Fourier Transform Multiple Quantum Nuclear Magnetic Resonance

The excitation and detection of multiple quantum transitions in systems of coupled spins offers, among other advantages, an increase in resolution over single quantum n.m.r. since the number of lines decreases as the order of the transition increases. This paper reviews the motivation for detecting multiple quantum transitions by a Fourier transform experiment and describes an experimental approach to high resolution multiple quantum spectra in dipolar systems along with results on some protonated liquid crystal systems. A simple operator formalism for the essential features of the time development is presented and some applications in progress are discussed.

The energy level diagram of a system of coupled spins 1/2 in high field is shown schematically in fig.I.The eigenstates are grouped according to Zeeman quantum number mi with smaller differences in energy within a Zeeman manifold due to the couplings between spins and the chemical shifts.For any eigenstate 10 of the spin Hamiltonian H (in frequency units) The single quantum selection rule 1 of the low power c.w. experiment and the one dimensional Fourier transform experiment arises because <i I lx I j) vanishes unless (2)   Simple combinatorial considerations show that the number of eigenstates decreases as lmi I increases and the number of transitions decreases as lqiil increases.The highest order transition possible is the single transition with lql = 21 where I is the total spin.For a system of N spins 1/2, transitions up to order N are possible.
Detection of multiple quantum transitions in c.w. experiments is well known. 2-4 Extension to high order transitions is not promising, since the transitions observed are a sensitive function of r.f.field strength.This leads to difficult spectral simulations and experimental problems of saturation and sample heating.
-7 This work treats a class of such multidimensional experiments in which the irradiation consists of pulses at the Larmor frequency.Time proportional incrementation of the r.f.phase (TPPI) allows separate determination of the spectra of all orders free from effects of magnet inhomogeneity.

EXPERIMENTAL
The spectrometer is of pulsed Fourier transform design with super-conducting magnet (Broker) operating in persistent mode at a proton frequency of 185 MHz.Phase shifting was performed at 185 MHz by a digitally controlled device (DAICO 10000898) under control of the pulse programmer.
Samples were approximately 400 mg sealed in 6 mm glass tubing after degassing by repeated freezing and evacuation.All observations are in the nematic phase.Synthesis of 4-cyano-4'-[ 2 H] 11 pentyl-biphenyl was by the procedure of Gray and Mosley. 8

RESULTS
The spectrum of benzene dissolved in a liquid crystal served as a prototype in the development of the single quantum n.m.r. of complex spin systems in ordered phases. 9he multiple quantum spectrum of ordered benzene is shown in fig. 2. The resolution is limited by magnetic homogeneity and the inhomogeneous linewidth is proportional to jqj.The increment in rp was 29.5 degrees and the increment in 1 1 was 10 JlS for each of 1024 points.The magnitude spectra for eight values of r between 9 and 12.5 JlS were added.The magnetization was sampled at t = r.
The spectrum of fig. 3 demonstrates the use of the spin echo to remove inhomogeneous line broadening and the use of time proportional phase increments (TPPI) to restore the offset.Resolution is limited by truncation of the multiple quantum free induction decay and scale of reproduction.The actuallinewidth is less than 2 Hz for all orders and suffices to resolve all allowed transitions of all orders.
An application of the TPPI method to the eight proton system of an alkyl-deuterated cyano biphenyl liquid crystal is shown in fig.The increments are 22.5 degrees in cp and 1.5 fJS in t 1 for each of 1024 points.The timer took five values between 0.5 and 1.0 ms and the magnetization was sampled at 64 intervals of 5 ps starting at t2 = (r + 0.1) ms.

DISCUSSION THREE PULSE EXPERIMENT
The time development of the spin system during the pulse sequences of fig. 2 and 3 is conveniently discussed in terms of a spherical tensor operator expansion of the density matrix.For any time, where T~ is the qth component of a spherical tensor operator of rank k. 10 The label ex.completes the specification of a complete basis of tensor operators.
The initial equilibrium density matrix is (4)   and immediately after a n/2y pulse (w 1 ~ H, w 1 tp 1 = n/2) (5) A solution to this dilemma is to view eqn ( 9) not as the effect of a frequency offset, but as a shift of the r.f.phase.In particular, consider the effect of preparing the system with pulses of phase rp and ip = rp + n at time zero and r, respectively, with IlfJ = lx cos rp --ly sin rp.Then The result is that again pq(r, rp) = exp (iqrp)pq(r) (10)   but this modulation is now an artifact of the r.f.phase of the first two pulses and does not depend on the evolution of the system during t 1 .However, since each point t 1 of the signal is collected separately, we may set rp = ~wn for some parameter ~w.thereby recovering an apparent offset.
The actual evolution during t 1 may include an echo pulse to remove the effect of field inhomogeneity, as in fig.3, or a train of n pulses to remove the effect of small chemical shifts and heteronuclear couplings from the dipolar spectra.This TPPI technique may be compared to the method of phase Fourier transformation (PFT) discussed elsewhereY• 12 The PFT method generates a signal array S( r, 1 1 , t 2 , rp) by repeating the experiment for each t 1 and rp, where again rp describes the phase of the preparation pulses as in eqn (10).Fourier transformation with respect to phase with q as the conjugate variable separates the orders.A second Fourier transformation with respect to t 1 gives the spectra.In the TPPI experiment the phase and time dimensions are collapsed into one by the relation rp = ~wt 1 • A single time Fourier transformation gives the spectrum of each order with apparent offset q~w as in fig. 3 and fig. 4.

MULTIPLE QUANTUM SPECTROSCOPY OF LIQUID CRYSTALS
Although the spectroscopy discussed is of general applicability, it is worthwhile to note the particular suitability of the m.f.t.n.m.r.method to the study of liquid crystals.The diffusion present in such systems sharply reduces the intermolecular dipolar couplings.This allows one to obtain resolved spectra reflecting only the intramolecular couplings without dilution of the spin system.Combinatorial arguments suggest that it suffices in general to analyse only the jqj = N -2 and N -l spectra to determine all couplings in a system of N spins 1/2.Since these transitions involve only the relatively small lml = ~';-1 and~-2 Zeeman manifolds of the basis of kets, the diagonalizations are simpler than those needed for single quantum spectroscopy.Should the study of mixtures be of interest, partial deuteration of the background species will reduce its contribution to the high order spectra more rapidly than to the single quantum spectrum and thus the requirement for isotopically pure synthesis is reduced.
Applications in progress include the configurational analysis of both ring and chain regions of liquid crystals, the study of relaxation effects in multiple quantum spectra FIG. I.-Schematic representation of the high field energy level diagram of coupled spins 1/2.Broken arrows indicate the forbidden types of transition observed in Fourier transform multiple quantum experiments.P 1 Pz FIG.3.--Multiplequantum spectrum of benzene at 22 oc and TPPI pulse sequence.The sample is the same as in fig.2.The pulses are Pl = 1t/21f1, P2 = 1t/2iP and P3 = 1t/2x, where rp = /).wt 1 • The increment in rp was 29.5 degrees and the increment in 1 1 was 10 JlS for each of 1024 points.The magnitude spectra for eight values of r between 9 and 12.5 JlS were added.The magnetization was sampled at t = r.

4 .
All eight orders are observed.Resolution is limited by truncation.Actual linewidths are < 100 Hz in a spectral width of ~ 40 kHz for each order.