Introduction

Quantitative analysis lies in the core of nuclear magnetic resonance (NMR) spectroscopy. The quantitative NMR measurements can provide accurate information of, either relative or absolute, content of components or species in the sample. It's required in a reliable quantitative NMR experiment that the signal intensity of each magnetically in-equivalent site is proportional to the number of corresponding spins. The simplest and commonly used method in quantitative NMR for 1/2-spin nuclei is the single pulse (SP) experiment, where the spin magnetization is directly excited via a 90° pulse followed by acquisition. To make sure all the spin magnetizations undergo near complete recovery characterized by the longitudinal relaxation with time constant of T₁, the SP experiment must require sufficient recycle delay (> 5 T₁). However, for dilute spin with excessively long T₁, the SP experiment would be very inefficient, and extremely time-consuming. Even with the small angle excitation^[1], the poor acquisition efficiency is hardly to be overcome.

Cross polarization (CP) is one of the most important techniques in solid-state NMR spectroscopy, and it has been commonly used for obtaining spectra of dilute spins under both static and magic-angle spinning (MAS) conditions^[2]. By virtue of the polarization transfer from abundant spins, especially proton, to the coupled dilute spins, the recorded CP NMR spectra have higher signal intensity than that by direct excitation. Besides, because the CP starts from the excitation of abundant spins, the recycle delay depends on the longitudinal relaxation of abundant spins that is usually much shorter than that of dilute spins. As a result, CP has much higher acquisition efficiency than the SP experiment.

In CP NMR experiment, simultaneous spin-locks are applied on both abundant spin (I) and dilute spin (S) channels. The strength of the two radio frequency (RF) irradiations, ${{\omega }_{S}}$ and ${\omega _I}$, should match Hartman-Hann (HH) condition under MAS rotation^{[2, 3]}:

$ {\omega _I} - {\omega _S} = \pm n{\omega _r} $

(1)

where ${\omega _r}$ denotes the spinning frequency and n is an integer. With the HH condition matched, I spin polarization is transferred to S spin which signal is therefore enhanced. The buildup rate of the S spin signal is governed by I-S dipolar coupling. Stronger the I-S dipolar coupling, faster the S spin signal builds up. The ideal maximum enhancement factor is around ${\gamma _I}/{\gamma _S}$^[2]. However, the polarization decays of I and S spins in the rotating frame, characterized by T_{1ρ, I} and T_{1ρ, S} respectively, counteract the buildup process of the signal^[4]. In the most occasions, such polarization decay during the CP spin lock will render non-uniform enhancement for S spins at different local fields^[5]. Moreover, CP with invariant RF power, i.e. flat-CP, is vulnerable to RF mismatches and the offset of resonance frequencies^{[6, 7]}. As a consequence, the conventional CP/MAS technique is not suitable in quantitative NMR experiment. During the past decades, numerous efforts^[8-20] have been made to obtain quantitative CP/MAS spectroscopy, where the uniform enhancement for all chemically non-equivalent sites can be achieved.

The following review introduces the quantitative CP methods chronologically, according to the main characteristics of each one of them. Specifically, the CP with variable amplitude broadens the HH matching condition under MAS conditions; the multiple-contact CP accomplishes successive accumulation of S signal under the repetitively restored I polarization; quantification of CP (QCP) compensates the effect of T_1ρ relaxation in the cross polarization/cross depolarization (CP/CDP) reciprocity relation; Lee-Goldburg frequency modulated CP (LG-FMCP) prolongs the T_1ρ relaxation times during the spin lock period; quantitative CP (QUCP), being independent to T₁, can reach uniform enhancement no matter what the recycle delay is.

1 Variable amplitude CP

As mentioned above, the flat-CP [Fig. 1(a)] often results in non-uniform enhancement factor for each magnetically non-equivalent site, in addition, the overall enhancement efficiency is very sensitive to RF mismatches and the offset of resonance frequency, especially under MAS condition. In the early 1990s, in order to improve the robustness of flat-CP, the variable-amplitude CP [VACP, Fig. 1(b)] was proposed by Peersen et al.^[21] The VACP is basically a train of flat-CPs with different amplitude and constant phase in one channel, while a spin-lock irradiation with constant amplitude and phase is applied in another channel. It could suppress the influence of resonance offset and broaden the CP HH match conditions. Then the idea of variable amplitude was further extended with the bringing up of ramped-amplitude CP (RAMP-CP)^{[10, 22, 23]}. Instead of the discrete steps with varied amplitudes in VACP, the irradiation in RAMP-CP contains smoother, monotonically varying power [Fig. 1(c)], which makes the HH matching profile much broader than the flat-CP NMR experiment [Fig. 1(d) and 1(e)].

The RAMP-CP can only obtain quantitative NMR spectra in spin systems where the longitudinal relaxation rate in rotating frame, T_{1ρ, I}, is much longer than the cross relaxation time, T_IS, which is determined by I-S dipolar coupling^[10]. The signal of S spins builds up quickly while the decay of I spin polarization being slow. It suggests that the quantitative RAMP-CP experiment is more suited for the spin systems with strong I-S dipolar coupling, more specifically, the systems with dilute S spins "immersed" in the abundant I spins. Consequently, there exists a time of spin locking sufficiently long for the signals of S spins to build up to the same factor close to ${\gamma _I}/{\gamma _S}$, while short enough for the T_{1ρ, I} relaxation to cause negligible decay.

However, for systems with relatively weak I-S dipolar coupling, T_IS will be comparable to, or even larger than T_{1ρ, I}. In this circumstance, significant decay could occur prior to the corresponding S signals reach the maximum enhancement factor of ${\gamma _I}/{\gamma _S}$^[11]. Unfortunately, this is the mostly encountered case in practical systems, and moreover, we lack the information of the relaxation parameters at most of time, which severely limits the applications of RAMP-CP in quantitative NMR.

2 Multiple-contact CP

In the regular flat-CP experiments, the magnetization of I spins is transferred to the closest S spins that match the HH conditions, then transferred among S spins via I-I and S-S spin diffusion^[24]. Usually the spin diffusion rate of dilute S spin is slow due to the dilute S-S spin network and relatively weak S-S dipolar couplings. It makes the signal of S spins only partially enhanced after a single flat-CP, and the portion of enhanced S spins strongly depends on CP HH matching condition. As a result, the uniform enhancement is hampered.

One solution is to perform multiple-contact CP whose concept was first introduced in 1985^[25]. In 1990, Zhang et al.^[8] applied the multiple-contact CP on the model compounds such as glycine, to investigate the CP dynamics during the multiple CP contacts. The multiple-contact CP consisted of successive CP contacts, and each CP contact was separated by 90° pulses on both I and S channels as shown in Fig. 2(a). After the first CP, the magnetization of I and S (denoted M_I and M_S) spins were both converted back to z direction by 90° pulse, then the spin system underwent a delay τ. The τ was set shorter than the spin-lattice relaxation time of S spins T_{1, S}, but longer than T_{1, I}, and it's sufficient for S spins to reach a quasiequilibrium state via spin diffusion. During the delay τ, the M_I was recovered via T₁ relaxation, and the M_S was stored on z direction without significant loss. At the end of τ, M_I and M_S were both converted to transversal magnetization by 90° pulses, and then followed by spin lock, i.e. the second CP. As it turned out, with the increasing number of CP contacts, the multiple-contact CP would render accumulation of M_S. As demonstrated in the examples of (NH₄)H₂PO₄ and glycine, the signal of ³¹P and ¹³C built up and almost reach their maximum enhancement factors^[8]. For the short spin-locking contact time, the mismatched multiple-contact CP could lead to faster signal build up than that with matched HH conditions. It's conclusive that the mismatched multiple-contact CP is more suitable for the samples with fast T_1ρ decay, where the repetitive short CP contacts would be able to establish uniform enhancement. However, multiple-contact CP is not robust to RF mismatch, and also, the performance can be hampered by the imperfection of 90° pulse^[26].

By incorporating with RAMP-CP, the performance of multiple-contact CP can be further improved, and such modified multiple-contact CP technique is called as multiCP^{[18, 19]} [Fig. 2(b)]. As mentioned above, the RAMP-CP with linearly varying amplitude could broaden the HH matching condition, and enlarge the portion of enhanced S spins in a single CP contact. The two-stage process therefore becomes obscure, and a single RAMP-CP contact won't result in quasiequilibrium state. As shown in Fig. 2(b), the signals of S spins keep growing with the increasing number of RAMP-CP contacts. The dynamics during the following delay in multiCP is similar to the multiple-contact CP. For instance, in ¹H-¹³C multiCP experiments for natural abundant L-alanine^[19], the ¹³C signal would reach a plateau after the third contact. Due to the shorter T_1ρ and weaker ¹H-¹³C dipolar coupling of COOH group, the corresponding polarization factor of the plateau was noticeably less than those of CH and CH₃ groups, and less than the theoretical limit (normalized to 1), albeit with good agreement to the simulation. The multiCP NMR results showed that three carbon signals had a disparity of ±6.5% at the third contact, and after the fifth contact, this signal disparity could be reduced to ±4.7%. However, due to the intrinsic T_1ρ distributions in the sample, more contact number resulted in no further improvement (Fig. 3). Theoretically, the efficient compensation on the effect caused by T_1ρ distribution could lower the signal disparity to ±0.5%, suggesting other approach to accomplish quantitative CP, which will be discussed in the next section.

For systems with strongly coupled I-S spin pairs, the multiCP can provide the quantitative information with higher accuracy. For example, multiCP played a crucial part in the structural characterization of a group of organic polymers known as copolymerized conjugated microporous polymers (CP-CMPs), as reported by Brownbill et al.^[27] The CP-CMPs have three-dimensional networks that constructed by pyrene and benzene (Fig. 4), and have high activity as photocatalysts in decomposing water to generate hydrogen. However, the amorphism and insolubility make it challenging to investigate the structure of CP-CMP, obscuring the relation between synthesis and the polymer composition. Brownbill et al. proved that the structure of CP-CMPs was directly related to the feed ratio of monomers, using dynamic nuclear polarization (DNP) enhanced ¹H-¹³C CP NMR experiments. Furthermore, by virtue of DNP enhanced ¹H-¹³C multiCP NMR technique, they investigated the relation between the feed ratio and the composition of final polymers (Table 1). The quantitative NMR spectra was accomplished by integration through all resonances in multiCP ¹³C NMR spectrum, then dividing the sum area by the number of carbon atom in the corresponding monomer unit. The results revealed that the final polymers had the identical, within error, pyrene/benzene ratio with the feed ratio in monomers, indicating near complete uptake of raw materials during the synthesis. This finding was instructive for alternating the CP-CMP structure therefore modulating the band gap^[27]. With the multiCP NMR method, one could also obtain the absolute content of multiple components in system of interest. By means of calibration curve, King et al.^[28] determined the purity of chitin in the ground shrimp shells. The results agreed well with that by other established method. More importantly, the quantitative analysis by multiCP was quicker, more accurate, and not harmful to the sample. Therefore it's suitable for fast purity measurement during chitin purification form shrimp shells.

The deviation by multiCP from theoretical limit can be more obvious when it is applied to the samples containing weak I-S dipolar coupling. Sam et al.^[29] investigated a class of silicate-siloxane copolymers called POSiSils via ¹H-²⁹Si multiCP technique. The T₁ of ²⁹Si in these copolymers were in the range of 50~160 s, indicating excessively long recycle delay of 800 s for quantitative NMR experiments by the direct excitation, while the recycle delay in regular RAMP-CP MAS NMR experiment was significantly shortened due to T_{1, H} < 1 s. As shown in the buildup curves of CP dynamics [Fig. 5(a)], the ²⁹Si signal from Q3 site, in which Si atom is connected to one O-H and three O-Si groups, would start decaying due to the short T_1ρ when the contact time is longer than 2 ms, and the maximum normalized enhancement factor was only 0.5. By performing the multiCP approach, the enhancement factor of Q3 could be improved close to 1.0, where totally 16 CP contacts were applied with individual CP contact time of 1.1 ms and t_z delay of 2.5 s [Fig. 5(a)]. However, an enhancement factor larger than 0.2 couldn't be achieved for the ²⁹Si bonded to four O-Si groups, i.e. Q4 site, neither by regular RAMP-CP nor multiCP technique. As shown in Fig. 5(b), no much difference was observed in the CP dynamics plots of ²⁹Si Q4 site, which is mainly due to very weak ¹H-²⁹Si dipolar couplings of Q4 with H in surroundings. These results demonstrated the limitation of multiCP applications in the systems with long T_IS (or weak I-S dipolar network). In addition, multiCP technique can be used for sensitivity enhancement in multi-dimensional NMR experiments, for instance, 2D ²⁹Si-²⁹Si DQ-SQ correlation spectroscopy with multiCP excitation but not a regular RAMP-CP^[29].

3 T_1ρ compensated quantitative CP

The HH CP acts as a forward polarization transfer, and in fact there is a reverse process, called as CDP^{[14, 30]}, which is conformed to a reciprocity relation:

$ Sum(t) = CP(t) + CDP(t) = 1 $

(2)

together with CP^{[14, 31, 32]}. The reciprocity relation suggests that, in an ideal system without relaxation, the sum of the polarization transferred forwards and backwards should be a constant that can be normalized to 1. Obviously, in the practical systems, the relaxation will make Sum(t) time-dependent and decay (Fig. 6), i.e.,

$ Sum'(t) = CP'(t) + CDP'(t) = f(t) $

(3)

As the decay of Sum'(t) is dominated by T_1ρ, it can be assumed:

$ f\left( t \right){\rm{ }} \approx {\rm{ exp }}( - t/{T_{1\rho }}){\rm{ }} \approx {\rm{ }}1 - t/{T_{1\rho }},{\rm{for}}\;\;t < < {T_{1\rho }} $

(4)

Once T_1ρ is measured by fitting the Sum'(t) curve, the CP'(t) and CDP'(t) can be compensated via dividing by Sum'(t). As reported by Shu et al.^{[14, 20]}, the reciprocity relation could be revised by compensating the relaxation effect.

With the relaxation effect compensated, the signal intensity of S spins is relatively independent on T_1ρ relaxation, which would allow for achieving quantitative information in the samples (QCP)^{[14, 20]}. For instance, there are equal numbers of ¹³C in CO, CH and CH₃ groups in alanine, thus uniform enhancement in a quantitative CP/MAS spectrum should give the same peak intensity for all three groups. However, by the regular CP, the deviation of ¹³C CP/MAS signals with a contact time of 0.1 ms is ±60.4% from the uniform enhancement. On the other hand, the deviation by QCP was reduced to ±3.4% even without T_1ρ compensation, and could be further reduced ±0.5% with the T_1ρ compensation, which indicated that uniform enhancement was achieved. To further demonstrate the performance of the QCP method, a sample of glycine/alanine mixture with molar ratio of 2.765 was prepared. As demonstrated in ¹³C QCP NMR experiment with a contact time of 0.5 ms, the ratio of ¹³CH₂ (glycine) and ¹³CH (alanine) signal intensities was 2.770 with an error of ±0.005^[12]. Notably, the QCP method is also suitable for the isotope labeled systems. In a mixture of ¹³C-labelled alanine and natural abundant glycine, the molar ratio of the two components determined by the QCP method was 10.62, in good agreement with 10.82 by single pulse NMR experiment^[20].

In comparison, the QCP method could improve the uniformity of enhancement, and significantly decrease the experimental error in quantitative NMR measurement. However, T_1ρ measurement would be required to obtain sufficient accuracy, where it's recommended that at least four NMR experiments are needed^[14], and more accurate measurement of T_1ρ requires more experiment data for both CP and DCP. Considering the practical systems with various I-S dipolar couplings, the QCP experiment is somewhat time-consuming especially when the systems contain multiple sites.

4 Quantitative CP with prolonged T_1ρ relaxation time

As seen, the quantitative CP methods discussed above were established under the shadow of the relaxation rates that is "intrinsic" characteristics of the spin system in practical spin systems. However, as it turned out, these characteristics could also be manipulated, with the application of decoupling or recoupling techniques^{[11, 33]}.

It was discovered that the Lee-Goldburg (LG) sequence^[34] could significantly elongate the proton T_1ρ by spin-locking the spins at the magic angle^[33], as the homonuclear dipolar couplings were suppressed efficiently during the irradiation. Fu et al.^[33] compared the relaxation times in rotating frame with spin lock in xy plane and at magic angle, i.e. normal on-resonance spin lock and LG spin lock, on a ¹⁵N-1, 3, 5, 7-labeled polypeptide, gramicidin A. This polypeptide was encapsulated by DPMC bilayers and dissolved in water. As a liquid crystal, the bilayer retained some structural anisotropy, and it was orientated parallel to the direction of B₀ field. The ¹H spin-lattice relaxation time in the rotating frame with the regular spin lock in xy plane was 3.51 ms, and prolonged to 7.37 ms under LG spin lock at the magic angle. It showed that the spin-lattice relaxation in the rotating frame along the magic angle was much slower than that in the xy plane. The LG spin lock was accomplished by firstly applying a 54.7° pulse, then followed by an off-resonance RF field irradiation. When the resonance offset Ω and RF power ω₁ satisfy the equation:

$ \omega_1 = \sqrt 2 \Omega $

(5)

an effective field:

$ \omega_{\rm eff} = \sqrt {3/2} \omega_1 $

(6)

would lie on the magic angle. The high-order homonuclear Hamiltonian contributes to the T_1ρ relaxation of ¹H during the spin lock in xy plane, especially under relatively weak RF irradiation. While applying the LG spin lock, served as a homonuclear decoupling sequence, could completely suppress the first-order ¹H-¹H homonuclear dipolar Hamiltonian terms. As a result, the spin-lattice relaxation in the rotating frame at magic angle was less affected by the internal spin interactions, showing a slower T_1ρ relaxation^[33]. As discussed above, the polarization transfer during the CP contact is always benefiting from a slower T_1ρ relaxation especially in the spin system with weak I-S dipolar coupling.

Fu et al.^[11] modified the conventional CP by applying the LG spinlock on proton channel, while the carrier frequency of ¹⁵N RF field irradiation was modulated by a sine wave (LG-FMCP) as shown in Fig. 7(a), and the modulated carrier frequency on ¹⁵N channel could broaden the CP HH matching condition. On the sample of ¹⁵N-L-histidine, T_1ρ of non-protonated ¹⁵N site were 13.6 ms and 28.9 ms for the regular on-resonance spin lock and LG spin lock along magic angle, respectively, indicating the LG irradiation slower the T_1ρ relaxation by a factor of 2. Higher RF power had little effect for the on-resonance spin lock, but could result in longer T_1ρ relaxation time in the LG-FMCP experiment. By applying the LG-FMCP method with a contact time of 8 ms, the intensities of protonated and non-protonated ¹⁵N signals could be improved by 20% and 27%, respectively, comparing to the regular CP/MAS spectrum. It should be noted that the buildup rates of ¹⁵N signal in LG-FMCP were slower than that in regular CP, as shown in Fig. 7(b) and 7(c), the corresponding scale-down factor was (1-λ) where

$ \lambda = {T_{{\rm{NH}}}}/{T_{1\rho ,{\rm{H}}}} $

(7)

In other words, the prolonged T_1ρ relaxation time increased the efficiency of polarization transfer while simultaneously slower the transfer rate^[11].

5 Spin diffusion driven quantitative CP

As mentioned above, there exist two processes during the CP period, polarization transfer from abundant spin I to dilute spin S, and the spin diffusion among dilute spins. Since the polarization transfer rate is much higher than the spin diffusion rate, and also, the polarization buildup has to suffer from the relaxation effect, usually a non-uniform enhancement is obtained in the conventional CP spectroscopy. Series of additional delays were introduced in multiple-contact CP, letting the S spins diffuse the polarization to reach the same spin temperature^[8]. Under MAS rotation, S-S homonuclear dipolar couplings are averaged out, and the spin diffusion might be slower in dilute spin S. Therefore, it is necessary to reintroduce the S-S homonuclear dipolar couplings by appropriate sequences.

Following the CP (either flat-CP or RAMP-CP) contact, an additional period with broadband homonuclear recoupling is applied prior to the acquisition, in the meanwhile the CP enhanced S signals can evolve to the final uniform enhancement [Fig. 8(a)], this method is named as QUCP^[12]. The broadband homonuclear recoupling is accomplished by dipolar-assisted rotational resonance (DARR)^[35]. Considering the rotational resonance recoupling^[36] in ¹³C spin system for instance, the ¹³C-¹³C homonuclear recoupling can be reintroduced only when the MAS frequency ${\omega _r}$ satisfies the recoupling condition:

$ n{\omega _r} = \Delta {\omega _{\rm{C}}} $

(8)

where $\Delta {\omega _{\rm{C}}}$ is the chemical shift difference (in Hz) between two coupled ¹³C spins and n=±1, ±2, etc. Apparently, it's difficult to match the rotational resonance condition and recouple the dipolar coupling for all the ¹³C spins, especially for the systems containing multiple resonance sites. To overcome this band-selective recoupling limitation, one can apply an RF irradiation with the amplitude of ${\omega _I} = {\omega _r}$, i.e. the n=1 rotary resonance condition^[37], on ¹H channel. With this low power continuous wave irradiation, both ¹H-¹³C and ¹H-¹H dipolar couplings are reintroduced, which results in seriously line-broadened NMR spectrum under MAS rotation. As long as the powder patterns of spins are broadened enough to overlap with each other, the broadband homonuclear recoupling among all spins would be obtained^[35]. DARR has been extensively used for the determination of spatial proximity and molecular structure, under low-to-moderate MAS rates.

In the QUCP sequence, heteronuclear polarization transfer is accomplished via the conventional CP, and then the non-uniformly enhanced magnetization of S spins is tilted to the z direction, followed by an additional mixing period for spin diffusion among S spins. During the mixing time, a DARR irradiation is applied on I channel to reintroduce the homonuclear dipolar couplings among the S spins. The redistribution of polarization occurs among these non-uniformly enhanced S spins, rendering a quasiequilibrium state therefore consistent enhancement, which is significant for the quantitative NMR measurements. Although the final enhancement factor depends on the efficiency of the prior CP contact, the following spin diffusion driven by homonuclear recoupling sequence would always redistribute the magnetization of S spins to uniform enhancement. In other words, given sufficient mixing time, the consistency of enhancement in QUCP would be independent on the experimental parameters in CP^{[12, 13]}.

Even with the very sparse S spins, the uniform enhancement still can be reached, albeit longer recoupling time is needed. For instance, ¹³C single pulse, ¹H-¹³C CP and QUCP NMR experiments had been carried out on natural abundant DL-alanine at a MAS frequency of 12 kHz^[13]. The enhancement factors in each NMR experiment, normalized by the intensity of single pulse excitation with long pulse delay, were summarized in Table 2. No surprisingly, the enhancement factors for carbonyl/methyne/methyl carbons by CP deviated very much from the consistence. The QUCP NMR spectrum with mixing time of 5 s showed less deviations, but still non-uniform enhancement, which is mainly due to the very weak ¹³C-¹³C dipolar couplings in low ¹³C density system. The slow polarization transfer would require longer spin diffusion time to reach the quasi-equilibrium state. As demonstrated, QUCP with mixing time of 10 s resulted in the enhancement factors of 1.56, 1.57, 1.57 for three carbon groups, and a quantitative CP/MAS NMR spectrum was achieved (Table 2). In comparison to the quantitative NMR measurement by the direct excitation, an efficiency gain of 50 times was achieved by the QUCP approach^[13]

In most quantitative solid-state NMR experiment, usually it's required that the optimized recycle delay is no less than 5 times of T₁, where T₁ denotes the spin-lattice relaxation time of dilute S spin when using the single pulse method, or that of abundant I spin when using CP-based quantitative NMR methods. Generally, T₁ of the abundant I spin is much shorter than that of dilute S spin, but the NMR experiments with recycle delay of 5T₁ can be time-consuming, and it should be noted that the spin-lattice relaxation time is required to be optimized prior to collecting quantitative NMR data.

It was discovered that, the feature of uniform enhancement in QUCP NMR experiments was not affected at all by the setting of recycle delay^[15]. As demonstrated in ¹³C NMR spectra collected on ¹³C-labelled L-tyrosine spun at MAS frequency of 14 kHz, an uniform enhancement could not be achieved by the regular CP method whatever the recycle delay was (Fig 9). On the contrary, using the QUCP sequence with a mixing time of 1 s, the enhancement uniformity could be always achieved, although the recycle delay affected the value of final uniform enhancement factor, as shown in Fig. 9. The NMR results showed that the deviation of the enhancement factors was within ±0.02, allowing for quantitative analysis with high accuracy. Similarly, in natural abundant DL-alanine, the consistency of enhancement was also independent of the recycle delay, although a longer mixing time of 20 s was required^[15]. The results indicated that the quantitative NMR measurements by QUCP would not be constrained by the relaxation time. The idea of breakthrough of T₁ constraint in quantitative CP/MAS spectroscopy can be expanded into the quantitative NMR measurement by the direct excitation. Homonuclear dipolar recoupling technique, i.e. DARR, could also be placed prior to the 90° pulse excitation, which would accelerate the spin diffusion of spin system and recover in a uniform rate. Therefore, the recycle delay could be shortened in quantitative single pulse (QUSP) experiments, and the acquisition efficiency was improved^[15].

During the last two decades, tremendous development has been made on fast and ultra-fast MAS rotation technique in solid-state NMR^[38]. With the availability of fast MAS frequencies up to 110 kHz, the spectral resolution can be improved greatly, and also, the proton reverse detection with high sensitivity would accelerate the data acquisition of multi-dimensional correlation spectroscopy. However, the implement of fast MAS technique brings up the NMR researchers new challenges, the demand for developing novel recoupling techniques suited for fast and ultra-fast MAS conditions^{[32, 39-42]}. For instance, numerous pulse sequences have been proposed for achieving broadband homonuclear recoupling under fast MAS rates^{[32, 43-46]}. In order to obtain quantitative CP/MAS spectrum under moderate-to-fast MAS rates (< 30 kHz), Takeda et al.^[17] modified the QUCP sequence with phase-modulated SHA irradiation (Second-order Hamiltonian among Analogous Nuclei Generated by Hetero-nuclear Assistance Irradiation, SHANGHAI) as the replacement of DARR irradiation during the mixing period. As demonstrated, the phase modulated RF irradiation could more efficiently drive the spin diffusion among S spins to reach the final uniform enhancement.

6 Conclusion and outlook

CP-based quantitative NMR methods provide us a powerful tool for obtaining accurate analysis of contents, component, and chemical sites in the practical systems with much higher sensitivity and acquisition efficiency. To achieve the quantification of CP/MAS spectroscopy, it is indispensable to accomplish consistent/uniform enhancement. The developed quantitative NMR methods started from multiple contact CP, including multiple-contact CP and multiCP, share the similar pulse sequence consists of a train of CP contacts. Within each CP contact, the magnetization of S spins is enhanced non-uniformly. An additional interval is applied between CP contacts, during the intervals I spins undergo the spin-lattice relaxation recovery and S spins undergo the spin diffusion process. Therefore, the reiteratively replenished polarization of I spins can further enhance the transfer to S spins, and the S spins evolve to a quasiequilibrium state. Repeated interval CP contacts could render the accumulation of signal intensity of detected spins, and reach, in theoretically, the maximum enhance factor of ${\gamma _I} = {\gamma _S}$. Therefore, the quantitative NMR experiment by the multiple CP method requires the enhancement factor as close to the maximum as possible. However, it is difficult to achieve in practical systems, especially when the spin-lattice relaxation time in the rotating frame of S spins T_{1ρ, S} is comparable to the cross relaxation time T_IS. Although the cross relaxation dominates the CP and CDP processes, the relaxation in the rotating frame brings about an exponential delay during the transfers. With the compensation of T_1ρ relaxation, CP and CDP processes would obey the reciprocity relation which allows for the quantitative NMR measurement, i.e. dubbed as QCP. On the other hand, additional NMR experiments for accurate measurement of T_1ρ relaxation times would be required for calculating the T_1ρ effect compensation. Alternatively, the T_1ρ relaxation during CP buildup can be prolonged greatly, when applying LG spin lock during CP contact to suppress the influence from homonuclear dipolar couplings. With LG-FMCP method, a longer contact time without noticeable T_1ρ relaxation effect would allow for obtaining quantitative NMR spectra. Another approach for establishing consistent enhancement is to introduce an additional mixing period after CP contact, where broadband homonuclear recoupling sequence irradiation is applied, dubbed as QUCP. During the mixing period, the S-S homonuclear couplings are reintroduced along with I-S heteronuclear dipolar couplings, which would speed up the polarization transfer among the coupled S spins, driving the S spins to the quasiequilibrium state. As long as the mixing time is sufficient, the polarization of S spins would finally reach the uniform enhancement regardless of CP matching condition, contact time, and even the recycle delay. It should be noted that QUCP experiments for the dilute spin systems in natural abundance require longer mixing time due to the low spin density. In the methodological aspect, the quantitative NMR methods discussed in this review showed distinct contemplation for accomplishing uniform enhancement in CP/MAS spectroscopy. The development of these quantitative NMR techniques certainly expanded the depth and breadth of solid-state NMR spectroscopy, making NMR more flexible in dealing with more complicate samples, though further improvement is still required for more extensive applications.