## 1 Introduction

^{13}C NMR spectroscopy offers a high potential for the quantitative analysis of complex mixtures, due to the broad spectral range and, usually, an absence of interference between peaks. It suffers, however, from poor sensitivity due to the low abundance and small magnetic moment of the ^{13}C nucleus. This sensitivity is further diminished if the conditions of quantitative NMR − inverse-gated-decoupling procedure [1] and a recycling delay between 5- and 10- times the longest spin–lattice relaxation time [2–4] – are employed. In fact, the use of these acquisition conditions may lead to unacceptably long accumulation times to obtain a spectrum with an acceptable signal/noise ratio for a solute in small concentration. The alternative is to add a paramagnetic relaxation reagent [5], such as triacetylacetonatochromium (III) Cr(AcAc)_{3}, which shortens the relaxation times. The unwanted side-effect of adding Cr(AcAc)_{3} is that it leads to line-broadening. Furthermore, adding a paramagnetic relaxation reagent may alter the compound under investigation. Alternative methods have been employed, detecting the signal in non-quantitative conditions for comparing concentrations of molecules with similar relaxation properties [6]. Also, the DEPT sequence [7] can be applied for quantitative analysis when optimised [8], but only to compare carbons of the same nature, which resonate in a small spectral range. Finally, when an accurate ^{13}C NMR quantitative spectrum is needed for a sample with different carbons in a 200 ppm range, the inverse-gated-decoupling sequence is still employed with a recycling delay of about seven times the larger T_{1}, for the different carbons of interest [9]. We propose here to improve the inverse-gated-decoupling sequence so that the recycling delay may be significantly reduced while the results remain quantitative.

## 2 Theory

### 2.1 General description of the magnetisation time course during the recovery delay

We consider a simple IS spin system (I = insensitive: ^{13}C; S = sensitive: ^{1}H), coupled by intramolecular dipole–dipole relaxation. All the development will be done for ^{13}C and ^{1}H, but similar results could be deduced for other nuclei.

The standard inverse-gated-decoupling sequence for quantitative measurements has been developed by Freeman et al. [1]. It is [T − D_{on} – 90° − FID – D_{off}]_{n}, where D_{on} and D_{off} symbolise the switching of the proton decoupler, T is a waiting time allowing the system to return to thermal equilibrium, usually chosen between 5 and 10 T_{1}(^{13}C) [2–4].

During acquisition, I magnetisation is detected in the transverse plane while S is saturated for obtaining a decoupled spectrum. In the same time, 〈I_{z}〉 increases and would reach the steady state value I_{0}(1+ η) if the acquisition time was very long. This may increase the delay T necessary for 〈I_{z}〉 to go back to I_{0} before the next detection. That is why the acquisition time T_{AQ} is usually chosen short and at the end of the acquisition 〈I_{z}〉 = ${\langle {I}_{z}\rangle}_{\text{AQ}}$ is shorter than I_{0}, depending on the length of the acquisition time.

At the end of the acquisition, S is no longer saturated. The behaviour of I and S is described by the two differential equations, first derived by Solomon [10]:

$$\begin{array}{c}\text{d}\langle {I}_{z}\rangle \text{/d}t\text{=}-{R}_{\text{I}}\phantom{\rule{0ex}{0ex}}\text{(}\langle {I}_{z}\rangle \text{\u2212}{I}_{0}\text{)}-{\sigma}_{\text{I}\text{S}}\text{(}\langle {S}_{z}\rangle -{S}_{0})\\ \text{d}\langle {S}_{z}\rangle \text{/d}t\text{=}-{R}_{\text{S}}\phantom{\rule{0ex}{0ex}}\text{(}\langle {S}_{z}\rangle \text{\u2212}{S}_{0})-\phantom{\rule{0ex}{0ex}}{\sigma}_{\text{I}\text{S}}\text{(}\langle {I}_{z}\rangle -{I}_{0}\text{)}\end{array}$$ | (1) |

_{z}and S

_{z}are the z components of the

^{13}C and

^{1}H magnetisations; R

_{I}and R

_{S}are the inverse of the carbon and

^{1}H longitudinal relaxation times respectively. The quantities σ

_{IS}and σ

_{SI}are cross-relaxation terms that, in the case of two spins 1/2, are equal: σ

_{IS}= σ

_{SI}= σ. From these coupled equations, it is possible to find I

_{z}magnetisation at any time t during the delay T:

$$\langle {I}_{Z}\text{(}t\text{)}\rangle {=I}_{\text{0}}{+C}_{+}\text{exp[}\text{\u2212}{\lambda}_{\text{+}}t]{+C}_{-}\text{exp[\u2212}{\lambda}_{-}t\text{]}$$ | (2) |

$${\lambda}_{\pm}=\text{1/2}\{{R}_{\text{I}}\text{+}{R}_{\text{S}}\pm \text{[}({R}_{\text{S}}-{R}_{\text{I}}{\text{)}}^{\text{2}}+4{\sigma}^{\text{2}}{\text{]}}^{\text{1/2}}\text{}}$$ | (3) |

$${C}_{\mp}=\pm \left\{\frac{{S}_{0}\text{}\sigma}{{\lambda}_{+}-{\lambda}_{-}}\right\}\pm \text{(}\text{\u2329}{I}_{z}\text{(}0\text{)}\text{\u232a}-{I}_{0}\text{)}\left(\frac{{\lambda}_{\pm}-{R}_{\text{I}}}{{\lambda}_{+}-{\lambda}_{-}}\right)$$ | (4) |

_{z}(0)〉 is the I

_{z}value at the beginning of the recycle delay T, when S is no more saturated.

Eq. (3) means that I_{z} goes back to equilibrium biexponentially. The relaxation rates λ_{+} and λ_{−} are both functions of R_{I}, R_{S} and σ, with λ_{+} larger than λ_{−}. This means that the corresponding exponential term exp[−λ_{+} t] should vanish more rapidly than exp[−λ_{−} t].

Then, I_{z} returns to equilibrium more rapidly if C_{−} << C_{+}. In that case, equilibrium should be reached for t = 5/λ_{+}.

It is interesting to determine the value of 〈I_{z}(0)〉 for which C_{−} would become null. It is called ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$:

$${\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}={I}_{0}-\left(\frac{{S}_{0}\sigma}{{\lambda}_{+}-{R}_{I}}\right)$$ | (5) |

This equation can be simplified in two extreme situations.

If σ >> |R_{S} − R_{I}|, λ_{+} − R_{I} ≅ σ and${\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}$ ≅ I_{0} − S_{0}.

If σ << |R_{S} − R_{I}|, λ_{+} ≅ R_{S}. The nuclear Overhauser enhancement factor η is defined by the relationship:

$$\text{\eta}=\frac{{S}_{0}\sigma}{{I}_{0}{R}_{\text{I}}}$$ | (6) |

In that case, ${\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}$ can be written:

$$\begin{array}{c}{\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}={I}_{0}\left(\text{1}-\frac{{R}_{I}}{{\lambda}_{+}-{R}_{I}}\text{\eta}\right)\\ \end{array}$$ | (7) |

$$\approx {I}_{0}\left(\text{1}-\frac{{R}_{I}}{{R}_{S}-{R}_{I}}\text{}\text{\eta}\right)\approx {I}_{0}$$ | (8) |

This shows that ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ can vary in a wide range.

In most cases, it should be possible to adapt the NMR sequence so that at the beginning of the delay T, 〈I_{z}〉 = ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ instead of ${\langle {I}_{z}\rangle}_{\text{AQ}}$. This would enable 〈I_{z}〉 to go back to equilibrium faster. The modification will depend on the value of ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ compared to ${\text{\u2329}{I}_{z}\text{\u232a}}_{\text{AQ}}$.

### 2.2 Optimisation of the sequence

Three cases can be considered.

(i) If ${\langle {I}_{z}\rangle}_{\text{AQ}}$ < ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ < I_{0}, it is possible to saturate S during a delay T_{sat} after the acquisition until 〈I_{z}〉 = ${\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}$. In that case, T_{sat} is adjusted so that:

$${I}_{0}\left(\text{1}+\text{\eta}\right)\left[\text{1}-\text{exp}\left(-{R}_{I}({T}_{\text{AQ}}+{\text{T}}_{\text{sat}})\right)\right]={\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}$$ | (9) |

(ii) If − ${\langle {I}_{z}\rangle}_{\text{AQ}}$ < ${\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}$ < ${\langle {I}_{z}\rangle}_{\text{AQ}}$, _{•} I_{z}_{•} can be reduced to the desired value by applying a RF pulse of angle θ.

(iii)If − I_{0}(1 + η) < ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ < −${\langle {I}_{z}\rangle}_{\text{AQ}}$, 〈I_{z}〉 could be amplified by saturating S after the acquisition until 〈I_{z}〉 = − ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$. An inversion pulse would then enable one to obtain the desired value for 〈I_{z}〉.

Carbons with different relaxation parameters may be present in a same sample. In that case, a global optimisation could still be obtained by saturating S during a delay T_{sat} sufficient for increasing ^{13}C magnetisation with the larger value of ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$, then replacing the RF pulse of angle θ by series of selective pulses with adequate flip angles for each carbon.

The corresponding sequence is represented in Fig. 1 for two types of carbons. As this sequence consists in Inverse-Gated Optimised Decoupling, we propose to call it ‘InGOD’.

## 3 Material and methods

N,N-Dimethylacetamide, obtained from Sigma Aldrich, was diluted to 10% (v/v) in deuterated water and analysed in a 5-mm sample tube. All NMR spectra were recorded at 303 K, on a Bruker Avance 400 DPX spectrometer operating at 9.4 Teslas. ^{13}C spectra were recorded at a spectrometer frequency of 100.62 MHz with a 5-mm dual probe. Free induction decays (FIDs) were accumulated in 64-K channels, with a spectral width of 19.38 kHz, an acquisition time of 1.69 s, using 90° read pulses of length 6.8 μs, and eight scans. Proton decoupling was obtained with the WALTZ16 mode. Selective pulses were obtained by applying an attenuated RF field during 133 μs. An exponential apodisation function of 5.0 Hz was applied to the FID prior to Fourier transformation. Phase and baseline were corrected manually. The areas of the peaks were measured by Lorentzian deconvolution with the Bruker program X-WinNMR 2.6. Nuclear Overhauser enhancement factors (η) for the different carbon sites were obtained by comparison of the corresponding peak areas I_{sat} and I_{0} when ^{1}H were decoupled and inverse-gated decoupled respectively [1], waiting 320 s between the end of the acquisition and the read pulse. They are defined as: I_{sat}/I_{0} = 1 + η. The recovery of ^{13}C magnetisation was studied by applying either an inverse-gated-decoupling sequence or the optimised sequence (Fig. 1) and waiting delays from 3 μs to 320 s before the read pulse.

Longitudinal relaxation times of ^{13}C were determined by using an inversion–recovery sequence with the decoupler on continuously. An inversion–recovery sequence was employed for ^{1}H longitudinal relaxation times determination. There, the baseline was corrected automatically and the areas of the peaks were analysed within the Bruker software.

## 4 Results and discussion

At the temperature considered, N,N-dimethylacetamide is composed of three non-equivalent CH_{3} groups and of a carbonyl group. As the relaxation properties of the two N–CH_{3} groups are quite close, we considered only three different spin systems: CH_{3}–C, CH_{3}–N and C=O. Their relaxation times are given in Table 1, together with their nuclear Overhauser enhancement η at steady state while ^{1}H were saturated. For each carbon, the longitudinal relaxation time of the closer ^{1}H group was also reported. The cross-relaxation coefficients σ were deduced from these values via Eq. (6). This enabled the calculation of λ_{+} via Eq. (3). Finally, the value of ${\langle {I}_{z}\left(0\right)\rangle}_{\text{opt}}$ was calculated via Eq. (7) and is reported in Table 1. The value of 〈I_{z}〉 at the end of the acquisition was measured by applying an inverse-gated-decoupling sequence where the delay T is null. It is called ${\langle {I}_{z}\rangle}_{\text{AQ}}$ and is also reported. From these results, and in order to increase the carbon relaxation, we have chosen to enhance all ^{13}C magnetisation during a delay T_{sat} so that for C=O $\langle {I}_{z}\rangle ={\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ at the end of the delay T_{sat.} The length of T_{sat} was deducted from Eq. (9). It is 17.7 s. Then, the ^{13}C magnetisation of the CH_{3} was significantly reduced by applying a selective 90° pulse at their average frequency. Fig. 2 represents the ^{13}C magnetisation time course as a function of T for the optimised pulse sequence (■) compared to the standard inverse-gated-decoupling sequence (○) for the carbonyl group (Fig. 2a) and for a CH_{3} carbon bound to the carbonyl (Fig. 2b). Similar curves as in Fig. 2b were obtained for the other carbons CH_{3}. One observes that, in both cases, magnetisation recovers faster during the delay T when the InGOD sequence is employed. For a given carbon, if one compares this delay T to the T_{1} of the corresponding carbon, it is in a range from 0.17 T_{1,} for the carbonyl, to 3.7 T_{1}, for the carbon C-2, which is much lower than the delay usually recommended for the inverse-gated-decoupling sequence. As a matter of fact, the additional delay T_{sat} makes the experiment longer. In order to assess the efficiency of our optimised version against the standard inverse-gated-decoupling sequence, we compared the global delay T_{sat} + T necessary for magnetisation relaxation to the larger relaxation time T_{1}^{max} (Table 2). Finally, the delay needed for complete recovery of the magnetisation of all carbons is 50.7 s, which is less than 1.8 T_{1}^{max}, thus much less than the delay of 7 T_{1}^{max}, usually recommended for allowing magnetisation of quaternary carbons to return to equilibrium after the inverse-gated-decoupling sequence [9].

**Table 1**

Relaxation properties for the different carbons and for the protons dipolar coupled to them. Comparison of the magnetisation ${\langle {I}_{z}\rangle}_{\text{AQ}}$ at the end of the acquisition to the optimal value ${\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}$ deduced from the relaxation parameters

Carbon | T_{1}(^{13}C) | T_{1}(^{1}H) | η | $\frac{{\langle {I}_{z}\rangle}_{\text{AQ}}}{{I}_{0}}$ | $\frac{{\langle {I}_{z}\text{(}0\text{)}\rangle}_{\text{opt}}}{{I}_{0}}$ |

C-3 (C=O) | 28.8 s | 2.81 s | 0.85 | 0.18 | 0.91 |

C-2 (CH_{3}–N) | 9.04 s | 2.99 s | 1.63 | 0.59 | 0.22 |

C-1 (CH_{3}–C) | 8.62 s | 2.81 s | 1.70 | 0.63 | 0.21 |

**Table 2**

Recovery delay of N,N-dimethylacetamide ^{13}C magnetisation after the saturating delay and the selective pulse for which 〈I_{z}〉 = I_{0} with a 1% precision

Carbon | T | T_{sat} + T |

C-3 (C=O) | 5 s | 22.7 s (0.79 T_{1}^{max}) |

C-2 (CH_{3}–N) | 33 s | 50.7 s (1.8 T_{1}^{max}) |

C-1 (CH_{3}–C) | 21.5 s | 39.2 s (1.4 T_{1}^{max}) |

One should note that for the sample studied, the quaternary carbon is well separated from the CH_{3} groups, which have similar relaxation properties and are in a small range. Therefore, it is very easy to apply a selective pulse on the CH_{3} groups without disturbing the carbonyl magnetisation. If protonated carbons would be in a more wide range, it should be possible to apply a series of selective pulses with optimal angles at the desired frequencies. For a better selectivity, shaped pulses as gaussians should then be used instead of low power rectangular pulses.

## 5 Conclusion

We have enhanced the inverse-gated-decoupling sequence for quantitative analysis to be performed faster. This optimisation can be performed accurately if the longitudinal relaxation times of the cross-relaxing nuclei are known, together with the NOE coefficients. It is particularly efficient in a sample containing quaternary carbons. The quantitative analysis of other carbons can be achieved provided that a selective pulse is applied at their resonance frequency. If this cannot be done, quantitative analysis could be performed in two separate experiments optimised for quaternary and fast relaxing carbons respectively. This method may be applied each time precise quantitative NMR is needed despite the presence of dipolar couplings.