Chemical Physics Letters 381 (2003) 634–641

www.elsevier.com/locate/cplett

High-resolution NMR spectroscopy in inhomogeneousB0 and B1 fields by two-dimensional correlation

Sasa Antonijevic, Stephen Wimperis *

School of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK

Received 29 July 2003; in final form 8 September 2003

Published online: 4 November 2003

Abstract

Recently, there has been much interest in methods for obtaining high-resolution NMR spectra in inhomogeneous B0

and B1 fields and in their application to so-called �ex situ� spectroscopy, where the sample and magnet/probe assembly

are spatially separated. Here we discuss the implementation of the well-known two-dimensional nutation experiment as

a method for correlating B0 and B1 inhomogeneities and, hence, achieving a high-resolution NMR spectrum. The

advantages of this approach lie in its simplicity, its spatial (�depth�) resolution, and in its not requiring a linear cor-

relation of fields, i.e., B1 ¼ aB0 þ k, across the sample.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

High-resolution NMR spectroscopy is normallyperformed in a B0 field that is spatially homoge-

neous. It has always been recognised, however,

that the use of such a magnetic field is not always

feasible. One example that has aroused interest

recently is that of so-called �ex situ� NMR spec-

troscopy, where the sample and magnet/probe as-

sembly are spatially separated [1]. An inevitable

feature of this experimental arrangement is thatthe radiofrequency or B1 field will also be spatially

inhomogeneous.

* Corresponding author. Fax: +44-1392-263434.

E-mail address: s.wimperis@exeter.ac.uk (S. Wimperis).

0009-2614/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.cplett.2003.09.116

Pines and coworkers have advocated an ap-

proach to recording high-resolution NMR spectra

that exploits a spatial correlation of the inhomo-geneous B0 and B1 fields [1–3]. Applications to the

NMR �logging� of oil wells [4] and in mobile, sur-

face-scanning NMR spectrometers [5] have been

suggested. One implementation involves the use of

a train of composite z-rotation pulses interleaved

with data sampling and allows direct acquisition of

a high-resolution spectrum [1–3]. However, a sec-

ond implementation uses two-dimensional NMRcorrelation of the frequency dispersions produced

by the inhomogeneous B0 and B1 fields [2] and is

identical to the well-known two-dimensional nu-

tation experiment [6–8] or, equivalently, to the so-

called �rotating-frame� imaging experiment [9].

The purpose of this Letter is to present a de-

tailed discussion of the two-dimensional nutation

ed.

S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641 635

experiment as an approach to high-resolution

NMR spectroscopy in correlated inhomogeneous

B0 and B1 fields. Although slower than direct ac-

quisition, the method will be shown to have the

same sensitivity and to possess a number advan-

tages, including simplicity and not requiring alinear correlation of fields, i.e., B1 ¼ aB0 þ k,across the sample.

2. Experimental details

Experiments were performed in a 100-mm ver-

tical bore superconducting NMR magnet gener-ating a magnetic field of B0 ¼ 4:7 T (m0 ¼ 200:06MHz for 1H). To mimic the B0 inhomogeneity

encountered in �ex situ� NMR, a B0 gradient was

produced along the laboratory-frame x axis by

passing a constant 0–10 A current through an x-gradient coil wound as part of a room-temperature

imaging gradient/shim set. Up to 60 W of power

was dissipated and generous quantities of coolingair were applied to the bore, probe and shims.

A home-built NMR probehead was used, tuned

to the 1H NMR frequency. The radiofrequency

coil (a 3-turn solenoid) and sample holder are

shown in the schematic and photograph in Fig. 1.

Fig. 1. Schematic and photograph of the radiofrequency coil, forme

NMR spectroscopy in inhomogeneous B0 and B1 fields.

The liquid sample was contained in a standard

5-mm o.d. high-resolution NMR tube (cut down

to �30 mm in length), which was sealed with a

Teflon plug. Again to mimic �ex situ� NMR, the

sample was positioned entirely outside the radio-

frequency coil, thereby ensuring a large B1 gradi-ent across the sample. The long axes of the coil and

sample tube were aligned with the laboratory-

frame x axis, thus aligning the B0 and B1 gradients.

A 100 W radiofrequency amplifier generated the1H pulses.

3. Two-dimensional nutation experiment

The pulse sequence for the two-dimensional

nutation experiment is shown in Fig. 2a. The du-

ration of the radiofrequency pulse is incremented

as the t1 evolution period of the two-dimensional

experiment, while the free induction decay is re-

corded during the t2 acquisition period.

During t1 the spins nutate in the spectrometerrotating frame at a frequency m1 ¼ cB1=2p, whileduring t2 they precess in the rotating frame at

m2 ¼ mCS þ m0 � mrf , where mCS is the chemical shift

frequency, m0 ¼ cB0=2p is the Larmor frequency,

and mrf is the rotating-frame frequency. Spatial

r and sample used in this work to demonstrate high-resolution

Fig. 2. The two-dimensional nutation experiment as a method for obtaining high-resolution NMR spectra in inhomogeneous B0 and

B1 fields: (a) pulse sequence, and (b) schematic two-dimensional spectrum showing how high-resolution cross sections and projections

can be extracted.

636 S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641

inhomogeneity of the B0 and B1 fields produces a

distribution of Larmor (m0) and nutation (m1)frequencies across the sample and the NMR peaksare broadened. If these two fields are perfectly

correlated within the sample volume, however

(i.e., each B1 value corresponds to a single B0

value), then each chemically distinct species

appears in the final two-dimensional nutation

spectrum as a �ridge� lineshape [10] and a high-

resolution NMR spectrum can be extracted by

either projection or cross section, as shown sche-matically in Fig. 2b.

The following points are worth noting about

this two-dimensional nutation experiment:

(1) The experiment yields an amplitude(sine)-modulated signal in t1. Therefore, a hypercomplex

two-dimensional Fourier transform yields pure

absorption-mode lineshapes [11,12]. (A complex

two-dimensional transform would yield so-called

�phase-twist� lineshapes, which have undesirable

properties.) Normally in two-dimensional NMR,

discrimination of the signs of the signal frequencies

in t1 is achieved by recording a separate cosine-modulated signal but this is not possible using the

S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641 637

pulse sequence in Fig. 2a. However, this is not a

problem here since we can be confident that the

nutation frequency m1 is always positive (in fact, we

only display this part of the spectrum).

(2) An amplitude-modulated signal can be

viewed as the sum of counter-rotating echo andantiecho signals [11,12]. If we assume a linear

correlation of the two inhomogeneous fields,

B1 ¼ aB0 þ k, then the two inhomogeneities are

refocussed in the echo signal during the acquisition

period at t2 ¼ at1 [2]. The antiecho signal does not

refocus and is weak but is nevertheless an essential

part of the overall modulation and allows pure

absorption-mode lineshapes to be obtained [11,12].(3) It is not advisable to use �delayed acquisi-

tion� [11] in a nutation experiment, i.e., to redefine

the t1 and t2 periods so that t01 ¼ ð1þ aÞt1 and

t02 ¼ t2 � at1. If this is done, the effect is to �shear�the echo signal such that the inhomogeneous

broadening lies only in the m2 frequency dimension

and a high-resolution spectrum can be obtained

simply by projection onto the m01 axis. The weakantiecho signal, however, is sheared in the wrong

direction by this procedure and still lies across

both dimensions. Worse still, because the echo and

antiecho signals are no longer coincident in the

frequency domain, both signals consist of unde-

sirable phase-twist lineshapes.

(4) Similarly, shearing the nutation experiment

data set after the Fourier transformation with re-spect to t2 [11,13] also produces echo and antiecho

signals that are noncoincident. This problem is

normally dealt with in two-dimensional NMR by

separating the echo and antiecho signals and (ef-

fectively) shearing them individually in opposite

senses. However, this separation is not possible

with the experiment in Fig. 2a because a cosine-

modulated signal is not available.(5) If shearing of the two-dimensional nutation

experiment data set is required, so that a high-res-

olution NMR projection can be obtained, then the

optimum way to achieve it is to record pure ab-

sorption-mode lineshapes and to shear these in the

frequency domain, using interpolation if necessary.

(6) It is possible to convert the experiment in

Fig. 2a into one which yields phase modulation int1, rather than amplitude modulation, by adding a

90� pulse with orthogonal phase in the rotating

frame to the end of the t1 nutation pulse [9]. There

is a factor of 2 increase in signal intensity and,

after a complex Fourier transformation (as ap-

propriate for phase-modulated data [11]), a factor

of 2p2 increase in signal-to-noise ratio compared

with the hypercomplex amplitude-modulated ex-periment. The absence of an antiecho signal,

however, means that this experiment yields unde-

sirable phase-twist lineshapes [9].

(7) By using a second, complementary phase-

modulated experiment, where the phase of the

additional 90� pulse is shifted by a further 180�,the weak antiecho signal can be recorded [9]. Pure

absorption-mode lineshapes can now be obtainedvia hypercomplex Fourier transformation of the

two data sets. The gain in signal-to-noise relative

to the amplitude-modulated experiment is reduced

to a factor of 2.

(8) The severe drawback of these phase-modu-

lated experiments is the difficulty of producing a

90� pulse that is uniform across the sample when,

by the very nature of the experiment, the B1 field ishighly inhomogeneous. Both composite [14] and

adiabatic 90� [15,16] pulses could possibly be used

to alleviate this problem, but in this Letter we will

concentrate on the implementation of the simpler,

if less sensitive, amplitude-modulated nutation

experiment in Fig. 2a.

4. Results and discussion

Conventional 1H NMR spectra and two-di-

mensional 1H nutation spectra of a 6-mm long

sample of ethanol (CH3CH2OH) are shown in

Fig. 3. In Fig. 3a, no B0 gradient was applied and

the three ridge lineshapes (corresponding to the

three peaks in the conventional 1H spectrum) areparallel to the m1 (or B1) axis, ranging from m1 ¼ 2

kHz (at the point in the sample furthest away from

the radiofrequency coil) to m1 ¼ 17 kHz (at the

point in sample closest to the coil). In Fig. 3b–d,

currents of 2.0, 4.0 and 8.0 A were passed through

the x-gradient coil generating estimated B0 gradi-

ents of 0.11, 0.22 and 0.40 mT cm�1, respectively.

These gradients are sufficient to destroy the reso-lution of the three peaks in the conventional

spectra but, owing to the spatial correlation of the

Fig. 3. Conventional 1H NMR spectra and two-dimensional 1H nutation spectra of a 6-mm long sample of liquid ethanol

(CH3CH2OH) recorded with currents of (a) 0 A, (b) 2.0 A, (c) 4.0 A and (d) 8.0 A passing through the x-gradient B0 coil. Two-di-

mensional experiments were recorded by averaging 4 transients (using CYCLOPS phase cycling) for each of 512 t1 increments of 20 ls.The recycle interval was 10 s.

638 S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641

B0 and B1 gradients, resolution of the three ridge

lineshapes is retained in the two-dimensional nu-

tation spectra.

Off-resonance phenomena are very noticeable in

Fig. 3, especially when the stronger B0 gradients

are used. If the rotating-frame frequency of a peak,m2, is not equal to zero then the effective nutation

frequency is meff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffim21 þ m22

pand the nutation axis

is tilted away from the rotating-frame x axis to-

wards the z axis by h ¼ tan�1 ðm2=m1Þ. Three major

effects of this are apparent. Firstly, not all the bulk

equilibrium magnetization will nutate: some will

be spin-locked along the tilted effective field andwill give rise to the unwanted peak at m1 ¼ 0.

Fig. 4. Two-dimensional 1H nutation spectrum from Fig. 3b

�banana sheared� so that the ridge lineshapes are parallel to the

m1 axis. As shown, the high-resolution projection may now be

calculated or, alternatively, high-resolution cross sections may

be extracted (as from the unsheared spectrum).

S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641 639

Secondly, the tilting of the nutation axis means

that the signal is now partly phase modulated as a

function of t1 and the lineshapes can no longer be

fully absorptive. And thirdly, meff may be much

larger than m1 in regions of the sample where m1 issmall and/or m2 large, yielding ridge lineshapes thatdisplay unusual shifts and curvature. Note that,

although these off-resonance effects are very evi-

dent in these two-dimensional spectra, particularly

in Fig. 3d, they will occur to an equal extent in the

direct-acquisition implementation [1–3] of the

method.

If the two inhomogeneous fields are linearly

correlated, B1 ¼ aB0 þ k, then, as shown sche-matically in Fig. 2b, the ridge lineshapes in the

nutation spectrum will be straight, with a constant

gradient a. The direct-acquisition implementation

requires such a linear correlation as it corresponds

to obtaining the high-resolution projection of the

two-dimensional nutation spectrum in a single free

induction decay. However, it is clear in the ex-

perimental spectra in Fig. 3b–d that the ridgelineshapes are curved rather than straight, indi-

cating a nonlinear correlation of B0 and B1. This is

not unexpected as we are using a B0 gradient that

is linear along the x axis but have a B1 field that,

outside the radiofrequency coil, decreases along

the x axis in an approximately exponential man-

ner. In contrast with the case of direct acquisition,

in the full two-dimensional nutation spectrum thisnonlinear correlation of fields is no barrier to ex-

tracting a high-resolution one-dimensional spec-

trum: either a m2 cross section may be taken, as

shown in Fig. 2b, or, if the full signal intensity

needs to be recovered, a more sophisticated �ba-nana� projection procedure can be envisaged that

sums intensity along the curved ridges into a one-

dimensional spectrum.Fig. 4 shows the two-dimensional nutation

spectrum of Fig. 3b (gradient current 2.0 A) after a

frequency-domain banana-shearing according to

the formula

m02 ¼ m2 � a0m1; ð1Þwhere, for each m1 value, a0 is chosen by the shearing

algorithm to ensure the resulting ridge lineshapes

are vertical. A high-resolution projection is now

easily calculated as shown.Alternatively, albeitwith

lower signal-to-noise ratio, cross sections may be

taken as in the unsheared spectrum.

Both the B0 and B1 dimensions of the two-

dimensional nutation spectrum provide spatial or

�depth� resolution [2,9]. Fig. 5 shows the conven-

tional 1H NMR spectra and two-dimensional 1Hnutation spectra of a two-compartment �phantom�consisting of (in order from the bottom of the

NMR tube) 3 mm of ethanol (CH3CH2OH), a

1-mm Teflon partitioning plug, 4 mm of toluene

(C5H6CH3), and a Teflon sealing plug. The con-

ventional spectrum in Fig. 5a was recorded in the

absence of a B0 gradient. The two toluene peaks

are relatively weak as a consequence of the dis-tance of this compartment from the radiofrequency

coil and difficult to resolve from the stronger etha-

nol peaks. In Fig. 5b and c, currents of 5.0 and 10.0

A were passed through the x-gradient coil generat-ing estimated gradients of 0.26 and 0.50 mTcm�1,

respectively. Three ridge lineshapes, corresponding

to the three peaks in the conventional 1H spectrum

Fig. 5. Conventional 1H NMR spectra and two-dimensional 1H nutation spectra of a two-compartment �phantom� consisting of (in

order from the bottom of the 5-mm o.d. NMR tube) 3 mm of liquid ethanol (CH3CH2OH), a 1-mm Teflon partitioning plug, 4 mm of

liquid toluene (C5H6CH3), and a Teflon sealing plug. The spectra were recorded with gradient currents of (a) 0 A, (b) 5.0 A, (c) 10.0 A

and (d) 5.0 A. The two ridge lineshapes from toluene are resolved (both spatially and spectroscopically) at low B1 fields (below m1 ¼ 6

kHz) in (b) and (c). In (d) the B0 and B1 gradients have been deliberately misaligned to show that the absence of a unique correlation

between the two fields leads to a complete loss of resolution. Two-dimensional experiments were recorded by averaging 4 transients

(using CYCLOPS phase cycling) for each of 820 t1 increments of 12.5 ls. The recycle interval was 10 s.

640 S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641

of ethanol, are resolved at high B1 fields (between

m1 ¼ 7 kHz and m1 ¼ 23 kHz) while the two ridge

lineshapes from toluene are resolved at low B1 fields

(below m1 ¼ 6 kHz), although off-resonance effectsare severe.

Finally, the importance of correlation, whether

linear or nonlinear, between the inhomogeneous

B0 and B1 fields is emphasised in Fig. 5d, which

shows the result of rotating the NMR probeheadso that the B1 gradient is no longer parallel to the

S. Antonijevic, S. Wimperis / Chemical Physics Letters 381 (2003) 634–641 641

B0 gradient. Once B1 is no longer a single-valued

function of B0 across the sample volume, there is a

complete loss of resolution in the two-dimensional

nutation spectrum.

5. Conclusions

As suggested by Pines and coworkers [2], the

two-dimensional nutation experiment can be used

to obtain high-resolution NMR spectra in corre-

lated inhomogeneous B0 and B1 fields. The method

is very simple (avoiding the need for sophisticated

pulses interleaved with data acquisition), providesspatial resolution in one dimension, and has the

great advantage of not requiring a linear correla-

tion of the two fields, making it perhaps more

generally applicable than the direct-acquisition

approach [1–3]. The latter has the advantage of

speed (i.e., the minimum experiment duration is

much shorter) but, per unit experiment time, will

have the same sensitivity as a projection of thetwo-dimensional nutation spectrum. To date, both

methods have only been demonstrated using one-

dimensional B0 and B1 gradients and the lifting of

this restriction is likely to be an important goal in

future work.

Acknowledgements

We are grateful to EPSRC for their generous

support from October 2000 to September 2003

(grant no. GR/N07622) and to Mike Jones and

Neville England for technical assistance.

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