A single-scan fourier transform method for measuring spin-lattice relaxation times

JOURNAL

OF MAGNETIC

RESONANCE

%$2%--3~

(1976)

A Single-ScanFourier Transform Method for Measuring Spin-Lattice Relaxation Times Measurements of nuclear spin-lattice relaxation times of molecules in solution can be very time consuming. The most commonly used method for Fourier transform spectroscopy is the inversion-recovery technique first proposed by Vold et al. (I). This method is based upon the pulse sequence [180”-r-90” (FID)-T],. The partially relaxed free induction decays (FID’s) for various times z are collected after inversion by a 180” pulse. A waiting time T > ST,(max) is required to allow restoration of equilibrium, where T,(max) is the longest expected TI. Other schemes, such as the saturation-recovery method (2), where the magnetization recovers after a burst of 90” pulses, or its homospoil equivalent (3), circumvent the problem of a long waiting time. A similar time saving holds for the progressive saturation method (4), based on the sequence [90” (FID)-r],. The apparent time saving in these methods, however, is offset to a large extent by a loss in sensitivity (dynamic range) of a factor of 2 compared to the inversion-recovery (IRFT) method, as the magnetization is inverted in IRFT rather than nulled (4). The same loss of dynamic range occurs in the recently proposed fast inversion-recovery method (5) (IRFT with short waiting time) for long T,‘s. This latter method has the advantage that it retains full sensitivity for lines with short T,‘s. The problem of long experimental times is especially acute in systems with very long TX’s (e.g., 13C lines of quaternary carbons). Also, the available time may be limited by sample instability and, in the case of CIDNP lines, by difficulty in maintaining steadystate levels of polarization for long periods. We wish here to propose a fast, single-scan, Fourier transform (SSFT) method which is suitable for such situations, although in practice it may be somewhat less accurate than other methods. This method can be used when a fast data storage device such as a disk is available. The SSFT method The method is described by the pulse sequence {180”-[e(FID)--z&T}, (see Fig. 1). After the inverting 180” pulse the recovery of the magnetization is sampled by a series of N equally spaced rf pulses of flip angle 8 (0 < 90”; typically 30”) and pulse spacing z sec. The N FID’s are then stored on disk immediately after acquisition. Homospoil pulses after the 180” pulse and after each aquisition destroy the transverse magnetization and reduce echo formation. If signal averaging is necessary the FID’s are summed into accumulation files on disk in a time T, and the entire sequence is repeated x times. Note that the time Tis included for data handling only, and that no waiting time is necessary, Since during a repetitive series of pulses the steady-state magnetization is different from the equilibrium magnetization, data collection is started after the first passage in which the steady state is established. The choice of 13is a compromise: small values of 8 give low signal intensity, while flip angles that are’too large reduce the z magnetization by too large an amount (clearly Copyright Q 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

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8 cannot be 90”). We shall now show that for intermediate flip angles the recovery to a steady state is exponential, but with an apparent relaxation time Tf’ which depends upon 8, z, and the real Tl. This points to a limitation of the method: its dependence upon an accurate knowledge of the flip angle. In the following analysis we shall follow the treatment of Look and Locker (6), who considered the behavior of the magnetization under a periodic train of rf pulses.

n FIG.

1. Pulse sequence for SSFT method showing rf pulse sequence and homospoil

sequence.

Let M,,,- and M,,,+ be the magnetizations along the z axis before and after the nth rf pulse, respectively. The pulse is assumed to be centered at the average Larmor frequency and to produce a nutation of the magnetization by an angle 13.Then iv,+ = Al,- COS0.

PI

Assuming that the transverse magnetization is zero before the pulse, the signal is proportional to M Y”+ = M n- sin 07 PI and hence proportional to A&,-. According to the Bloch equations the magnetization relaxes exponentially between pulses as Al,- = (Mnml+ - AI,,) exp (--z/T,) + Mea,

[31

where Meq is the equilibrium magnetization. The effect of the successivepulses can now be described using the notation c(= cos 8, /I = exp(-r/T1), and MO- is the z magnetization after the 180” pulse. Then we have Ml- = Alo-a/3 + A&,( 1 - j?), M2- = Al,-a*p* + A&,( 1 - /I) (1 + a/?), . . . . . . . . . . . . . . . . . . . . . . . . ..*......

and in general, n-1

= &fo-a”B” + &fq (1-- I9 (1 - av. 1 - a/I.

141

Thus, for large n, a steady state is reached : &f,-=j,f

(l--B -=

eq (1 - CC/?)

M

-1

eq 1 -.

- expWTl) cos

8 exp (-Z/T,) ’

[51

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and the approach to this steady state is given by AI,- - n/r,- = (MO- - Mm-)rRB”.

WI

M - Mm = 6% - MA exp (-dTl),

171

This can be written as with t = nz and 1 -=--T:T,

1

1ncos.O z

1 =T;+ry 1

1

c

181

which shows that the recovery is exponential with an apparent relaxation time T:. True T,‘s can be obtained from a semilogarithmic plot and by applying the correction time T, = -r/in cos 13according to Eq. [8]. The behavior of the z magnetization under the pulse sequence of Fig. 1 is sketched in Fig. 2.

FIG. 2. Recovery of z component of magnetization

under SSFT pulse sequence.

Experimental details. The pulse sequence described above has been implemented on a Bruker HX-360 spectrometer equipped with a 20 K BNC-12 computer and Diablo disk. The software is a modification of the Nicolet Tl program. Accurate pulse width settings for intermediate flip angles proved to be a difficulty in this system because of imperfect pulse shapes. We solved this problem by generating a string of exactly equal pulses separated by about 20 psec. If, for example, a 30” pulse is required, its width can be determined by obtaining a null signal under the application of a string of twelve pulses (i.e., a series comprising an effective 360” pulse, provided that the total length of the composite pulse is short compared to T,, T,). Under our experimental conditions, minimum r values for a 4000 Hz spectral width are 2.5 set for 8K FID’S (1 set acquisition time + 1.5 set disk-writing time) and I .5 set for 4K FID’s (0.5 set + 1 set). This sets a lower limit of about 3-4 sec.’ for the T,‘s that can be measured. Figure 3 shows an application to proton spin-lattice relaxation in a solution of acetone and DSS in DzO (99.7% enriched). The following parameters were used: spectral width = 4000 Hz, 0 = 30”, and z = 2.5 sec. (This yields T, = 17.38 sec.) After 1 We are currently implementing

a fast disk-writing

procedure to lower this limit to about 2 sec.

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SSFT

FIG. 3. SSFT measurement of mixture of proton spin-lattice DSS in D20. Spectral width = 4000 Hz, 0 = 30”, z = 2.5 sec.

T,

relaxation

in mixture of acetone and

the first passage to establish a steady state, four accumulations were made. The diskhandling time T was 52 sec. Very good exponential recovery was observed. Table 1 shows the results of this experiment for the three lines observed (HDO, acetone, and DSS) together with the results from the IRFT method. The measured T: are corrected to Eq. [8] with T, = 17.38 sec. The agreement between the two methods is excellent. TABLE T1 VALUES

MEASURED

BY IRFT

1

SSFT (0=. 30”, z = 2.5 set)

AND

METHODS

Proton line

T: (~6

T, (I3SFT)

HDO Acetone DSS

8.6 6.7 4.0

17.0

16.6

10.9

10.5

5.2

4.8

TI (1-T)

Discussion. Since in the present method relatively more time is spent in data collection and less in waiting than in the other methods, it is faster than the others. This is particularly obvious when the signal-to-noise ratio is good enough that no signal averaging is necessary. The total measuring time of the SSFT method is then 5T:, whereas that of the IRFT method would be approximately n6T,(max), where II is the number of pulse separations used. However, in the weak-signal case we must take into account the fact that the sensitivity is lower by a factor of 2 to 3, because of the smaller flip angles (sin 30” = +) and because the steady-state signal has a reduced intensity (cf. Eq. [S]). This can be seen in Fig. 3 for the HDO line. Nevertheless, taking into account the

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greater number of accumulations needed, it is estimated that the present method is approximately two to three times as fast as the IRFT method. The main source of error is associated with inaccurate setting of the 8 pulse (the method is not sensitive to missetting of the 180” pulse). The pulse width may be calibrated by the pulse train method described above or by measuring a known T1. Because of the flip angle dependence one might expect errors to occur for lines that are not exactly on resonance, because of the finite strength of the Hi field. However, since relatively small flip angles are used this limitation is not as serious as it would be in methods that depend upon an accurate setting of 90” pulses (4). The dependence on frequency offset from resonance was tested for the proton-decoupled 13C line of benzene (solution with 10 % C,D,) which has a T, of 24 sec. Table 2 shows experimental values of T1 and Tf (average of two experiments) measured with 30” pulses and r = 6 sec. It can be concluded that no serious systematic errors are introduced for lines within -7 kHz of the carrier frequency in spite of our relatively weak H1 field (30 psec for a 90” pulse). TABLE

2

OFFSET DEPENDENCE OF 13C T, OF BENZENE MEASURED BY THE SSFT METHOD WITH @=30”ANDZ=6StX

Offset (Hz)

Tf (set)

600 2600 4600 6600 8600 12600

15.5 15.6 16.2 15.2 18.7 22.2

24.1 24.9 26.5 23.9 33.9 47.5

Furthermore, it should be noted from the form of Eq. [8] that experimental errors in

Tf are exaggerated to some extent by making the correction to obtain Tl. To minimize this effect the parameters 8 and r should be chosen such that T, < T,(max). For the measurement of long T1’s in multiline NMR spectra the present method seems to be the fastest currently available, although the accuracy may be somewhat less than that of conventional methods. The major limitation of this method is that it requires an accurate knowledge of the flip angle 8. The method should be most useful in cases where time is limited by sample instability and for the measurement of less abundant nuclei such as 13C and 15N. ACKNOWLEDGMENT This work was supported by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO). REFERENCES I. R. L. VOLD, J. S. WAUGH, M. P. KLEIN, AND D. E. PHELPS, J. Chem. Whys. 48,383l (1968). 2. J. L. MARKLEY, W. H. HORSLEY, AND M. P. KLEIN, J. Chem. Phys. 55, 3604 (1971).

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3. G. G. MCDONALD AND J. S. LEIGH, J. Magn. Resonance 9, 358 (1973). 4. (a) R. FREEMAN AND H. D. W. HILL, J. Chem. Phys. 54,3367 (1971). (b) R. FREEMAN, AND R. KAPTEIN, J. Magn. Resonance 7,82 (1972). 5. D. CANET, G. C. LEVY, AND I. R. PEAT, J. Mugn. Resonance 18,199 (1975). 6. D. C. Loon AND D. R. LOCKER, Rev. Sci. Znstrum. 41,250 (1970).

H. D. W. HILL,

R. KAPTEIN K. DLIKSTRA C. E. TARR* Department of Physical Chemistry University of Groningen Groningen, The Netherlands Received July 20,1976 * Permanent

address:

Department

of Physics,

University

of Maine,

Orono,

Maine

04473.