Double-quantum homonuclear correlation MAS NMR spectroscopy of dipolar-coupled quadrupolar nuclei Gregor Mali∗ National Institute of Chemistry, Hajdrihova 19, SI-1001 Ljubljana, Slovenia Gerhard Fink and Francis Taulelle RMN & Chimie du Solide, Tectonique Moleculaire du Cristal, UMR CNRS 7140, Universite Louis Pasteur, 4 Rue Blaise Pascal, F-67070 Strasbourg, France
Abstract A double-quantum homonuclear correlation NMR experiment for dipolar-coupled half-integer quadrupolar nuclei (DQ-DCQ) in solids is presented. The experiment is based on rotary resonance dipolar recoupling and uses ‘bracketed’ spin-lock pulses to excite double-quantum coherence and later to convert it to the zero-quantum one. A central-transition-selective π pulse at the beginning of the t1 evolution period differentiates coherence transfer pathways of double-quantum coherences arising from coupled spins and from a single spin, so that the later can be efficiently filtered out by phase cycling. The experiment was tested on an aluminophosphate molecular sieve AlPO4 -14, a material with a variety of aluminum quadrupolar coupling constants, isotropic chemical shifts and homonuclear distances. In a two-dimensional spectrum aluminum dipolar couplings with internuclear distances between 2.9 and 5.5 ˚ A were resolved. Although the experiment requires an application of weak rf fields, frequency offsets didn’t affect its performance crucially.
∗
Electronic address:
[email protected]
1
I.
INTRODUCTION
Solid-state nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for the investigation of polycrystalline and amorphous materials. Its basic method is magic angle sample spinning (MAS), which provides high resolution spectra of spin-1/2 nuclei by removing broadening due to chemical shift anisotropy and dipolar couplings. The dipolar couplings, however, often contain valuable information about the local structure of solids and to extract this information in the presence of MAS, dedicated techniques that reintroduce hetero-1–3 or homonuclear4–9 dipolar interactions are needed. In most of these techniques the reintroduction of the dipolar interactions is achieved by manipulation of nuclear spins by pulses of radio-frequency magnetic field (rf pulses). Although quadrupolar nuclei, i.e. nuclei with spin I > 1/2, are frequently found in solids, NMR spectroscopy of these nuclei is less widespread and attempts to reintroduce the dipolar interactions among them are rare10–14 . The main reason for this is a strong quadrupolar interaction, which exceeds the strength of the interaction between a nuclear spin and an rf field by orders of magnitude. Because of the dominant quadrupolar interaction the response of a quadrupolar nucleus to rf pulses is different from the response of a spin-1/2 nucleus, and most of the methods developed for spin-1/2 systems cannot be directly translated to systems of quadrupolar nuclei. In the case of half-integer quadrupolar nuclei, however, the application of weak (selective) rf pulses affects only their central transition coherence, while satellite transition coherences remain intact. Manipulation and observation of central transitions only has two advantages; first, the central-transition spectrum is not affected by first-order quadrupolar interaction, and second, a restriction to the two-level subspace allows one to describe half-integer quadrupolar nuclei as fictitious spins 1/2. Consequently, in experiments, in which weak rf pulses are employed, behavior of half-integer quadrupolar and spin-1/2 nuclei is expected to be similar. For example, cross polarization (CP) between two different types of half-integer quadrupolar nuclei or between half-integer quadrupolar and spin-1/2 nuclei, within the limitation of selective rf fields during the spin-lock, leads to Hartmann-Hahn sideband matching conditions and CP dynamics in full analogy to pure spin-1/2 systems15,16 . Weak rf pulses and limitation to central transition coherences is not only useful for studying heteronuclear dipolar couplings, but might be promising also for analyses of homonuclear 2
correlations. Recently a rotary resonance dipolar recoupling among
23
Na nuclei (I = 3/2)
was achieved13,14 by applying a weak rf pulse, during which nutation frequency of the
23
Na
central transition was equal to one half of the sample rotation frequency. This approach was used to build a two-dimensional homonuclear correlation experiment13 , in which the transverse magnetization associated with the central transition coherence was first excited and allowed to evolve for a time t1 . Before the detection of the signal, a mixing period with low amplitude rf pulse enabled the exchange of magnetization among coupled nuclei. The resulting two-dimensional spectrum thus contained cross-peaks between dipolar-coupled nuclei in addition to a strong diagonal spectrum arising from those spins, for which no magnetization exchange occurred during the mixing period. Because of the strong diagonal, an exchange between two spatially proximal nuclei that occupy crystallographically equivalent sites cannot be detected by this, otherwise elegant, experiment. Recently two other experiments using either a 4I-quantum17 or a double-quantum18 filter were proposed for the investigation of homonuclear correlations in the system of half-integer quadrupolar nuclei. Because the 4I-quantum coherence can arise only from a coupled pair of half-integer quadrupolar nuclei with spin I (and with much lower probability from more than two nuclei), the resulting spectra are easy to analyze even for the dipolar couplings between nuclei that occupy crystallographically equivalent sites. However, the excitation efficiency of the 4I-quantum coherences is poor and represents a serious drawback of the experiment. On the other hand, the excitation of the double-quantum coherence is more efficient, but it can arise from a coupled pair of nuclei as well as from a single nucleus. Therefore a special care has to be taken to maximize the first and minimize the second contribution and an analysis of the dipolar couplings between nuclei that occupy crystallographically equivalent sites is again not so clear. In this work we present a homonuclear correlation experiment, in which we extend the rotary resonance approach to build a HORROR6 -like double-quantum filtered experiment for half-integer quadrupolar nuclei and with which double-quantum coherences that arise from a single spin can be very efficiently suppressed. We apply the experiment to two aluminophosphate molecular sieves to investigate the aluminum (I = 5/2) subnetworks within them and to demonstrate and investigate the properties of the proposed experiment.
3
II.
THEORETICAL BACKGROUND
In this section we shall briefly describe the spin dynamics of a dipolar-coupled pair of spin-5/2 nuclei under low amplitude rf irradiation and MAS conditions. The discussion will be limited to the cases that satisfy the relation νrf < νR ¿ νQ = 3e2 qQ/2I(2I − 1)h. Here νrf = γBrf , νR is the sample rotation frequency, and νQ is the quadrupole frequency. In the rotating frame the Hamiltonian of an IS spin system can be written as H = νeQI (t)(3Iz2 − I(I + 1)) + νeQS (t)(3Sz2 − S(S + 1)) − νrf Ix − νrf Sx + νeD (t)(2Iz Sz − Ix Sx − Iy Sy ).
(1)
In an axially symmetric electric field gradient (EFG) the time-dependent quadrupole frequency is νeQ (t) =
1 ν (3 cos2 θQ (t) 12 Q
− 1) =
1 ν (G1 12 Q
cos(2πνR t) + G2 cos(4πνR t)),
(2)
with 0 0 G1 = 23 sin 2θm sin 2θQ , G2 = − 32 sin2 θm sin2 θQ .
(3)
0 The polar angles θQ (t) and θQ describe the orientation of EFG principal axes system within
laboratory and MAS frames, respectively, and θm is the magic angle. In a similar way a time-dependent magnitude of the dipolar coupling can be expressed as νeD (t) = νD (G1 cos(2πνR t) + G2 cos(4πνR t)),
(4)
3 0 with the dipole frequency νD = µ0 γ 2 h ¯ /16π 2 rIS . Of course, a different polar angle θD enters
the expressions for G1 and G2 . Magic angle spinning causes νeQ to oscillate between positive and negative values. This has a profound effect on the spin locking of central-transition coherence of quadrupolar nuclei. The nature of the spin locking is, namely, determined by the rate of the passages from positive to negative νeQ 15 . The rate of the passages is described by the parameter α = ν12 /νQ νR . When α À 1 the zero crossings are slow and described as adiabatic. In this limit the central-transition coherence is periodically demagnetized and remagnetized, giving rise to an echo-like time dependence of the spin-locking signal. When zero crossings are fast such that α ¿ 1, the passages are characterized as sudden, and do not affect the central-transition coherence. In the intermediate regime the spin-locking efficiency is greatly reduced. 4
For the forthcoming analysis it is useful to express the Hamiltonian in Eq. (1) with fictitious spin-1/2 operators19 . When central-transition coherence is efficiently spin-locked (α ¿ 1) and away from rotary-resonance conditions ((I + 1/2)νrf 6= νR , 2νR , ...) we can keep only terms associated with the central transition and ignore the terms that connect the central-transition coherence to the satellite-transition coherences13 : ³
´
³
´
H ≈ ν1CT Ix34 + Sx34 + νeD (t) 2Iz34 Sz34 − 9Ix34 Sx34 − 9Iy34 Sy34 .
(5)
Here ν1CT = (I +1/2)νrf = 3νrf is central-transition nutation frequency in selective-excitation 34 34 regime, and Ix,y,z and Sx,y,z are fictitious spin-1/2 operators associated with the central
transition of spin-5/2 nuclei. Let us now suppose that the low amplitude spin-lock pulse is preceded and followed by selective π/2 pulses, out of phase with the spin-lock pulse by ±π/2. The dynamics of the IS spin system can then be described by the evolution of a density matrix, which is governed by the propagator ³
´
³
´
³
´
Ue = exp i π2 (Iy34 + Sy34 ) exp −i2πHτmix exp −i π2 (Iy34 + Sy34 ) ³
(6)
´
f = exp −i2π Hτ mix .
Here τmix is the duration of the spin-lock (mixing) pulse that is usually set to a multiple of f is the effective (tilted) Hamiltonian of the form the sample rotation period τR , and H ³
´
³
´
f = exp i π (I 34 + S 34 ) H exp −i π (I 34 + S 34 ) H y y 2 y 2 y ³
= ν1CT (Iz34 + Sz34 ) − νeD (t) 9Iz34 Sz34 − 74 (I+34 S−34 + I−34 S+34 ) −
11 34 34 (I+ S+ 4
´
+ I−34 S−34 ) .
(7)
The dipolar coupling part in the above Hamiltonian is composed of zero- and doublequantum terms. The terms respond differently to the transformation into an interaction representation, which is realized by an operator ³
³
´
´
U (i) = exp iν1CT Iz34 + Sz34 τmix .
(8)
While the zero-quantum terms remain unchanged, the double-quantum terms are multiplied by factors exp(±2iν1CT τmix ). If the rf amplitude of the spin-lock pulse satisfies the relation ν1CT =
νR , 2
(9)
the above time-dependent factors cancel with time-dependent factors exp(±iνR τmix ) from νeD (t) and, in the interaction frame, we get a time-independent double-quantum Hamiltonian f(i) = H
11 ν G 8 D 1
³
´
I+34 S+34 + I−34 S−34 .
5
(10)
Eq. 9 and Eq. 10 thus represent a condition and a resulting Hamiltonian for an efficient rotary-resonance homonuclear recoupling between spin-5/2 nuclei. To investigate the dynamics of the IS spin system, we have to describe the time-evolution of the density matrix under the above double-quantum Hamiltonian. Let |mI i and |mS i be the eigenstates of fictitious spin-1/2 operators Iz34 and Sz34 , respectively, and let us denote 4 possible product functions |mI i|mS i as |1i = | 12 i| 12 i |2i = | 12 i|− 12 i
(11)
|3i = |− 12 i| 12 i |4i = |− 12 i|− 12 i.
In the vector space that is spanned by the above product functions, we can express the double-quantum Hamiltonian with a single fictitious spin-1/2 operator as f(i) = aJ 14 . H x
Here a =
11 ν G. 4 D 1
(12)
Because the Hamiltonian affects only the central-transition coherence,
the initial reduced density matrix can be written as σ(0) = σ (i) (0) = Iz34 + Sz34 = 2Jz14 .
(13)
A commutation relation that introduces Jy14 , i.e. [Jz14 , Jx14 ] = iJy14 , is cyclic, therefore the evolution of the density matrix in time is σ (i) (τmix ) = exp (−iaJx34 τmix ) 2Jz34 exp (iaJx34 τmix ) = 2Jz34 cos(aτmix ) − 2Jy34 sin(aτmix ) ³
´
(14)
= (Iz34 + Sz34 ) cos(aτmix ) + i I+34 S+34 − I−34 S−34 sin(aτmix ). As we can see, a ‘bracketed’ spin-lock pulse at the rotary-resonance condition of Eq. 9 can excite double-quantum coherences in a system of dipolar-coupled spin-5/2 nuclei. It can, of course, also convert double-quantum coherences to the zero-quantum one. Note that the rf field satisfying a rotary-resonance condition for spin-5/2 nuclei is relatively weak for moderate spinning frequencies (10 - 15 kHz), which can lead to sensitivity of the recoupling efficiency to frequency offsets. In the above calculation we have neglected this effect but we will discuss it along with experimental results. The evolution of the double-quantum coherence in an experiment is also affected by relaxation and by multi-spin interactions. 6
III.
PULSE SEQUENCES
We can use two ‘bracketed’ spin-lock pulses to build a double-quantum homonuclearcorrelation experiment for spin-5/2 nuclei. The sequence of pulses for the proposed experiment is shown in Fig. 1 (a). The first ‘bracketed’ spin-lock pulse excites double-quantum coherences in a system of dipolar-coupled spin-5/2 nuclei. The double-quantum coherences are allowed to evolve for a time t1 before they are converted to zero-quantum coherences with the second ‘bracketed’ pulse. The signal is acquired after the selective π/2 pulse. Note that in the calculation, phases of selective π/2 pulses that preceded and followed the spin-lock pulse were of the opposite sign, while in the experiment they are equal. This slightly simplifies the experiment, but doesn’t affect its performance. Test measurements showed that the performance of the experiment was better if the phase of the spin-lock field was shifted by π in the middle of the pulse. In this way single-quantum terms that appear because of offset effects (due to chemical and quadrupolar shifts) are removed from the average Hamiltonian and thus the excitation of the double-quantum coherences is made more efficient. The validity of the calculation in the previous section was limited to the case of α ¿ 1. If this condition was violated, for example for nuclei with small quadrupolar coupling, terms that connect the central transition with the first satellite transition could no longer be excluded from the Hamiltonian in Eq. 5. As a consequence, the density matrix in Eq. 14 could contain satellite-transition single-quantum terms I±23 and I±45 and double-quantum terms I±24 and I±35 . In the following, let us refer to the former terms as satellite-transition coherences and to the later terms as single-spin double-quantum coherences. With the above-presented experiment we cannot distinguish between the single-spin double-quantum coherences and double-quantum coherences arising from a coupled pair of identical nuclei. Therefore the presence of the former prevents the analysis of dipolar couplings among identical spins. Recently Kwak and Gan demonstrated, that a selective π pulse can very efficiently transform a satellite-transition coherence to a single-spin double-quantum coherence20 . They used such a pulse to prepare double-quantum-filtered STMAS experiment. We can use similar approach to edit the double-quantum homonuclearcorrelation experiment. Selective to the central transition, a π pulse inverts the occupancies of | ± 1/2i states and leaves other spin states undisturbed. This can be very conveniently seen in the matrix representation in Fig. 2. In the language of fictitious spin-1/2 operators, 7
the action of a selective π pulse can be described as I+34 S+34 ↔ I−34 S−34 I±24 ↔ I±23
,
S±24 ↔ S±23
I±35 ↔ I±45
,
S±35 ↔ S±45 .
(15)
Complete description of the action of a selective π pulse can be deduced from commutation relations presented by Vega19 . Eq. 15 shows, that a selective π pulse interchanges a two-spin +2Q coherence with a −2Q coherence, whereas a single-spin ±2Q coherences are interchanged with ±1Q satellite-transition coherences. As a consequence, by adding a selective π pulse to a homonuclear-correlation double-quantum experiment coherence transfer pathways of the single- and two-spin contributions become resolved. An adequate phase cycling can then suppress the unwanted pathways and select just the required one. It should be noted here that selectivity is in fact never exact in case of a polycrystalline material. Since the frequency of the satellite transition depends substantially on the orientation of the EFG principal axes system, in a polycrystalline sample there are always crystallites for which satellite and central transitions are close in frequency and for which both transitions are thus affected by a ‘selective’ π pulse. However, the fraction of such crystallites is small and in practice the deviation from the exact selectivity is usually not problematic. The pulse sequence with a selective π pulse and coherence transfer pathways for two-spin and single-spin contributions are presented in Fig. 1 (b - d). The single-spin contributions can be eliminated if pathways 0 → ±2 → ∓2 → 0 → −1 are selected and pathways 0 → ±2 → ±1 → 0 → −1 and 0 → ±1 → ±2 → 0 → −1 are suppressed. The later coherence transfer pathway reminds us that because of the π pulse we can get single-spin double-quantum coherences from the satellite-transition coherences, i.e. that we can get unwanted double-quantum coherences even if they were not excited by the ‘bracketed’ spinlock pulse.
IV.
EXPERIMENTAL RESULTS
The performance of the proposed experiments was demonstrated on aluminophosphate molecular sieves AlPO4 -31 and AlPO4 -14. Aluminum MAS spectra with sketches of framework structures for both materials are shown in Fig. 3 and values of isotropic chemical 8
shifts and quadrupolar coupling constants for individual resonance lines are listed in Table I. Aluminum nuclei within calcined AlPO4 -31 occupy a single 4-coordinated crystallographic site21 . Each nucleus ‘sees’ three identical nuclei at the range of distances between 4 and 5 ˚ A, 8 identical nuclei at the range of distances between 5 and 6 ˚ A, etc. Because of a sharp and strong resonance line and because of short aluminum spin-lattice relaxation time, which enables fast repetition of experiments with the repetition delay of 0.1 s, calcined AlPO4 -31 is a convenient probe material for initial tests. Aluminum nuclei within as-synthesised AlPO4 14 occupy two 4-, one 5- and one 6-coordinated crystallographic sites22 and have relatively large differences in their isotropic chemical shifts and quadrupolar coupling constants. Along with Al–O–P–O–Al connectivities, which are common for all aluminophosphate molecular sieves, in AlPO4 -14 there are also some Al–O–Al connectivities due to edge sharing AlO6 and vertex sharing AlO5 and AlO6 polyhedra. The variety of quadrupolar coupling constants, isotropic chemical shifts and distances among nearest aluminum nuclei make AlPO4 -14 a very demanding test material for a newly proposed experiment.
A.
Initial tests on AlPO4 -31
Efficient excitation of double-quantum coherences with the above presented pulse sequences relies on an adjustment of the central-transition nutation frequency to the rotaryresonance condition and on an optimization of the mixing time. The easiest and the fastest way to determine the ‘resonant’ central-transition nutation frequency in a real sample is to use a simple spin-lock experiment13 , very insensitive to the selection of the spin-lock time. In the experiment, in which a selective π/2 pulse is immediately followed by a weak spin-lock pulse and a detection period, we monitor the amplitude of the central-transition coherence as a function of the spin-lock rf field. When the nutation frequency is away from the rotary-resonance condition the central-transition coherence is efficiently spin-locked and the amplitude of the recorded signal is high. However, when the nutation frequency approaches the double-quantum recoupling condition, central-transition coherence is partly transformed to the double-quantum coherence and thus the signal amplitude drops. Position of the minimum precisely determines the ‘resonant’ central-transition nutation frequency. Results of spin-lock experiments on AlPO4 -31 at several sample rotation frequencies are shown in Fig. 4. In all curves at least two minima can be observed. The first minimum 9
at ν1CT = νR /2 is indicative of the double-quantum homonuclear recoupling, whereas the second, deeper minimum at ν1CT = νR is due to the recoupling of homo- and heteronuclear dipolar interactions, chemical shift anisotropy and second order quadrupolar interaction14 . The rotary-resonance condition associated with the later minimum is not convenient for a homonuclear correlation experiment, because the efficiency of the double-quantum-coherence excitation could depend severely on chemical shift and quadrupolar parameters. Results presented in Fig. 4 also show that in general the intensity of the spin-locked signal increases and the depth of the first minimum decreases with rotation frequency. In other words, the spin-lock efficiency increases and the efficiency of rotary-resonance recoupling decreases with higher sample rotation frequency. Once the ‘resonant’ nutation frequencies were determined, pulse sequences presented in Fig. 1 (a) and (b) could be tested. First the evolution of double-quantum coherences with mixing time τmix was monitored at several sample rotation frequencies. For this purpose t1 was fixed to a small value, e.g. zero or one rotation period, π/2 and π pulses were kept constant, whereas ‘resonant’ nutation frequency followed the rotation frequency. The results obtained by two pulse sequences are shown in Fig. 5 (a) and (b). Let us denote the pulse sequences without and with the selective π pulse as pulse sequences 1 and 2, respectively. From Fig. 5 we can learn immediately that pulse sequence 2 performs better than pulse sequence 1. The time evolution recorded with sequence 2, namely, always starts at zero amplitude for τmix = 0 (as it should according to Eq. 14) and is nearly independent of the sample rotation frequency. The ‘irregular’ evolution of double-quantum coherences recorded with pulse sequence 1 is induced most probably by the presence of unwanted single-spin contributions. Experiments employing a fixed mixing time and varying nutation frequency (Fig. 5 (c) and (d)) demonstrate even more clearly that pulse sequence 2 is superior to pulse sequence 1. Experiments are similar to constant-time spin-lock experiments presented in Fig. 4, only that they show the amplitude of the double-quantum coherence instead of the centraltransition coherence as a function of the central-transition nutation frequency. As we can see, with pulse sequence 2 the signal, which is proportional to the amount of the excited double-quantum coherences, is detected only when nutation frequency is close to the rotaryresonance condition. At higher or lower nutation frequencies double-quantum coherence is not excited. On the contrary, pulse sequence 1 can excite double-quantum coherence even far 10
away from the rotary-resonance condition, which means that it excites also unwanted singlespin double-quantum coherences. The amount of the unwanted coherences is increasing with the sample rotation frequency and at rotation frequency of 15 kHz the signal of the two-spin contributions is hardly detected in the strong ‘background’ of single-spin contributions. The mixture of single-spin and two-spin double-quantum coherences is also the source of phasing problems in spectra recorded with pulse sequence 1, whereas spectra recorded with sequence 2 are easily phased and maintain the same phase for the whole range of central-transition nutation frequencies.
B.
2D experiments on AlPO4 -14
In case of AlPO4 -31 we were dealing with aluminum nuclei occupying only one crystallographically distinct site, the quadrupolar coupling was relatively weak, and we were always working on resonance. In case of AlPO4 -14, however, values of quadrupolar coupling constants range between 1.57 and 5.61 MHz, while isotropic chemical shifts in magnetic field of 14.1 T are spread over 7 kHz. Measurements on AlPO4 -14 will thus provide a true test, whether the pulse sequence 2 can efficiently excite double-quantum coherences in a coupled-spins system, in which spins have much different quadrupolar coupling constants and isotropic chemical shifts. But before discussing the performance of pulse sequence 2, let us first take a look at Fig. 6 that shows results of 9 spin-lock experiments on AlPO4 -14. Between successive experiments the carrier frequency was increased in steps of one kHz. Fig. 6 thus presents the aluminum central-transition coherences in AlPO4 -14 as functions of central-transition nutation frequency and frequency offset. The sample rotation frequency was 10 kHz, therefore we shall be interested in the local minimum at ν1CT = 5 kHz. As we can see, with an increasing offset the position of the minimum moves to lower values of ν1CT . This is understandable, q
because now it is
(ν1CT )2 + (∆ν0 )2 rather than ν1CT that has to satisfy the rotary-resonance
condition. When an offset becomes so large that the rotary-resonance condition cannot be fulfilled any longer, the minimum disappears. The overall signal amplitude decreases and at even larger offsets eventually becomes negligible. In Fig. 6 we can compare results of the spin-lock experiments for peaks 1, 3, and 4 belonging to aluminum nuclei on sites Al1 , Al3 , and Al4 . Because isotropic positions of peaks 1 and 2 differ only slightly, their behavior is 11
very similar and thus the experimental results for peak 2 are not shown. Presented results indicate that if the carrier frequency was in the center of the spectrum, a (‘bracketed’) spinlock pulse could perhaps excite double-quantum coherences between aluminum nuclei at all four crystallographic sites. If, however, the carrier frequency was offset by few kHz in either direction, certainly not all double-quantum coherences could be excited. At this point it is worth noting that the effect of an offset ∆ν0 depends on the ratio ∆ν0 /ν1CT or, since through the rotary-resonance condition ν1CT is determined by νR , on the ratio ∆ν0 /νR . This means that one could reduce problems connected with frequency offsets by using faster sample rotation. Unfortunately, the series of spin-lock experiments presented in Fig. 4 showed, that the recoupling efficiency decreases with faster rotation, which implies that the selection of the sample rotation frequency is compromised between the sufficient efficiency of homonuclear recoupling and tolerable offset effects. A two-dimensional double-quantum homonuclear correlation spectrum of aluminum in AlPO4 -14 was recorded with pulse sequence 2 at 12.5 kHz rotation frequency (Fig. 7). The spectral width in the indirectly detected dimension was equal to the rotation frequency. If the steps in t1 were not synchronized with rotation, the sensitivity of the measurements would reduce, while numerous rotation sidebands would very much complicate the qualitative analysis of the spectrum. The aluminum central-transition signals of AlPO4 -14 cover about 8 kHz wide region. This means that the signals of the two-dimensional double-quantum spectrum will cover about 16 kHz wide region along indirectly detected dimension. Since our MAS probehead used with 14.1 T magnet doesn’t enable that fast sample rotation, our decision about synchronous measurement necessarily implied that there will be folded signals present in the two-dimensional spectrum. However, as we can see in Fig. 7, this is not a serious obstacle, because the resolution of the spectral lines is very good and we can clearly identify the folded lines. Starting at 6 kHz and going to lower frequencies along the indirectly detected dimension of the two-dimensional spectrum in Fig. 7, we can assign cross-peaks to pairs of coupled nuclei occupying sites 2–2, 4–4 (folded signal), 1–3, 2–3, 1–4, 2–4, 3–4, 1–1 (folded signal), 1–2 (folded signals). By considering Table II, in which distances among nearest aluminum nuclei are collected, we can see that in this measurement we detected homonuclear dipolar coupling between aluminum nuclei that are up to about 5.5 ˚ A apart. The dipolar coupling between two nearest Al3 nuclei, which are 6.7 ˚ A apart, was not detected. The absence of 12
any 3–3 cross-peak confirms, that the applied pulse sequence 2 indeed efficiently filters out single-spin double-quantum coherences. The intensities of cross-peaks are not all equal and thus potentially provide a qualitative information about the distances among coupled aluminum nuclei. We can take a look at traces through the two-dimensional spectrum presented in Fig. 8 and at distances listed in Table II. Although cross-peak intensities do not reflect small differences between larger internuclear distances, they do clearly show that Al3 –Al4 and Al4 –Al4 distances are by far the shortest and thus suggest that there are Al3 –O–Al4 and Al4 –O–Al4 connectivities present in AlPO4 -14. Often such a qualitative information can already be very helpful to someone, who wants to determine connectivities within the aluminophosphate framework of a molecular sieve. Related to the problem of folding, we can highlight a very nice property of pulse sequence 2, namely, that it can apparently scale the magnitude of the chemical shift and quadrupolar interaction in the indirectly detected dimension by an arbitrarily selected factor between 1 and 0. Let us limit to an IS spin pair that is subjected only to isotropic chemical shift interaction and let us only take a look at what happens to a +2Q coherence that was excited by a ‘bracketed’ spin-lock pulse. Schematically we can describe the action of a selective π pulse and evolution in t1 under the action of isotropic chemical shift as π
t
1 I+ S+ −→ I− S− −→ I− S− ei(νI +νS )t1 .
(16)
Now suppose that after the ‘bracketed’ spin-lock pulse we first let double-quantum coherence to evolve for time kt1 , then apply a selective π pulse, and finally let double-quantum coherence to evolve for another time (1 − k)t1 (Fig. 9). In analogy with the above expression we now get kt
π
1 I+ S+ −→ I+ S+ e−i(νI +νS )kt1 −→ I− S− e−i(νI +νS )kt1
(1−k)t1
−→ I− S− e−i(νI +νS )kt1 ei(νI +νS )(1−k)t1 = I− S− ei(νI +νS )(1−2k)t1 .
(17)
As we can see, by applying a selective π pulse after a k-th part of t1 evolution we scale isotropic chemical shift by a factor 1−2k. An experimental proof is presented in Fig. 9. Two double-quantum homonuclear correlation spectra were recorded under the same conditions (except for the number of scans) as the spectrum in Fig. 7, but with k = 1/8 and k = 1/4. As we can see, the spectral width along the indirectly detected dimension is scaled by a 13
factor of 3/4 and 1/2, respectively. The folding is avoided in this way, but the resolution is lowered as well. The final example deals with the effect of frequency offset again. Fig. 10 shows two doublequantum homonuclear correlation spectra that were recorded under similar conditions as the spectrum in Fig. 7, but this time the carrier frequency was offset by ±3250 Hz from the middle of the aluminum spectrum. As a result, in one case only signals that correspond to nuclei on two 4-coordinated sites, Al1 and Al2 , can be detected, and in the other case only signals that correspond to nuclei on 5- and 6- coordinated sites, Al3 and Al4 , can be detected. It seems that with rotation frequency of 12.5 kHz and the corresponding centraltransition nutation frequency of 6.25 kHz, frequency offsets in excess of about 4 kHz cannot be ‘overcome’.
V.
CONCLUSIONS
In this contribution we have presented a double-quantum homonuclear correlation experiment for dipolar-coupled half integer quadrupolar nuclei. The experiment is based on rotary resonance dipolar recoupling and uses ‘bracketed’ spin-lock pulses to excite the doublequantum coherence and later to convert it to the zero-quantum one. A selective π pulse resolves coherence transfer pathways of double-quantum coherences arising from dipolarcoupled spins and from a single spin, so that the later can be efficiently suppressed by phase cycling. The experiment was tested on an aluminophosphate molecular sieve AlPO4 -14, a material with a diversity of aluminum quadrupolar coupling constants, isotropic chemical shifts and homonuclear distances. In a two-dimensional spectrum aluminum dipolar couplings with internuclear distances between 2.9 and 5.5 ˚ A were detected. A qualitative analysis of cross-peak intensities allowed the discrimination of homonuclear aluminum pairs with largest dipolar couplings, i.e. with smallest internuclear distances, and confirmed the existence of some direct Al–O–Al connectivities in addition to common Al–O–P–O–Al connectivities. A qualitative agreement of cross-peak intensities with actual distances also suggests that the dipolar couplings acting here are most probably direct couplings with negligible relay through other dipolar couplings. In the presented two-dimensional experiment it is useful to equalize the discrete steps along t1 dimension with the sample rotation period. This can sometimes lead to signal14
folding, which can, however, be avoided by an apparent scaling of interactions by an arbitrarily selected factor between 1 and 0. The apparent scaling is obtained simply by shifting the selective π pulse from the beginning of the t1 evolution period to the selected fraction of the evolution period. Although the above experiments used weak rf fields, frequency offset didn’t affect their performance crucially. With aluminum central-transition nutation frequency of 6.25 kHz, double-quantum coherences were efficiently excited in an 8 kHz wide spectrum. If wider spectra were expected, larger nutation frequency and thus faster sample rotation should be employed. With faster rotation the efficiency of the rotary resonance recoupling decreases, which means that experiments may become more time consuming but still operating. Smaller internuclear distances and thus larger dipolar couplings should allow an application of higher sample rotation frequencies; the offset effects would thus be reduced. The experiment should be less exposed to offset problems also in a case of spin-3/2 nuclei, where the ‘resonant’ frequency ν1 is a quarter of the sample rotation frequency instead of a sixth part as for spin-5/2 nuclei. Perhaps the most serious problem that one can encounter, when applying the experiment to a real sample, is very fast relaxation of double-quantum coherences during spin-lock pulses. If this was the case, spin-lock pulses should be shortened and the sensitivity would decrease. Apart from these possible limitations, however, the proposed experiment presents most qualities needed - it efficiently excites and reconverts double-quantum coherences, its optimization is quite simple, and it is not highly dependent on the exact rotary resonance condition.
Acknowledgments
We would like to thank Alenka Risti´c and Nataˇsa Novak Tuˇsar for preparation of the samples. The Slovenian Ministry of Education, Science, and Sport is acknowledged for the financial support through research projects Z1-3277 and P0-0516-0104. The French and Slovenian Ministries of Science have funded the collaborative project between laboratories in Ljubljana and Strasbourg under the Integrated Action Program 03821 WE.
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16
Tables
TABLE I: Isotropic chemical shifts and quadrupolar coupling constants of aluminum sites in AlPO4 31 and AlPO4 -14. Notation of aluminum sites in AlPO4 -14 agrees with notation of resonance lines in Fig. 3 (b). Al site
iso [ppm] δCS
CQ [MHz]
AlPO4 -31
1
37.3
1.57
AlPO4 -14
1
42.7
1.72
2
43.5
3.90
3
27.1
5.61
4
-1.3
2.55
TABLE II: Distances (in ˚ A) among nearest aluminum neighbors in AlPO4 -14. Al1
Al2
Al3
Al4
Al1
4.2
4.3
4.2
5.3
Al2
4.3
4.4
5.1
5.5
Al3
4.2
5.1
6.7
3.6
Al4
5.3
5.5
3.6
2.9
17
Figure captions
FIG. 1: Pulse sequences and coherence transfer pathways for double-quantum homonuclearcorrelation experiments on spin-5/2 nuclei. Both pulse sequences, (a) and (b), use ‘bracketed’ spin-lock pulses for the excitation of double-quantum coherences and for their conversion to zero-quantum coherences. All π/2 pulses are central-transition selective pulses (typical length of a π/2 pulse in our experiments was 22 µs). The central-transition nutation frequency during spin-lock pulses satisfies the rotary-resonance condition of Eq. 9. In pulse sequence (a) doublequantum coherences during t1 evolution are selected by 4-step phase cycling (φ1 = 0, 90, 180, 270◦ , φ2 = φ1 + 90◦ , φR = 0, 180◦ ). In pulse sequence (b) the phase of the selective π pulse is additionally incremented in 90◦ steps resulting in a total phase cycle of 16 steps (optionally, the phase of the read-out π/2 pulse can also be incremented in 180◦ steps). This phase cycling selects coherence transfer pathways of two-spin double-quantum coherences (c) and rejects coherence transfer pathways of single-spin contributions (d).
FIG. 2: Effect of a selective π pulse on different terms (coherences) of a spin-5/2 density matrix. Coherences in bold are those that are excited prior to selective π pulse with the largest probability. Double arrows indicate that the π pulse interchanges satellite-transition single-quantum and double-quantum coherences but doesn’t mix them with central-transition coherences.
18
FIG. 3: Aluminophosphate frameworks and aluminum MAS spectra of (a) AlPO4 -31 and (b) AlPO4 -14. In (a) aluminophosphate framework built of alternating AlO4 and PO4 tetrahedra is shown, whereas in (b) PO4 tetrahedra are omitted for clarity. In AlPO4 -14 aluminum sites Al1 and Al2 are 4-, site Al3 is 5-, and site Al4 is 6-coordinated. Note the presence of Al3 –O–Al4 and Al4 –O–Al4 connectivities due to vertex and edge sharing AlOn polyhedra. MAS NMR spectra were recorded in the external magnetic field of 14.1 T. In case of AlPO4 -14 individual lines of the decomposed spectrum are also shown. Resonance lines (and corresponding aluminum sites) are numbered according to the order in which they appear in the MAS spectrum. Note that this numbers don’t correspond to crystallographic notation of aluminum sites as usually found in literature. The line at about 8 ppm, which is not numbered, belongs to an impurity.
FIG. 4: Aluminum central-transition coherence in AlPO4 -31 at different rotation frequencies as a function of central-transition nutation frequency. Vertical dashed and dotted lines indicate the position of the rotary-resonance condition ν1CT = νR /2 and ν1CT = νR , respectively. Experiments were performed at external magnetic field of 11.7 T using constant spin-lock time of 3.2 ms.
FIG. 5: Aluminum double-quantum coherence in AlPO4 -31 at different rotation frequencies as a function of mixing time (a,b) and central-transition nutation frequency (c,d). Measurements were performed with double-quantum homonuclear recoupling experiments without (a,c) and with (b,d) a selective π pulse in the external magnetic field of 14.1 T. Vertical dashed lines indicate the position of the rotary-resonance condition ν1CT = νR /2.
FIG. 6: Results of spin-lock experiments on AlPO4 -14 providing aluminum central-transition coherences as functions of central-transition nutation frequency and frequency offset. Experiments were performed at constant spin-lock time of 3.2 ms and at sample rotation frequency of 10 kHz. The carrier frequency for 9 successive experiments was increased each time for 1 kHz - positions of the carrier frequency with respect to the spectrum are indicated by arrows in the left column. Each spectrum in this column was recorded by a spin-lock experiment using a different carrier frequency. Intensities of central-transition peaks 1, 3 and 4 as functions of rf amplitude and offset are shown in columns 2–4. Thick curves indicate (close to) on-resonance cases.
19
FIG. 7: Double-quantum homonuclear correlation spectrum of aluminum in AlPO4 -14. The spectrum on the left was recorded at 14.1 T magnetic field and 12.5 kHz rotation frequency using mixing time of 560 µs. Hypercomplex approach was used to obtain pure absorption two-dimensional spectrum. Carrier frequency was set to the middle of the spectrum. The number of scans was 2400, repetition delay 0.25 s, and the number of increments along indirectly detected dimension 28. Total acquisition time was 10 hours. The spectrum on the right is not a result of an experiment, but is artificially obtained from the spectrum on the left by moving the folded lines to positions, at which they would appear if spectral width along indirectly detected dimension was larger and there was no folding.
FIG. 8: Traces through a two-dimensional double-quantum homonuclear correlation spectrum of AlPO4 -14. Numbers on the left of each trace indicate which coupled sites it corresponds to.
FIG. 9: Pulse sequence for double-quantum homonuclear-correlation experiment that apparently scales chemical shift and quadrupolar interaction along the indirectly detected dimension (a) and corresponding two-dimensional spectra of aluminum in AlPO4 -14 (b,c). If the selective π pulse doesn’t follow the ‘bracketed’ spin-lock pulse immediately as in pulse sequence 2 but is applied yet at kt1 , the interactions along the indirectly detected dimension are apparently scaled by a factor 1 − 2k. In experiments (b) and (c) k was equal to 1/8 and 1/4, respectively, what resulted in scaling factors 3/4 and 1/2. Other experimental conditions, except for the number of scans, were equivalent to conditions, under which the spectrum in Fig. 7 was recorded. The number of scans was 1200 and the total experimental time was 5 hours.
FIG. 10: Double-quantum homonuclear correlation spectra of aluminum in AlPO4 -14 that were recorded under similar experimental conditions as the spectrum in Fig. 7, but with the carrier frequency that was offset by ±3250 Hz from the middle of the aluminum spectrum. Zero in the directly detected dimension indicates the position of the carrier frequency.
20
φ1 φ1 φ2 φ2+π
X
Y −Y
XX φR
(a) π/2 τmix π/2
π/2
t1
φ1 φ1 φ3 φ2 φ2+π
X
Y −Y
t2
XX φR
(b) τmix
π
t1
t2 +2 +1 0
(c)
-1 -2
+2 +1 0
(d)
-1 -2
G. Mali et al., Fig. 1
21
| 23 i
| 21 i
| − 21 i | − 23 i | − 52 i
ST2
2Q ⇔
3Q
4Q
..
ST ⇔
2Q
3Q
..
CT
2Q m
.
.
m
| 52 i . h 52 | . . h 32 | 1 h2| h− 12 | 3 h− 2 | PSfrag replacements 5 h− 2 |
CT∗
..
.
ST ..
G. Mali et al., Fig. 2
22
.
5Q 4Q 3Q m 2Q ST2 .. .
(a) P Al
P Al
P
Al Al
P
P
Al
Al P
Al P
Al
P
50
40
30
20
10
0
-10
ppm (b) Al1
Al2 Al2
4
Al1
1
Al4 Al3
2 3
Al3 Al4 Al1 Al2 Al2
Al1
50
40
30
G. Mali et al., Fig. 3
23
20 ppm
10
0
-10
νR = 5 kHz
10 kHz
Intensity
15 kHz
20 kHz
25 kHz
0
5
10
15 CT
ν1
20
[kHz]
G. Mali et al., Fig. 4
24
25
30
νR = 5 kHz
(a)
0
1
(c)
0
2 τmix [ms]
10 kHz
10 kHz
15 kHz
15 kHz
3
4 0
νR = 5 kHz
5 ν1CT [kHz]
νR = 5 kHz
(b)
1
2 τmix [ms]
3
4
νR = 5 kHz
(d)
10 kHz
10 kHz
15 kHz
15 kHz
10
0
5 ν1CT [kHz]
G. Mali et al., Fig. 5
25
10
Al1
1 2
4
3
2
0 -2 kHz
Al3
Al4
4
-4
0
5 10 ν1CT [kHz]
0
5 10 ν1CT [kHz]
G. Mali et al., Fig. 6
26
0
5 10 ν1CT [kHz]
4
1
3 2
- folded
F1 [kHz] -4 -3 -2 -1 0 1 2 3 4 5 6
- folded 4
3
2
F1 [kHz] -4 -3 -2 -1 0 1 2 3 4 5 6
1 0 -1 -2 -3 -4 -5 F2 [kHz]
G. Mali et al., Fig. 7
27
2
1 0 -1 -2 -3 -4 -5 F2 [kHz]
4-4 3-4 2-4 1-4 3-3 2-3 1-3 2-2 1-2 1-1
4
2
0 kHz
-2
G. Mali et al., Fig. 8
28
-4
(a) kt1
F1 [kHz] -4 -3 -2 -1 0 1 2 3 4 5 6
(1-k)t1
(b)
4
t2
F1 [kHz] -4 -3 -2 -1 0 1 2 3 4 5 6 3
2
1 0 -1 -2 -3 -4 -5 F2 [kHz]
(c)
4
G. Mali et al., Fig. 9
29
3
2
1 0 -1 -2 -3 -4 -5 F2 [kHz]
F1 [kHz] -4 -3 -2 -1 0 1 2 3 4 5 6
F1 [kHz] -4 -3 -2 -1 0 1 2 3 4 5 6 1
0 -1 -2 -3 -4 -5 -6 -7 -8 F2 [kHz]
8
7
G. Mali et al., Fig. 10
30
6
5
4 3 2 F2 [kHz]
1 -0 -1
