonance maximum, o~ is the excitation frequency, T] and T 2 are the spin lattice and transverse relaxation times, Mo is the equilibrium magnetisation. u and v are thc dispersion and absorption mode signal strengths. For observation of a Lorentzian lineshape with a half width v,5 given by / V1/2 =
~rT~
(4)
the condilion 7 B~ T~ T,. < < 1
(5)
364 must apply. For the maximum observable intensity in the absorption mode it can be readily shown that the radiofrequency field B l should be a I~
r(Y,T2)1/2
=
(6)
with an intensity I = 0.5 Mo (T= /
T1)
(7)
However such a field strength would result in a broadening of the resonance line so that in practice a less than nmximum value must be used if the narrowest possible lines are to be observed. In the dispersion mode the signal intensity grows with B 1 to a maximum value of 0.5 Mo but with an increase in the width of the dispersion. The combination of Equations 6 and 1 suggest that for maxinmm intensiU' under adiabatic conditions the criterion must be that dBo
<
1
<
dt
7 T, T2
(8)
If the scan rate is expressed in Hertz / second this becomes dv dt
<
1
<
2n" T, T~_
(9)
Table I lists some results based on the right hand side of Equation 9. Table 1 ' Scan times under a variety of conditions for continuous wave NMR when the condition in Equation 9 is replaced by an equality. In practice, therefore, scan times would have to bc considerably increased to meet the inequalib' in Equation 9. Sweep Width (Hz) 600 3000 600 600 600
T2(s) 1.59 1.59 1.00 O. 10 0.032
Half Width (Hz) 0.20 0.20 ().32 3.20 11).()
T 1(s) 3.0 3.0 1.0 1.0 0.5
Scan Rate (HzJs) 0.067 0.067 0.32 3.20 10.00
Scan Time (s) 9000 45000 1887 187 30
365 Since the tcrm l/(2rt TIT 2 ) in Equation 9 is larger than the required rate the actual scan times must be much longer than this. It is immediately apparent that in order to obtain true lineshapes, very long scan times must be used even for a sweep width of 600 Hz i.e for a proton spectrum in a 60Ml-Iz spectrometer. For a 300 MHz proton spectrum impractically long times are required. For the low field case the scan times become much more attractive when line width increases and T l decreases. It is clear from the Table that for high qualily, high resolution spectra of the whole spectral range Fourier transform methods are to be prefcrred. However. when relaxation times become shorter the advantage of Fourier transform methods is not so great. In many practical situations of quality control only one or two components are of interest and may have short relaxalion times. Under these circumstances the CW method may prove more attractive. 3 RAPID SCAN METHODS It is clear from Table 1 that a major disadvantage of the CW approach using the criterion set out in Equation 9 is the time taken to obtain a single scan spectrum. This precludes signal averaging and makes wide spectral scans veD' long. In order to increase the scan rate it is necessaD, to relax the adiabatic criterion given in Equation I. The effect of this is well known. It results in the formation of tinging or beats following the peak maximum during the scan. An illustration is given in Figure IA. Such a spectn~m may be usable, and indeed, in many of the older publications which make use of CW spectra the effects of rapid passage are apparent (see for example Abraham and Loftus. 1978) but do not preclude spectral interpretalion. However, it is clear that if spectra are to be interpreted by eye scan rates must be such that no overlap between the ringing and adjacen! lines occurs. This will limit scan rates. In order to make general use of rapid passage methods it is necessary to disentangle the beats from the underlying line shape. The beats arise because violation of the adiabatic passage criterion results in residual transverse magnetisation after the field has swept through the peak maximum. For a pure absorption spectrum this magnetisation oscillates with a frequency Cos (bt2/2)" where b is the frequency sweep rate in radians per second and t is the time elapsed after passing through the peak maximum. The oscillating magnetisalion decays with a rate determined by the transverse relaxation rate. The shape of the filnction with an exponential transverse relaxation term is shown in Figure 2. In general, the spectrum will contain real and imaginaD' parts in which case the beats are described by a term of the form exp (-ibl2/2). Two approaches are possible for the disentanglement process. The first is to follow the original suggestion of Dadok and Sprecher (1974) and cross correlate the observed spectrum with the spectrum, obtained under identical conditions, of a single reference line. The alternative is to deconvolute the ringing by the use of reverse Fourier transform. In practice the cross correlation process amounts to complex con.jugatc multiplicalion in the Fourier domain whilst deconvolution is a process of s~mplc multiplication in the Fourier domain.
366 Inverse Fourier transformation of the observed spectrum under rapid passage conditions can be shown to yield the following (Gupta. Ferretti and Becker. 1974) time domain result. F(T) = h(T) exp (ibT~/ 2)
(10)
T is the variable in the time domain. It is important to realise that although the observed spectrum is time dependent it is essentially in the frequency domain and its time scale t should not be confused with the timescale T in the time domain, h(T) is simply the free induction decay of the spins and represents the lineshapes unaffected by ringing. A simple division therefore yields the required free induction decay, which, on for~vard Fourier transformation, will give the frequency domain spectrum. In Figure 1B is shown the result of this process when applied to Figure 1A.
r
m
B
a"--~
--:~
-----6 Fig. 1 9 A spcctntm of 5% ethyl benzene. IA acquired under rapid scan conditions. IB after dcconvolution. Reproduced with permission from Bariat. Bclton and Goodfellow. 1993a.
367
I/it,v ........
Fig. 2 9 Simulated lincshapc undcr conditions of rapid scan with an exponential relaxation fi~nction. If a rcfcrencc spcctmm is uscd the proccss is to reverse Fourier transform both the reference spectrum and the obscn'cd spectrum and mulliply the complex conjugate of the transformed rcfcrence spectrum with the transformed observed spectrum. The result obtained on forward Fouricr transformation is a frequency, spectrum in which the line widths are the sum of lhe reference line widths with the true line width. In practice, cross correlation does not seem to have any advantages, and, although it has lent its name to the technique, does not seem to be preferable to the direct Fourier transformation method using the calculated fi~nction. A summary of the steps involved in the direct Fourier transformation method is given in Figure .3. 3.1 Practical considerations Combination of Equations l and 6 suggest that if the adiabatic criterion is ignored then dBo > yB~ dt
(11)
however saturation mus! still be avoided so that the criterion given in Equation 6 still applies and thcrcfore
368 dBo dt
>
1
YT, T:
(12) OBSERVED SPECTRUM
1
I
REVERSE FT
DIVIDE BY exp(-ibT2/2)
l APODISATION I
I
FORWARD FT I
Ill
I II
I
1 PHASE ADJUSTMENT
I
I
Fig. 3 9Block diagram of the process of obtaining an undistorted spectrum from a rapid scan spectrum The upper limit of the sweep rate is in principle, therefore arbitrary. In practice electronic considerations will represent an absolute upper limit to sweep rates, but a likely upper limit is also to be found in digitisation problems. As sweep rate is increased, the maximum sampling rate required is increased. Applying the Nyquist criterion that at least two points must be sampled each cycle, the maximum digitisation frequency required is bt/2n. Since the intensity of the ringing will decay by transverse relaxation, it may be assumed to have reached zero intensity bv 3T2" where T2* is the effective exponential transverse relaxation time.
369 Hence the sampling frequency in Hcnz required (H) is given by H
m
3bT; 2z
(13)
A typical upper limit might be for a scan rate of 10 kHz/s and a linewidth of 0.2 Hz. Under these conditions the maximum digitisation frequency required is 47.7 kHz. This can be readily achieved. With a line width of 10 Hz and the same scan rate the required maximum rate is 960 Hz. In order to properly characterise the spectrum it will be necessary to accumulate data for a time at least 3T2" after the last peak maximum. For an unknown spectrum it must be assumed that peaks can occur throughout the spectral frequency range. Therefore, there will be an additional time required, in excess of that required to scan the spectrum, in order to allow the decay of the ringing of the peaks that occur near the end of the spectral range. For a peak width at half height of 0.2 Hz this corresponds to a time of about 4.85 S. If the scan rate is 10 kHz/s and the minimum sampling frequency is used this will rcquirc accumulalion of 229k of data points in addition to those used in scanning the spectral range of interest. This number of points is achievable, but is extravagant for a simple one dimensional spectrum. In addition, for rapid scan rates, most of the acquisition lime is spent in acquiring the transverse decay which is independent of scan rale or spectral width. A reduction of the scan rate to 200 Hz/s vr require lhc addition of only 4580 additional data points and for a 600 Hz spectral width would require a total acquisition time of 10.8 seconds. This is a very modest increase in time for a major saving of memory and suggests that more modest scan rates may be of considerable value despite the apparent attractions of high scan rates. A usefi~l way of considering the practical exploitation of rapid scan methods is to compare with direct Fourier transform methods using pulse excitation. In its simplest form the pulse method excites the whole spectrum. As a result there can be problems of dynamic range when a ver)' large signal, typically soh,ent, coexists with a small signal. These problems ma.v be overcome by the use of selective excitation methods but these require high levels of sophistication in hardware and software. By contrast CW and rapid scan methods require only that the region of interest of the spectrum be scanned. No additional sophistication is required. An analysis of the relative signal to noise ratios of pulse and rapid scan methods has been carried out (Gupta. Fcrrelli and Becker, 1974). For a situation in which the total time available is fixed and conditions are optimised for each type of experiment the relative signal to noise ratios are expressed as follows 9 Sp
[ (F/b' + 3T')(! - E~)(I + Ek)] ';2
(14)
The subscripts R and P refer to Rapid Scan and Pulsed methods respectively, S is the signal to noise ratio. F is the sweep width in Hertz.
370 b' = b / 2 z
(15)
Ep = exp (-3 T; / T,)
(16)
Ea = exp(-(F/b' + 3 T~) / T,)
(17)
Care should ahvays be taken with comparisons of signal to noise ratios from theoretical considerations as they represent ideal cases. However, Equation 14 can be used to give indications of trends. It is clear that when F/b' is small compared to T2* the scanning experiment becomes comparable to the pulsed experiment. In the limit of infinitely fast scan they become equivalent. Indeed in this limit the time will be spent acquiring what is effectively a free induction decay convoluted with the tinging function. This has all the disadvantages of rapid digitisation with none of the advantages of the direct pulsed method and therefore a straightforward increase in scan rate has little to offer. If, however, the sweep width is kept small and a modest scan rate employed the ratio F/b' is reduced and the advantages of pulsed methods are less apparent. Typically for T2* = 0.155 and T 1 = Is and a scan rate of 200 Hz/s across a sweep of 200 Hz the ratio from Equation 14 is close to unity. This situation is one that might well be encountered in the analysis of industrial fluids where panicles and dissolved macromolecules reduce T2* and where the requirement is to monitor the intensi b' of an identified spectral peak for analytical purposes. Undcr these circumstances the intensity of the major components may well be a problem and would rcquirc elimination by a multipulse method if the pulsed Fourier transform tcchniquc were used. In the next section examples of this type of application are given. 4 APPLICATIONS OF RAPID SCAN C O R R E L A T I O N N M R
It has been stressed in this chapter that one of the major advantages of the use of the rapid scan method is when it is used to scan limited spectral regions. Figure 4 illustrates this when applied to a sample of vinegar. The main components of this material are water and acetic acid. A 6() MHz proton spectrum is illustrated. In the lower spectrum the fidl range is covcrcd and only peaks from water and the methyl group of the acid are observcd. The rcst of the spccln~m, except for spinning side bands of the water, is lost because of insufficient signal for digitisation. When a smaller region is scanned (upper spectrum), at a higher gain level small features become apparent. A critical featurc in any tcchniquc that employs signal averaging is that co-addition of signals results in an improvement in signal to noise ratios. This requires that the noise co-adds to a zero sum and that the signal remains constant. In the analysis of ethanol content of alcoholic drinks using proton NMR the intensity of the water can represent a problem when the ethanol content is low. However, by confining the region scanned to the methyl triplet which is most remote from the water peak, good linear calibrations for ethanol content may be obtained.
371 Ethanol contents as low as ().()095'Y,, in water were quantitatively measured. Other quantitative applications include glucose in water and levels of unsaturation in oils (Barjat. Belton and Goodfeliow. 1993a. b). Figure 5 shows the effects of accumulating scans of proton spectra on a spectrometer with no field drift correction (Barjat. Belton and Goodfellow, 1993a). Clearly the expected gain in signal to noise is not observed. However, as shown in Figure 6 when field drift effects arc eliminated the expected signal to noise improvement is observed.
A
B
Fig. 4 : Partial spectrum of vincgar showing the effects of dynamic range. Peak A is water, peak B is from Ihc methyl protons of acetic acid. The upper spectrum is from a reduced ~vidlh spectrum obtained at a higher amplifier gain.
372
50
E! o
40
o3
E!
El
-,--4
o
30
o
E!
t~
20
10
~ 0
n I 1
i
I 2
L
I 3
,
1 4
,
, 5
, 6
Square root of n u m b e r of scans
Fig. 5 9 Variation of signal to noise ratio on coaddition of signals obtained without field correction.
5 CONCLUSIONS If continuous wave NMR methods are to be used they are best used in the rapid scan mode. Even in this mode the advantages of the technique over pulsed Fourier transform methods are very limited. When wide spectral ranges are to be examined the pulsed method is clearly superior and continuous wave methods do not permit the wide variety of multidimensional experiments available on pulsed Fourier transform instruments. However, for limited applications where the required sweep width is small, the rapid scan method has a small or negligible signal to noise disadvantage and offers the potential advantages of selective excitation of spectral regions of interest and low cost.
373 200
0
op.-g
r o0 0
o
I00
ra
-,-.4
0 0
2 4 6 Square root of number of scans
8
Fig. 6 9As for Figure 5 but with field correction. The straight line is a linear regression fit to the data.
6 REFERENCES Abragam, A., 1978. The Principles of Nuclear Magnetism, 39-56, Oxford University Press, Oxford. Abraham, R.J. and Loftus, P., 1978. Proton and Carbon-13 NMR Spectroscopy, 34-39, Heyden, London. Barjat, H., Belton, P.S. and Goodfellow, B.J., 1993a. The use of Rapid Scan Correlation NMR as a Quantitative Analytical Method. Analyst, 18, 73-77. Barjat H., Belton, P.S. and Goodfellow, B.J., 1993b. Rapid Scan Correlation NMR Spectroscopy for Food Analysis. Food Chem., 48, 307-312. Dadok, J. and Sprecher, R.F., 1974. Correlation NMR Spectroscopy. d. Magn. Reson., 13,243-248. Gupta, R.K., Ferretti, J.A. and Becker, E.D., 1974. Rapid Scan Fourier Transform NMR Spectroscopy. d. Magn. Reson., 13,275-290. Martin, M.L., Delpuech, J-J. and Martin, G.J., 1980. Practical NMR Spectroscopy Heyden and Son, London.
Signal Treatment and Signal Analysis in NMR Ed. by D.N. Rutledge 9 1996 Elsevier Science B.V. All rights reserved.
Chapter 17 C O N T I N U O U S W A V E A N D RAPID SCAN CORRELATION NMR P.S. Belton
Institute of Food Research, Norwich Laboratory, Norwich Research Park, Colney, Norwich, NR4 7UA, UK.
1 INTRODUCTION Historically, time domain and frequency domain NMR diverged very early in the history of NMR, they were only reunited in the 1970's by the development of Fourier transform methods. Continuous wave (CW) methods were conceptually analogous to other spectroscopic methods in use, such as dispersive infrared or ultraviolet spectroscopy, and in the early stages of development of NMR were the only practical means of obtaining spectra. With the development of Fourier transform instruments they have largely fallen out of favour except for some low cost applications in low field magnets. However for many applications outside the research laboratory, there is a need to trade off cost against performance. In addition high fields may not be beneficial in an industrial setting where stray field problems can lead to difficulties. Under some circumstances, therefore, there can be applications for continuous wave NMR. In order to see what thesc may be it is nccessar3.' to undcrstand the constraints on the technique, but also to examine where it might have advantages. In the remainder of the chapter this is done, first by outlining the principles of CW NMR and then considering the routes to optimum signal to noise ratio and spcctral acquisition time and finally by outlining some areas where application of the techniques may be useful. 2 THE PRINCIPLES OF CW NMR A good account of the principles and practice of CW NMR is given in the books by Martin. Delpuech and Martin (198(I) and Abragam (1978). This section draws heavily on these. Conceptually a non-multiplexing approach to spectroscopy will involve the sequential measurements of intensities associated with frequency elements. In NMR the resonance
363 frequency is linearly related to the magnetic field strength so that it is practically easier to observe at one frequency and linearly vau, the field strength. In this case resonance lines with different chemical shifts come sequentially in to resonance. If the lineshape is to be undistoned by saturation or scan rate effects, two criteria must be met. These are : 1) The scanning process must be adiabatic, that is the nuclear magnetisation maintains a constant angle with respect to the effective magnetic field (Abragam, 1980). 2) The radio frequency strength must be sufficiently small so that saturation does not occur. The definition of criterion 1 is given by Abragam (1980) as dBo dt
<
< yB~ 2
(l)
Where Bo is the main magnetic field, t is time, g is the magnetogyric ratio and B 1 is the radiofrequency field. The criterion for condition 2 may be deduced (Martin, Delpeuch and Martin, 1980) from the Bloch equations for adiabalic passage. For the absorption mode signal these give U=
"Y B~ A (o T2 2 Mo 1 + T22 A ( 0 2 +
7"2 T~ T2
(2)
and 7" B~ T2 Mo
v= 1 +
T2 2 A
(_z)+ 7" 2 B, T, T2
(3)
A ~..~= (o.~-o~o) e~o is the resonance fiequcncy, that is the frequency of the resonance maximum, o~ is the excitation frequency, T] and T 2 are the spin lattice and transverse relaxation times, Mo is the equilibrium magnetisation. u and v are thc dispersion and absorption mode signal strengths. For observation of a Lorentzian lineshape with a half width v,5 given by / V1/2 =
~rT~
(4)
the condilion 7 B~ T~ T,. < < 1
(5)
364 must apply. For the maximum observable intensity in the absorption mode it can be readily shown that the radiofrequency field B l should be a I~
r(Y,T2)1/2
=
(6)
with an intensity I = 0.5 Mo (T= /
T1)
(7)
However such a field strength would result in a broadening of the resonance line so that in practice a less than nmximum value must be used if the narrowest possible lines are to be observed. In the dispersion mode the signal intensity grows with B 1 to a maximum value of 0.5 Mo but with an increase in the width of the dispersion. The combination of Equations 6 and 1 suggest that for maxinmm intensiU' under adiabatic conditions the criterion must be that dBo
<
1
<
dt
7 T, T2
(8)
If the scan rate is expressed in Hertz / second this becomes dv dt
<
1
<
2n" T, T~_
(9)
Table I lists some results based on the right hand side of Equation 9. Table 1 ' Scan times under a variety of conditions for continuous wave NMR when the condition in Equation 9 is replaced by an equality. In practice, therefore, scan times would have to bc considerably increased to meet the inequalib' in Equation 9. Sweep Width (Hz) 600 3000 600 600 600
T2(s) 1.59 1.59 1.00 O. 10 0.032
Half Width (Hz) 0.20 0.20 ().32 3.20 11).()
T 1(s) 3.0 3.0 1.0 1.0 0.5
Scan Rate (HzJs) 0.067 0.067 0.32 3.20 10.00
Scan Time (s) 9000 45000 1887 187 30
365 Since the tcrm l/(2rt TIT 2 ) in Equation 9 is larger than the required rate the actual scan times must be much longer than this. It is immediately apparent that in order to obtain true lineshapes, very long scan times must be used even for a sweep width of 600 Hz i.e for a proton spectrum in a 60Ml-Iz spectrometer. For a 300 MHz proton spectrum impractically long times are required. For the low field case the scan times become much more attractive when line width increases and T l decreases. It is clear from the Table that for high qualily, high resolution spectra of the whole spectral range Fourier transform methods are to be prefcrred. However. when relaxation times become shorter the advantage of Fourier transform methods is not so great. In many practical situations of quality control only one or two components are of interest and may have short relaxalion times. Under these circumstances the CW method may prove more attractive. 3 RAPID SCAN METHODS It is clear from Table 1 that a major disadvantage of the CW approach using the criterion set out in Equation 9 is the time taken to obtain a single scan spectrum. This precludes signal averaging and makes wide spectral scans veD' long. In order to increase the scan rate it is necessaD, to relax the adiabatic criterion given in Equation I. The effect of this is well known. It results in the formation of tinging or beats following the peak maximum during the scan. An illustration is given in Figure IA. Such a spectn~m may be usable, and indeed, in many of the older publications which make use of CW spectra the effects of rapid passage are apparent (see for example Abraham and Loftus. 1978) but do not preclude spectral interpretalion. However, it is clear that if spectra are to be interpreted by eye scan rates must be such that no overlap between the ringing and adjacen! lines occurs. This will limit scan rates. In order to make general use of rapid passage methods it is necessary to disentangle the beats from the underlying line shape. The beats arise because violation of the adiabatic passage criterion results in residual transverse magnetisation after the field has swept through the peak maximum. For a pure absorption spectrum this magnetisation oscillates with a frequency Cos (bt2/2)" where b is the frequency sweep rate in radians per second and t is the time elapsed after passing through the peak maximum. The oscillating magnetisalion decays with a rate determined by the transverse relaxation rate. The shape of the filnction with an exponential transverse relaxation term is shown in Figure 2. In general, the spectrum will contain real and imaginaD' parts in which case the beats are described by a term of the form exp (-ibl2/2). Two approaches are possible for the disentanglement process. The first is to follow the original suggestion of Dadok and Sprecher (1974) and cross correlate the observed spectrum with the spectrum, obtained under identical conditions, of a single reference line. The alternative is to deconvolute the ringing by the use of reverse Fourier transform. In practice the cross correlation process amounts to complex con.jugatc multiplicalion in the Fourier domain whilst deconvolution is a process of s~mplc multiplication in the Fourier domain.
366 Inverse Fourier transformation of the observed spectrum under rapid passage conditions can be shown to yield the following (Gupta. Ferretti and Becker. 1974) time domain result. F(T) = h(T) exp (ibT~/ 2)
(10)
T is the variable in the time domain. It is important to realise that although the observed spectrum is time dependent it is essentially in the frequency domain and its time scale t should not be confused with the timescale T in the time domain, h(T) is simply the free induction decay of the spins and represents the lineshapes unaffected by ringing. A simple division therefore yields the required free induction decay, which, on for~vard Fourier transformation, will give the frequency domain spectrum. In Figure 1B is shown the result of this process when applied to Figure 1A.
r
m
B
a"--~
--:~
-----6 Fig. 1 9 A spcctntm of 5% ethyl benzene. IA acquired under rapid scan conditions. IB after dcconvolution. Reproduced with permission from Bariat. Bclton and Goodfellow. 1993a.
367
I/it,v ........
Fig. 2 9 Simulated lincshapc undcr conditions of rapid scan with an exponential relaxation fi~nction. If a rcfcrencc spcctmm is uscd the proccss is to reverse Fourier transform both the reference spectrum and the obscn'cd spectrum and mulliply the complex conjugate of the transformed rcfcrence spectrum with the transformed observed spectrum. The result obtained on forward Fouricr transformation is a frequency, spectrum in which the line widths are the sum of lhe reference line widths with the true line width. In practice, cross correlation does not seem to have any advantages, and, although it has lent its name to the technique, does not seem to be preferable to the direct Fourier transformation method using the calculated fi~nction. A summary of the steps involved in the direct Fourier transformation method is given in Figure .3. 3.1 Practical considerations Combination of Equations l and 6 suggest that if the adiabatic criterion is ignored then dBo > yB~ dt
(11)
however saturation mus! still be avoided so that the criterion given in Equation 6 still applies and thcrcfore
368 dBo dt
>
1
YT, T:
(12) OBSERVED SPECTRUM
1
I
REVERSE FT
DIVIDE BY exp(-ibT2/2)
l APODISATION I
I
FORWARD FT I
Ill
I II
I
1 PHASE ADJUSTMENT
I
I
Fig. 3 9Block diagram of the process of obtaining an undistorted spectrum from a rapid scan spectrum The upper limit of the sweep rate is in principle, therefore arbitrary. In practice electronic considerations will represent an absolute upper limit to sweep rates, but a likely upper limit is also to be found in digitisation problems. As sweep rate is increased, the maximum sampling rate required is increased. Applying the Nyquist criterion that at least two points must be sampled each cycle, the maximum digitisation frequency required is bt/2n. Since the intensity of the ringing will decay by transverse relaxation, it may be assumed to have reached zero intensity bv 3T2" where T2* is the effective exponential transverse relaxation time.
369 Hence the sampling frequency in Hcnz required (H) is given by H
m
3bT; 2z
(13)
A typical upper limit might be for a scan rate of 10 kHz/s and a linewidth of 0.2 Hz. Under these conditions the maximum digitisation frequency required is 47.7 kHz. This can be readily achieved. With a line width of 10 Hz and the same scan rate the required maximum rate is 960 Hz. In order to properly characterise the spectrum it will be necessary to accumulate data for a time at least 3T2" after the last peak maximum. For an unknown spectrum it must be assumed that peaks can occur throughout the spectral frequency range. Therefore, there will be an additional time required, in excess of that required to scan the spectrum, in order to allow the decay of the ringing of the peaks that occur near the end of the spectral range. For a peak width at half height of 0.2 Hz this corresponds to a time of about 4.85 S. If the scan rate is 10 kHz/s and the minimum sampling frequency is used this will rcquirc accumulalion of 229k of data points in addition to those used in scanning the spectral range of interest. This number of points is achievable, but is extravagant for a simple one dimensional spectrum. In addition, for rapid scan rates, most of the acquisition lime is spent in acquiring the transverse decay which is independent of scan rale or spectral width. A reduction of the scan rate to 200 Hz/s vr require lhc addition of only 4580 additional data points and for a 600 Hz spectral width would require a total acquisition time of 10.8 seconds. This is a very modest increase in time for a major saving of memory and suggests that more modest scan rates may be of considerable value despite the apparent attractions of high scan rates. A usefi~l way of considering the practical exploitation of rapid scan methods is to compare with direct Fourier transform methods using pulse excitation. In its simplest form the pulse method excites the whole spectrum. As a result there can be problems of dynamic range when a ver)' large signal, typically soh,ent, coexists with a small signal. These problems ma.v be overcome by the use of selective excitation methods but these require high levels of sophistication in hardware and software. By contrast CW and rapid scan methods require only that the region of interest of the spectrum be scanned. No additional sophistication is required. An analysis of the relative signal to noise ratios of pulse and rapid scan methods has been carried out (Gupta. Fcrrelli and Becker, 1974). For a situation in which the total time available is fixed and conditions are optimised for each type of experiment the relative signal to noise ratios are expressed as follows 9 Sp
[ (F/b' + 3T')(! - E~)(I + Ek)] ';2
(14)
The subscripts R and P refer to Rapid Scan and Pulsed methods respectively, S is the signal to noise ratio. F is the sweep width in Hertz.
370 b' = b / 2 z
(15)
Ep = exp (-3 T; / T,)
(16)
Ea = exp(-(F/b' + 3 T~) / T,)
(17)
Care should ahvays be taken with comparisons of signal to noise ratios from theoretical considerations as they represent ideal cases. However, Equation 14 can be used to give indications of trends. It is clear that when F/b' is small compared to T2* the scanning experiment becomes comparable to the pulsed experiment. In the limit of infinitely fast scan they become equivalent. Indeed in this limit the time will be spent acquiring what is effectively a free induction decay convoluted with the tinging function. This has all the disadvantages of rapid digitisation with none of the advantages of the direct pulsed method and therefore a straightforward increase in scan rate has little to offer. If, however, the sweep width is kept small and a modest scan rate employed the ratio F/b' is reduced and the advantages of pulsed methods are less apparent. Typically for T2* = 0.155 and T 1 = Is and a scan rate of 200 Hz/s across a sweep of 200 Hz the ratio from Equation 14 is close to unity. This situation is one that might well be encountered in the analysis of industrial fluids where panicles and dissolved macromolecules reduce T2* and where the requirement is to monitor the intensi b' of an identified spectral peak for analytical purposes. Undcr these circumstances the intensity of the major components may well be a problem and would rcquirc elimination by a multipulse method if the pulsed Fourier transform tcchniquc were used. In the next section examples of this type of application are given. 4 APPLICATIONS OF RAPID SCAN C O R R E L A T I O N N M R
It has been stressed in this chapter that one of the major advantages of the use of the rapid scan method is when it is used to scan limited spectral regions. Figure 4 illustrates this when applied to a sample of vinegar. The main components of this material are water and acetic acid. A 6() MHz proton spectrum is illustrated. In the lower spectrum the fidl range is covcrcd and only peaks from water and the methyl group of the acid are observcd. The rcst of the spccln~m, except for spinning side bands of the water, is lost because of insufficient signal for digitisation. When a smaller region is scanned (upper spectrum), at a higher gain level small features become apparent. A critical featurc in any tcchniquc that employs signal averaging is that co-addition of signals results in an improvement in signal to noise ratios. This requires that the noise co-adds to a zero sum and that the signal remains constant. In the analysis of ethanol content of alcoholic drinks using proton NMR the intensity of the water can represent a problem when the ethanol content is low. However, by confining the region scanned to the methyl triplet which is most remote from the water peak, good linear calibrations for ethanol content may be obtained.
371 Ethanol contents as low as ().()095'Y,, in water were quantitatively measured. Other quantitative applications include glucose in water and levels of unsaturation in oils (Barjat. Belton and Goodfeliow. 1993a. b). Figure 5 shows the effects of accumulating scans of proton spectra on a spectrometer with no field drift correction (Barjat. Belton and Goodfellow, 1993a). Clearly the expected gain in signal to noise is not observed. However, as shown in Figure 6 when field drift effects arc eliminated the expected signal to noise improvement is observed.
A
B
Fig. 4 : Partial spectrum of vincgar showing the effects of dynamic range. Peak A is water, peak B is from Ihc methyl protons of acetic acid. The upper spectrum is from a reduced ~vidlh spectrum obtained at a higher amplifier gain.
372
50
E! o
40
o3
E!
El
-,--4
o
30
o
E!
t~
20
10
~ 0
n I 1
i
I 2
L
I 3
,
1 4
,
, 5
, 6
Square root of n u m b e r of scans
Fig. 5 9 Variation of signal to noise ratio on coaddition of signals obtained without field correction.
5 CONCLUSIONS If continuous wave NMR methods are to be used they are best used in the rapid scan mode. Even in this mode the advantages of the technique over pulsed Fourier transform methods are very limited. When wide spectral ranges are to be examined the pulsed method is clearly superior and continuous wave methods do not permit the wide variety of multidimensional experiments available on pulsed Fourier transform instruments. However, for limited applications where the required sweep width is small, the rapid scan method has a small or negligible signal to noise disadvantage and offers the potential advantages of selective excitation of spectral regions of interest and low cost.
373 200
0
op.-g
r o0 0
o
I00
ra
-,-.4
0 0
2 4 6 Square root of number of scans
8
Fig. 6 9As for Figure 5 but with field correction. The straight line is a linear regression fit to the data.
6 REFERENCES Abragam, A., 1978. The Principles of Nuclear Magnetism, 39-56, Oxford University Press, Oxford. Abraham, R.J. and Loftus, P., 1978. Proton and Carbon-13 NMR Spectroscopy, 34-39, Heyden, London. Barjat, H., Belton, P.S. and Goodfellow, B.J., 1993a. The use of Rapid Scan Correlation NMR as a Quantitative Analytical Method. Analyst, 18, 73-77. Barjat H., Belton, P.S. and Goodfellow, B.J., 1993b. Rapid Scan Correlation NMR Spectroscopy for Food Analysis. Food Chem., 48, 307-312. Dadok, J. and Sprecher, R.F., 1974. Correlation NMR Spectroscopy. d. Magn. Reson., 13,243-248. Gupta, R.K., Ferretti, J.A. and Becker, E.D., 1974. Rapid Scan Fourier Transform NMR Spectroscopy. d. Magn. Reson., 13,275-290. Martin, M.L., Delpuech, J-J. and Martin, G.J., 1980. Practical NMR Spectroscopy Heyden and Son, London.